3.1. Non-dimensionalisation

It is convenient to redefine the governing equation (3.1) in terms of dimensionless length and time scales. Therefore, we scale the spatial $x$- and $z$-coordinates by the particle diameter $d_p$,

(3.2a,b)\begin{equation} x^{*} = \frac{x}{d_{p}},\quad z^{*} = \frac{z}{d_{p}}. \end{equation}

We further scale the time $t$ by the shear rate at the initial liquid concentration peak location near the free surface, ${{{\dot {\gamma }}_{c}}^{s}}$ (evaluation shown in appendix A),

(3.3)\begin{equation} {t^{*}} = t{{{\dot{\gamma}}_{c}}^{s}}. \end{equation}

All other variables are scaled accordingly, with superscript $^{*}$ denoting the scaled variables. Working with the non-dimensional variables, (3.1) is re-written as

(3.4)\begin{equation} \frac{\partial{Q}}{\partial t^{*}} = {C_{liq}}^{*}\left({\frac{\partial^{2}{(\dot{\gamma}^{*}Q)}}{\partial {x^{*}}^{2}} + \frac{\partial^{2}{(\dot{\gamma}^{*}Q)}}{\partial {z^{*}}^{2}}}\right). \end{equation}

All the variables henceforth are non-dimensionalised and we omit the superscript $^{*}$ subsequently.

3.2. Numerical scheme

Numerical methods suitable for the solution of the fluid transport equations are a matter of extensive research in computational fluid dynamics. Their application is predominant in various research fields such as geophysical fluid dynamics (Durran & Mobbs Reference Durran and Mobbs2001; Baumgarten & Kamrin Reference Baumgarten and Kamrin2019), hydrological processes (Igboekwe & Achi Reference Igboekwe and Achi2011), reactor flow (Hastaoglu & Abba Reference Hastaoglu and Abba1996) etc. In the Eulerian solution of equations, difficulties arise because of the dual advective–diffusive nature of the transport equation. When the transport is advection or drift dominated, the equation behaves as a first-order hyperbolic equation, but when the transport is diffusion dominated, the equation behaves as a second-order parabolic equation. To accurately model the drift-diffusion transport, the numerical scheme must be able to handle the mixed parabolic–hyperbolic character of the systems. Eulerian models that have grids fixed in space have a number of difficulties when transport is drift dominated. These include numerical diffusion, oscillations, instabilities and peak clipping because of the numerical representation of advection terms in the transport equation. To avoid instabilities, we choose the finite volume method (FVM) with semi-implicit time stepping (Patankar Reference Patankar1980) to solve the liquid transport equations in this paper. The details of the numerical methods and discretisation are described in appendix B.

The grid sizes are chosen as $400$ in the $x$-direction and $100$ in the $z$-direction, which is equivalent to a grid spacing of ${\textrm {d}x} = \textrm {d} z \approx 0.08$. The solutions are checked for different grid sizes (the finest resolution tested is ${\textrm {d}x} = \textrm {d} z \approx 0.008$) and the trend is maintained both qualitatively and quantitatively. The time step is chosen as $\textrm {d} t = 10^{-4}$. These values meet the necessary Courant–Friedrichs–Lewy (CFL) condition for the stability of the solutions with $\mathrm {CFL} = 0.59< \mathrm {CFL}_{max}$. The details of the calculation of the CFL number is elaborated in appendix C. It is important to emphasise that this is not a sufficient condition for stability and other stability conditions are generally more restrictive than the CFL condition.

4.1. Liquid concentration profile

Initially, the liquid concentration $Q\mid _{t = 0} = Q_{0} = 6.9\times 10^{-3}$ is homogeneous; thus, the time derivative of the liquid concentration $Q$ is proportional to the second spatial derivative of $\dot {\gamma }$ i.e. $\nabla ^{2}\dot {\gamma }$. Thus, the liquid concentration initially decreases in the shear band, where the second derivative of $\dot {\gamma }$ is negative, and a liquid concentration peak is formed at the edge of the shear band, where the second derivative of $\dot {\gamma }$ is largest.

4.2. Liquid concentration peak

The location of the liquid concentration peak could be defined as the location of the maximum of the liquid concentration profile. However, this has the following disadvantages: as the grid resolution is finite, the liquid concentration peak can only be located with a limited accuracy, resulting in an undesirable stepwise definition. To avoid these effects, the location of the liquid concentration peak in the right half of the shear cell is defined as the centroid of the liquid concentration profile above the initial value $Q_{0}$. Thus the definition of $x_{c}$ is given as

(4.1)\begin{equation} x_{c} = \frac{\int_0^{L/2}{x\tilde{Q}{{\rm d}x}}}{\int_0^{L/2}{\tilde{Q}{{\rm d}x}}} \quad {\rm where}\ \tilde{Q} = {max}(Q - Q_{0}, 0), \end{equation}

where $L = 60$ is the width of the shear cell in non-dimensional form. The location of the liquid concentration peak is well approximated by this definition for the continuum model. However, if $Q_{0}$ is more scattered, as in the case of data from DPM simulation, we define $\tilde {Q} = {max}(Q - (1+\epsilon )Q_{0}, 0)$, where $\epsilon \ll 1$ is the standard deviation of $Q_{0}$ in the data.

4.3. Accumulated liquid concentration

The concentration of liquid accumulated in the edge of the shear band is given as the integral of the liquid concentration profile lying above the value $Q_{0}$ as follows:

(4.2)\begin{equation} \phi = \int_0^{L/2}{\tilde{Q}{\textrm{d}x}}. \end{equation}

Ideally, there is no loss of liquid from the system (e.g. due to vaporisation). The conservation of liquid volume requires that the volume of liquid accumulated in the edge of the shear band equals the volume of liquid drained from the centre of the shear band. Figure 2 shows a typical liquid concentration profile $Q$ as a function of $x$ at a fixed height, $z = 3.6$ ($W = 3$) for the full liquid migration model given by solving (3.4) after $t = 3.6$.

