2.1. Definitions and scaling

A Newtonian fluid with uniform density $\rho$ and kinematic viscosity $\nu$ is bounded by a flat wall located in the $({\boldsymbol {e}}_{\boldsymbol {1}},{\boldsymbol {e}}_{\boldsymbol {2}})$ plane. The fluid flows towards the wall in the form of an axisymmetric linear straining flow (so-called biaxial straining flow) with a radial (respectively axial) strain rate $B$ (respectively $-2B$). As this inviscid solution does not satisfy the no-slip condition at the wall, a boundary layer with characteristic thickness $\delta =(\nu /B)^{1/2}$ exists along the wall. We consider a neutrally buoyant spherical particle with radius $a$ standing on the axis of the straining flow and entrained by the fluid towards the wall. At time $T$, the gap between the particle and the wall is $h(T)$, so that the distance separating the particle centre from the wall is $\ell (T)=h(T)+a$ (see figure 1). We make use of a coordinate system ${\boldsymbol {X}}=(X_1,X_2,X_3)$ translating with the particle and having its origin at its centre. Then we normalize distances by the particle radius, $a$, whereas time is normalized by a characteristic time scale, $\tau _c$, to be defined later. Velocities are normalized by the unknown slip velocity between the particle and fluid, $V_c$, so that the characteristic Reynolds number is $Re=aV_c/\nu$, the dimensionless strain rate is $\alpha =aB/V_c$ (hence the product $\alpha Re$ is the strain-based Reynolds number), and forces are normalized by $\rho \nu aV_c$. Beyond the boundary layer, the local fluid velocity with respect to the wall is, in dimensionless form,

(2.1)\begin{equation} {\boldsymbol {U}}_0({\boldsymbol {x}},t)\approx {\boldsymbol {U}}_0({\boldsymbol {x}}={\boldsymbol{0}},t)+ \alpha({{\boldsymbol {x}}}-3x_3{\boldsymbol {e}}_{\boldsymbol{3}}),\end{equation}

where ${\boldsymbol {x}}=(x_1,x_2,x_3)=a^{-1}(X_1,X_2,X_3)$ denotes the dimensionless local position with respect to the current position of the particle centre, $t=T/\tau _c$ is the dimensionless time and ${\boldsymbol {e}}_{\boldsymbol {3}}$ is the unit normal to the wall directed into the fluid. In the momentum balance, the above normalization implies that the advective acceleration is of $\mathcal {O}(Re)$ compared to the viscous term. Similarly, the temporal acceleration is of $\mathcal {O}(ReSt)$, with $St=a/V_c\tau _c$ the Strouhal number comparing the advective time scale $a/V_c$ to the characteristic time $\tau _c$ of the flow. In the specific problem considered here, apart from the possible transient following the release of the particle in the flow, unsteadiness arises because of the non-uniformity of the carrying flow, which transforms into a time-varying flow in the particle reference frame. It is therefore relevant to select $\tau _c=B^{-1}$ as the characteristic time scale, which implies $St\equiv \alpha$. This is why, compared to viscous effects, time-rate-of-change terms are of $\mathcal {O}(\alpha Re)$.

Figure 1. Sketch of the flow configuration. The particle radius $a$, wall–particle separation, $\ell$, and boundary-layer characteristic thickness, $\delta$, yield the dimensionless length ratios $\kappa =a/\ell$, $\varDelta =\delta /a$ and $\varLambda =\delta /k_\delta \ell$ used throughout the paper (the boundary-layer shape parameter $k_\delta$ is defined in § 2.2).

2.2. A rough model for the boundary-layer flow

The viscous axisymmetric stagnation-point flow problem is governed by a third-order differential equation supplemented by suitable boundary conditions (Homann Reference Homann1936). Its exact self-similar solution cannot be obtained in closed form and must be determined numerically. To keep the problem tractable analytically, a simple algebraic approximation of this solution is desirable. Rather than trying to fit the full numerical solution with detailed quadratures, we sought a straightforward algebraic divergence-free expression of the velocity field satisfying the no-slip condition at the wall and tending toward (2.1) at large distances from it, with a thickness of the transition layer independent from the particle size. Defining the inverse of the dimensionless separation, $\kappa (t)=a/\ell (t)$, we found the simplest base flow satisfying these requirements to be

(2.2)\begin{align} {\boldsymbol {U}}_0({\boldsymbol {x}},t)& = {\boldsymbol {U}}_0({\boldsymbol {x}}={\boldsymbol{0}},t)\nonumber\\ &\quad + \alpha\left\{({{\boldsymbol {x}}}_\parallel-2x_3 {\boldsymbol {e}}_{\boldsymbol{3}})-\frac{{\boldsymbol {x}}_\parallel}{(1+ \mathcal{K}_\delta(\kappa^{-1}+x_3))^{2}}-\frac{2\mathcal{K}_\delta^{-1} {\boldsymbol {e}}_{\boldsymbol{3}}}{1+\mathcal{K}_\delta(\kappa^{-1}+x_3)}\right\},\end{align}

with ${\boldsymbol {x}}_\parallel =x_1{\boldsymbol {e}}_{\boldsymbol {1}}+ x_2{\boldsymbol {e}}_{\boldsymbol {2}}$ and $\mathcal {K}_\delta =k_\delta (\alpha Re)^{1/2}$, $k_\delta$ denoting an adjustable shape parameter to be discussed below. The first term within braces is the linear straining flow considered in (2.1), while the other two contributions represent a rough model of the flow modification within the boundary layer. In the reference frame translating with the particle, the wall is located at $x_3=-\kappa ^{-1}(t)$. Therefore the no-slip condition ${\boldsymbol {U}}_0({\boldsymbol {x}}_\parallel ,x_3=-\kappa ^{-1},t)={\boldsymbol {0}}$ implies that the fluid velocity at the current position of the particle centre is ${\boldsymbol {U}}_0({\boldsymbol {x}}={\boldsymbol {0}},t)= 2\alpha (\mathcal {K}_\delta ^{-1}-\kappa ^{-1}){\boldsymbol {e}}_{\boldsymbol {3}}$.

Since $\alpha Re=a^{2}B/\nu \equiv a^{2}/\delta ^{2}$, the dimensionless characteristic boundary-layer thickness $\varDelta$ obeys the relation $\varDelta =(\alpha Re)^{-1/2}$, which implies $\mathcal {K}_\delta =k_\delta \varDelta ^{-1}$. Hence the second term within curly braces in (2.2) reduces to $-{\boldsymbol {x}}_\parallel (1+k_\delta )^{-2}$ when the particle stands a distance $\kappa ^{-1}=\varDelta$ from the wall. With $k_\delta =2$, the tangential velocity $\alpha {\boldsymbol {x}}_\parallel (1-(1+k_\delta )^{-2})$ reaches approximately $90\,\%$ of its free-stream value at this position, a percentage that increases to $98\,\%$ for $\kappa ^{-1}=3\varDelta$. These features are in good agreement with the actual velocity profile of the HH flow displayed in figure 2 of Li et al. (Reference Li, Abbas, Morris, Climent and Magnaudet2020). Thus (2.2) with $k_\delta \approx 2$ is expected to represent well the variation of the carrying flow in the part of the boundary layer close to its outer edge. However, the approximate base flow must also correctly estimate the curvature $\mathcal {C}=\partial^2({\boldsymbol {U}}_0\boldsymbol{\cdot}{\boldsymbol {e}}_{\boldsymbol{3}})/\partial x_3^2 $ of the normal velocity ${\boldsymbol {U}}_0 \boldsymbol {\cdot }{\boldsymbol {e}}_{\boldsymbol {3}}$ in the limit $x_3\rightarrow -1/\kappa$, since this curvature governs the variation of all three velocity components within the inner part of the boundary layer, say for $0\le x_3+1/\kappa \lesssim \varDelta$. In this limit, the velocity field (2.2) reduces to the nearly parallel distribution ${\boldsymbol {U}}_0({\boldsymbol {x}},t) \approx 2\mathcal {K}_\delta \alpha (\kappa ^{-1}+x_3)\{{{\boldsymbol {x}}}_\parallel - (\kappa ^{-1}+x_3){\boldsymbol {e}}_{\boldsymbol {3}}\}$, so that (2.2) predicts $\mathcal {C}\approx -4\mathcal {K}_\delta \alpha =-4k_\delta \alpha /\varDelta$. Figure 2 shows how this model approaches the variation of ${\boldsymbol {U}}_0 \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$ encountered near the wall in the actual HH flow. It turns out that the above value $k_\delta =2$ significantly over-estimates $\mathcal {C}$, hence $-{\boldsymbol {U}}_0 \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$, throughout this region and even beyond. A much better agreement with the actual profile is obtained with $k_\delta =1$. Nevertheless, with this lower $k_\delta$, the tangential velocity reaches $98\,\%$ of its free-stream value only for $\kappa ^{-1}=6\varDelta$. Hence it appears that a single value of $k_\delta$ does not allow (2.2) to fit closely the actual near-wall flow throughout the boundary layer. This is not unexpected since the velocity field in (2.2) is not an exact solution of the Navier–Stokes equation. Indeed, the corresponding vorticity, $\boldsymbol \omega _\delta ({\boldsymbol {x}})=-2\alpha \mathcal {K}_\delta (x_2{\boldsymbol {e}}_{\boldsymbol {1}}- x_1{\boldsymbol {e}}_{\boldsymbol {2}})(1+\mathcal {K}_\delta (\kappa ^{-1}+x_3))^{-3}$, does not satisfy the vorticity transport equation, except in the region closest to the wall ($\kappa ^{-1}+x_3\ll 1$). Nevertheless, since the influence of boundary-layer effects on the particle dynamics is expected to be large essentially within the $\mathcal {O}(\varDelta )$-thick region next to the wall, it is likely that $k_\delta =1$ is the optimal choice to be used in conjunction with the simple model (2.2). Comparisons of slip velocities predicted by the present theory with results of fully resolved simulations will later confirm this conclusion (see figure 3$b$). However, to keep the results more general, $k_\delta$ will be left unspecified throughout the developments performed in the next sections.

Figure 2. Near-wall profile of the wall-normal velocity in the base flow; the velocity and distance to the wall are normalized using boundary-layer quantities, i.e. $B\delta /V_c=\alpha \varDelta$ and $\delta /a=\varDelta$, respectively. Blue line, theoretical solution (Homann Reference Homann1936); dotted line, numerical solution (Li et al. Reference Li, Abbas, Morris, Climent and Magnaudet2020); red and green lines, model (2.2) with $k_\delta =2$ and $k_\delta =1$, respectively.

Figure 3. Slip velocity profile as a function of the gap $\epsilon =\kappa ^{-1}-1$ for a particle with relative radius $\varDelta ^{-1}=0.3$ compared to the characteristic boundary-layer thickness. $(a)$ Comparison between simulation results (green dashed line) and predictions of the GMR equation using the undisturbed flow (2.2) with $k_\delta =1$ (green line); $(b)$ comparison between simulation results (green dashed line) and predictions of (3.6) with $k_\delta =2$ (black line) and $k_\delta =1$ (green line).

