3.1. Leading-order solution ${O}(1)$

By substituting (2.12) in (2.1a,b) and (2.2a,b), the ${O}(1)$ governing equations and the corresponding boundary conditions can be obtained. The leading-order solution is reported and analysed in Chaithanya & Thampi (Reference Chaithanya and Thampi2019), so below we just provide the final expressions.

Using the standard technique of superposition of vector harmonics (Leal Reference Leal2007; Chaithanya & Thampi Reference Chaithanya and Thampi2019), we may write the velocity and pressure fields in the outer fluid as a linear combination of $\boldsymbol{\mathsf{E}}$ and $\boldsymbol{\varOmega }$ as

(3.1)\begin{gather} p^{(0)}=c_{1} \boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:} \boldsymbol{\mathsf{E}}, \end{gather}

(3.2)\begin{gather} \boldsymbol{u}^{(0)}=(\boldsymbol{\mathsf{E}}+\boldsymbol{\varOmega}) \boldsymbol{\cdot} \boldsymbol{x}+\tfrac{1}{2}p^{(0)}\boldsymbol{x}+c_{2}\boldsymbol{\mathsf{D}}_{(1)} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}}+c_{3} \boldsymbol{\mathsf{D}}_{(3)} \boldsymbol{:} \boldsymbol{\mathsf{E}}, \end{gather}

where $\boldsymbol{\mathsf{D}}_{(n)}$ represents the decaying spherical harmonics of order $n$, defined as $n$th-order gradients of fundamental solution of Laplace's equation, $1/r$. Similarly, the pressure and velocity fields in the inner fluid are

(3.3)\begin{gather} \hat{p}^{(0)}=d_{1} \boldsymbol{\mathsf{G}}_{(2)} \boldsymbol{:} \boldsymbol{\mathsf{E}}+e_{1} \boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:} \boldsymbol{\mathsf{E}}, \end{gather}

(3.4)\begin{gather}\hat{\boldsymbol{u}}^{(0)}=\frac{\lambda}{2}\boldsymbol{x}\hat{p}^{(0)}+\boldsymbol{\varOmega} \boldsymbol{\cdot} \boldsymbol{x}+d_{2}\boldsymbol{\mathsf{G}}_{(1)} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}}+d_{3}\boldsymbol{\mathsf{G}}_{(3)} \boldsymbol{:} \boldsymbol{\mathsf{E}}+e_{2}\boldsymbol{\mathsf{D}}_{(1)} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}}+e_{3} \boldsymbol{\mathsf{D}}_{(3)} \boldsymbol{:} \boldsymbol{\mathsf{E}}, \end{gather}

where, $\boldsymbol{\mathsf{G}}_{(n)}$ represents the growing spherical harmonics of order $n$ and it is related to the decaying harmonics as, $\boldsymbol{\mathsf{G}}_{(n)}=r^{2n+1}\boldsymbol{\mathsf{D}}_{(n)}$. Here, $c_i$, $d_i$ and $e_i$ are the constants that are linear function of shape parameter $b_1$, which is related to the leading-order shape function as,

(3.5)\begin{equation} f^{(0)}=b_{1}\boldsymbol{\mathsf{E}}\boldsymbol{:}\frac{\boldsymbol{xx}}{r^2}. \end{equation}

The expressions for the constants $c_i$, $d_i$ and $e_i$ are given in Appendix A. Using (3.5) and leading-order form of (2.11), we obtain the temporal evolution of the shape parameter,

(3.6)\begin{equation} b_{1}(t)=b_{1}^{*} \left(1-\exp\left(-\frac{t}{t_{c}}\right)\right), \end{equation}

indicating that the relaxation process of the confining interface is exponential with $b_1^{*}$ as the steady state shape parameter and $t_{c}$ as the relaxation time scale.

3.2. First-order solution ${O}(Ca)$

In this section, we solve for the ${O}(Ca)$ pressure and velocity fields. The governing equations at ${O}(Ca)$ are given by

(3.7)\begin{equation} \left. \begin{array}{c} (1/\lambda) \nabla^{2}\boldsymbol{\hat{u}}^{(1)}-\boldsymbol{\nabla} \hat{p}^{(1)}=0; \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\hat{u}}^{(1)}=0,\\ \nabla^{2}\boldsymbol{u}^{(1)}-\boldsymbol{\nabla} p^{(1)}=0; \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}^{(1)}=0. \end{array} \right\} \end{equation}

The corresponding boundary conditions are,

(3.8)\begin{equation} \boldsymbol{u}^{(1)} \rightarrow 0 \text{ as } \boldsymbol{x} \rightarrow \infty, \end{equation}

the no-slip boundary condition on the interface of the encapsulated particle,

(3.9)\begin{equation} \boldsymbol{\hat{u}}^{(1)}=0 \text{ at } r=1/\alpha, \end{equation}

the continuity of normal and tangential velocities on the confining interface,

(3.10)\begin{equation} \left. \begin{array}{c} ( (\boldsymbol{u}-\boldsymbol{\hat{u}}) \boldsymbol{\cdot} \boldsymbol{n} )^{(1)}_{r= ( 1+Ca f^{(0)})} =0, \\ (( \boldsymbol{\mathsf{I}}-\boldsymbol{n} \boldsymbol{n}) \boldsymbol{\cdot} (\boldsymbol{u} -\boldsymbol{\hat{u}} ))^{(1)}_{r= ( 1+Ca f^{(0)})} =0, \end{array} \right\} \end{equation}

and the stress balance at the interface,

(3.11a)\begin{gather} \left(\left(\boldsymbol{\sigma} -\frac{1}{\lambda}\boldsymbol{\hat{\sigma}} \right) \boldsymbol{\colon} \boldsymbol{n n} \right)^{(1)}_{r= ( 1+Ca f^{(0)})} = (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{n})^{(2)}, \end{gather}

(3.11b)\begin{gather}\left( \left( ( \boldsymbol{\mathsf{I}}-\boldsymbol{n} \boldsymbol{n} ) \boldsymbol{\cdot} \left( \boldsymbol{\sigma}-\frac{1}{\lambda} \boldsymbol{\hat{\sigma}} \right) \right) \boldsymbol{\cdot} \boldsymbol{n} \right)^{(1)}_{r= ( 1+Ca f^{(0)})}=0. \end{gather}

Similar to the previous section, (3.7) are solved using the technique of superposition of vector harmonics. The pressure and velocity fields in the outer fluid are at most quadratically dependent on $\boldsymbol{\mathsf{E}}$ and $\boldsymbol{\varOmega }$. Thus,

(3.12) \begin{align} p^{(1)}&=c_{4}\boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:} \boldsymbol{\mathsf{E}}+\,c_{5}\boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:} (\boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}})+c_{6}(\boldsymbol{\mathsf{D}}_{(4)}\boldsymbol{:}\boldsymbol{\mathsf{E}})\boldsymbol{:}\boldsymbol{\mathsf{E}} +\,c_{7}(\boldsymbol{\mathsf{E}}\boldsymbol{:}\boldsymbol{\mathsf{E}}) \nonumber\\ &\quad +c_{8}\boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:} (\boldsymbol{\varOmega}\boldsymbol{\cdot}\boldsymbol{\mathsf{E}})+c_{9}\boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:}(\boldsymbol{\mathsf{E}}\boldsymbol{\cdot} \boldsymbol{\varOmega})+c_{10}(\boldsymbol{\varOmega: \varOmega}),\end{align}