Figure 2. Liquid concentration $Q$ as a function of $x$ at $z \approx 3.6$ ($W \approx 3$) by solving (3.4) after $t = 3.6$. The black solid line indicates the initial location of the liquid concentration peak ${x_{c}^{0}}$ and the black dashed line indicates the location of the liquid concentration peak ${x_{c}}$ after time $t = 3.6$. Here, $\phi$ represents the accumulated liquid concentration after time $t$.

We distinguish the two main features of the liquid concentration profile, namely the peak concentration location $x_{c}$ and the accumulated liquid concentration $\phi$. Further, the dynamic characteristics of these features are the subject of our discussion as and when we simplify the governing equation for liquid migration in the following sections §§ 5, 6 and 7.

5.1. Comparison of the full and simplified models

Figure 3(a,b) shows contour plots of liquid concentration as a function of space $x$ and $z$ after $t = 14.4$ solved for the full model and the simplified model, respectively. As observed from the figures, both the models have a minimum liquid concentration at the shear-band centre, i.e. at $x = 0$, corresponding to the region with dark blue colour. A high liquid concentration is developed at the edges of the shear band for both the models, corresponding to the narrow region with dark red colour. The region close to the boundary, represented by the cyan colour, is not yet affected by the liquid migration and the liquid concentration is unchanged. A height-wise gradient of the liquid concentration is observed inside the peak location for the full model, represented in figure 3(a). This is due to the liquid diffusion in the $z$-direction. Unlike the full model, the simplified model shows a rather uniform liquid concentration inside the peak location, represented in figure 3(b). Starting to shear from a uniform concentration of liquid $Q_{0}$, the initial location of the liquid concentration peak after a single time step $t = { \textrm {d} t}$, is given by ${x_{c}^{0}}$. This location is obtained analytically for the simplified model from (5.1) as ${x_{c}^{0}} = \sqrt {1.5}W$ where $\partial ^{2}{\dot {\gamma }}/\partial {{x}^{2}}$ is maximum. Note that, for the full model, ${x_{c}^{0}}$ is at the location where $\nabla ^{2}\dot {\gamma }$ is maximum. The red solid line in figure 3(b) shows the locus of ${x_{c}^{0}}$ at different heights for the simplified model. The liquid concentration propagates away from the shear band with time and the red dashed lines in figure 3(a,b) represent the location of the peak after $t = 14.4$ for the two models. Note that this is an intermediate time chosen to show the liquid concentration profile when the initial liquid redistribution phase has ended and the liquid migration phenomenon has started.

Figure 3. Contour plot of liquid concentration in Cartesian shear cell after time $t = 14.4$ for (a) full model and (b) simplified model. The red solid line in (b) indicates the initial locus of the liquid concentration peak ${x_{c}^{0}}$ at different heights. The red dashed lines in (a,b) denote the liquid concentration peak locus ${x_{c}}$ obtained at different heights from (4.1).

Next, we do a quantitative analysis of the liquid concentration peak ${x_{c}}$ and the accumulated liquid concentration $\phi$ as a function of time at different heights picked from figure 3(a,b). The results of ${x_{c}}$ and $\phi$ at $z = 3.6$ ($W = 3$) (blue lines) and $z = 5.4$ ($W = 3.5$) (red lines) are shown in figure 4(a,b), respectively. The key parameters ${x_{c}}$ and $\phi$ both increase with time, indicating that the process has not reached a steady state. The solutions of the full and the simplified models are represented by the solid and the dash-dotted lines, respectively. While there is only a difference of less than $5\,\%$ in ${x_{c}}$ between the full and the simplified models, $\phi$ is significantly affected by the diffusion in the $z$-direction. The accumulated liquid concentration increases by $33\,\%$ more for the simplified model as compared to the full model after $t = 72$, closer to the base of the shear cell (blue lines), where the vertical shear gradient is stronger. The location of the liquid peak position ${x_{c}}$ is insignificantly affected by diffusion in the $z$-direction as only the horizontal diffusion shifts the liquid peak away from the shear band. Thus, $x_c$ is captured with up to more than $95\,\%$ accuracy through the simplified model and is the key parameter that we want to focus on by further analysis in this paper. Thus, an approach towards a simplified model solution is likely to deviate quantitatively from the full model in the first place, especially close to the bottom of the shear cell. However, the location of liquid concentration peak ${x_{c}}$ can be readily captured, which in itself is an important feature of the liquid concentration profile. The location of the liquid concentration peak deviates for the simplified model as compared with the full model by less than $5\,\%$. Although the horizontal $x$-diffusion is primarily shifting the liquid concentration peak, the component of the $z$-diffusion that is perpendicular to the liquid concentration profile is also contributing to the process. Hence, a difference is observed between the location of the liquid concentration peak between the full and the simplified models.

Figure 4. (a) Location of the liquid concentration peak ${x_{c}}$ and (b) accumulated liquid concentration $\phi$ as a function of $t$ as obtained from (3.4) for full model and (5.1) for simplified model, respectively, at two different heights $z = 5.4$ and $z = 3.6$.

Next, we investigate the velocity of propagation of the liquid concentration peak ${v_{c}}$ as a function of the peak location ${x_{c}}$. Figure 5 shows the dependence of ${v_{c}}$ on ${x_{c}}$ at three different heights for the full model (solid lines) and simplified model (dashed lines). The propagation velocity of the peak position decreases with increasing ${x_{c}}$ as the peak moves away from the shear band. Note that the initial peak locations ${x_{c}^{0}}$ are different for the full and the simplified models and hence the ${x_{c}}$ ranges are also different for the two models. Initially, the liquid peak is not well developed (for small $x_c$), resulting in a subtle peak whose location is difficult to analyse. These data for the initial time steps are eliminated from our analysis. The velocity of the simplified model (dashed lines) at a lower height, close to the split position, deviates from the behaviour of the full model (solid lines) by up to $20\,\%$ at $z = 2.7$. This is due to the absence of diffusion in the $z$-direction, which is more prominent at a lower height, close to the split position. The effect of diffusion in the $z$-direction is weak near the surface and thus the velocity profiles for the full and the simplified models almost collapse close to the free surface at $z = 5.4$.