Returning to (2.2) and defining

(2.3a,b)\begin{equation} {\boldsymbol {U}}_{0}^{0}(t)={\boldsymbol {U}}_0({\boldsymbol {x}} ={\boldsymbol{0}},t)\quad \mbox{and}\quad\varLambda(t)= \frac{\kappa(t)}{\mathcal{K}_\delta}=\frac{\kappa(t)\varDelta}{k_\delta},\end{equation}

the carrying flow close to the particle (formally within the region $|x_3| \ll (1+\varLambda )/\kappa$) may be expanded in the form

(2.4)\begin{equation} {\boldsymbol {U}}_0({\boldsymbol {x}},t)= {\boldsymbol {U}}_0^{0}(t)+\alpha_b(t)({{\boldsymbol {x}}}- 3x_3{\boldsymbol {e}}_{\boldsymbol{3}})+\alpha_c(t)x_3({{\boldsymbol {x}}}- 2x_3{\boldsymbol {e}}_{\boldsymbol{3}})+\cdots,\end{equation}

with

(2.5a–c)\begin{align} {\boldsymbol {U}}_0^{0}(t)=-2\alpha\frac{1}{\kappa(t)(1+\varLambda(t))} {\boldsymbol {e}}_{\boldsymbol{3}},\quad \alpha_b(t)=\alpha\frac{1+ 2\varLambda(t)}{(1+\varLambda(t))^{2}},\quad \alpha_c(t)=2\alpha\kappa(t) \frac{\varLambda^{2}(t)}{(1+\varLambda(t))^{3}}.\end{align}

The inviscid base flow (2.1) is recovered in the limit $\varLambda \rightarrow 0$, for which $\alpha _b\rightarrow \alpha$ and $\alpha _c\rightarrow 0$. For finite $\varLambda$, the leading influence of the boundary layer is to reduce the effective strain rate at the position of the particle to an $\mathcal {O}((1+2\varLambda )/(1+\varLambda )^{2})$-fraction of its free-stream value, and to introduce a quadratic component of the flow with an $\mathcal {O}(\kappa \varLambda ^{2}/(1+\varLambda )^{3})$-magnitude. The quantity $\varLambda ^{-1}=k_\delta (\kappa \varDelta )^{-1}$ may be thought of as the distance separating the particle from the wall normalized by the effective boundary-layer thickness $6\varDelta /k_\delta$, the distance to the wall at which the tangential velocity reaches $98\,\%$ of its free-stream value. For reasons to be discussed later, the asymptotic approach developed in the next sections will be restricted to particles much smaller than the boundary-layer thickness, which implies $\varDelta \gg 1$. For such particles, $\varLambda$ varies from near-zero values when the particle is far from the boundary layer $(\kappa \rightarrow 0$) to large $\mathcal {O}(\varDelta )$-values (since $1\lesssim k_\delta \lesssim 2$) when it gets very close to the wall.

2.3. Reciprocal theorem

Forces acting on a spherical buoyant drop with an arbitrary viscosity immersed in a linear flow bounded by a single flat wall and translating with velocity ${\boldsymbol {V}}$ in an arbitrary direction with respect to that wall were considered in M1. In a preliminary step, a general expression for the force balance, valid whatever the magnitude of unsteadiness and inertia effects, was obtained by making use of the reciprocal theorem. It is straightforward to extend this force balance to the quadratic flow (2.4), and consider the particular case of a neutrally buoyant rigid particle. For the sake of self-consistency, the main steps of the derivation are provided in appendix A. As is well known, evaluating wall-normal forces with the help of the reciprocal theorem requires the determination of the solution of the ‘auxiliary’ problem corresponding to a spherical particle translating perpendicularly to the wall with unit velocity in a fluid at rest. Let $\hat {{\boldsymbol {U}}}$ and $\hat {\boldsymbol {\varSigma }}$ be the fluid velocity and stress fields associated with this problem, respectively. Then let ${\boldsymbol {u}}({\boldsymbol {x}},t)$ and ${\boldsymbol {V}}_{S0}(t)={\boldsymbol {V}}(t)-{\boldsymbol {U}}_0^{0}(t)$ be the velocity disturbance and time-dependent slip velocity between the particle and fluid involved in the actual (‘direct’) problem, respectively.

Using the scalings established in § 2.1, the derivation in appendix A provides the exact dimensionless force balance on a rigid neutrally buoyant spherical particle moving perpendicular to the wall in the form (A13). This result being valid for an arbitrary carrying flow, the force balance in a quadratic flow such as that defined by (2.4) becomes

(2.6)\begin{align} &Re\left(\frac{4}{3}{\rm \pi}\alpha \frac{ \textrm{d} {\boldsymbol {V}}}{ \textrm{d} t}-\int_\mathcal{V_A} \frac{\textrm{D}{\boldsymbol {U}}_0}{\textrm{D}t}\, \textrm{d} \mathcal{V}\right) \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}=\hat{{\boldsymbol {F}}}_D \boldsymbol{\cdot} {\boldsymbol {V}}_{S0}-\hat{{\boldsymbol {T}}}_D:\nabla^{0} {\boldsymbol {U}}_0-\frac{1}{2}\hat{{\boldsymbol {S}}}_D {\vdots}\nabla^{0} \boldsymbol{\nabla} {{\boldsymbol {U}}_0}\nonumber\\ &\quad-Re\int_\mathcal{V}(\hat{{\boldsymbol {U}}}+ {\boldsymbol {e}}_{\boldsymbol{3}}) \boldsymbol{\cdot} \left(\alpha \frac{\partial{\boldsymbol {u}}}{\partial t}+{\boldsymbol {u}} \boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol {U}}_0+({\boldsymbol {U}}_0- {\boldsymbol {U}}_0^{0}) \boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol {u}}+({\boldsymbol {u}}- {\boldsymbol {V}}_{S0}) \boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol {u}}\right)\, \textrm{d} \mathcal{V}, \end{align}

where $\mathcal {V_A}$ and $\mathcal {V}$ refer to the volume occupied by the particle and the fluid, respectively, and $\hat {{\boldsymbol {F}}}_D=\int _\mathcal {A}\hat {\boldsymbol {\varSigma }} \boldsymbol {\cdot } {\boldsymbol {n}}\, \textrm {d} \mathcal {A}$ is the drag force on the particle in the auxiliary problem, ${\boldsymbol {n}}$ denoting the unit normal to the particle surface $\mathcal {A}$ directed into the fluid. The gradient $\nabla ^{0}{\boldsymbol {U}}_0= \boldsymbol {\nabla } {\boldsymbol {U}}_0({\boldsymbol {x}}= \boldsymbol {0})=\alpha _b({\boldsymbol {I}}-3{\boldsymbol {e}}_{\boldsymbol {3}} {\boldsymbol {e}}_{\boldsymbol {3}})$ and Hessian $\nabla ^{0} \boldsymbol {\nabla } {\boldsymbol {U}}_0= \boldsymbol {\nabla } ( \boldsymbol {\nabla } {\boldsymbol {U}}_0) ({\boldsymbol {x}} =\boldsymbol {0})=\alpha _c{\boldsymbol {e}}_{\boldsymbol {3}}({\boldsymbol {I}}- 2{\boldsymbol {e}}_{\boldsymbol {3}}{\boldsymbol {e}}_{\boldsymbol {3}})$ of the undisturbed velocity (2.4) at the centre of the particle being non-zero, they provide additional contributions to the force through the first- and second-order moments of the surface stress, $\hat {{\boldsymbol {T}}}_D=\int _\mathcal {A}{\boldsymbol {x}} \hat {\boldsymbol {\varSigma }} \boldsymbol {\cdot } {\boldsymbol {n}}\, \textrm {d} \mathcal {A}$ and $\hat {{\boldsymbol {S}}}_D=\int _\mathcal {A}{\boldsymbol {x}} {\boldsymbol {x}}\hat {\boldsymbol {\varSigma }} \boldsymbol {\cdot } {\boldsymbol {n}}\, \textrm {d} \mathcal {A}$, with ${\boldsymbol {x}}$ the local position with respect to the particle centre and ${\boldsymbol {I}}$ the Kronecker delta. In (2.6), $\textrm {d}/ \textrm {d} t$ is the time derivative following the particle motion, while $\textrm {D}{\boldsymbol {U}}_0/\textrm {D}t$ is the acceleration of the undisturbed carrying flow. In the reference frame translating with the particle, this acceleration reads $\textrm {D}{\boldsymbol {U}}_0/\textrm {D}t=\alpha \textrm {d}{\boldsymbol {U}}_0/\textrm {d}t+({\boldsymbol {U}}_0- {\boldsymbol {V}}) \boldsymbol {\cdot } \boldsymbol {\nabla } {\boldsymbol {U}}_0$, the $\alpha$-pre-factor resulting from the scaling of unsteady effects as discussed in § 2.1.

Beyond the boundary layer, the carrying flow is linear, implying $\nabla ^{0} \boldsymbol {\nabla } {{\boldsymbol {U}}}_0=\boldsymbol {0}$ and making the undisturbed fluid acceleration uniform. Hence the left-hand side of (2.6) is proportional to the relative acceleration $\alpha \, \textrm {d}{\boldsymbol {V}}/ \textrm {d} t-\textrm {D}{\boldsymbol {U}}_0/\textrm {D}t$. Since $\alpha _b=\alpha$, $\boldsymbol {\nabla } {\boldsymbol {U}}_0=\alpha ({\boldsymbol {I}}-3 {\boldsymbol {e}}_{\boldsymbol {3}}{\boldsymbol {e}}_{\boldsymbol {3}})$ is of ${\mathcal {O}}(\alpha )$ there, and all terms in (2.6) involving the fluid and particle accelerations are of $\mathcal {O}(\alpha Re)$. The left-hand side of (2.6) then yields a net inertial force ${\boldsymbol {F}}_0$ on the particle

(2.7)\begin{equation} {\boldsymbol {F}}_0 \boldsymbol{\cdot} {{\boldsymbol {e}}_{\boldsymbol{3}}}= \frac{4}{3}{\rm \pi} \alpha Re\left(\frac{ \textrm{d} {\boldsymbol {V}}_{S0}}{ \textrm{d} t}- 2{\boldsymbol {V}}_{S0}\right) \boldsymbol{\cdot} {{\boldsymbol {e}}_{\boldsymbol{3}}}.\end{equation}

Within the boundary layer, the local strain rates $\alpha _b(t)$ and $\alpha _c(t)$ in (2.4) vary with the position of the particle with respect to the wall. Then an additional force proportional to $\alpha _c(t)$ takes place, owing to the $-\frac {1}{2}\hat {{\boldsymbol {S}}}_D {\vdots }\nabla ^{0} \boldsymbol {\nabla } {{\boldsymbol {U}}_0}$ contribution in the right-hand side of (2.6). Moreover, the body force $\int _\mathcal {V_A}({\textrm {D}{\boldsymbol {U}}_0}/{\textrm {D}t})\, \textrm {d} \mathcal {V}$ includes quadratic corrections proportional to $\textrm {d} \alpha _c/ \textrm {d} t$ and $\alpha _b(t)\alpha _c(t)$ which modify (2.7) into

(2.8)\begin{equation} {\boldsymbol {F}}_0 \boldsymbol{\cdot} {{\boldsymbol {e}}_{\boldsymbol{3}}}= \frac{4}{3}{\rm \pi} Re \left\{\left(\alpha\frac{ \textrm{d} {\boldsymbol {V}}_{S0}}{ \textrm{d} t}- 2\alpha_b{\boldsymbol {V}}_{S0}\right) \boldsymbol{\cdot} {{\boldsymbol {e}}_{\boldsymbol{3}}} - \frac{1}{5}\left(6\alpha_b\alpha_c-\frac{ \textrm{d} \alpha_c}{ \textrm{d} t}\right)\right\}.\end{equation}

2.4. Solving the auxiliary problem

To make practical use of (2.6), a key step is to solve the auxiliary problem. An exact solution of this problem based on bipolar coordinates, valid until the particle touches the wall, was derived independently by Brenner (Reference Brenner1961) and Maude (Reference Maude1961). Nevertheless making use of the corresponding solution to compute inertial terms involved on the right-hand side of (2.6) is non-trivial. A more tractable approach consists in assuming formally that the separation between the particle and the wall is large and seeking the solution in the form of a series of ‘reflections’ of the fundamental solution corresponding to a particle translating in an unbounded fluid. To this end, it is customary to expand the solution with respect to the small parameter $\kappa =a/\ell =(1+\epsilon )^{-1}$, where $\epsilon =h/a$ is the dimensionless gap. An approximate solution truncated at $\mathcal {O}(\kappa ^{4})$ was obtained in M1 and M2 using this technique. The main steps involved in the elaboration of this solution are summarized in appendix B, together with the explicit expressions for $\hat {{\boldsymbol {F}}}_D$, $\hat {{\boldsymbol {T}}}_D$ and $\hat {{\boldsymbol {S}}}_D$ required to evaluate the first three contributions on the right-hand side of (2.6). This appendix also discusses the limit of validity of this approximate solution, determined by comparing its predictions for the drag force with exact solutions and computational results. The conclusion is that this truncated solution is valid approximately up to $\kappa =0.5$, i.e. down to $\epsilon \approx 1$. Clearly, lubrication effects that take place when $\kappa \rightarrow 1$ ($\epsilon \rightarrow 0$) cannot be captured and stay beyond the capabilities of the present asymptotic theory.