(3.13)\begin{align} \boldsymbol{u}^{(1)}&=\tfrac{1}{2}\boldsymbol{x}p^{(1)}+c_{11}\boldsymbol{\mathsf{D}}_{(1)} \boldsymbol{\,\cdot\,\, \boldsymbol{\mathsf{E}}}+\, c_{12}\boldsymbol{\mathsf{D}}_{(3)} \boldsymbol{:}\boldsymbol{\mathsf{E}}\,+\,c_{13}( \boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}}) \boldsymbol{\cdot} \boldsymbol{\mathsf{D}}_{(1)}+c_{14}\boldsymbol{\mathsf{E}}\boldsymbol{\,\cdot}(\boldsymbol{\mathsf{D}}_{(3)}\boldsymbol{:}\boldsymbol{\mathsf{E}}) \nonumber\\ &\quad +c_{15}\boldsymbol{\mathsf{D}}_{(1)}(\boldsymbol{\mathsf{E}} \boldsymbol{:} \boldsymbol{\mathsf{E}})+c_{16}\boldsymbol{\mathsf{D}}_{(3)}\boldsymbol{:}(\boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}}) +c_{17}\boldsymbol{\varOmega \cdot}\boldsymbol{\mathsf{D}}_{(1)}+c_{18}(\boldsymbol{\varOmega}\boldsymbol{\cdot}\boldsymbol{\mathsf{E}}) \boldsymbol{\cdot}\boldsymbol{\mathsf{D}}_{(1)} \nonumber\\ &\quad +c_{19} (\boldsymbol{\mathsf{E}}\boldsymbol{\cdot}\boldsymbol{\varOmega}) \boldsymbol{\cdot} \boldsymbol{\mathsf{D}}_{(1)}+\,c_{20}\boldsymbol{\varOmega \,\cdot}( \boldsymbol{\mathsf{D}}_{(3)}\boldsymbol{:}\boldsymbol{\mathsf{E}})+c_{21}(\boldsymbol{\varOmega \cdot \varOmega})\boldsymbol{\cdot} \boldsymbol{\mathsf{D}}_{(1)}+\,c_{22}(\boldsymbol{\varOmega : \varOmega} )\boldsymbol{\mathsf{D}}_{(1)}\nonumber\\ &\quad +c_{23}\boldsymbol{\mathsf{E}} \boldsymbol{:}(\boldsymbol{\mathsf{E}}\boldsymbol{:}\boldsymbol{\mathsf{D}}_{(5)})+c_{24} (\boldsymbol{\mathsf{D}}_{(3)} \boldsymbol{\,\cdot\, \varOmega} )\boldsymbol{: \varOmega}+\,c_{25} (\boldsymbol{\mathsf{D}}_{(3)} \boldsymbol{\,\cdot\, \varOmega} )\boldsymbol{:}\boldsymbol{\mathsf{E}}\!. \end{align}

Similarly, the pressure and velocity fields in the inner fluid are,

(3.14)\begin{align} \hat{p}^{(1)}&=d_{4}\boldsymbol{\mathsf{G}}_{(2)}\boldsymbol{:}\boldsymbol{\mathsf{E}}+\,d_{5}\boldsymbol{\mathsf{G}}_{(2)}\boldsymbol{:} (\boldsymbol{\mathsf{E}}\boldsymbol{\cdot}\boldsymbol{\mathsf{E}})+d_{6}(\boldsymbol{\mathsf{G}}_{(4)}\boldsymbol{:}\boldsymbol{\mathsf{E}})\boldsymbol{:}\boldsymbol{\mathsf{E}}+\,d_{7}(\boldsymbol{\mathsf{E}}\boldsymbol{:}\boldsymbol{\mathsf{E}})+d_{8}\boldsymbol{\mathsf{G}}_{(2)}\boldsymbol{:} (\boldsymbol{\varOmega}\boldsymbol{\cdot}\boldsymbol{\mathsf{E}})\nonumber\\ &\quad +d_{9}\boldsymbol{\mathsf{G}}_{(2)}\boldsymbol{:}(\boldsymbol{\mathsf{E}}\boldsymbol{\cdot} \boldsymbol{\varOmega})+d_{10}(\boldsymbol{\varOmega: \varOmega})+e_{4}\boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:}\boldsymbol{\mathsf{E}}+\,e_{5}\boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:} (\boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}})\nonumber\\ &\quad +e_{6}(\boldsymbol{\mathsf{D}}_{(4)}\boldsymbol{:}\boldsymbol{\mathsf{E}})\boldsymbol{:}\boldsymbol{\mathsf{E}} +\,e_{7}(\boldsymbol{\mathsf{E}}\boldsymbol{:}\boldsymbol{\mathsf{E}})+e_{8}\boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:} (\boldsymbol{\varOmega}\boldsymbol{\cdot}\boldsymbol{\mathsf{E}})\nonumber\\ &\quad +e_{9}\boldsymbol{\mathsf{D}}_{(2)}\boldsymbol{:}(\boldsymbol{\mathsf{E}}\boldsymbol{\cdot} \boldsymbol{\varOmega})+e_{10}(\boldsymbol{\varOmega: \varOmega}), \end{align}