Figure 5. Liquid concentration peak propagation velocity ${v_{c}}$ as a function of the liquid concentration peak location ${x_c}$ for different heights $z = 2.7$, $3.6$ and $5.4$, denoted by different colours for the full model (solid lines) and simplified model (dash-dotted lines).

It is observed that the trajectory of the location of the liquid concentration peak ${x_{c}}$ can be predicted from the simplified model with an accuracy of $95\,\%$ as compared with that of the full model. The change of velocity of propagation of the liquid concentration peak location ${v_{c}}$ as a function of the local shear rate is closely predicted by both the full and the simplified models near the free surface, but deviates significantly near the bottom of the shear cell. The variation of the accumulated liquid concentration for the full and the simplified models as a function of time is closer near the free surface, but deviates by approximately $20\,\%$ near the bottom where the shear rate is higher. To summarise, the deviation of predictions of accumulated liquid concentration given by the simplified model from the full model is higher where the local shear rate is higher. Also, the liquid peak propagation velocity is proportional to the local shear rate, with a zero velocity corresponding to zero shear rate, indicating that the liquid migration is a dynamic process which is solely shear driven.

The simplification of the model allows further analysis and the development of analytical solutions. We propose to show analytical solutions for the simplified model with suitable transformations of the equation in § 6.1. We choose an intermediate height $z = 3.6$, where the effect of the local shear rate is moderate and thus the results of analytical predictions are closer to the results of the full continuum model.

6.1. Transformation of the equation: drift and diffusion

Next, we aim to transform (5.1) into a form that is analytically more treatable. We follow the approach of Risken (Reference Risken1989) and apply a coordinate transformation,

(6.1)\begin{equation} \xi(x) = \int_0^{x}\frac{1}{\sqrt{{C_{liq}}{\dot{\gamma}(x')}}}{\textrm{d}x}'. \end{equation}

The liquid distribution in the transformed coordinate, $Q'(t,\xi )$, is thus given by

(6.2)\begin{equation} Q^{\prime}=\frac{{ \textrm{d}}x}{{ \textrm{d}}\xi} Q= \sqrt{C_{liq}\dot\gamma}Q. \end{equation}

Applying this change of variables to (5.1) yields a diffusion and a drift term,

(6.3)\begin{equation} \frac{\partial{{Q'}}}{\partial{t}} ={-}\underbrace{\frac{\partial{{D'}{Q'}}}{\partial{\xi}}}_{Drift} + \underbrace{\frac{\partial^{2}{{Q'}}}{\partial{{\xi}^{2}}}}_{Diffusion}. \end{equation}

This equation has a constant diffusion coefficient (equal to 1) and a variable drift coefficient,

(6.4)\begin{equation} {D'} = \frac{{ \textrm{d}}^{2}\xi}{{{ \textrm{d}}x}^{2}}{C_{liq}}{\dot{\gamma}}. \end{equation}

Applying the coordinate transformation used in the paper has the advantage of a constant diffusion coefficient, which in our opinion results in a split that is better to analyse and understand. Moreover, this form of decomposition adopted by us to analyse the liquid migration phenomenon is rather a novel approach compared with the conventional chain rule decomposition. In order to see the individual contributions of the drift and diffusion terms on the overall liquid transfer, we separate the drift and diffusion processes and show analytical solutions for each in §§ 6.1.1 and 6.1.2, respectively.

6.1.1. Drift

In order to measure the contribution of the drift term to the liquid transport, we neglect the diffusion term in (6.3), obtaining the simpler equation

(6.5)\begin{equation} \frac{\partial{{{Q_{drift}}'}}}{\partial{t}} ={-}\frac{\partial {D'}{{Q_{drift}}'}}{\partial{\xi}}. \end{equation}

Using (6.2) and introducing $E(x)=D'\sqrt {C_{liq}\dot \gamma (x)}$, we write (6.5) in terms of the original $x$-coordinate

(6.6)\begin{equation} \frac{\partial{{{Q_{drift}}}}}{\partial{t}}={-} \frac{\partial{E {Q_{drift}}}}{\partial{x}}. \end{equation}

Next, we define a new variable $R(t,x) = E(x){Q_{drift}}(t,x)$, which yields

(6.7)\begin{equation} \frac{\partial{R}}{\partial{t}} ={-}E \frac{\partial{R}}{\partial{x}}. \end{equation}

The general solution of (6.7) is given by

(6.8)\begin{equation} R(t, x) = {R_{0}}\left(A(x)-t\right), \end{equation}

where $A$ is an anti-derivative,

(6.9)\begin{equation} A(x)=\int \frac{1}{E(x)} {\textrm{d}x}, \end{equation}

and ${R_{0}}$ is defined such the initial condition, $R(0,x)=E(x)Q_0$, is satisfied,

(6.10)\begin{equation} R_0(x)=E(A^{{-}1}(x))Q_0. \end{equation}

Transforming back to the original variable ${Q_{drift}}$, we obtain the analytic solution

(6.11)\begin{equation} {Q_{drift}}(t,x) = \frac{R(t,x)}{E(x)} = \frac{E(x_0)}{E(x)}Q_0 \quad \mbox{with }x_0=A^{{-}1}(A(x)-t). \end{equation}

Note, this analytic solution is valid for any shear-rate profile $\dot \gamma (x)$. Substituting $\dot {\gamma }$ from (2.2) into the definitions of $A$ and $E$, (6.9) and (6.12a,b), we obtain