3.1. Wall- and curvature-induced Faxén forces

In this section we totally disregard inertial effects, which in particular implies that the contributions of the volume integrals on the left- and right-hand sides of (2.6) are neglected. The total force acting on the particle is then merely the sum of the contributions resulting from the slip velocity ${\boldsymbol {V}}_{S0}$, and the successive gradients of the carrying flow at the position of the particle, $\nabla ^{0}{{\boldsymbol {U}}_0}$ and $\nabla ^{0} \boldsymbol {\nabla } {{\boldsymbol {U}}_0}$.

Inserting the explicit expression for $\hat {{\boldsymbol {T}}}_D$ provided by (B2) in (2.6), with $\nabla ^{0}{{\boldsymbol {U}}_0}$ derived from (2.4), reveals that in the present axisymmetric straining flow the force moment $\hat {{\boldsymbol {T}}}_D=\int _\mathcal {A}{\boldsymbol {x}}(\hat {\boldsymbol {\varSigma }} \boldsymbol {\cdot } {{\boldsymbol {e}}_r})\, \textrm {d} \mathcal {S}$ yields a net force on the particle

(3.3)\begin{equation} {{\boldsymbol {F}}}_{F} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}= \tfrac{45}{4}{\rm \pi}\alpha_b\kappa^{2}(1+\tfrac{9}{8}\kappa+\cdots).\end{equation}

This force tends to repel the particle from the wall, i.e. to make it lag behind the impinging straining flow (2.1). With reference to the well-known Faxén force resulting from the inhomogeneity of the undisturbed velocity field in quadratic flows, this contribution may be thought of as a wall-induced Faxén force. Its origin is made clear by considering the fundamental solution of the ‘direct’ problem in the unbounded case. As the particle is neutrally buoyant, this solution is merely the sum of a stresslet and an irrotational quadrupole. Since the disturbance induced by the stresslet decays as $r^{-2}$, with $r=\|{\boldsymbol {x}}\|$ the distance to the particle centre, its reflection on the wall induces a velocity correction proportional to $\alpha _b\kappa ^{2}\boldsymbol {e}_{\boldsymbol{3}}$ in the vicinity of the particle, yielding an $\mathcal {O}(\kappa ^{2})$-repelling force. Rallabandi et al. (Reference Rallabandi, Hilgenfeldt and Stone2017) made use of bipolar coordinates to evaluate the drag force acting on a spherical particle translating perpendicularly to a curved wall along the axis of an arbitrary non-uniform axisymmetric flow. They found that the linear variation of the flow induces a normal force, say ${\boldsymbol {F}}_{RA} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$, which in the present notations reads $-6{\rm \pi} \mathcal {B}{\boldsymbol {e}}_{\boldsymbol {3}} \boldsymbol {\cdot } \nabla ^{0} {\boldsymbol {U}}_0 \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$. In the limit of large gaps and weak wall curvature, $\mathcal {B}\rightarrow \frac {15}{16}\epsilon ^{-2}$ (their (5.4$a$)). Since $\kappa \approx \epsilon ^{-1}$ in that limit and ${\boldsymbol {e}}_{\boldsymbol {3}} \boldsymbol {\cdot } \nabla ^{0}{\boldsymbol {U}}_0= -2\alpha _b{\boldsymbol {e}}_{\boldsymbol {3}}$ in the present flow, their result may be re-written in the form ${\boldsymbol {F}}_{RA} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}} \rightarrow \frac {45}{4}{\rm \pi} \kappa ^{2}\alpha _b$ in this specific situation, which is exactly the leading-order contribution in (3.3). For $\epsilon =1$ ($\kappa =1/2$), the $\mathcal {O}(\kappa ^{3})$-approximation of ${{\boldsymbol {F}}}_{F}$ provided by (3.3) and the exact solution of Rallabandi et al. (Reference Rallabandi, Hilgenfeldt and Stone2017) differ by less than $13\,\%$.

Evaluating now the contribution of the quadratic flow component $\nabla ^{0} \boldsymbol {\nabla } {{\boldsymbol {U}}_0}$ in (2.6) with the aid of (B3), we find that the corresponding force is

(3.4)\begin{align} {{\boldsymbol {F}}_{F\delta}} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}&= {\rm \pi}\big(1+\tfrac{9}{8}\kappa+\tfrac{81}{64}\kappa^{2}+\tfrac{217}{512} \kappa^{3}\big)(\nabla^{2})^{0}{{\boldsymbol {U}}_0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}+ \tfrac{15}{8}{\rm \pi}\kappa^{3}{\boldsymbol {e}}_{\boldsymbol{3}}\nonumber\\ & \quad \boldsymbol{\cdot} \nabla^{0}({\boldsymbol {e}}_{\boldsymbol{3}} \boldsymbol{\cdot} \boldsymbol{\nabla} {{\boldsymbol {U}}_0}) \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}+\mathcal{O}(\kappa^{4}), \end{align}

where $(\nabla ^{2})^{0}{{\boldsymbol {U}}_0}$ denotes the Laplacian of the carrying velocity field at the position of the particle centre. The corresponding term in (3.4) is the classical Faxén force originating in the curvature of the carrying flow. In the present context, this force is zero when the particle stands in the outer flow region, but increases as it approaches the wall once it is immersed within the boundary layer. A similar force component was computed by Rallabandi et al. (Reference Rallabandi, Hilgenfeldt and Stone2017) who, in present notations, wrote it in the form $3{\rm \pi} \mathcal {D}(\nabla ^{2})^{0}{{\boldsymbol {U}}_0} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$. Figure 3 in their paper indicates that $\mathcal {D}\rightarrow 1/3$ for $\kappa \rightarrow 0$ and increases to $0.65$ for $\kappa =1/2$. The prediction (3.4) fully agrees with this variation, with less than $1\,\%$ difference for $\kappa =1/2$.

The contribution proportional to $\nabla ^{0}({\boldsymbol {e}}_{\boldsymbol {3}} \boldsymbol {\cdot } \boldsymbol {\nabla } {{\boldsymbol {U}}_0}) \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$ in (3.4) results from the anisotropy introduced by the wall at $\mathcal {O}(\kappa ^{3})$ in the solution of the auxiliary problem (see the discussion in appendix B). The force resulting from this contribution was also computed by Rallabandi et al. (Reference Rallabandi, Hilgenfeldt and Stone2017) ($\mathcal {C}$-term in their (4.12) and figure 3). In the present context, the quadratic velocity component in (2.4) is of $\mathcal {O}(\alpha _c)$, hence of $\mathcal {O}(\kappa )$ for a given $\varLambda$ according to (2.5a–c), so that the $\mathcal {O}(\kappa ^{3})$-terms in (3.4) have to be neglected to remain consistent with the general $\mathcal {O}(\kappa ^{4})$-truncation discussed in § 2.4. With ${{\boldsymbol {U}}_0}$ given by (2.4), (3.4) then yields

(3.5)\begin{equation} {{\boldsymbol {F}}_{F\delta}} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} \approx-2{\rm \pi}\big(1+\tfrac{9}{8}\kappa+\tfrac{81}{64}\kappa^{2}\big)\alpha_c. \end{equation}

Finally, taking into account (B1), (3.3) and (3.5) and the definitions of $\alpha _b$ and $\alpha _c$ in (2.5a–c), the zero-$Re$ force balance resulting from (2.6) is found to be

(3.6)\begin{align} 24\left(1+\frac{9}{8}\kappa+\cdots\right){\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}&\approx\alpha\kappa \left\{45\frac{1+2\varLambda}{(1+\varLambda)^{2}}\left(1+\frac{9}{8}\kappa+\cdots\right) \kappa\right.\nonumber\\ &\quad -16\left.\left(1+\frac{9}{8}\kappa+\cdots\right) \frac{\varLambda^{2}}{(1+\varLambda)^{3}}\right\}. \end{align}

The wall-induced force (3.3) resulting from the gradients of the carrying flow is responsible for the first contribution within the curly brackets. It tends to produce a positive slip velocity growing quadratically as the separation decreases. The curvature-induced Faxén force (second term within the curly brackets) acts to reduce this positive slip. However, the resulting behaviour is not entirely intuitive. In the limit of large separations, i.e. $\varLambda \rightarrow 0$, the right-hand side of (3.6) is positive only if $\kappa \lesssim \frac {45}{16}k_\delta ^{2}\varDelta ^{-2}(1-\frac {135}{16}k_\delta \varDelta ^{-1})^{-1}$. So, at a given separation such that $\kappa \ll \varDelta ^{-1}$, only sufficiently large particles experience a positive slip. For instance, with $k_\delta =1$, the slip of a particle 20 times smaller than the boundary-layer characteristic thickness (i.e. such that $\varDelta =20$) is found to be positive for $\kappa \lesssim 0.014$ but is then negative until $\kappa \approx 0.089$ before it becomes positive again for smaller separations. Very close to the wall, $\varLambda$ is large for small particles. Therefore both terms on the right-hand side of (3.6) behave as $1/\varLambda$ in that limit but the large pre-factor of the first of them ensures that the positive driving force dominates. For instance, still with $k_\delta =1$, $\varLambda =2.5$ (respectively $5$) when $\kappa =1/2$ (respectively $1$) for particles corresponding to $\varDelta =5$, so that the positive force is approximately $4.5$ (respectively $7.5$) times larger than the negative one. That the slip velocity predicted by (3.6) is positive whatever the particle size in the limit $\kappa \rightarrow 1$ is of physical interest, although the present theory is not expected to apply in that limit. Since the fluid velocity is still negative (i.e. directed towards the wall) at the position of the particle centre, but the velocity of the particle has to vanish when the latter touches the wall, the actual slip velocity is undoubtedly positive. Obviously, lubrication effects not accounted for in the present theory contribute to slow down the particle as it gets very close to the wall (Li et al. Reference Li, Abbas, Morris, Climent and Magnaudet2020). Nevertheless, what (3.6) reveals is that the longer-range hydrodynamic forces considered here contribute to this slowing down, as they force the slip velocity to be positive and to increase with $\kappa$ for $\kappa \lesssim 1$.

3.2. Comparison with numerical results

Li et al. (Reference Li, Abbas, Morris, Climent and Magnaudet2020) reported results of fully resolved numerical simulations carried out with particles released from rest on the stagnation streamline of a HH flow. Although analyses in their paper focus on ‘large’ particles, some of which with radii of the order of the total boundary-layer total thickness (up to $\varDelta ^{-1}=3.2$), other simulations were run with smaller particles, corresponding to relative sizes $\varDelta ^{-1}$ down to $0.1$ (Q. Li, private communication 2019). Technical details about these simulations are provided in appendix C. Here, we select some of these results obtained with ‘small’ particles to discuss several features of the near-wall variations of the slip velocity ${\boldsymbol {V}}_{S0}$ with the position of the particle, and compare present zero-Reynolds-number predictions (which are in principle only valid for $\varDelta ^{-1}\ll 1$) with those of the full Navier–Stokes equations. In figures 3–5, slip profiles are plotted vs the dimensionless gap $\epsilon =\kappa ^{-1}-1$ to make the physical interpretation easier.