(3.15)\begin{align} \boldsymbol{\hat{u}}^{(1)}&=\frac{\lambda}{2}\boldsymbol{x}\hat{p}^{(1)}+d_{11}\boldsymbol{\mathsf{G}}_{(1)} \boldsymbol{\cdot}\boldsymbol{\mathsf{E}}+ d_{12}\boldsymbol{\mathsf{G}}_{(3)} \boldsymbol{:}\boldsymbol{\mathsf{E}}+d_{13}( \boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}}) \boldsymbol{\cdot} \boldsymbol{\mathsf{G}}_{(1)}+d_{14}\boldsymbol{\mathsf{E}}\boldsymbol{\cdot}(\boldsymbol{\mathsf{G}}_{(3)}\boldsymbol{:}\boldsymbol{\mathsf{E}})\nonumber\\ &\quad +d_{15}\boldsymbol{\mathsf{G}}_{(1)}(\boldsymbol{\mathsf{E}}\boldsymbol{:}\boldsymbol{\mathsf{E}})+d_{16}\boldsymbol{\mathsf{G}}_{(3)}\boldsymbol{:}(\boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}})+d_{17}\boldsymbol{\varOmega \cdot}\boldsymbol{\mathsf{G}}_{(1)}+d_{18}(\boldsymbol{\varOmega}\boldsymbol{\cdot}\boldsymbol{\mathsf{E}}) \boldsymbol{\cdot} \boldsymbol{\mathsf{G}}_{(1)}\nonumber\\ &\quad +d_{19} (\boldsymbol{\mathsf{E}}\boldsymbol{\cdot}\boldsymbol{\varOmega}) \boldsymbol{\cdot} \boldsymbol{\mathsf{G}}_{(1)}+\,d_{20}\boldsymbol{\varOmega \cdot}( \boldsymbol{\mathsf{G}}_{(3)}\boldsymbol{:} \boldsymbol{\mathsf{E}})+d_{21}(\boldsymbol{\varOmega \cdot \varOmega} )\boldsymbol{\cdot} \boldsymbol{\mathsf{G}}_{(1)}+\,c_{22}(\boldsymbol{\varOmega : \varOmega} )\boldsymbol{\mathsf{G}}_{(1)}\nonumber\\ &\quad +d_{23}\boldsymbol{\mathsf{E}}\boldsymbol{:}(\boldsymbol{\mathsf{E}}\boldsymbol{:} \boldsymbol{\mathsf{G}}_{(5)})+d_{24} (\boldsymbol{\mathsf{G}}_{(3)} \boldsymbol{\,\cdot \,\varOmega} )\boldsymbol{: \varOmega}+\,d_{25} (\boldsymbol{\mathsf{G}}_{(3)} \boldsymbol{\,\cdot \,\varOmega} )\boldsymbol{:}\boldsymbol{\mathsf{E}}+\,e_{11}\boldsymbol{\mathsf{D}}_{(1)} \boldsymbol{\cdot}\boldsymbol{\mathsf{E}}\nonumber\\ &\quad +e_{12}\boldsymbol{\mathsf{D}}_{(3)} \boldsymbol{:}\boldsymbol{\mathsf{E}}+\,e_{13}( \boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}}) \boldsymbol{\cdot} \boldsymbol{\mathsf{D}}_{(1)}+e_{14}\boldsymbol{\mathsf{E}}\boldsymbol{\cdot}(\boldsymbol{\mathsf{D}}_{(3)}\boldsymbol{:}\boldsymbol{\mathsf{E}})+e_{15}\boldsymbol{\mathsf{D}}_{(1)}(\boldsymbol{\mathsf{E}}\boldsymbol{:}\boldsymbol{\mathsf{E}})\nonumber\\ &\quad +e_{16}\boldsymbol{\mathsf{D}}_{(3)}\boldsymbol{:}(\boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}})+e_{17}\boldsymbol{\varOmega \cdot}\boldsymbol{\mathsf{D}}_{(1)}+e_{18}(\boldsymbol{\varOmega}\boldsymbol{\cdot}\boldsymbol{\mathsf{E}}) \boldsymbol{\cdot}\boldsymbol{\mathsf{D}}_{(1)}+\,e_{19} (\boldsymbol{\mathsf{E}}\boldsymbol{\cdot}\boldsymbol{\varOmega}) \boldsymbol{\cdot}\boldsymbol{\mathsf{D}}_{(1)}\nonumber\\ &\quad +e_{20}\boldsymbol{\varOmega \cdot}( \boldsymbol{\mathsf{D}}_{(3)}\boldsymbol{:} \boldsymbol{\mathsf{E}})+e_{21}(\boldsymbol{\varOmega \cdot \varOmega} )\boldsymbol{\cdot}\boldsymbol{\mathsf{D}}_{(1)}+\,e_{22}(\boldsymbol{\varOmega : \varOmega} )\boldsymbol{\mathsf{D}}_{(1)}+e_{23}\boldsymbol{\mathsf{E}}\boldsymbol{:}(\boldsymbol{\mathsf{E}}\boldsymbol{:}\boldsymbol{\mathsf{D}}_{(5)})\nonumber\\ &\quad +e_{24} (\boldsymbol{\mathsf{D}}_{(3)} \boldsymbol{\,\cdot \, \varOmega} )\boldsymbol{: \varOmega}+\,e_{25} (\boldsymbol{\mathsf{D}}_{(3)} \boldsymbol{\,\cdot\, \varOmega} )\boldsymbol{:}\boldsymbol{\mathsf{E}}, \end{align}

where $c_{i}$, $d_{i}$ and $e_{i}$ for $i=4 \text { to } 25$ are the constants determined from the boundary conditions as follows. As earlier, the shape function $f^{(1)}$ may be expressed as quadratic combinations of $\boldsymbol{\mathsf{E}}$ and $\boldsymbol{\varOmega }$ as,

(3.16) \begin{align} f^{(1)}&=b_{2}\frac{\boldsymbol{x} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{x}}{r^2}+b_{3}\boldsymbol{\mathsf{E}}\boldsymbol{:}\boldsymbol{\mathsf{E}}+b_{4}\frac{\boldsymbol{x} \boldsymbol{\cdot} (\boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}}) \boldsymbol{\cdot} \boldsymbol{x}}{r^2}+b_{5}\frac{(\boldsymbol{x} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{x})^{2}}{r^{4}} \nonumber\\ & \quad +b_{6} \frac{\boldsymbol{x \cdot (\varOmega \cdot \boldsymbol{\mathsf{E}}) \cdot x}}{r^2} +b_{7} \boldsymbol{\varOmega : \varOmega}+b_{8}\frac{\boldsymbol{x \cdot (\varOmega \cdot \varOmega) \cdot x}}{r^2}, \end{align}

where $b_{j}$, $j=2 \text { to } 8$ are unknown constants. In other words, we have seven additional constants along with $b_{1}$ discussed in the previous section to describe the deformed drop shape. As the volume of the inner fluid in the compound particle should remain constant,

(3.17)\begin{equation} \frac{1}{3} \int_{S_{D}} (1+Ca f^{(0)}+Ca^2 f^{(1)})^{3} \, {\rm d} \varOmega-\frac{1}{3} \int_{S_{P}} (1/\alpha)^{3} \, {\rm d} \varOmega = \frac{4{\rm \pi}}{3}-\frac{4{\rm \pi} (1/\alpha)^{3}}{3}, \end{equation}

where $S_{D}$ and $S_{P}$ are the surfaces of the confining drop and the encapsulated particle, respectively. Simplifying (3.17), we get

(3.18)\begin{equation} \int_{S_{D}} (1+Ca f^{(0)}+Ca^2 f^{(1)})^{3} \, {\rm d} \varOmega =4{\rm \pi} , \end{equation}

and this gives the following relations between the constants describing the interface

(3.19a,b)\begin{equation} b_{3}+\frac{b_{4}}{3}+\frac{2 b_{5}}{15}+\frac{2 b_{1}^{2}}{15}=0, \quad b_{7}+\frac{b_{8}}{3}=0. \end{equation}

Enforcing the equation of continuity on the velocity field results additional relations between the unknown constants,

(3.20)\begin{align} \left. \begin{aligned} c_{6}={-}c_{14},\enspace c_{7}=0,\enspace c_{10}=0, \enspace c_{11}=0, \enspace c_{13}=0, \enspace c_{18}=c_{19},\enspace c_{21}=0,\\ \displaystyle\lambda d_{4}={-}\dfrac{126 d_{12}}{5},\enspace \lambda d_{6}={-}\dfrac{110 d_{23}}{7},\enspace \lambda d_{9}={-}\dfrac{42d_{24}}{5}, \enspace -\dfrac{5\lambda d_{5}}{2}+6d_{14}+21d_{16}=0,\\ \displaystyle\dfrac{3 \lambda d_{7}}{2}+d_{13}+3d_{15}=0, \enspace \dfrac{3 \lambda d_{10}}{2}+3d_{22}-d_{21}=0, \enspace \dfrac{5 \lambda d_{8}}{2}+14d_{20}+21d_{25}=0,\\ \lambda e_{6}={-}e_{14},\enspace e_{7}=0, \enspace e_{10}=0, \enspace e_{11}=0, \enspace e_{13}=0, \enspace e_{18}=e_{19},\enspace e_{21}=0. \end{aligned} \right\} \end{align}

In order to impose the continuity in velocity and stress boundary conditions on the deformed interface, we evaluate those quantities at $r = (1+Ca f^{(0)})$ as,

(3.21)\begin{align} \left. \begin{aligned} (\boldsymbol{u}|_{r= (1+Ca f^{(0)})})^{(1)}=\boldsymbol{u}^{(1)}|_{r=1}+ f^{(0)} \boldsymbol{\nabla} \boldsymbol{u}^{(0)} \boldsymbol{\cdot} \boldsymbol{x}|_{r=1} ,\\ (\boldsymbol{\sigma} \boldsymbol{\cdot} \boldsymbol{n} )^{(1)}_{r= (1+Ca f^{(0)})} =\boldsymbol{\sigma}^{(1)}|_{r=1} \boldsymbol{\cdot} \boldsymbol{n}^{(0)}+\boldsymbol{\sigma}^{(0)}|_{r=1} \boldsymbol{\cdot} \boldsymbol{n}^{(1)} + f^{(0)}( \boldsymbol{\nabla} \boldsymbol{\sigma}^{(0)} \boldsymbol{\cdot} \boldsymbol{x}|_{r=1} ) \boldsymbol{\cdot} \boldsymbol{n}^{(0)}. \end{aligned} \right\} \end{align}