(6.12a,b)\begin{equation} E(x)= \frac{C}{2} x \exp\left(-\frac{x^{2}}{W^{2}}\right),\quad A(x)= \frac{{Ei}({x^{2}}/{W^{2}})}{C}, \end{equation}

where $C=2 C_{liq}V W^{-3}\sqrt {2/{\rm \pi} }$, and ${Ei}$ is the exponential integral (Chiccoli, Lorenzutta & Maino Reference Chiccoli, Lorenzutta and Maino1988), a special function satisfying $\textrm {d}\,{Ei}(x)/{\textrm {d}x} = \exp (x)/x$. Thus,

(6.13)\begin{equation} {Q_{drift}}(t,x) = \frac{{x_{0}}\exp(-(x_{0}/W)^{2})}{x\exp{(-{(x/W)}^{2})}}Q_0\quad \mbox{with }{x_{0}} = W\sqrt{Ei^{{-}1}\left(Ei({x^{2}}/{W^{2}})-C t\right)}. \end{equation}

The plots of $Q_{drift}^{\prime }(t,\xi )$ in the transformed coordinate and $Q_{drift}(t,x)$ in the original coordinate are shown in figures 6(a) and 6(b), respectively. The initial condition, $Q^{\prime }_{drift} = Q_0\sqrt {C_{liq}\dot {\gamma }(x)}$ is simply a Gaussian function of $x$. The initial peak location of $Q^{\prime }_{drift}$ at $\xi = 0$ is as shown in figure 6(a). One can see from the inset of figure 6(b) that, initially, $x_0=x$, hence $Q=Q_0$. For large values of $t$, $x_0\approx 0$ for small $x$ (hence $Q\approx 0$) and $x_0\approx x$ for large $x$ (hence $Q\approx Q_0$); in between, the facts that $x_0 < x$ and $E(x)$ has a maximum ensures there is a peak value. As observed from figure 6(b), drift induces the liquid concentration peak $x_c$ to move further away from the shear-band centre, leading to complete rapid drying of the shear band and surroundings by pushing the liquid to the peak region. As a result, we observe the liquid concentration profile forming a dry shear band that is surrounded by a wet region, with a sharp liquid concentration peak between the dry and wet regions, as shown in figure 6(b). We have verified that the analytical solution $Q_{drift}(t,x)$ given by (6.13) agrees with the numerical solutions of (6.7) (results not shown here). Note that (6.1)–(6.5) are valid for any shear-rate profile $\dot \gamma (x)$. We only apply the specific value of $\dot \gamma (x)$ in (6.6)–(6.12a,b) to obtain an analytical solutions.

Figure 6. (a) Solutions $Q_{drift}^{\prime }(t,\xi )$, inset: $E(x)$ and (b) solutions $Q_{drift}(t,x)$, inset: $x_0(x)$ to the drift equation, given by (6.13).

Based on this analysis, we can derive an estimate, $x_e(t)$, of the peak position. First, we define we define $x_e^{0}$ as the peak location of $E(x)$, thus $E'(x_e^{0})=0$, with $'$ denoting the derivative with respect to the variable $x$. Next, we define $x_e(t)$ for $t>0$ such that $x_0(t,x_e)=x_e^{0}$. Note that $x_e>x_e^{0}$, since $x_0$ is monotonically increasing in $x$, see the inset of figure 6(b). Further, note that $E'(x_e^{0})<0$, since $E$ is monotonically decreasing for $x>x_e^{0}$, see the inset of figure 6(a). We will now show that $x_c^{0} < x_e < x_c$: substituting $x=x_e$ into (6.13) we get

(6.14)\begin{equation} {Q_{drift}}(t,x_e) = \frac{E(x_e^{0})}{E(x_e)}Q_0. \end{equation}

Since $E(x_e^{0})$ is the maximum value of $E$, we get ${Q_{drift}}(t,x_e)>Q_0$, and thus $x_e>x_0^{c}$. Next, we show $x_e < x_c$: Differentiating (6.13) and substituting $x=x_e$, we get

(6.15)\begin{equation} \frac{\partial {Q_{drift}}}{\partial x}(t,x_e) = \frac{E'(x_e^{0})x_0'(t,x_e)E(x_e)-E(x_e^{0})E'(x_e)}{E(x_e)^{2}}. \end{equation}

The first term in the numerator is zero, since $E_0(x_e^{0}) = 0$. The second term in the numerator is positive, since $E_0(x_e) < 0$ and $E(x) > 0$ for all $x \in R$. Thus, $Q_{drift}'(t,x_e)>0$, which implies $x_e < x_c$. Therefore, $x_e\in [x_c^{0},x_c]$. Thus, $x_e$ yields a (lower) estimate of the peak location, in particular for large times, as we observe that the peak narrows over time. Note that $x_e^{0}=x_0(t,x_e)=A^{-1}(A(x_e)-t)$, thus we get the analytic expression $x_e=A^{-1}(t+A(x_e^{0}))$. Thus, $x_e(t)$ has the shape of the inverted exponential integral, shifted to the right by a constant. A plot of $x_e$ is given in figure 7. As observed in the figure, the peak $x_e$ moves away from the shear band with increasing time, leading to drying of the shear band. Thus, the final behaviour of the solution for the drift operator is to push the dry front to infinity, resulting in a completely dry domain everywhere, however, this is a very slow process, as apparent from the figure.

Figure 7. Plot of the peak estimate $x_e$.