First of all, figure 3$(a)$ compares the numerical slip velocity profile typical of a small particle (with a radius ten times smaller than the total boundary-layer thickness $3\varDelta$) with the prediction of the GMR model. In this case, the strain Reynolds number is $0.09$ and the maximum slip-based Reynolds number is less than $0.03$, so that inertial effects are expected to be negligibly small throughout the particle trajectory. Hence the GMR model (e.g. (48) in Maxey & Riley (Reference Maxey and Riley1983)) reduces to a balance between the viscous drag linearly proportional to ${\boldsymbol {V}}_{S0}$ and the curvature-induced Faxén force proportional to $(\nabla ^{2})^{0}{\boldsymbol {U}}_0$, both of which evaluated as if the particle motion were taking place in an unbounded fluid. In the notations of (2.6), this balance results in

(3.7)\begin{equation} \hat{{\boldsymbol {F}}}_D^{\infty} \boldsymbol{\cdot} {\boldsymbol {V}}_{S0} \approx\tfrac{1}{2}\hat{{\boldsymbol {S}}}_D^{\infty} {\vdots} \nabla^{0}( \boldsymbol{\nabla} {{\boldsymbol {U}}_0}), \end{equation}

with, following (B1) and (B3), $\hat {{\boldsymbol {F}}}_D^{\infty }\equiv \hat {{\boldsymbol {F}}}_D(\kappa \rightarrow 0) =-6{\rm \pi} {\boldsymbol {e}}_{\boldsymbol {3}}$ and $\hat {{\boldsymbol {S}}}_D^{\infty }\equiv \hat {{\boldsymbol {S}}}_D(\kappa \rightarrow 0) =-2{\rm \pi} {{\boldsymbol {I}}}{\boldsymbol {e}}_{\boldsymbol {3}}$. According to (2.4), $\nabla ^{0}( \boldsymbol {\nabla } {\boldsymbol {U}}_0)=(\nabla ^{2})^{0}{\boldsymbol {U}}_0= -2\alpha _c{\boldsymbol {e}}_{\boldsymbol {3}}$ is negative throughout the near-wall region and increases as the wall is approached through the rise of $\alpha _c$. Hence (3.7) predicts that the slip velocity is negative (i.e. the particle leads the fluid) and increases as the gap goes to zero. This is in contradiction with the numerical profile displayed in figure 3$(a)$ which shows that, starting from zero far from the wall, the slip velocity becomes increasingly positive down to the wall.

Obviously the shortcoming of the GMR model in the present context is due to the omission of wall interaction effects. In the present theory, when the particle stands within the boundary layer, the magnitude of these effects is influenced by the shape parameter $k_\delta$ involved in the approximate flow model (2.4). The discussion in § 2.2 suggested that the value $k_\delta =2$ properly describes the outer part of the boundary layer (where the particle stands when the separation distance is larger than $\varDelta$, i.e. $\epsilon >\varDelta -1$), whereas $k_\delta =1$ much better describes the flow profile in the inner region relevant when $\epsilon \lesssim \varDelta -1$. Figure 3$(b)$ shows the predictions of (3.6) for the same small particle obtained with these two values of $k_\delta$; particles with a smaller or larger size behave similarly. First of all, it must be noticed that, unlike the GMR prediction in figure 3$(a)$, both predictions are in qualitative agreement with the numerical slip velocity profile. This emphasizes the crucial role of the repelling wall-induced Faxén force (3.3) in the particle dynamics. Moreover, in line with the earlier discussion in § 2.2, the figure confirms that the predicted profile obtained with $k_\delta =2$ agrees slightly better with numerical data for $\epsilon \gtrsim 2$, while a much better agreement is obtained with $k_\delta =1$ for $\epsilon \lesssim 1.5$. Hence the latter value is to be selected to obtain reliable predictions in the near-wall region, where the slip velocity exhibits large variations with the distance to the wall.

Last, figure 4 compares predictions based on (3.6) (with $k_\delta =1$) with numerical results for three different particle sizes corresponding to $\varDelta ^{-1}=0.3,\,0.4$ and $0.5$, i.e. $\alpha Re=0.09,\,0.16$ and $0.25$, respectively. For each particle, the slip velocity is found to increase sharply as the particle approaches the wall. Moreover, the larger the particle the larger ${\boldsymbol {V}}_{S0}$ is when the dimensionless gap becomes small enough, typically $\epsilon \lesssim 1.5$. These trends are well captured by the viscous prediction. However, (3.6) starts to under-predict ${\boldsymbol {V}}_{S0}$ when the gap is such that $\epsilon \lesssim \varDelta$. More precisely, for an increasing particle size, the viscous theory is found to under-estimate the actual slip velocity at $\epsilon =1$ by $5\,\%$, $17\,\%$ and $22\,\%$, respectively. Therefore, the larger the particle, the stronger the under-estimate of ${\boldsymbol {V}}_{S0}$ is, a clear indication that inertial effects become responsible for an increasing fraction of the slip velocity as the particle size increases. At smaller gaps, the smallest particle displays a peculiar behaviour, since the slight under-estimate observed for $\epsilon \gtrsim 1$ almost vanishes. However, this agreement is presumably fortuitous since the asymptotic expressions involved in (3.6) are barely accurate for such small gaps. We rather suspect that the corresponding simulation is slightly under-resolved in this case, owing to a marginally sufficient number of grid points per particle radius (see appendix C).

Figure 4. Slip velocity in the near-wall region for particles with increasing relative size $\varDelta ^{-1}=0.3$ (green lines), $0.4$ (purple lines), $0.5$ (black lines). Dashed lines show simulation results; solid lines show creeping-flow prediction (3.6) using the undisturbed flow (2.2) with $k_\delta =1$.

4.1. General considerations

The above discussion sheds light on the limitations of the purely viscous force balance (3.6) when the particle size increases and the wall is approached. To extend the validity of the theory toward larger particles, it is mandatory to include inertial corrections. Strictly speaking, only the limit of small-but-finite inertial effects can be tackled theoretically, which keeps the condition (3.2) unchanged. Nevertheless, in practice one may hope the results of such a weakly inertial theory to apply within an extended range of particle sizes satisfying the less restrictive condition $Re\ll \varDelta ^{-2}\lesssim 1$. This is the goal of the developments summarized in the present section.

The force balance (2.6) is valid without any restriction regarding the magnitude of inertial effects. It provides the contribution of the velocity disturbance to these effects in the form of a volume integral over the entire flow domain. Examining the momentum equation for the disturbance under condition (3.1) reveals that inertial terms become comparable to viscous terms at distances of $\mathcal {O}((\alpha Re)^{-1/2})$ from the particle. Hence, provided the latter is close enough to the wall for the condition

(4.1)\begin{equation} \kappa^{-1}\lesssim(\alpha Re)^{-1/2}\iff\kappa^{2}\gtrsim \alpha Re,\end{equation}

to be satisfied, the flow field is properly approximated by the quasi-steady Stokes solution throughout the wall–particle gap. As recognized by Cox & Brenner (Reference Cox and Brenner1968), this in turn implies that in the outer region corresponding to distances $r\gtrsim (\alpha Re)^{-1/2}$ from the particle centre, the disturbance decays faster than in an unbounded domain, owing to the influence of the ‘image’ field that cancels the disturbance at the wall. Because of this faster decay, Cox & Brenner (Reference Cox and Brenner1968) and Cox & Hsu (Reference Cox and Hsu1977) showed that, within a large class of carrying flows, including the family of quadratic flows of interest here, the leading-order inertial corrections can be obtained through a regular perturbation procedure provided the particle is sufficiently close to the wall for (4.1) to hold. Their argument was extended to unsteady situations in M1. Nevertheless, second-order inertial corrections of $\mathcal {O}(Re^{2})$, $\mathcal {O}(\alpha Re^{2})$ and $\mathcal {O}((\alpha Re)^{2})$ remain associated with a singular perturbation, similar to the classical Oseen problem (Proudman & Pearson Reference Proudman and Pearson1957). Therefore, a consistent description of small-but-finite inertial effects may be obtained solely via a regular perturbation procedure only if the leading-order contributions are larger than the second-order ones. Provided (3.1) holds, all the above second-order corrections are smaller than the $\mathcal {O}(\alpha Re)$-terms involved in the volume integral on the right-hand side of (2.6). This is why we concentrate on the first three contributions to this volume integral in what follows.

4.2. Effects of unsteadiness

The inertial force associated with unsteady effects, namely ${{\boldsymbol {F}}}_U=-\alpha Re\int _\mathcal {V}(\hat {{\boldsymbol {U}}}+{\boldsymbol {e}}_{\boldsymbol {3}}) \boldsymbol {\cdot } (\partial {\boldsymbol {u}}/\partial t)\, \textrm {d} \mathcal {V}$ in (2.6), was computed in M1 in the case where unsteadiness arises solely through time variations of the slip velocity. As far as $\alpha$ does not vary (i.e. $\alpha _b=\alpha$ and $\alpha _c=0$ in (2.4)) this contribution does not depend on the specific spatial structure of the carrying flow. Consequently, results derived in M1 apply directly to the present problem. In particular, (17b) of M1 provides the ${\boldsymbol {e}}_{\boldsymbol {3}}$-component of the unsteady contribution ${{\boldsymbol {F}}}_U$ in the form

(4.2)\begin{equation} {{\boldsymbol {F}}}_U \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} =-\frac{9}{4}{\rm \pi} \alpha Re\left(\kappa^{-1}-\frac{13}{108}+ \mathcal{O}(\kappa)\right)\frac{ \textrm{d} {\boldsymbol {V}}_{S0}}{ \textrm{d} t} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}.\end{equation}

This result only holds if the condition (4.1) is satisfied, which makes the limit $\kappa \rightarrow 0$ irrelevant. To understand the physical origin of this force, it is useful to evaluate its order of magnitude at the maximum wall–particle distance for which (4.2) is valid, i.e. $\kappa ^{-1}\sim (\alpha Re)^{-1/2}$. In this situation, the leading-order term in (4.2) is of $\mathcal {O}((\alpha Re)^{1/2})$. This is reminiscent of the magnitude of the ‘unsteady Oseen force’ computed by Lovalenti & Brady (Reference Lovalenti and Brady1993) in the case of a particle with a finite slip Reynolds number accelerating or decelerating in an unbounded flow domain with the fluid at rest at infinity. Indeed, these authors found the unsteady Oseen force to be of $\mathcal {O}((St Re)^{1/2})$. Since $St\equiv \alpha$ here, the magnitude of ${{\boldsymbol {F}}}_U$ predicted by (4.2) for $\kappa ^{-1}\sim (\alpha Re)^{-1/2}$ is similar to that of the inertial force they computed. This is a strong indication that ${\boldsymbol {F}}_U$ is not a force that originates from the wall, but is merely what is left from the unsteady Oseen force as the wall is approached. Starting from a magnitude of $\mathcal {O}((\alpha Re)^{1/2})$ for large separation distances ($\kappa \rightarrow 0$), the unsteady Oseen force is gradually weakened by the wall as $\kappa$ increases and becomes of $\mathcal {O}(\alpha Re)$ for small separations ($\kappa \rightarrow 1$). The prediction (4.2) expresses this near-wall variation for moderate-to-small separation distances such that $\kappa \gtrsim (\alpha Re)^{1/2}$. Lovalenti & Brady (Reference Lovalenti and Brady1993) showed that the unsteady Oseen force primarily results from the time variations of the wake structure due to the particle acceleration or deceleration. Any disturbance originating in a time variation of ${\boldsymbol {V}}_{S0}$ requires a finite time to diffuse away from the particle surface and reach the wake region. For this reason, the expression for this force in the case of an unbounded fluid domain involves a convolution integral. The corresponding kernel, inertial by nature, is distinct from that associated with the Basset–Boussinesq force, which originates in the unsteady diffusion of vorticity close to the particle. The near-wall situation considered here, combined with the slow evolution implied by the restriction $ReSt\equiv \alpha Re\ll 1$, drastically reduces the above finite memory effect. Indeed, these slow variations imply that the leading-order contribution to the disturbance ${\boldsymbol {u}}$ is governed by the quasi-steady Stokes equation at distances less than $(ReSt)^{-1/2}$. Since the dominant contribution to the near-wall unsteady effects is provided by a regular perturbation procedure, only this quasi-steady disturbance is involved, making the resulting force only dependent on the current acceleration $\textrm {d} {\boldsymbol {V}}_{S0}/ \textrm {d} t$. The same happens with the contribution due to the time rate-of-change of the near-particle disturbance, which usually yields the Basset–Boussinesq force and is here also encapsulated in the $\mathcal {O}(\kappa ^{-1})$-term of (4.2), while the added-mass contribution and second-order corrections associated with the unsteady Oseen force form the $\mathcal {O}(\kappa ^{0})$-term. Hence the entire contribution of unsteady effects at any time is expressible solely in terms of the current acceleration $\textrm {d} {\boldsymbol {V}}_{S0}/ \textrm {d} t$ when the particle gets close enough to the wall and time variations are slow enough for the condition $\alpha Re\ll 1$ to be satisfied.