From (2.9), (2.10) and (3.16), we obtain the normal vector $\boldsymbol{n}^{(1)}$ and the curvature of the interface $(\boldsymbol {\nabla } \boldsymbol{\,\cdot\, n})^{(2)}$ at ${O}(Ca)$ as

(3.22)\begin{equation} \boldsymbol{n}^{(1)}={-}\boldsymbol{\nabla} f^{(0)}=2b_{1}\left(\frac{\boldsymbol{x}(\boldsymbol{x}\boldsymbol{\cdot} \boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{x} )}{r^{4}}-\frac{\boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{x}}{r^{2}} \right)_{r=1}, \end{equation}

and

(3.23) \begin{align} (\boldsymbol{\nabla} \boldsymbol{\,\cdot\, n})^{(2)}&= 2((f^{(0)})^{2}-f^{(1)})-( \nabla^{2} f)^{(1)}-(\boldsymbol{\nabla} f^{(0)} )^{2} -\boldsymbol{x \,\cdot\,}( \boldsymbol{\nabla} f^{(0)} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{\nabla} f^{(0)} ) \nonumber\\ &=({-}10b_{1}^{2}+18b_{5})( \boldsymbol{x} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{x} )^{2}+4b_{2} \boldsymbol{x} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{x} -(2b_3+2b_4) \boldsymbol{\mathsf{E}} \boldsymbol{:}\boldsymbol{\mathsf{E}} \nonumber\\ &\quad + (4b_{4}-8b_{5}) \boldsymbol{x \,\cdot } (\boldsymbol{\mathsf{E}} \boldsymbol{\cdot} \boldsymbol{\mathsf{E}})\boldsymbol{ \cdot\, x}+\,4b_{6} \boldsymbol{x \,\cdot } (\boldsymbol{\varOmega}\boldsymbol{\cdot}\boldsymbol{\mathsf{E}})\boldsymbol{ \cdot\, x}\,-(2b_{7}+2b_{8})\boldsymbol{\varOmega : \varOmega} \nonumber\\ &\quad +4b_{8} \boldsymbol{x \,\cdot } (\boldsymbol{\varOmega \cdot \varOmega})\boldsymbol{ \,\cdot \, x}, \end{align}

respectively. The system of equations obtained by imposing the boundary conditions are given in Appendix B. In the limit $\alpha \to \infty$, the expressions for the constants (refer (B16)–(B30) in Appendix B) are consistent with the ${O}(Ca)$ calculations for a drop (Ramachandran & Leal Reference Ramachandran and Leal2012). As done earlier, the kinematic boundary condition may be used to evaluate the shape function, $f^{(1)}$ as,

(3.24)\begin{equation} \frac{\partial f^{(1)}}{\partial t}=(\boldsymbol{u} \boldsymbol{\,\cdot\, n})^{(1)}_{r=(1+Ca f^{(0)})}, \end{equation}

which further reduces to the description of temporal evolution of the deformed interface as,

(3.25a)\begin{gather} \frac{\partial b_{2}}{\partial t}=\frac{c_{4}}{2 }+ 9c_{12} , \end{gather}

(3.25b)\begin{gather}\frac{\partial b_{4}}{\partial t}={-}\frac{3c_{5}}{2 }- 24c_{6} + 9c_{16} - 300c_{23} -b_{1}(2- 12c_{3} ), \end{gather}

(3.25c)\begin{gather}\frac{\partial b_{5}}{\partial t}=\frac{75c_{6}}{2 }+ 525c_{23} +b_{1}(3-c_{1} - 48c_{3} ), \end{gather}

(3.25d)\begin{gather}\frac{\partial b_{6}}{\partial t}={-}\frac{3c_{8}}{2 }- 6c_{20} -9c_{25} +2b_{1}, \end{gather}

(3.25e)\begin{gather}\frac{\partial b_{8}}{\partial t}={-}\frac{3c_{9}}{2 }-9c_{24}. \end{gather}

Here, the variables $c_i$ are linear functions of $b_j$ where $j$ varies from $2$ to $8$. We solve the above set of (3.25a)–(3.25e) along with (3.19a,b) to obtain the shape parameters $b_{j}$. The temporal evolution of the shape is illustrated using plots in the next section and the steady state shape values of $b_{j}$ are given in Appendix C. The expressions for the shape functions in the limit $\alpha \to \infty$ are also given in Appendix C (refer (C2)–(C7)), they are consistent with the calculations for a drop (Ramachandran & Leal Reference Ramachandran and Leal2012).

Hence, using a domain perturbation approach and standard technique of superposition of vector harmonics, we calculated the pressure and velocity fields up to ${O}(Ca)$ in both inner and outer fluids for a compound particle when subjected to an imposed linear flow. Along with these, we determined the time evolution of the deforming interface of the confining drop up to ${O}(Ca^2)$. These solutions are illustrated in the next section for various linear flows.

4.1. Simple shear flow

Consider a simple shear flow of the form $\boldsymbol{u}= x_{2} \boldsymbol {i}_{1}$, where $x_{2}$ is the component of position vector $\boldsymbol{x}$ and $i_{1}$ is the unit normal vector associated with $x_{1}$ in the chosen coordinate system. The symmetric and anti-symmetric parts of the velocity gradient tensor of the imposed flow are,

(4.1a,b)\begin{equation} \boldsymbol{\mathsf{E}}^{S}=\frac{1}{2}\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}, \quad \boldsymbol{\varOmega}^{S}=\frac{1}{2}\begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}. \end{equation}

Using (4.1a,b), the shape of the confining drop may be obtained as,

(4.2)\begin{align} r(\theta, \phi)&=\left(1+ b_{1} Ca\left(\frac{x_{1}x_{2}}{r^{2}} \right)+Ca^{2}\left(b_{2}\frac{x_{1}x_{2}}{r^{2}}+\frac{b_{3}-b_{7}}{2} +\frac{b_{4}-b_{8}}{4}\left(\frac{x_{1}^{2}+x_{2}^{2}}{r^{2}}\right) \right.\right.\nonumber\\ & \quad \left.\left.+b_{5}\frac{x_{1}^{2}x_{2}^{2}}{r^{4}}+\frac{b_{6}}{4} \left(\frac{x_{1}^{2}-x_{2}^{2}}{r^{2}}\right)\right)+O(Ca^3)\right), \end{align}

where $x_{1}=r\sin {\theta }\cos {\phi }$, $x_{2}=r\sin {\theta }\sin {\phi }$ and $x_{3}=r\cos {\theta }$, $r$ is the magnitude of the position vector $\boldsymbol{x}$, $\theta$ is the polar angle measured from the $x_{3}$ axis ($0\leq \theta \leq {\rm \pi}$) and $\phi$ is the azimuthal angle measured around the $x_3$ axis ($0 \leq \phi \leq 2{\rm \pi}$).