6.1.2. Diffusion

In order to measure the contribution of the diffusion term to the liquid transport, we neglect the drift term in (6.3)

(6.16)\begin{equation} \frac{\partial Q_{diffusion}'}{\partial{t}} = \frac{\partial^{2} Q_{diffusion}'}{\partial{{\xi}^{2}}}. \end{equation}

An analytical solution of (6.16) is given by the convolution of the initial condition, $Q_{0}' = Q_{0}\sqrt {C_{liq}\dot {\gamma }}$, with a kernel function $l$,

(6.17)\begin{equation} Q_{diffusion}' = l \otimes {Q_{0}'} = \int_0^{\infty} l(t,\xi-y) Q_{0}'(y)\,{\textrm{d}y}, \end{equation}

where $l$ is defined as

(6.18)\begin{equation} l(t,\xi) = \frac{1}{\sqrt{4{\rm \pi} t}}\exp{\left(-\frac{{\xi}^{2}}{4t}\right)}. \end{equation}

To prove that (6.17) satisfies (6.16), it is sufficient to show that

(6.19)\begin{equation} \frac{\partial l}{\partial t}= \frac{\xi^{2}-2t}{8{\rm \pi} t^{5/2}}\exp{\left(-\frac{{\xi}^{2}}{4t}\right)} = \frac{\partial^{2} l}{\partial \xi^{2}}. \end{equation}

Thus,

(6.20)\begin{equation} \frac{\partial Q_{diffusion}'}{\partial{t}}= \frac{\partial l}{\partial t} \otimes {Q_{0}'} = \frac{\partial^{2} l}{\partial \xi^{2}} \otimes {Q_{0}'}= \frac{\partial^{2} Q_{diffusion}'}{\partial{\xi}^{2}}. \end{equation}

We solve (6.17) numerically in Matlab to get the final solution. The plots of $Q_{diffusion}^{\prime}(t,\xi )$ in the transformed coordinate and ${Q_{diffusion}}(t,x)$ in the original coordinate are shown in figures 8(a) and 8(b), respectively. Figure 8(a) shows a typical solution of the diffusion equation in the transformed coordinate, where the peak of the Gaussian solution decreases and broadens with increasing time. The corresponding solution in the original coordinate is shown in figure 8(b). Unlike drift, figure 8(b) shows that diffusion induces a smooth liquid concentration peak $x_c$ instead of a sharp interface and slowly leads to the drying of the shear band and its surroundings. As a result, we observe a smooth liquid concentration profile with a nearly dry shear band and its periphery and a subtle liquid concentration peak that moves away from the shear band, shown in figure 8(b). We have verified that the analytical solution $Q_{diff}(t,x)$ given by (6.17) agrees with the numerical solutions of (6.16) (results not shown here). So far, we have simplified the full model for the liquid migration process and obtained analytical solutions for the simplified forms. In § 6.2, we use the analytical solutions for the drift and diffusion terms and find their significance, individually, in the overall liquid transfer process.

Figure 8. (a) Solutions $Q_{diffusion}^{\prime}(t,\xi )$ to the diffusion equation, given by (6.17). (b) Solutions $Q_{diffusion}(t,x)$ in the transformed coordinate system.

6.2. Significance of drift and diffusion

In this section, we explore the significance of the transformed drift and diffusion processes, respectively, as a part of the overall liquid transport process as given by the one-dimensional simplified model equation. The results of the drift and diffusion contributions are obtained from (6.13) and (6.17), respectively, by analytical solutions which are considered as new mathematical tools of great significance. While both drift and diffusion contributions change over time, they behave qualitatively the same, i.e. having a minimum at the centre $x = 0$ in the original coordinate system, a propagating liquid concentration peak at the edges of the shear band ${x_{c}}$ and a constant liquid concentration near the boundary region. Furthermore, the liquid concentration peak location as well as the accumulated liquid concentration changes over time for the solutions of all the aforementioned models. The new mathematical tools are used to investigate further the relative significance of drift and diffusion in the liquid migration process. In the following subsections, we explore the contribution of drift and diffusion processes, individually, to the velocity of propagation of the liquid concentration peak ${v_{c}}$ and to the accumulated liquid concentration $\phi$ at a given height of the shear cell $z = 3.6$ ($W = 3$).

6.2.1. Local Péclet number and velocity of propagation of liquid concentration peak

Among the associated dimensionless numbers, we identify the Péclet number for the study of liquid transport in granular media. The Péclet number is a class of dimensionless numbers which measures the rate of advection of a physical quantity by the flow to the rate of diffusion of the same. This dimensionless number is relevant for the study of transport phenomena in fluid flows, signifying the transport by drift and diffusion processes. We define the local Péclet number $Pe$ for liquid transport, obtained from (6.3) in the transformed coordinate, as $Pe = {{D'}(\xi )\xi }$, where ${D'}(\xi )$ is the drift coefficient, the diffusion coefficient is constant (equal to 1) and ${\xi }$ is the characteristic length in the transformed coordinate system. Physically, this dimensionless number signifies the ratio of mass transfer by drift to that by diffusion processes in the $\xi$-direction. Note that, although the Péclet number is defined here in the transformed coordinate, we analyse the corresponding results in the original coordinate system. Figure 9(a) shows the variation of the local Péclet number in the domain. The local $Pe$ is independent of time since it is only varying with the shear rate $\dot {\gamma }$ and space $x$, which are independent of time. There is an inner region in the vicinity of the shear band centre where $Pe \ll 1$, thus diffusion dominates liquid transport and drift is insignificant. Simultaneously, there is an adjoining outer region where the effects of drift and diffusion become comparable ($Pe \approx 1$).

Figure 9. (a) Contour plot of local Péclet number $Pe = {{D'}(\xi )\xi }/{D_{c}}$, where $D_{c} = 1$. Note: here, the local Péclet number is calculated in the transformed coordinate $\xi$ and is plotted against the original coordinate system. (b) Liquid concentration peak velocity ${v_{c}}$ as a function of ${x_{c}}$ for the simplified model (red line), drift (blue line) and diffusion (green line) by solving (5.1), (6.13) and (6.17), respectively, at $z \approx 3.6$ ($W \approx 3$). The cyan line represents the sum of the drift and the diffusion velocities. The relative significance of the drift and diffusion velocities in (b) is comparable with the corresponding local Péclet number in (a) at a given height $z = 3.6$.