Note that, since $\kappa$ is less than 1 by definition, the $\kappa ^{-1}$-term is always dominant in (4.2). Hence ${{\boldsymbol {F}}}_U$ always tends to lower the relative acceleration $\textrm {d} {\boldsymbol {V}}_{S0}/ \textrm {d} t$, just as the familiar added-mass effect does.

When the particle stands within the boundary layer, other sources of unsteadiness arise through the time-dependent strain rates $\alpha _b(t)$ and $\alpha _c(t)$. Since $\textrm {d}\kappa / \textrm {d} t=-\alpha ^{-1}\kappa ^{2}{\boldsymbol {V}} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$, the definitions of $\alpha _b$ and $\alpha _c$ in (2.5a–c) imply that $\textrm {d} \alpha _b/ \textrm {d} t=2\kappa ({\varLambda ^{2}}/{(1+\varLambda )^{3}}) {\boldsymbol {V}} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$ and $\textrm {d} \alpha _c/ \textrm {d} t=-6\kappa ^{2}({\varLambda ^{2}}/{(1+\varLambda )^{4}}) {\boldsymbol {V}} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$. To express the corresponding contributions to the force, it is convenient to split the particle velocity in the form ${\boldsymbol {V}}={\boldsymbol {V}}_{S0}+{\boldsymbol {U}}_0^{0}$, with ${\boldsymbol {U}}_0^{0}$ as given in (2.5a–c). Keeping in mind that $\alpha Re\varLambda ^{2}=k_\delta ^{-2}\kappa ^{2}$ and that $\varLambda =\mathcal {O}(1)$ for $\kappa =\mathcal {O}(\varDelta ^{-1})$, variations of $\alpha _b(t)$ are found to contribute to generate a non-zero slip through an $\mathcal {O}(\kappa ^{2})$-source term (since ${\boldsymbol {U}}_0^{0}\propto \kappa ^{-1}$), and an $\mathcal {O}(\kappa ^{3})$-correction to the pre-factor of the force contribution proportional to ${\boldsymbol {V}}_{S0}$, i.e. to the drag coefficient. Variations of $\alpha _c(t)$ provide contributions smaller by an $\mathcal {O}({\kappa }/({1+\varLambda }))$-factor. Let us first consider the force resulting from $\alpha _b(t)$-variations. The procedure employed to compute this contribution and all those to come in this section is summarized in appendix D. According to (D5) and the considerations that follow, this force is found to be

(4.3)\begin{align} {{\boldsymbol {F}}}_{U\delta} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} &\approx-\frac{15}{4}{\rm \pi}\alpha Re\frac{ \textrm{d} \alpha_b}{ \textrm{d} t}\left(1+\frac{9}{8} \kappa\right)({\boldsymbol {U}}_0^{0}+{\boldsymbol {V}}_{S0}) \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}\nonumber\\ &\approx\frac{15{\rm \pi}}{2k_\delta^{2}}\frac{\kappa^{2}}{(1+\varLambda)^{3}} \left\{\frac{2\alpha}{1+\varLambda}\left(1+\frac{9}{8}\kappa\right)-\kappa {\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}\right\}, \end{align}

where the second approximation is obtained by incorporating the explicit expressions for $\textrm {d} \alpha _b/ \textrm {d} t$ and ${\boldsymbol {U}}_0^{0}$.

The source term in (4.3) (first term within braces) is positive, contributing to make the particle lag behind the fluid. That a body translating steadily perpendicular to a wall generates a non-zero normal force directly through the time variation of its position is not uncommon. In particular, this is the case in the inviscid limit, where the increase of the fluid volume entrained by the body as it gets closer to the wall results in a repulsive force, just as in (4.3) (Milne-Thomson Reference Milne-Thomson1962).

At this point it is useful to compare the magnitude of the $\mathcal {O}(\kappa ^{3})$-terms in (4.3) with those involved in the zero-$Re$ approximation (3.6), keeping in mind that $\varLambda$ becomes large when $\kappa \rightarrow 1$. To fix ideas, let us consider a particle $10$ times smaller than the boundary-layer thickness, i.e. $\varDelta =10$, standing at the position corresponding to $\kappa =1/2$. With $k_\delta =1$ one then has $\varLambda =5$. Consequently the ratio of the $\mathcal {O}(\kappa ^{3})$-source term in (4.3) to its counterpart in the curvature-induced Faxén term in (3.6) is of $\mathcal {O}({1}/{\varLambda ^{2}(1+\varLambda )})\approx 0.007$. Similarly, the ratio of the $\mathcal {O}(\kappa ^{3})$-drag correction in (4.3) (second term within braces) to the corresponding term in (3.6) is of $\mathcal {O}({1}/{(1+\varLambda )^{3}})\approx 0.005$. These estimates indicate that $\mathcal {O}(\kappa ^{3})$-corrections weighted by a ${1}/ \varLambda^{n}$-factor with $n\ge 3$ are negligibly small at the present order of approximation. For this reason, such terms will be systematically dropped in what follows, and only the leading-order $\mathcal {O}(\kappa ^{2})$-source term present in (4.3) will be kept when ${{\boldsymbol {F}}}_{U\delta }$ will be inserted in the final force balance. As mentioned above, contributions involved in the force correction resulting from variations of $\alpha _c(t)$ are smaller than those induced by $\alpha _b(t)$-variations by an $\mathcal {O}({\kappa }/({1+\varLambda }))$-factor. Hence the previous argument shows that all of them are negligible at the present order of approximation. For the same reason, the last two terms within parentheses on the right-hand side of (2.8) also provide a negligible contribution to the inertial force ${\boldsymbol {F}}_0$.

4.3. Effects of advective transport

Within the framework of the above conditions, especially (3.1), the other contributions to be considered in the volume integral of the right-hand side of (2.6) are the advective terms proportional to $\alpha Re$, which result from the quasi-linear contribution ${\boldsymbol {u}} \boldsymbol {\cdot } \boldsymbol {\nabla } {{\boldsymbol {U}}}_0({\boldsymbol {x}},t) +({\boldsymbol {U}}_0({\boldsymbol {x}},t)-{\boldsymbol {U}}_0^{0}(t)) \boldsymbol {\cdot } \boldsymbol {\nabla } {\boldsymbol {u}}$ in the disturbance momentum equation. Due to the ambient strain, the leading-order contribution to the disturbance arises from a stresslet. For this reason, its advective transport by the linear flow component (and vice versa) yields a contribution of $\mathcal {O}(\alpha _b^{2}Re)$. Cox & Hsu (Reference Cox and Hsu1977) evaluated a similar term in the case of a uniformly sheared carrying flow, where it yields a net lift force on the particle; their prediction was later confirmed by Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994). Although the scaling of this force with respect to $\alpha _b$, $Re$ and $\kappa$ does not depend on the specific linear base flow under consideration, the pre-factor that determines its actual strength does. To the best of our knowledge, this contribution, say ${{\boldsymbol {F}}}_{I}$, has not been evaluated so far in the axisymmetric straining flow (2.1). Based on (D6) and the considerations that follow, the final result valid up to $\mathcal {O}(\kappa )$ is

(4.4)\begin{equation} {{\boldsymbol {F}}}_{I} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} \approx\big(1+\tfrac{9}{8}\kappa\big)\tfrac{75}{16}{\rm \pi}\alpha_b^{2} Re. \end{equation}

The force ${{\boldsymbol {F}}}_{I}$ arises due to the asymmetry created by the wall in the transport of the stresslet by the straining flow and vice versa. In a linear shear flow, the counterpart of ${{\boldsymbol {F}}}_{I}$ involves a pre-factor $\frac {55}{96}{\rm \pi}$ instead of $\frac {75}{16}{\rm \pi}$ (Cox & Hsu Reference Cox and Hsu1977). Consequently, the magnitude of ${{\boldsymbol {F}}}_{I}$ is approximately $8.2$ times larger in the present axisymmetric straining flow than in a uniform shear with strength $\alpha _b$. Similar to that of ${{\boldsymbol {F}}}_U$, the above prediction for ${{\boldsymbol {F}}}_{I}$ only holds up to a maximum separation of $\mathcal {O}((\alpha _b Re)^{-1/2})$. For larger separations, ${{\boldsymbol {F}}}_{I}$ must tend to zero as $\kappa \rightarrow 0$ but this decay cannot be captured by the regular expansion procedure employed here.

Within the boundary layer, several additional contributions arise, due to the presence of the quadratic flow component in (2.4). A detailed examination of their respective magnitudes reveals that the largest one is provided by the transport of the leading $\mathcal {O}(\alpha _b)$-stresslet by the quadratic $\mathcal {O}(\alpha _c)$-flow component and vice versa. This mechanism results in an $\mathcal {O}(\kappa ^{-1}\alpha _b\alpha _cRe)$-force, the formal expression of which takes the form (D7). As outlined in appendix D, numerical evaluation of this expression and truncation considerations based on the argument discussed at the end of § 4.2 lead to

(4.5)\begin{equation} {{\boldsymbol {F}}}_{I\delta} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} \approx\frac{85}{8}\alpha_b\alpha_cRe\kappa^{-1}=\frac{85}{4}\alpha \frac{1+2\varLambda}{(1+\varLambda)^{5}}\kappa^{2}, \end{equation}

where the last equality results from the definitions of $\alpha _b$ and $\alpha _c$ in (2.5a–c) and the relation $\varLambda ^{2}=\kappa ^{2}/(k_\delta ^{2}\alpha Re)$.

Another inertial effect results from the transport of the Stokeslet associated with the slip velocity by the ambient straining flow and vice versa. This advective process yields a force whose leading-order contribution is proportional to $\alpha _b Re\kappa ^{-1}{{\boldsymbol {V}}}_{S0}$. Since the zero-$Re$ force balance (3.6) suggests that the slip velocity is of $\mathcal {O}(\kappa ^{2}\alpha _b)$, this force correction is expected to be of $\mathcal {O}(\kappa \alpha _b^{2}Re)$, i.e. smaller than ${{\boldsymbol {F}}}_I$ by an $\mathcal {O}(\kappa )$-order of magnitude. Nevertheless, for $\kappa =\mathcal {O}(\alpha _b Re)^{1/2}$, $\alpha _b Re\kappa ^{-1}{{\boldsymbol {V}}}_{S0}=(\alpha _b Re)^{1/2}{{\boldsymbol {V}}}_{S0}$. Hence this effect provides a correction to the drag which is for instance larger than the second term in the inertial force ${{\boldsymbol {F}}}_0$ in (2.7) and must be included for consistency. Details regarding the computation of this contribution are also provided in appendix D (see (D8) and the comments that follow). Its final expression is found to be

(4.6)\begin{equation} {{\boldsymbol {F}}}_{D\alpha} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} \approx\frac{\rm \pi}{16}\alpha_b Re\left(45\kappa^{-1}-\frac{1861}{60}\right) {\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}. \end{equation}

For similar reasons, the contribution resulting from the transport of the Stokeslet associated with the slip velocity by the quadratic flow and vice versa must also be considered. The corresponding force is proportional to $\alpha _c Re\kappa ^{-2}{{\boldsymbol {V}}}_{S0}\sim ({\kappa }/{k_\delta ^{2}(1 +\varLambda )^{3}}){{\boldsymbol {V}}}_{S0}$. As outlined in appendix D, evaluating the corresponding volume integral and truncating the result in line with the discussion in § 4.2 yields

(4.7)\begin{align} {{\boldsymbol {F}}}_{D\alpha\delta} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} \approx\frac{3}{4}{\rm \pi}\alpha_cRe\kappa^{-2}\left(1+\frac{9}{4}\kappa\right) {\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}= \frac{3}{2}{\rm \pi}\frac{\kappa}{k_\delta^{2}(1+\varLambda)^{3}}\left(1+\frac{9}{4}\kappa\right) {\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}.\end{align}

It is worth noting that, although the boundary-layer contributions (4.3), (4.5) and (4.7) are inertial by nature, the strain Reynolds number $\alpha Re$ no longer appears in their final expression once ${\boldsymbol {U}}_{0}^{0}$, $\alpha _b$ and $\alpha _c$ have been replaced by their definitions as given in (2.5a–c). This is because they are proportional to $\textrm {d} \alpha _b/ \textrm {d} t$ or $\alpha _c$, both of which are proportional to $\kappa \varLambda ^{2}\propto \kappa ^{3}\varDelta ^{2}$, and $\varDelta$ equals $(\alpha Re)^{-1/2}$.