The steady state flow fields in and around a deformed compound particle are illustrated in figure 2(a). Equations (A2), (A4), (B2), (B4) and (C1) have been used to calculate the velocity field and the corresponding confining drop shape. For comparison, ${O}(1)$ velocity field is shown in figure 2(b) (Chaithanya & Thampi Reference Chaithanya and Thampi2019). In the imposed simple shear flow, the encapsulated rigid particle rotates with an angular velocity $\mathcal {G}/2$, and the curved streamlines inside the confining drop illustrate the recirculating fluid flow around this rotating rigid particle. Similarly, the outer fluid close to the interface also has curved streamlines. However, compared to that of a spherical compound particle (figure 2b) or a particle without a confining drop (Sadhal et al. Reference Sadhal, Ayyaswamy and Chung1997; Leal Reference Leal2007), the streamlines around the compound particle show distortions corresponding to the deformed interface.

Figure 2. (a) Velocity field, represented as streamlines, in and around a compound particle in an imposed shear flow in the flow–gradient ($x_{1}$–$x_{2}$) plane, for $\alpha =2$, $Ca=0.1$ and $\lambda =1$. The ${O}(1)$ flow field is shown in (b) for comparison. The colour field shows the magnitude of velocity. The filled circle is the encapsulated solid particle and the solid red line shows the confining drop interface.

Now we proceed to analyse the confining drop shape and its evolution. The time evolution of the confining drop shape (up to ${O}(Ca^{2})$) in a simple shear flow for $\alpha =2$, $\lambda =1$ and $Ca=0.2$ is depicted in figure 3(a). At $t=0$, the confining drop is spherical, but the imposed flow deforms the interface with time. Finally, the confining drop attains a steady shape that is elongated in a direction close to the extensional axis of the imposed flow and compressed in the orthogonal direction.

Figure 3. The time evolution of (a) the confining drop shape in a simple shear flow for $\lambda =1$, $\alpha =2$ and $Ca=0.2$ (equations (3.6), (3.25) along with (3.19a,b)). The black patch at the centre indicates the encapsulated solid particle. (b) The deformation parameter $\mathcal {D}$ determined by (4.3), and (c) the alignment angle $\varphi$, for various size ratios $\alpha$ and viscosity ratios $\lambda$. The solid lines correspond to $\mathcal {D}$ (and $\varphi$) obtained for various $\alpha$ and the dotted lines correspond to those obtained for various $\lambda$.

Similar to the deformation parameter defined by Taylor (Reference Taylor1932) to analyse the shape of drops, we define a deformation parameter $\mathcal {D}$ as

(4.3)\begin{equation} \mathcal{D}=\frac{(r_{max}- a)-(r_{min}-a)}{(r_{max}-a)+(r_{min}-a)}, \end{equation}

that quantifies the extent of deformation of the compound particle as done in Chaithanya & Thampi (Reference Chaithanya and Thampi2019). Here, $r_{max}$ and $r_{min}$ are the longest and shortest dimensions of the deformed interface as shown in figure 1. The limits, $\mathcal {D}=0$ and $\mathcal {D}=1$ respectively correspond to the case of an undeformed spherical interface and the case where the confining interface touches the encapsulated solid particle. The latter case can be regarded as the onset of break-up of the confining drop since the interface comes into contact with the encapsulated solid particle. Therefore, unlike that of a simple drop, large values of deformation parameter do not necessarily mean a large deformation of the confining drop. This is especially the case in the limit $\alpha \to 1$. In other words, in the limit $\alpha \to 1$, breakup ($\mathcal {D} = 1$) may occur even with a weak deformation of the confining drop, and therefore the perturbation approach used in this work remains valid all the way up to the confining drop breakup.

The time evolution of deformation parameter $\mathcal {D}$ for $Ca=0.2$ but for various values of $\alpha$ and $\lambda$ are plotted in figure 3(b). In all cases, $\mathcal {D}$ increases with time indicating the progression of deformation. Finally $\mathcal {D}$ reaches a plateau corresponding to the steady state shape of the confining drop. In some cases, $\mathcal {D}$ reaches $1$ representing the break up of the confining interface. Of course the progression of deformation and the final value of $\mathcal {D}$ depend upon the particular values of $\alpha$ and $\lambda$, and this dependence is discussed later in this section.

Another consequence of calculations at ${O}(Ca^2)$ is its ability to predict the orientation of the elongated interface in a simple shear flow. The ${O}(Ca)$ correction to the confining drop shape shows that the elongated direction of the confining drop aligns with the extensional axis of the flow, i.e. $45^\circ$ with the flow direction in the flow–gradient plane as reported in Chaithanya & Thampi (Reference Chaithanya and Thampi2019). However, ${O}(Ca^2)$ calculation takes into account the effect of vorticity of the imposed flow, which then predicts that the elongated interface does not orient along the extensional axis of the imposed flow. We define an alignment angle $\varphi$, as shown in figure 1, as the angle between the elongated direction of the confining drop and the flow direction ($x_1$) in a simple shear flow.

The time evolution of alignment angle $\varphi$ at $Ca=0.2$ for various values of $\alpha$ and $\lambda$ is plotted in figure 3(c). The alignment angle $\varphi$ decreases with time before reaching a steady value, indicating that the deformed drop rotates towards the flow direction as dictated by the imposed vorticity, finally attaining an orientation that is in between the flow direction and the extensional axis of the imposed shear flow.

The extent of deformation of the confining interface of the compound particle that resulted from the imposed shear flow is different in different directions. The steady state shape of the confining drop (see (4.2)), when viewed in three different planes, namely the flow–gradient ($x_{1}$–$x_{2}$), flow–vorticity ($x_{1}$–$x_{3}$) and gradient–vorticity ($x_{2}$–$x_{3}$) planes are shown in figures 4(a)–4(c). The interface deformation is largest in the flow–gradient plane with the confining drop elongated in a direction described by $\varphi$. Confining drop is lesser deformed in the flow–vorticity plane and the gradient–vorticity plane, with the interface elongated in the flow direction in the former and compressed along the vorticity direction in the latter.

Figure 4. The deformed shape (see (4.2)) of the confining drop viewed from various planes: (a) in flow–gradient ($x_{1}$–$x_{2}$) plane, (b) in flow–vorticity ($x_{1}$–$x_{3}$) plane and (c) in gradient–vorticity ($x_{2}$–$x_{3}$) plane for different capillary numbers. The dashed line represents the initial, undeformed shape of the confining drop and the filled circle is the encapsulated solid particle.

Figures 4(a)–4(c) also illustrate the deformed shape of the confining drop for various capillary numbers. Increase in $Ca$ signifies increasingly dominant shear force over the resisting interfacial tension and, therefore, the extent of deformation of the confining interface increases with increase in $Ca$. This is evident from the confining drop shapes shown in various planes (figures 4(a)–4(c)). For the largest capillary number $Ca = 0.3$ considered in this figure, dimples form on the interface, as evident in the flow–gradient plane (figure 4a). The extent of compression of the interface is not same in all directions, with the maximum compression observed along the vorticity direction. Incidentally, in this particular case, the deformed interface comes in contact with the encapsulated solid particle (figure 4c) and marks the break up of the confining drop.

The extent of deformation of the confining drop is mainly dependent upon the three non-dimensional numbers, namely the capillary number $Ca$, the size ratio $\alpha$ and the viscosity ratio $\lambda$, as illustrated comprehensively in figure 5. The steady state deformation parameter $\mathcal {D}$ as a function of $Ca$ for various $\alpha$ and $\lambda$ is shown in figures 5(a) and 5(b) respectively. The corresponding steady state shapes of the deformed interface of the compound particle at a fixed $Ca =0.1$ but for various $\alpha$ and $\lambda$ are shown in figures 5(c) and 5(d) respectively. The deformation parameter (or deformation of the confining drop) increases with increase in capillary number $Ca$, and decreases with increase in size ratio $\alpha$ and viscosity ratio $\lambda$. This dependency of steady state deformation parameter on $Ca$, $\alpha$ and $\lambda$ based on leading-order calculations is reported by Chaithanya & Thampi (Reference Chaithanya and Thampi2019), and they are shown as dotted lines in figure 5. The $O(Ca^2)$ calculations are shown as solid lines in figure 5.