In figure 9(b), we compare the liquid concentration peak velocity for the simplified model, drift and diffusion, respectively, at a given height $z = 3.6$ ($W = 3$). The red line corresponds to the velocity of the simplified model, ${{v_{c}}}_{simplified~model}$. The blue and the green lines correspond to the propagation velocity of liquid concentration peak for the drift and diffusion models, ${{v_{c}}}_{drift}$ and ${{v_{c}}}_{diffusion}$, respectively. The cyan line represents the sum of the velocities of propagation for the drift and diffusion models, individually, i.e. ${{v_{c}}}_{drift} + {{v_{c}}}_{diffusion}$. It is observed from figure 9(b) that the red line collapses with the cyan line. This indicates that net contribution to the velocity of propagation of the liquid concentration peak from drift and diffusion is approximately equal to the velocity given by the simplified model throughout the range of ${x_{c}}$ and therefore

(6.21)\begin{equation} {{v_{c}}}_{simplified\ model} = {{v_{c}}}_{drift} + {{v_{c}}}_{diffusion}. \end{equation}

Thus, we observe that the combination of drift and diffusion processes has an additive effect on the velocity of propagation of the liquid concentration peak for the simplified model. This also shows the validity of the simplified liquid migration model.

We also investigate the relative contribution of drift and diffusion to the overall liquid transfer and make a comparison with the local Péclet number. Focusing on the local Péclet number at a given height $z = 3.6$ in figure 9(a), one observes an inner dark blue region where $Pe < 1$. Simultaneously, there exists an outer dark red region in the vicinity where $Pe > 1$. This profile of the Péclet number can be related to the velocity profile described in figure 9(b). The contribution by diffusion is approximately $80\,\%$ of the total contribution by drift and diffusion at ${x_{c}} = 4.5$. This corresponds to the dark blue region in figure 9(a) where $Pe \approx 0.5$. The contribution by diffusion decreases with increasing ${x_{c}}$ and thus the Péclet number increases. The contributions by drift and diffusion are approximately equal at ${x_{c}} = 6$, corresponding to $Pe \approx 1$. On further increasing ${x_{c}}$, diffusion becomes less dominant and its contribution is only $33\,\%$ of the total contribution at ${x_{c}} = 8$. This corresponds to the dark red region in figure 9(a) where $Pe \approx 1.3$. The mass transfer is dominated by diffusion and is restricted to the inner region where the velocity of liquid concentration peak propagation ${v_{c}}$ is two orders of magnitude higher than that in the outer region. This is in similar philosophy with the steady-state mass transfer at low $Pe\ (Pe \ll 1)$ (Jones Reference Jones1973; Neale & Nader Reference Neale and Nader1974; Srivastava & Srivastava Reference Srivastava and Srivastava2006; Bell et al. Reference Bell, Byrne, Whiteley and Waters2014), a typical example of Brinkman and Darcy flow. Under conditions of low Reynolds number, for transport of liquid past granular materials, in the quasistatic state, the transverse diffusion flow is more important than the drift flow. Thus, we understand the significance of the Péclet number with respect to the propagation velocity of the liquid concentration peak. This shows how the simplified model is used to understand the essence of the liquid migration process.

In this subsection, we investigated the relative significance of drift and diffusion in the velocity of propagation of the liquid concentration peak. Thereby, we validated the simplified model, which is an important mathematical tool for studying the physics of the liquid migration process. The accumulated liquid concentration associated with the increment of the liquid concentration peak is also an important feature of the liquid concentration profile. Thus, in the following § 6.2.2, we investigate the relative contributions of drift and diffusion to the accumulated liquid concentration required for increase of the peak of the liquid concentration.

6.2.2. Accumulated liquid concentrations

The differential equation for the transport of liquid from the shear band is composed of two terms in the transformed coordinate, namely, diffusion and drift. While the drift is the directed flow of liquid by the bulk motion, diffusion leads to random spreading of liquid under the influence of shear from higher to lower shear rate. However, in spite of different significance of drift and diffusion on the process, the accumulated liquid concentrations contributed by the two are additive. It is expected that, if the two accumulated liquid concentrations are separately integrated over time, the resulting accumulated liquid concentrations should be comparable with those of the simplified model, at least for a small incremental time scale, when other effects, such as coupling, are negligible. We verify this from our results.

In this section, we analyse the incremental accumulated liquid concentrations ${\rm \Delta} {\phi _{tot}}^{{\rm \Delta} {t}}$ from the two contributions by drift and diffusion separately. The objective is to see if the resultant accumulated liquid concentrations from the two components of the drift and diffusive terms, together, contribute to the same incremental accumulated liquid concentrations as obtained by integrating (5.1). Moreover, we check the validity of this assumption for different initial conditions. We start running the drift and diffusion models from different initial conditions, such that they have an initial accumulated liquid concentration equal to that of the simplified model ${\phi _{{simplified\ model}}}$ at time $t$. The net accumulated liquid concentration ${\phi _{tot}}^{{\rm \Delta} {t}}$ in time ${\rm \Delta} {t}$ contributed by the drift and diffusion processes is given by

(6.22)\begin{equation} {\phi_{tot}}^{{\rm \Delta}{t}} = {\phi_{{drift}}}^{{\rm \Delta}{t}} + {\phi_{{diffusion}}}^{{\rm \Delta}{t}}, \end{equation}

where ${\phi _{{drift}}}^{{\rm \Delta} {t}}$ and ${\phi _{{diffusion}}}^{{\rm \Delta} {t}}$ are the contributions to the incremental accumulated liquid concentration by drift and diffusion, respectively. All the aforementioned accumulated liquid concentrations are obtained as described in (4.2).