4.4. Final force balance

All contributions computed in §§ 4.2 and 4.3 may finally be gathered to enhance (3.6) with effects of finite fluid inertia. The updated force balance can be expressed in the form

(4.8)\begin{align} ({{\boldsymbol {F}}}_0-{{\boldsymbol {F}}}_U) \boldsymbol{\cdot} {\boldsymbol{e}_3}- \hat{{\boldsymbol {F}}}_D \boldsymbol{\cdot} {\boldsymbol {V}}_{S0}-({\boldsymbol {F}}_{D\alpha}+ {\boldsymbol {F}}_{D\alpha\delta}) \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} =({{\boldsymbol {F}}}_{F}+{{\boldsymbol {F}}}_{F\delta}+{{\boldsymbol {F}}}_I +{{\boldsymbol {F}}}_{I\delta}+{{\boldsymbol {F}}}_{U\delta}) \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}},\end{align}

with ${{\boldsymbol {F}}}_0$ as given in (2.8), and ${{\boldsymbol {F}}}_U$, ${{\boldsymbol {F}}}_{U\delta }$, ${{\boldsymbol {F}}}_I$, ${{\boldsymbol {F}}}_{I\delta }$, ${\boldsymbol {F}}_{D\alpha }$ and ${\boldsymbol {F}}_{D\alpha \delta }$ taken from (4.2)–(4.7). We then define the ratios

(4.9a–d)\begin{align} A_\varLambda=\frac{1+2\varLambda}{(1+\varLambda)^{2}},\quad B_\varLambda=\frac{1}{k_\delta^{2}(1+\varLambda)^{3}},\quad C_\varLambda=\frac{1}{k_\delta^{2}(1+\varLambda)^{4}},\quad D_\varLambda=\frac{\varLambda^{2}}{(1+\varLambda)^{3}}.\end{align}

Boundary-layer effects become negligible in the double limit $\varLambda \ll 1$ (i.e. the separation $\kappa ^{-1}$ is very large compared to $\varDelta /k_\delta$) and $k_\delta \rightarrow \infty$ (i.e. the fluid layer within which the no-slip condition at the wall significantly influences the carrying flow is much thinner than $\varDelta$), in which case $A_\varLambda \rightarrow 1$ and $B_\varLambda$, $C_\varLambda$ and $D_\varLambda \rightarrow 0$. Nevertheless, condition (4.1) implies that the inertial corrections derived in §§ 4.2 and 4.3 are valid only for $\kappa \gtrsim \varDelta ^{-1}$, i.e. ${\varLambda \gtrsim k_\delta ^{-1}}$. Therefore predictions involving these corrections are not expected to be relevant for small values of $\varLambda$. As already mentioned, $\varLambda$ is large when $\kappa \rightarrow 1$ since we are considering small particles. Consequently all four ratios in (4.9a–d) go through $\mathcal {O}(1)$-values in some intermediate range of $\kappa$ and become small in the limit $\kappa \rightarrow 1$. The final approximate force balance (4.8) takes the form

(4.10)\begin{align} &9\alpha Re\left(\kappa^{-1}+\frac{17}{36}\right)\frac{ \textrm{d} {\boldsymbol {V}}_{S0}}{ \textrm{d} t} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} +24\left\{1+\frac{9}{8}\kappa+\frac{81}{64}\kappa^{2}+ \frac{473}{512}\kappa^{3}-\frac{1}{4}\kappa \left(1+\frac{9}{4}\kappa\right)B_\varLambda\right.\nonumber\\ &\quad-\left.\frac{15}{32}\alpha Re\left(\kappa^{-1}+\frac{4421}{2700}\right)A_\varLambda\right\} {\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} \approx\alpha \left\{\kappa^{2}A_\varLambda\left(45\left(1+\frac{9}{8}\kappa\right)+ 85B_\varLambda\right)\right.\nonumber\\ &\quad +60\kappa^{2}C_\varLambda-16\kappa\left.\left(1+\frac{9}{8}\kappa+ \frac{81}{64}\kappa^{2}\right) D_\varLambda+\frac{75}{4}\alpha Re\left(1+\frac{9}{8} \kappa\right)A_\varLambda^{2}\right\}. \end{align}

Inertial forces ${{\boldsymbol {F}}}_{I}$ and ${{\boldsymbol {F}}}_{I\delta }$ resulting from the advective transport of the stresslet by the linear and quadratic flow components, respectively, and ${{\boldsymbol {F}}}_{U\delta }$ resulting from the time variation of the straining rate about the particle, all provide positive contributions to the right-hand side of (4.10). Hence they all contribute to make the particle lag behind the fluid (since ${\boldsymbol {U}}_{0}^{0} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}<0$), similar to the wall-induced Faxén force ${{\boldsymbol {F}}}_{F}$. Only the curvature-induced Faxén force ${{\boldsymbol {F}}}_{F\delta }$ tends to make the particle lead the fluid; the smaller the particle the larger the relative influence of this force at a given distance from the wall. Consider for instance a particle standing a distance $\varDelta$ from the wall, i.e. $\varLambda =1$. The right-hand side of (4.10) then becomes negative only if ${\varDelta }^{-1}\lesssim 0.037$. Comparing with the prediction provided by the zero-$Re$ approximation (3.6) indicates that inertial effects lower the critical size of particles for which the driving force changes sign at this location by a factor of $1.6$. Alternatively, inertial effects may be said to enhance the positive slip between the particle and the fluid. Moreover, all inertial terms that contribute to the $B_\varLambda$- and $\alpha ReA_\varLambda$-terms in the pre-factor of the ${\boldsymbol {V}}_{S0}$-term, namely forces ${{\boldsymbol {F}}}_{D\alpha }$ and ${{\boldsymbol {F}}}_{D\delta }$ resulting from the transport of the Stokeslet by the linear and quadratic flow components, respectively, and the advective part of the force ${{\boldsymbol {F}}}_{0}$ due to the acceleration of the undisturbed flow, decrease the drag coefficient. Hence they all tend to enhance the slip velocity for a given value of the overall source term, reinforcing the role of inertia in the slip increase. Incidentally, this points out to the fact that, unlike the usual inertial increase of the drag coefficient encountered in the classical Oseen problem (Proudman & Pearson Reference Proudman and Pearson1957), inertial corrections in the HH flow lower the drag coefficient.

4.5. Comparison with numerical results

Unlike the purely viscous solution (3.6), predictions involving inertial corrections are only meaningful within a limited separation range, since (4.10) is expected to be valid only in the near-wall region such that $\kappa \gtrsim \varDelta ^{-1}$. Consequently, the larger the particle the smaller the separation range over which the comparison between predictions of (4.10) and results of fully resolved simulations is relevant. As (4.10) is a first-order differential equation with respect to ${\boldsymbol {V}}_{S0}$, an initial condition for the slip velocity is required. If the expressions obtained for the inertial corrections were valid up to large separations, ${\boldsymbol {V}}_{S0}(t=0)=\boldsymbol {0}$ in the limit $\kappa \rightarrow 0$ would be a natural choice. Given their limited range of validity, an alternative is required. Without results from fully resolved simulations available, the most obvious choice is to use the slip velocity provided by the viscous prediction (3.6) to initialize the determination of ${\boldsymbol {V}}_{S0}$ at a position $\kappa _i$ such that $\kappa _i=\mathcal {O}((\alpha Re)^{1/2})=\mathcal {O}(\varDelta ^{-1})$. Since figure 2 indicates that the carrying flow model (2.4) correctly fits the actual HH profile with $k_\delta =1$ up to a distance to the wall of approximately $1.5\varDelta$, we select $\kappa _i=(1.5\varDelta )^{-1}$, i.e. $\epsilon _i=3\varDelta /2-1$, a position at which the creeping-flow approximation (3.6) and the fully resolved simulation predict close values of the slip velocity. Based on this initialization protocol, figure 5$(a)$ compares predictions of (4.10) with simulation results for the three particles already considered in figure 4.

Figure 5. Predictions for the slip velocity in the near-wall region for particles with increasing relative size $\varDelta ^{-1}=0.3$ (green lines), $0.4$ (purple lines), $0.5$ (black lines); all predictions are based on the undisturbed flow (2.2) with $k_\delta =1$. $(a)$ initial gap $\epsilon _i=3\varDelta /2-1$; $(b)$ $\epsilon _i=\varDelta -1$. Dashed lines show simulation results; thick solid lines show finite-$Re$ prediction from (4.10); thin solid lines in $(a)$ show creeping-flow prediction (3.6) (the thick purple line and the thin black line almost overlap).

In all cases, inertial effects are seen to increase the slip velocity at a given separation distance (compare the predictions corresponding to the thin and thick solid lines for each particle). This is because all inertial terms on the right-hand side of (4.10) are positive, while all inertial corrections to the drag coefficient in the left-hand side are negative. Moreover, since $\alpha Re=\varDelta ^{-2}$ and all coefficients $A_\varLambda -D_\varLambda$ are decreasing functions of $\varLambda$ (hence of $\varDelta$), increasing the particle size, i.e. $\varDelta ^{-1}$, makes all inertial terms on the right-hand side increase at a given $\kappa$. Because of this, the larger the particle the stronger the inertial correction to the slip at a given distance from the wall is. Both features act to compensate for the deficiencies of the purely viscous force balance (3.6) analysed in § 3.2. This makes the weakly inertial prediction based on (4.10) significantly closer to the numerical solution for moderate-to-small gaps (the agreement deteriorates at small gaps for the smallest particle, owing to the peculiar behaviour of the numerical prediction mentioned in § 3.2).