Figure 5. The effect of $Ca$ on the steady state deformation parameter $\mathcal {D}$ (a) for various size ratios $\alpha$, at $\lambda =1$ and (b) for various viscosity ratios $\lambda$, at $\alpha =1.5$. The markers in (a, b) indicate $Ca_{crit}$, the capillary number at which $D = 1$, which signifies breakup of the confining drop as the interface comes in contact with the encapsulated rigid particle. The corresponding steady state confining drop shapes at $Ca=0.1$ and (c) for various $\alpha$ and (d) for various $\lambda$. The solid and the dotted lines correspond to ${O}(Ca^2)$ and ${O}(Ca)$ calculations respectively.

As expected ${O}(Ca)$ and ${O}(Ca^2)$ theory respectively show a linear and a quadratic dependence of $\mathcal {D}$ on $Ca$. More importantly, ${O}(Ca)$ calculations underpredict the deformation parameter. (However, at very large viscosity ratios, ${O}(Ca)$ calculations overpredict the value of $\mathcal {D}$.) Interestingly, the difference between ${O}(Ca^2)$ and ${O}(Ca)$ calculations decreases with decrease in $\alpha$, and with increase in $\lambda$ at a fixed capillary number. This is because, small $\alpha$ or large $\lambda$ correspond to cases where the deformed interface of the confining drop deviates least from the spherical shape. The degree of underprediction in $\mathcal {D}$ by linear theory is proportional to $Ca$, showing that ${O}(Ca^2)$ theory significantly deviates from ${O}(Ca)$ predictions as capillary number increases. This observation also assumes importance because, compared to ${O}(Ca^2)$ calculations the linear theory will also be overpredicting $Ca_{crit}$, the capillary number at which break up of the confining interface of the compound particle occurs.

Similar to $\mathcal {D}$, the alignment angle also depends upon the capillary number, the size ratio and the viscosity ratio. Figures 6(a) and 6(b) show the dependency of steady state $\varphi$ on $Ca$ for various size ratios and viscosity ratios. In all cases, the alignment angle decreases with increase in $Ca$, making the elongated interface to orient closer to the flow direction of the imposed shear flow. The vorticity associated with the imposed flow acts to rotate the deformed drop towards the flow axis, while the surface tension force acts to revert it back to the spherical shape, and this competition determines the alignment angle. If the surface tension forces dominate the shear forces (smaller $Ca$), the confining drop aligns with the extensional axis, otherwise the confining drop aligns more with the flow axis. Similarly, $\varphi$ decreases with decrease in $\alpha$ or $\lambda$ at a given $Ca$. Thus, the deformed drop aligns closer to the flow axis in strong shear flows, and when the encapsulated particle is bigger and the viscosity of the confining fluid is larger. This behaviour of alignment angle following that of deformation parameter occurs as larger deformation results in more anisotropic shapes that orient closer to the flow direction.

Figure 6. The effect of capillary number on the steady state alignment angle, $\varphi$ calculated from (4.2) in a simple shear flow (a) for various size ratios at $\lambda = 1$ and (b) for various viscosity ratios at $\alpha = 1.5$.

To summarise this section, we have analysed fluid flows in and around a compound particle as well as the consequent deformation of the confining interface when it is subjected to an imposed shear flow. The ${O}(Ca^2)$ calculation of the deformed shape of the confining drop shows that the deformation of the interface increases with increase in $Ca$ and this larger deformation results in orienting the deformed interface to align close to the flow axis in a simple shear flow. Further, the steady state deformation parameter and the critical capillary number increase while the alignment angle decreases with increase in $\alpha$ and $\lambda$ due to increased hydrodynamic interaction between the enclosed solid and the confining interface.

4.2. Extensional flows

In this section, we analyse the deformation of the confining drop when the compound particle is subjected to an extensional flow. This analysis is important since earlier investigations, both experimental (Taylor Reference Taylor1934; Bai Chin & Dae Han Reference Bai Chin and Dae Han1979; Grace Reference Grace1982; Mietus et al. Reference Mietus, Matar, Lawrence and Briscoe2002; Mulligan & Rothstein Reference Mulligan and Rothstein2011) and theoretical or numerical (Taylor Reference Taylor1934; Bai Chin & Dae Han Reference Bai Chin and Dae Han1979; Qu & Wang Reference Qu and Wang2012), have concluded that the deformation and breakup of a simple drop (without encapsulated particles) differ considerably when subjected to shear and extensional flows.

Consider the two extensional flows namely, the uniaxial and biaxial flows described by the strain rate tensor,

(4.4a,b)\begin{equation} \boldsymbol{\mathsf{E}}^{E}=\frac{1}{2}\begin{bmatrix} \mp1 & 0 & 0 \\ 0 & \mp1 & 0\\ 0 & 0 & \pm2 \end{bmatrix}, \quad \boldsymbol{\varOmega}^{E}=\boldsymbol{0}, \end{equation}

where the first and second signs indicate uniaxial and biaxial flows, respectively. We follow the same convention throughout this section. Then the shape of the confining drop for uniaxial and biaxial flows can be obtained as

(4.5)\begin{align} r(\theta, \phi)&= \left( 1+b_{1} Ca \left({\mp}\frac{1}{2}\frac{(x_{1}^{2}+x_{2}^{2})}{r^{2}} \pm\frac{x_{3}^{2}}{r^{2}}\right)+Ca^{2}\left(b_{2}\left({\mp} \frac{1}{2}\frac{(x_{1}^{2}+x_{2}^{2})}{r^{2}}\pm\frac{x_{3}^{2}}{r^{2}}\right)\right.\right.\nonumber\\ &\quad +\left.\left.\frac{3}{2}b_{3} +b_{4} \left(\frac{1}{4}\frac{(x_{1}^{2}+x_{2}^{2})}{r^{2}} +\frac{x_{3}^{2}}{r^{2}}\right) +b_{5} \left(\frac{1}{2}\frac{(x_{1}^{2}+x_{2}^{2})}{r^{2}} -\frac{x_{3}^{2}}{r^{2}}\right)^{2}\right)+O(Ca^3) \right). \end{align}

Figures 7(a) and 7(b) show the velocity fields in and around a compound particle obtained for uniaxial and biaxial flows, respectively. The deformed shape of the confining drop is also plotted (solid red line). Clearly, the velocity fields are similar in both cases with recirculating fluid flows appearing in each quadrant. The orientation of the elongated interface is along the extensional axis of the flow but the extent of the deformation is different for uniaxial and biaxial flows. For uniaxial flows, extensional axis of the imposed flow is along the $x_3$ axis while for biaxial flows extensional flow is in the radial direction in the $x_1\text {--}x_2$ plane. This difference results in the confining interface adopting a prolate spheroid-like shape for uniaxial flows and an oblate spheroid-like shape for biaxial flows. These shapes, for various $\alpha$, when viewed in two different planes, the $x_1\text {--}x_3$ plane and the $x_1\text {--}x_2$ plane are shown in figures 7(c)–7(f). The axisymmetry of the deformed shapes is apparent in the $x_1\text {--}x_2$ plane in all cases.