Figure 10(a) shows the comparison of the sum of the accumulated liquid concentrations contributed by drift and diffusion, indicated as ${\phi _{tot}}^{{\rm \Delta} {t}}$, with the accumulated liquid concentration for the simplified model, ${\phi _{{simplified\ model}}}$. We observe that, within a very small time interval ${{\rm \Delta} {t}} = 0.72$, ${\phi _{tot}}^{{\rm \Delta} {t}} \approx {\phi _{simplified\ model}}^{{\rm \Delta} {t}}$, signifying that the effects of drift and diffusion are additive within a very short duration of time. However, with further progress in time, the net contribution from the two effects ${\phi _{tot}}^{{\rm \Delta} {t}}$ over-predicts the accumulated liquid concentration ${\phi _{{simplified\ model}}}^{{\rm \Delta} {t}}$. The deviation of the accumulated liquid concentrations decreases at longer time (or shear) when the incremental drift and diffusion fluxes become weaker. This is shown in figure 10(b) where the simulations are run, starting at $t = 288$ and for ${\rm \Delta} t = 72$. The trends show that the two accumulated liquid concentrations ${\phi _{tot}}^{{\rm \Delta} {t}}$ and ${\phi _{{simplified\ model}}}^{{\rm \Delta} {t}}$ coincide, indicating that the incremental accumulated liquid concentrations are equal even after a long time. The contributions by drift and diffusion to the accumulated liquid concentrations are shown by the blue and the green lines respectively. While drift leads to a positive (increasing) contribution to accumulated liquid concentration, diffusion leads to a negative (decreasing) contribution and the net accumulated liquid concentration is constant relative to the simplified model. This shows another method of validating the simplified model and the mathematical tools used for predicting the drift and diffusion behaviour of liquid migration.

Figure 10. (a) The accumulated liquid concentration from the simplified model ${\phi _{{simplified\ model}}}$ (red dash-dotted line) as a function of time $t$ and the net contribution by the drift and diffusion processes ${\phi _{tot}}^{{\rm \Delta} {t}}$ (cyan lines) starting from different initial conditions plotted over time interval ${\rm \Delta} t = 0.72$ at $z = 3.6$ ($W = 3$) and (b) zoom into the simulation set as in (a) for a later initial condition at $t = 288$, ${\rm \Delta} {t} = 72$. The black dots represent the initial condition for drift and diffusion models.

In this subsection, we have seen that the liquid volume required for the increment of the liquid concentration peak from any initial condition is contributed from the sum of the liquid volumes from the drift and diffusion processes. This observation is valid for a small time interval and the sum deviates from the liquid volume of the simplified model for a longer time interval. Further, the deviation slows down as the local shear rate is weaker when the peak moves away from the shear-band centre. Depending on the initial condition, the contribution by diffusion to the accumulated liquid concentration might be incremental or decremental. In § 6.2.3, we analyse in detail the sign of the incremental accumulated liquid concentrations for drift and diffusion processes, which determines whether the shear band wets or dries.

One of the major results found in this subsection is that after reducing to a quasi-one-dimensional system and transforming variables, one can separately solve the equation without a diffusion term or without a drift term. The two solutions, when added together, agree with solutions of the unseparated equation when proper initial conditions are used. The fundamental approximation of this additive splitting can be shown mathematically for an incremental time step and this is described in appendix D.

6.2.3. Drift vs diffusion: drying and wetting

In this section we analyse the increase in accumulated liquid concentration by drift and diffusion individually, relative to the accumulated liquid concentration at any time $t$. We start simulations corresponding to drift and diffusion with an initial condition of an accumulated liquid concentration ${\phi _{simplified\ model}}$ corresponding to time $t$. The relative increase in accumulated liquid concentration by drift in time ${\rm \Delta} t$ with respect to the initial condition is given by

(6.23)\begin{equation} \eta_{drift} = \frac{{\phi_{drift}}^{{\rm \Delta}{t}}}{{\phi_{simplified\ model}}}, \end{equation}

and we express the relative increase in accumulated liquid concentration by diffusion in a similar way. Since the total liquid is conserved in the system, an increment in accumulated liquid concentration at the peak location indicates a decrement of equal amount of accumulated liquid concentration in the shear band. Thus, a positive value of $\eta$ (${\rm \Delta} {\phi } > 0$) signifies drying of the shear band with respect to the initial condition and a negative value of $\eta$ (${\rm \Delta} {\phi } < 0$) signifies wetting of the shear band.

Figure 11 shows the relative increment in the accumulated liquid concentration for the drift and diffusion processes as a function of the initial accumulated liquid concentration $\phi _{simplified\ model}$. Here, $\eta$ decreases for both drift and diffusion processes with increasing $\phi _{simplified\ model}$ until it reaches a condition such that the change in accumulated liquid concentration ratio $\eta$ becomes very small. As the liquid concentration peak propagates away from the shear band with increasing time, the local shear rate becomes smaller. With decreasing shear rate, the driving forces for liquid transport by both drift and diffusion mechanisms become slow enough, such that it requires very long time to transport liquid. Diffusion leads to wetting of the shear band under certain initial conditions when liquid is transported back to the shear band such that $\eta$ becomes negative. However, drift always leads to drying of the shear band only. This is yet another aspect of the physics of liquid migration which we are able to understand by disintegrating the simplified model into drift and diffusion components.

Figure 11. The relative increase in accumulated liquid concentration $\eta$ for drift (blue line) and diffusion (green line) as a function of the initial accumulated liquid concentration corresponding to that of the simplified model ${\phi _{simplified\ model}}$. We use the convention that a positive $\eta$ signifies drying and negative $\eta$ signifies wetting of the shear band.

Thus, the mechanism of liquid transport can now be separated and understood one by one, under varied initial conditions. While drift always leads to drying of the shear band (with rapid build up and narrowing of the liquid front), diffusion can sometimes lead to wetting of the shear band, depending on the initial condition. We studied in details the role of drift and diffusion in the liquid migration process in § 6.2. The origin of this theoretical studies begins with the model given by (3.4) which was originally proposed and validated by Mani et al. (Reference Mani, Kadau, Or and Herrmann2012). We compare this model once again with the DPM simulations as described in § 7.