Nevertheless, a closer look at the slip velocity profiles in figure 5$(a)$ shows that the slope $\textrm {d}({\boldsymbol {V}}_{S0} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}})/ \textrm {d}\kappa$ is under-estimated for $\epsilon \lesssim \epsilon _i$, which maintains the predicted values of ${\boldsymbol {V}}_{S0}$ slightly below those found in the simulations down to $\epsilon \approx \epsilon _i/3$. To get some insight into the origin of this shortcoming, it is of interest to consider the predictions of (4.10) obtained by selecting a smaller initial separation, $\kappa _i=\varDelta ^{-1}$, i.e. $\epsilon _i=\varDelta -1$. Since the viscous force balance (3.6) significantly under-estimates the actual slip velocity at this smaller separation (see figure 4), we employed the value ${\boldsymbol {V}}_{S0}(\epsilon _i=\varDelta -1)$ provided by the fully resolved simulations as initial condition in this case. As figure 5$(b)$ shows, the prediction resulting from (4.10) now closely agrees with the simulation results for $\epsilon \le \epsilon _i$, especially for the largest two particles. The agreement extends down to a dimensionless gap $\epsilon \approx 0.3$ $(\kappa \approx 3/4)$, significantly beyond the expected limit of validity ($\epsilon \approx 1$) of the truncated asymptotic expression of the ‘auxiliary’ solution. The reason why the slope $\textrm {d}({\boldsymbol {V}}_{S0} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}})/\textrm {d}\kappa$ is correctly predicted when $\epsilon _i=\varDelta -1$ but is under-estimated when $\epsilon _i=3\varDelta /2-1$ is readily identified in (4.10), keeping in mind that the term that absorbs the local variations of ${\boldsymbol {V}}_{S0}$ is the unsteady force proportional to $\textrm {d} {\boldsymbol {V}}_{S0}/ \textrm {d} t$. As discussed in § 4.2, the expression (4.2) for this contribution is dominated by a term proportional to $\kappa ^{-1}$. The growth of this term with the separation distance is only correct as far as the wall stands in the inner region of the disturbance. For larger separations, it becomes unphysical, since the entire contribution must tend toward the finite ‘unsteady Oseen force’ computed by Lovalenti & Brady (Reference Lovalenti and Brady1993) when $\kappa \rightarrow 0$. This unphysical growth makes this force over-estimated for $\kappa \lesssim (\alpha Re)^{1/2}$ and is responsible for the slight under-estimate of ${\boldsymbol {V}}_{S0}$ noticed for $\epsilon \lesssim \epsilon _i$ in figure 5$(a)$. This analysis leads to the conclusion that the technical bottleneck that restricts most the validity of (4.10) towards larger separations is the limited range of validity of (4.2). This calls for a specific study aimed at deriving the proper expression for the unsteady Oseen force in the case where the particle is already influenced by the wall but the latter stands in the outer region of the disturbance.

5.1. Preliminaries

Up to now, we constrained the particle to move along the symmetry axis of the HH flow. Although the simulations of Li et al. (Reference Li, Abbas, Morris, Climent and Magnaudet2020) only addressed this case, it represents a quite specific situation. The techniques used to obtain the various wall-normal forces in §§ 3 and 4 may also be applied to predict the wall-parallel slip velocity component and the modifications of the slip wall-normal component when the particle stands an arbitrary time-dependent radial distance from the axis, say $\rho _0(t)$, as sketched in figure 6. In order for the flow to satisfy the no-slip boundary condition at the wall whatever ${\boldsymbol {x}}_{0\parallel }=\rho _0(t){\boldsymbol {e}}_{\boldsymbol {1}}$, the radial position ${\boldsymbol {x}}_\parallel$ involved in (2.2) has to be changed into ${\boldsymbol {x}}_\parallel +{\boldsymbol {x}}_{0\parallel }$ (hence ${\boldsymbol {x}}$ into ${\boldsymbol {x}}+{\boldsymbol {x}}_{0\parallel }$). With this transformation, the undisturbed flow field in the vicinity of the particle ($|x_3| \ll (1+\varLambda )/\kappa$) takes the form

(5.1)\begin{align} {\boldsymbol {U}}_0({\boldsymbol {x}},t)&= {\boldsymbol {U}}_{0}^{\rho_0}(t)+\{\alpha_b(t)({{\boldsymbol {x}}}- 3x_3{\boldsymbol {e}}_{\boldsymbol{3}})+\alpha_c(t)x_3({{\boldsymbol {x}}}- 2x_3{\boldsymbol {e}}_{\boldsymbol{3}})\}\nonumber\\ &\quad +\rho_0(t)\{\alpha_c(t)x_3+ \alpha_d(t)x_3^{2}\}{\boldsymbol {e}}_{\boldsymbol{1}}+\cdots, \end{align}

with

(5.2a,b)\begin{equation} {\boldsymbol {U}}_{0}^{\rho_0}(t)={\boldsymbol {U}}_{0}^{0}(t)+ \alpha_b\rho_0{\boldsymbol {e}}_{\boldsymbol{1}}\quad \mbox{and}\quad \alpha_d(t)=-3\alpha\kappa^{2}\frac{\varLambda^{2}}{(1+\varLambda)^{4}},\end{equation}

with $\alpha _b(t)$, $\alpha _c(t)$ and ${\boldsymbol {U}}_{0}^{0}(t)$ being still as given in (2.5a–c). Compared to (2.4), (5.1) reveals that, at a radial position $\rho _0$ from the axis, the undisturbed flow comprises an additional shear component proportional to $\rho _0(t)\alpha _c(t)$, a parabolic component proportional to $\rho _0(t)\alpha _d(t)$ etc., all of which correspond to a radial flow whose intensity increases linearly with $\rho _0$. As time elapses, the particle is transported away from the axis $\rho _0=0$ by the carrying flow. Therefore $\rho _0(t)$ increases, which makes the radial component in (5.1) increase at the expanse of the axial wall-normal component. In other words, the flow in the vicinity of the particle looks more and more like a wall-parallel shear flow.

Figure 6. Sketch of the configuration with the particle released some distance from the axis of the HH flow.

Let us provisionally consider that the particle stands beyond the boundary layer. Compared to the axisymmetric configuration contemplated so far, there is no change in the strain-induced disturbance, since the straining motion is identical to that in (2.1). In particular, the disturbance does not depend on the radial position $\rho _0$. Consequently, all forces which only depend on the strain rate and distance to the wall are unchanged. This remark enables us to conclude that no source term for the parallel slip component can exist as far as the particle has not entered the boundary layer, even though inertial effects are taken into account. Indeed, the two contributions ${\boldsymbol {F}}_F$ in (3.3) and ${\boldsymbol {F}}_I$ in (4.4) result from the interaction of the $\rho _0$-independent stresslet with the wall, so that any non-zero ${\boldsymbol {e}}_{\boldsymbol {1}}$-component of one of these forces would be $\rho _0$-independent. Since no radial force component can exist when the particle stands on the flow axis, such a component remains null whatever $\rho _0$.

To obtain the various contributions to the radial force within the boundary layer, we need to project the reciprocal theorem onto the ${\boldsymbol {e}}_{\boldsymbol {1}}$-direction. The result is similar to (2.6), except that the unit vector ${\boldsymbol {e}}_{\boldsymbol {3}}$ has to be replaced with ${\boldsymbol {e}}_{\boldsymbol {1}}$ everywhere, and the relevant auxiliary problem now corresponds to a sphere steadily translating with unit velocity in the ${\boldsymbol {e}}_{\boldsymbol {1}}$-direction. Solving this problem with the techniques described in appendix B yields an approximation of the corresponding velocity field, $\hat {{\boldsymbol {U}}}_{\parallel }$, accurate up to terms of $\mathcal {O}(\kappa ^{3})$. The surface quantities $\hat {{\boldsymbol {F}}}_{D\parallel }$, $\hat {{\boldsymbol {T}}}_{D\parallel }$ and $\hat {{\boldsymbol {S}}}_{D\parallel }$ which are the counterparts of $\hat {{\boldsymbol {F}}}_{D}$, $\hat {{\boldsymbol {T}}}_{D}$ and $\hat {{\boldsymbol {S}}}_{D}$ in (2.6) may then be deduced; the corresponding evaluations result in (B4)–(B6). Last, the radial component of the inertial body force due to the relative acceleration between the particle and the undisturbed flow is

(5.3)\begin{align} {\boldsymbol {F}}_0 \boldsymbol{\cdot} {{\boldsymbol {e}}_{\boldsymbol{1}}}= \frac{4}{3}{\rm \pi} Re\left\{ \left(\alpha\frac{ \textrm{d} {\boldsymbol {V}}_{S0}}{ \textrm{d} t}+ \alpha_b{\boldsymbol {V}}_{S0}\right) \boldsymbol{\cdot} {{\boldsymbol {e}}_{\boldsymbol{1}}} \!+\! \rho_0\alpha_c{\boldsymbol {V}}_{S0}\boldsymbol{\cdot}{\boldsymbol {e}}_{\boldsymbol{3}} - \frac{1}{5}\left(\frac{ \textrm{d} (\rho_0\alpha_d)}{ \textrm{d} t}-3\rho_0\alpha_b\alpha_d\right)\right\}\!, \end{align}

with ${\boldsymbol {V}}_{S0}(t)={\boldsymbol {V}}(t)-{\boldsymbol {U}}_{0}^{\rho_0}(t)$.

5.2. Stokes-flow approximation

Applying (B5) and (B6) to (5.1), the ${\boldsymbol {e}}_{\boldsymbol {1}}$-projection of the reciprocal theorem indicates that the $\rho _0$-dependent radial component of the carrying flow generates a non-zero force such that

(5.4)\begin{equation} {\boldsymbol {F}}_{F\delta} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}} \approx-\frac{3}{8}{\rm \pi}\alpha_c\rho_0\kappa\left\{5\kappa+\frac{8}{1+ \varLambda}\left(1+\frac{9}{16}\kappa\right)\right\}.\end{equation}

Both terms on the right-hand side of (5.4) provide a negative contribution to ${\boldsymbol {F}}_{F\delta }$, making the particle lag behind the fluid in the ${\boldsymbol {e}}_{\boldsymbol {1}}$-direction. Balancing (5.4) with the drag force $-\hat {{\boldsymbol {F}}}_{D\parallel } \boldsymbol {\cdot } {\boldsymbol {V}}_{S0}$ evaluated with the aid of (B4), the creeping-flow approximation indicates that, for small $\kappa$, the radial slip velocity is primarily due to the first term on the right-hand side of (5.4). This yields

(5.5)\begin{equation} {\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}} \approx-\alpha\rho_0\kappa^{2}\frac{\varLambda^{2}}{(1+\varLambda)^{4}}.\end{equation}

Since $\varLambda =\kappa \varDelta /k_\delta$, the radial slip in (5.5), which originates from the curvature-induced Faxén force, is of $\mathcal {O}(\kappa ^{4}\varDelta ^{2})$ compared to the radial component of the primary straining flow. As the particle gets closer to the wall, $\varLambda$ becomes large. There, the dominant contribution to the right-hand side of (5.4) is provided by the second term, i.e. the wall-induced Faxén force associated with the radial shear flow $\rho _0\alpha _cx_3{\boldsymbol {e}}_{\boldsymbol {1}}$ in (5.1), and the relative slip becomes of $\mathcal {O}(\kappa ^{2}\varDelta ^{-1})$.

5.3. Inertial corrections

Similar to the route followed in § 4, we first compute inertial forces due to unsteadiness and then consider advective contributions.

First of all, the radial component of the force ${\boldsymbol {F}}_U$ due to possible time variations in the radial slip velocity was computed in M1 and was found to be

(5.6)\begin{equation} {{\boldsymbol {F}}}_U \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}} =-\frac{9}{4}{\rm \pi} \alpha Re\left(3\kappa^{-1}+\frac{217}{216}+ \mathcal{O}(\kappa)\right)\frac{ \textrm{d} {\boldsymbol {V}}_{S0}}{ \textrm{d} t} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}}. \end{equation}

This result still applies here, as it is independent of the background flow.

The argument provided in § 5.1 indicates that none of the inertial contributions resulting from the axisymmetric component of the carrying flow in (5.1) can have a non-zero radial component. Hence, only the radial flow $\rho _0(t)\{\alpha _c(t)x_3+\alpha _d(t)x_3^{2}\}{\boldsymbol {e}}_{\boldsymbol {1}}$ in (5.1) may provide non-zero radial forces arising from unsteadiness or advective transport. Moreover, contributions due to the parabolic component $\rho _0\alpha _dx_3^{2}{\boldsymbol {e}}_{\boldsymbol {1}}$ are smaller by a factor of $\mathcal {O}(\kappa (1+\varLambda )^{-1})$ than those due to the shear component $\rho _0\alpha _cx_3{\boldsymbol {e}}_{\boldsymbol {1}}$. Consequently, following the argument discussed in § 4.2, only the latter needs to be considered at the present order of approximation. To compute the corresponding inertial corrections, the relevant shear Reynolds number has to be small. As the strength of the shear in (5.1) is $\rho _0\alpha _c$ and the magnitude of $\alpha _c$ cannot exceed values of $\mathcal {O}(\alpha )$, this condition implies $\rho _0\alpha Re\ll 1$, i.e.