Figure 7. Velocity field in and around a compound particle in an imposed (a) uniaxial and (b) biaxial flow. Here, $\alpha =2$, $Ca=0.1$ and $\lambda =1$. The colour bar represents the magnitude of the velocity. Effect of the size ratio $\alpha$ on the deformation of a compound particle (c) in the $x_{1}$–$x_{3}$ plane and (e) in the $x_{1}$–$x_{2}$ plane for uniaxial flow and (d) in the $x_{1}$–$x_{3}$ plane and (f) in the $x_{1}$–$x_{2}$ plane for biaxial flow. The solid and dotted lines represent the drop shape at ${O}(Ca^2)$ and ${O}(Ca)$ (Chaithanya & Thampi Reference Chaithanya and Thampi2019) respectively.

As earlier, the deformation parameter $\mathcal {D}$ can also be calculated using (4.3) for uniaxial and biaxial flows as

(4.6)\begin{equation} \mathcal{D}=\frac{3\alpha}{4(\alpha-1)}\left( b_{1} Ca+Ca^{2}\left(b_{2}\pm\frac{b_{4}}{2}\pm\frac{b_{5}}{2}\mp\frac{\alpha b_{1}^{2}}{4(\alpha-1)} \right) \right) +O(Ca^3). \end{equation}

In the limit of $\lambda \rightarrow \infty$ and ($\alpha - 1) \rightarrow 0$ such that the hydrodynamic interaction between the encapsulated particle and the interface is a dominant effect, (4.6) reduces to give the deformation parameter as,

(4.7)\begin{equation} \mathcal{D}= \frac{630\alpha(\alpha-1)Ca\pm\alpha(671\alpha-986)Ca^{2}}{420 (\alpha-1)^2}. \end{equation}

Thus, $\mathcal {D}$ is same for uniaxial and biaxial flows up to ${O}(Ca)$ but differ at ${O}(Ca^2)$. This difference is also illustrated in figures 7(c)–7(f). The deformation calculated at ${O}(Ca^2)$ (solid lines) is more compared to that at ${O}(Ca)$ (dotted lines) for uniaxial flows. On the other hand, for biaxial flows, drop deformation and thus the predicted value $\mathcal {D}$ are less at ${O}(Ca^2)$ compared to that at ${O}(Ca)$. In other words, ${O}(Ca)$ calculation underpredicts deformation for uniaxial flows and overpredicts it for biaxial flows.

Similar to the case of a simple shear flow, the deformation of a confining drop in extensional flows increases with decreasing $\alpha$ or $\lambda$ for a particular $Ca$, and increases with increase in $Ca$. These dependencies can also be understood by monitoring critical capillary number, $Ca_{crit}$ beyond which the confining drop breaks up ($\mathcal {D} = 1$). Therefore, $Ca_{crit}$ is plotted as a function of $\lambda$ for various $\alpha$ and various imposed flows in figure 8. Irrespective of the flow type, $Ca_{crit}$ increases with increase in $\lambda$ or $\alpha$. For small viscosity ratios, $\lambda <1$, $Ca_{crit}$ for shear flows is smaller suggesting that shear flows are stronger in deforming the interface compared to the uniaxial and biaxial flows. This trend reverses for large $\lambda$. As mentioned earlier, due to the difference in the deformation characteristics, uniaxial flows show lower $Ca_{crit}$ compared to biaxial flows.

Figure 8. The variation of the critical capillary number $Ca_{crit}$ vs the viscosity ratio $\lambda$ for various size ratios $\alpha ~(=1.5, 2, 2.5)$ and in different imposed flows.

4.3. Generalised shear and extensional flows

In this section, we expand the analysis described in the two previous sections to generalised shear and generalised extensional flows. We may define a generalised shear flow (Graham Reference Graham2018) which has the velocity gradient tensor as,

(4.8a,b)\begin{equation} \boldsymbol{\mathsf{E}}^{GS}=\frac{(1+\beta)}{2}\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}, \quad \boldsymbol{\varOmega}^{GS}=\frac{(1-\beta)}{2}\begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix},\end{equation}

where the parameter $\beta$ takes values from $-1$ to $1$. The limiting cases of $\beta =-1$ and $\beta =1$ correspond to the pure rotational and pure straining flows, respectively. Simple shear flow is obtained when $\beta = 0$.

Figure 9 shows the steady state flow fields in and around a deformed compound particle for $\beta = -1$ and $\beta = 1$. As shown in figure 9(a), for $\beta = -1$ which corresponds to a pure rotational flow, the velocity field has only azimuthal component of velocity, there will not be any velocity normal to the interface, and therefore the confining drop remains spherical. As $\beta$ increases, the strength of the extensional component in the imposed flow increases (refer to figure 9b), which results in larger deformations, with the largest $D$ obtained for a purely extensional flow ($\beta = 1$).

Figure 9. Velocity field in and around a compound particle in an imposed flow; (a) $\beta =-1$, (b) $\beta =1$. Here, $\alpha =2$, $Ca=0.05$ and $\lambda =1$. The colour bar represents the magnitude of the velocity.

Figure 10(a) summarises the deformation behaviour of the confining drop of the compound particle by plotting the dependence of deformation parameter, $\mathcal {D}$, on $\beta$. In purely rotational flows ($\beta = -1$), the confining drop does not deform ($\mathcal {D} = 0$), since the imposed flow corresponds to deformation free (rigid-body-like) rotation of the fluid. As $\beta$ increases, the strength of extensional component in the imposed flow increases which results in larger deformations, with largest $\mathcal {D}$ obtained for $\beta =1$. Figure 10(a) also shows the predicted values of $\mathcal {D}$ at ${O}(Ca)$, shown as dotted lines for comparison. In the linear theory, $\mathcal {D}$ increases linearly with $\beta$ with rotational component playing no role. As mentioned earlier, the first effect of imposed vorticity is taken into account at ${O}(Ca^{2})$, and thus $\mathcal {D}$ calculated at ${O}(Ca^{2})$ exhibits a nonlinear behaviour with an increase in $\beta$, as shown. Thus, in the rotation dominated flows ($\beta < 1$), ${O}(Ca)$ calculation underpredicts the deformation of the confining drop. On the other hand, as $\beta \rightarrow 1$, ${O}(Ca)$ overpredicts $\mathcal {D}$. It is also worth noting that ${O}(Ca^{2})$ corrections are significant when the hydrodynamic interaction between the encapsulated particle and the confining drop interface becomes strong. Therefore, the case of $\alpha = 1.5$ shows largest deviation in the predicted values of $\mathcal {D}$ at ${O}(Ca^2)$ compared to predictions of linear theory.

Figure 10. The deformation parameter $\mathcal {D}$ (a) in generalised shear flows, as a function of $\beta$ and (b) in generalised extensional flows as a function of $m$. In these plots, $\lambda =1$ and $Ca=0.1$. Solid and dotted lines correspond to ${O}(Ca^2)$ and ${O}(Ca)$ calculations.

A similar analysis can be performed for extensional flows as well, where the generalised extensional flow may be defined with a velocity gradient tensor as,

(4.9a,b)\begin{equation} \boldsymbol{\mathsf{E}}^{GE}=\frac{1 }{\sqrt{2(1+m+m^2)}}\begin{bmatrix} 1 & 0 & 0 \\ 0 & m & 0\\ 0 & 0 & -(1+m) \end{bmatrix}, \quad \boldsymbol{\varOmega}^{GE} = \boldsymbol{0},\end{equation}

where, the parameter $m$ may take different values. For example, $m = -1/2$ corresponds to a uniaxial flow ($x_1$ being the extensional axis), $m =0$ corresponds to a planar extensional flow and $m =1$ corresponds to a biaxial flow ($x_3$ being the compressional axis).