7.1. Discrete to continuum

We use the coarse-graining post-processing tool MercuryCG (Thornton et al. Reference Thornton, Weinhart, Luding and Bokhove2012; Weinhart et al. Reference Weinhart, Thornton, Luding and Bokhove2012a,Reference Weinhart, Thornton, Luding and Bokhoveb, Reference Weinhart, Hartkamp, Thornton and Luding2013; Tunuguntla, Thornton & Weinhart Reference Tunuguntla, Thornton and Weinhart2016; Tunuguntla, Weinhart & Thornton Reference Tunuguntla, Weinhart and Thornton2017a,Reference Tunuguntla, Weinhart and Thorntonb) to translate DPM data from discrete to continuum scale, averaged over time and over the $y$-direction, at $x$ and $z$ positions on a $400\times 100$ grid over the whole shear cell. We use a Gaussian spatial coarse-graining function with a coarse-graining width (standard deviation) of one particle diameter. Since the liquid concentration profile is dynamic in nature, we temporally averaged over a short period of time of ${\rm \Delta} t = 0.25$ ($5$ snapshots) in order to get the dynamic behaviour of liquid transport. Thus, we obtained average values for the liquid concentration $Q$ present in the liquid bridges and liquid films of the DPM simulation. The details of the coarse-graining formulation are given in appendix E.3. Likewise, we obtained other continuum fields, such as the local pressure, concentration, velocity and the shear rate, in order to draw the shear-band profile. The velocity of particles ${v_y}$ in this geometry is typically fitted by an error function of the spatial length $x$ at a given height (Fenistein, van de Meent & van Hecke Reference Fenistein, van de Meent and van Hecke2004; Ries et al. Reference Ries, Wolf and Unger2007). Figure 13(a) shows the particle velocity ${v_y}$ as a function of $x$ at $z = 3.6$ and $z = 5.4$. The red solid and the dashed lines represent the fitting of the velocity profile at the two heights, respectively. We obtain the width of the shear band $W$ by fitting the particle velocity ${v_y}$ as a function of $x$ at different heights $z$. Thereby, we obtain the shear-band width $W$ as a function of $z$, as shown in figure 13(b). The red solid line in this figure corresponds to the fitting of the data by (2.1) with the fitting parameters ${W_{top}} \approx 4$ as the width of the shear band near the free surface, $H = 8$ as the filling height and $\alpha = 0.88$ as the power obtained by fitting DPM data as detailed by Singh et al. (Reference Singh, Magnanimo, Saitoh and Luding2014). Substituting (2.1) in (2.2), we obtain the shear-rate profile in the continuum models.

Figure 13. (a) Velocity of particles ${v_y}$ as a function of centre distance $x$ for different heights $z = 3.6$ and $z = 5.4$. The solid and the dashed lines are the corresponding fits obtained from Fenistein et al. (Reference Fenistein, van de Meent and van Hecke2004) and (b) width of the shear band as a function of height as obtained from the DPM ($\circ$) after $t = 72$. The corresponding line is the fit given by (2.1).

7.2. Comparison of liquid concentration profile

Figure 14(a,b) shows a comparison of the results from the continuum full model with those from the coarse-grained DPM simulations. The location of the peak liquid concentration of the continuum model is well aligned with the peak liquid concentration of the DPM model, as observed in figure 14(b). Liquid gets depleted from the region near the shear band and a liquid concentration peak appears in the neighbouring region. The liquid concentrations close to the right boundary remain unchanged for both the DPM and continuum models. The location of the peak liquid density for the continuum model, denoted by the magenta-coloured $\diamond$ markers in figure 14(b) is well aligned with the liquid density peak for the DPM model. The peak liquid concentration is decreasing with height for both the continuum and DPM models. However, certain differences still exist in the liquid concentration profiles of the two models: the curvature of the liquid concentration peak locus is nearly linear in the DPM simulations, but convex in the continuum model results, as observed in figure 14(a,b). Also, the width of the liquid concentration peak for the DPM model is slightly larger than that of the continuum model, as observed in figure 14(a,b).

Figure 14. Contour plot of liquid concentration $Q$ from (a) continuum full model, (b) DPM simulations after $t = 72$ with coarse-graining width of $d_p$. The magenta-coloured $\diamond$ markers represent the locus of the liquid concentration peak location ${x_{c}}$ obtained from the continuum full model at different heights.

Figure 15(a) shows a comparison of the liquid concentration profile as a function of space $x$ at two different heights $z = 3.6$ ($W = 3$) and $z = 5.4$ ($W = 3.5$) for the DPM (scattered points) and continuum models (solid lines) after time $t = 72$. The comparison shows that the liquid concentration profiles are in good qualitative agreement for the two models, although the liquid concentration peak is slightly lower for the continuum model at $z = 5.4$. This is also evident from the contour plots of the two models in figure 14(a,b). The peak location $x_c$ is well aligned for the two models at both heights. As observed in figure 15(b), the trajectories of the peak location $x_c$ for the two models are in good agreement. There are various possible explanations for some quantitative differences between the two models: firstly, we fitted the continuum model with the DPM data by using a constant value of $C_{liq}$. However, our soft particles in the DPM simulation are subjected to a confining pressure under gravity. The particles near the base are therefore more compressed than the particles at the free surface. Thus, the number of contacts per particle near the bottom is slightly higher than at the free surface (not shown). Secondly, in the continuum theory, we assume a steady-state shear-rate profile at the beginning of the simulation. However, the DPM simulations show an evolving shear rate inside the shear band, until it reaches a steady state. Thirdly, the liquid concentration profile in the DPM is not developed during the transient and is more reliable after a longer simulation time. So there is a subtle difference in the liquid concentration profile between the two during the initial transient phases.

Figure 15. Comparison of (a) liquid concentration as a function of $x$ after $t = 68$ from DPM (scattered points) and continuum models (solid lines) and (b) trajectory of liquid concentration peak ${x_{c}}$ as a function of $t$ from the DPM (marked limes) and continuum models (solid lines) at different heights in a Cartesian shear cell geometry.