(5.7)\begin{equation} \rho_0\ll\varDelta^{2}. \end{equation}

Due to the presence of the radial shear component in the carrying flow, the disturbance now comprises a stresslet and an irrotational quadrupole which are not present when the particle stands on the axis of the HH flow. Close to the particle, the velocity disturbance induced by this stresslet, say ${\boldsymbol {u}}_{str\parallel }$, has the form ${x_1x_3{\boldsymbol {x}}}/{r^{5}}$ while that induced by the stresslet associated with the primary axisymmetric strain, say ${\boldsymbol {u}}_{str\perp }$, has the form ${{\boldsymbol {x}}}/{r^{3}}-3({x_3^{2}{\boldsymbol {x}}}/{r^{5}})$.

Similar to (4.3), the evolution of the radial and wall-normal particle positions result in a net force, as it makes the strength of the ${\boldsymbol {u}}_{str\parallel }$-contribution vary over time through the time variations of $\rho _0\alpha _c$. Following the results and approximations discussed at the end of appendix D, the leading-order contribution to this force is found to be

(5.8)\begin{equation} {{\boldsymbol {F}}}_{U\delta} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}} \approx\frac{33}{4}{\rm \pi}\alpha\rho_0\kappa^{3}\frac{7+2\varLambda}{k_\delta^{2}(1+\varLambda)^{5}}.\end{equation}

Time variations of $\rho _0\alpha _d$ induce a qualitatively similar contribution, but it is negligible at the present order of approximation for the reason mentioned above.

Let us now consider advective contributions. Gradients of the axisymmetric disturbance ${\boldsymbol {u}}_{str\perp }$ are advected by the shear flow and vice versa, which yields a radial inertial force, say ${{\boldsymbol {F}}}_{I\delta } \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {1}}$. As reported in appendix D, evaluation of (D12) yields

(5.9)\begin{equation} {{\boldsymbol {F}}}_{I\delta} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}}= \frac{15}{16}{\rm \pi}\alpha\frac{\rho_0}{k_\delta^{2}}\kappa^{3}\left(1+\frac{9}{16}\kappa\right) \frac{(1+2\varLambda)}{(1+\varLambda)^{5}}. \end{equation}

Here also we disregard the $\mathcal {O}(\kappa /(1+\varLambda ))$-smaller contribution of the parabolic radial flow component in (5.1) to the advective transport of ${\boldsymbol {u}}_{str\perp }$.

Similar to (4.6) in the wall-normal direction, advection of the Stokeslet-type disturbance associated with the radial slip velocity ${\boldsymbol {V}}_{S0} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {1}}$ by the base straining flow (and vice versa) results in an inertial correction to the radial drag coefficient. According to (D11) and the comments that follow, evaluation of this contribution up to $\mathcal {O}(\kappa ^{0})$-terms yields

(5.10)\begin{equation} {{\boldsymbol {F}}}_{D\alpha} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}}=- \frac{\rm \pi}{32}\alpha Re\frac{(1+2\varLambda)}{(1+\varLambda)^{2}} \left(99\kappa^{-1}+\frac{29\,237}{120}+\mathcal{O}(\kappa)\right) {\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}}.\end{equation}

Similarly, we must consider the force resulting from the transport of the same disturbance by the radial shear flow and vice versa. However, the eigenvectors of the velocity gradient ${\boldsymbol {e}}_{\boldsymbol {3}}{\boldsymbol {e}}_{\boldsymbol {1}}$ corresponding to the radial shear flow are inclined by an angle of $\pm {\rm \pi}/4$ with respect to the $({\boldsymbol {e}}_{\boldsymbol {1}},\,{\boldsymbol {e}}_{\boldsymbol {3}})$ axes. For this reason, this advective transport results in a transverse force along the ${\boldsymbol {e}}_{\boldsymbol {3}}$-direction, not in a correction to the drag. For the same reason, the transport of the disturbance associated with the wall-normal slip ${\boldsymbol {V}}_{S0} \boldsymbol {\cdot } {\boldsymbol {e}}_{\boldsymbol {3}}$ by the shear flow yields a radial force along the ${\boldsymbol {e}}_{\boldsymbol {1}}$-direction. The first of these contributions was computed to leading order by Cox & Hsu (Reference Cox and Hsu1977), and to second order by Lovalenti in an appendix to Cherukat & McLaughlin (Reference Cherukat and McLaughlin1994). The second was computed in M1 and M2; its second-order term was amended by Magnaudet (Reference Magnaudet2004). Making use of these results and noting that the shear strength in (5.1) is $\rho _0\alpha _c$, the lift force resulting from both contributions may be written in the form

(5.11)\begin{equation} {{\boldsymbol {F}}}_{L\delta}=-\frac{9}{16}{\rm \pi}\frac{\rho_0}{k_\delta^{2}}\kappa^{2} \frac{1}{(1+\varLambda)^{3}}\!\left\{\left(5+\frac{253}{432}\kappa \right)({\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}) {\boldsymbol {e}}_{\boldsymbol{1}}+\left(\frac{11}{3}+\frac{443}{144}\kappa \right)({\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}}) {\boldsymbol {e}}_{\boldsymbol{3}}\right\}\!.\end{equation}

Last, in a shear flow, the stresslet ${\boldsymbol {u}}_{str\parallel }$ is known to induce an inertial force perpendicular to the streamlines, i.e. a lift force acting in the ${\boldsymbol {e}}_{\boldsymbol {3}}$-direction. With a shear rate $\alpha$ and a particle free to rotate as it is here, this contribution, first computed at leading order by Cox & Hsu (Reference Cox and Hsu1977), yields a force $\frac {55}{96}{\rm \pi} \alpha ^{2}Re{\boldsymbol {e}}_{\boldsymbol {3}}+\mathcal {O}(\kappa )$. Considering again that the shear rate in (5.1) is $\rho _0\alpha _c$ and taking into account the $1+\frac {9}{8}\kappa$ multiplicative factor resulting from the reflection of the Stokeslet at stake, this lift force, say ${\boldsymbol {F}}_{L\alpha ^{2}}$, is here

(5.12)\begin{equation} {\boldsymbol {F}}_{L\alpha^{2}} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}} \approx\frac{55}{24}{\rm \pi}\alpha \left(\frac{\rho_0}{k_\delta}\right)^{2}\kappa^{4}\left(1+ \frac{9}{8}\kappa\right)\frac{\varLambda^{2}}{(1+\varLambda)^{6}}.\end{equation}

Although (5.12) reveals a $\kappa ^{4}$-dependence of ${\boldsymbol {F}}_{L\alpha ^{2}}$, $\rho _0$ may become large, which makes this force potentially significant when $\kappa$ increases, as discussed below.

5.4. Final force balance

The contributions derived in §§ 5.2 and 5.3 may finally be gathered to obtain the differential equation governing the evolution of the radial slip. Defining $E_\varLambda ={1}/({1+\varLambda })$ and $F_\varLambda =({7+2\varLambda })/{(1+\varLambda )^{2}}$ and applying the same truncation rules as in § 4, this force balance may be recast in the form

(5.13)\begin{align} &9 \alpha Re\left(3\kappa^{-1}+\frac{115}{72}\right)\frac{ \textrm{d} {\boldsymbol {V}}_{S0}}{ \textrm{d} t} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}}+24\left\{1+\frac{9}{16}\kappa+ \frac{81}{256}\kappa^{2}+ \frac{217}{4096}\kappa^{3}\right.\nonumber\\ & \quad +\left.\frac{33}{64}\alpha Re\left(\kappa^{-1}+ \frac{34357}{11880}\right)A_\varLambda\right\}{\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}}\approx 3\alpha\rho_0\kappa^{2}\left\{\kappa B_\varLambda \left(11F_\varLambda+\frac{5}{4}A_\varLambda\right)\right.\nonumber\\ &\quad \left.-D_\varLambda \left(5\kappa+8E_\varLambda\left(1+\frac{9}{16}\kappa\right)\right)\right\}- \frac{45}{4}\rho_0\kappa^{2}B_\varLambda{\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{3}}, \end{align}

with $A_\varLambda$, $B_\varLambda$ and $D_\varLambda$ as defined in (4.9a–d).

Moreover, (5.12) and the ${\boldsymbol {e}}_{\boldsymbol {3}}$-projection of (5.11) represent lift contributions which alter the evolution of the wall-normal slip velocity. More specifically, at an arbitrary radial position $\rho _0(t)$, the right-hand side of (4.10) is supplemented by the $\rho _0$-dependent inertial contribution

(5.14)\begin{equation} F_{L3\rho_0}=\rho_0\kappa^{2}B_\varLambda\left\{\frac{55}{6}\alpha \rho_0\kappa^{2}D_\varLambda\left(1+\frac{9}{8}\kappa\right) -\frac{9}{4} \left(\frac{11}{3}+\frac{443}{144}\kappa\right){\boldsymbol {V}}_{S0} \boldsymbol{\cdot} {\boldsymbol {e}}_{\boldsymbol{1}}\right\}.\end{equation}

Terms involving the slip velocity on the right-hand side of (5.13) and (5.14) couple the evolution of the slip along the ${\boldsymbol {e}}_{\boldsymbol {1}}$- and ${\boldsymbol {e}}_{\boldsymbol {3}}$-axes. In a given direction, they tend to produce a slip with opposite sign in the perpendicular direction. This is similar to the familiar Saffman lift force (Saffman Reference Saffman1965) which drives a particle leading the fluid toward the low-velocity side in a shear flow. Unlike the situation noticed in (4.10), the inertial correction to the drag coefficient is positive in (5.13), similar to the usual Oseen correction. Inertial effects proportional to $\alpha \rho _0$ in (5.13) and (5.14) provide positive source terms that tend to make the particle lead the fluid. However, present expressions for the inertial corrections are valid only for separations such that $\kappa \gtrsim \varDelta ^{-1}$, so that $\varLambda$ is of $\mathcal {O}(1)$ or larger. Because of this, negative (i.e. inward) zero-Reynolds-number effects corresponding to the two types of Faxén forces already present in (5.4) always dominate on the right-hand side of (5.13), and inertial forces (5.8) and (5.10) are only able to reduce the relative inward motion between the particle and the fluid.

In contrast, the first term on the right-hand side of (5.14), which results from the lift force (5.12), may become large when the radial distance increases, owing to its $\rho _0^{2}$-dependence. Since it behaves as $(\rho _0/\varDelta ^{2})^{2}$ very close to the wall ($\varLambda \gg 1$), it is of $\mathcal {O}(\varDelta ^{-1})$ for $\rho _0\sim \varDelta ^{3/2}$, similar to the two Faxén contributions that dominate the right-hand side of the wall-normal force balance (4.10). It even becomes the dominant source term if $\rho _0$ stands in the range $\varDelta ^{3/2}\ll \rho _0\ll \varDelta ^{2}$. Indeed, at such large radial distances, the shear flow component in (5.1) has become larger than the base straining flow. For this reason, the particle motion in the ${\boldsymbol {e}}_{\boldsymbol {3}}$-direction is dominated by lift effects associated with the shear, rather than by the interaction of the axisymmetric straining flow with the wall. In other terms, what (4.10) supplemented with (5.14) describes is the wall-normal dynamics of a particle in a carrying flow which gradually evolves from a bi-axial straining flow at small $\rho _0$ to a nearly wall-parallel uniform shear flow at large $\rho _0$. While this wall-normal dynamics is initially primarily governed by the wall-induced and curvature-induced Faxén forces (3.3) and (3.5), it becomes eventually dominated by the inertial shear-induced lift force (5.12).