Figure 10(b) shows the dependency of deformation parameter $\mathcal {D}$ on $m$. Clearly, the deformation parameter is only weakly dependent on $m$, with $\mathcal {D}$ varying only slightly between uniaxial ($m = -1/2$), planar extensional ($m = 0$) and biaxial ($m = 1$) flows. However, irrespective of the type of imposed flow, $\mathcal {D}$ increases with decrease in $\alpha$. Again, predictions from ${O}(Ca)$ calculations are shown as dotted lines in the same figure for comparison. It is interesting to note that, the linear theory shows a monotonic increase of $\mathcal {D}$ with $m$ while this is not the case at ${O}(Ca^2)$. The implication is that linear theory falsely suggests that biaxial flows ($m = 1$) are stronger than uniaxial flows ($m = -1/2$) in deforming the confining interface (Chaithanya & Thampi Reference Chaithanya and Thampi2019), while the situation reverses by taking into account of ${O}(Ca^2)$ corrections. In other words, $\mathcal {D}$ at $m = 1$ is smaller than $\mathcal {D}$ at $m = -1/2$, similar to the calculations for a simple drop (Davis & Brenner Reference Davis and Brenner1981), and therefore, uniaxial flows are stronger in deforming a compound particle. This effect may be understood as follows. Biaxial flow compresses the confining drop towards an oblate shape while a uniaxial flow stretches it towards a prolate shape. As deformation increases, the prolate shaped interface meets the solid inclusion before the corresponding oblate. Therefore, though biaxial flow is obtained by reversing the direction of uniaxial flow, the latter is stronger than the former to cause breakup of the confining drop. The other observation to note is that, as observed for generalised shear flows, the difference that arises due to ${O}(Ca^2)$ corrections from the linear theory also increases with decrease in $\alpha$, making the ${O}(Ca^2)$ calculations more relevant for compound particles.

5.1. Shear viscosity

Subjecting the dilute dispersion of compound particles to an imposed flow, $\boldsymbol{u}^{imp}= x_{2} \boldsymbol {i}_{1}$, the effective shear viscosity of the dispersion can be calculated from (5.4) as,

(5.5)\begin{equation} \mu_{eff}= \frac{\langle\sigma\rangle_{12}}{2E_{12}} = \left(1-\phi \frac{c_{1} }{2} \right)+{O}(\phi^{2}, \phi^2 Ca, \phi Ca^2 ). \end{equation}

Thus, the first correction to the shear viscosity is ${O}(\phi )$ and it depends only on the leading-order stress field. In other words, ${O}(Ca)$ stress field generated due to deformation of the confining drop does not contribute to shear viscosity for dilute dispersions. However, hydrodynamic interaction between compound particles can lead to further corrections, which will be accounted at ${O}(\phi ^2)$. Taking appropriate limits the expression derived above for shear viscosity (5.5) can also be deduced from the works of Davis & Brenner (Reference Davis and Brenner1981), Stone & Leal (Reference Stone and Leal1990), Mandal et al. (Reference Mandal, Ghosh and Chakraborty2016) and Das et al. (Reference Das, Mandal and Chakraborty2020).

It is interesting to look at the limiting case of (5.5) first. (i) In the limit, $\alpha \rightarrow 1$ (or $\lambda \rightarrow 0$), (5.5) reduces to the Einstein relation for the effective viscosity of a dilute suspension of spherical rigid particles $\mu _{eff} = (1+\frac {5 }{2} \phi )$ (Einstein Reference Einstein1906, Reference Einstein1911). (ii) In the limit of $\alpha \rightarrow \infty$, (5.5) reduces to Taylor's relation for the effective viscosity of a dilute emulsion $\mu _{eff} = (1+ (({5+2\lambda })/({2+2\lambda }))\phi )$ (Taylor Reference Taylor1932; Oldroyd Reference Oldroyd1953). (iii) In the limit of $\alpha \to 1$, but retaining ${O}(\alpha - 1)$ terms, the effective shear viscosity of a suspension of particles that are coated with a thin fluid film of inner fluid can be calculated as,

(5.6)\begin{equation} \mu_{eff} \simeq \left(1+\phi \left( \tfrac{5}{2}-\tfrac{15}{8} (\alpha-1) \lambda+{O}(\alpha-1)^{2} \right) \right). \end{equation}

Therefore, ${O}(\alpha -1)$ correction to the effective viscosity of a dispersion is negative, indicating that the viscosity of a dilute suspension of spheres can be reduced by providing a thin coating of a viscous fluid. (iv) In the limit of $\lambda \rightarrow \infty$, (5.5) reduces to the effective viscosity of a dilute emulsion of bubbles $\mu _{eff} = (1+ \phi )$, where the positive viscosity correction arises purely due to the interfacial tension of the confining interface.

Figure 11(a) shows the complete picture, where the enhancement factor in the effective shear viscosity $(\mu _{eff}-1)/\phi$ is plotted as a function of size ratio $\alpha$ and viscosity ratio $\lambda$. At a given viscosity ratio, $\mu _{eff}$ decreases with increase in $\alpha$ from Einstein's to Taylor's limit. Similarly, for a given size ratio, the effective viscosity decreases with increase in viscosity ratio. This behaviour can be understood as follows. As the resistance to the fluid flow increases, an increase in the effective viscosity may be anticipated. The resistance to the imposed flow is maximum in the case of rigid solid spheres. This resistance decreases with increase in the thickness of the coating (confining drop size) on the rigid spheres. Therefore, the effective viscosity decreases with increase in the size of the confining drop ($\alpha$). Similarly, as the viscosity of the inner fluid decreases, the resistance to the imposed flow decreases, thus, the outer fluid can easily slip past the confining interface. Therefore, the effective viscosity decreases with increase in the viscosity ratio. It is worth noting that, as mentioned earlier, $Ca$ does not appear at ${O}(\phi )$ in (5.4) and therefore the stress resulting from the deformation of the confining drop does not have any effect on these calculations.

Figure 11. Effect of the viscosity ratio $\lambda$ and size ratio $\alpha$ on the shear viscosity, plotted as contours of the enhancement factor, (a) $(\mu _{eff}-1)/\phi$ (see (5.5)) and (b) $(\mu _{eff}^{S}-1)/\phi$ corresponding to the case of $Ca\rightarrow \infty$ (see (5.7)).

Another useful rheological measure that can be extracted from (5.4) is the shear viscosity of the compound particles when the confining interface has zero surface tension or $Ca\rightarrow \infty$. This corresponds to a situation where the kinematic boundary condition (2.8) is not respected, $b_{1} = 0$ and therefore the effective shear viscosity is obtained as,

(5.7)\begin{equation} \mu_{eff}^{S}= \left(1-\frac{\phi c_{1}^{S}}{2}\right), \end{equation}

where the expression for $c_{1}^{S}$ is given in Appendix A (see (A16)). In the limit of $\alpha \rightarrow \infty$, the $\mu _{eff}^{S} = \lambda \mu (1+\phi ({5(1-\lambda )}/{(2+3\lambda )}))$, a result obtained by Batchelor & Green (Reference Batchelor and Green1972) for a dilute emulsion of drops.

Figure 11(b) illustrates the dependence of $\mu _{eff}^{S}$ on the size ratio and viscosity ratio of the compound particles in the dispersion. The dependence of $\mu _{eff}^{S}$ on $\alpha$ and $\lambda$ is similar to that of $\mu _{eff}$, however, with the important difference that the effective viscosity $\mu _{eff}^{S}$ is smaller than the suspending fluid viscosity $\mu$ for a range of viscosity ratios. As seen in the figure, this reduction in viscosity occurs only beyond a viscosity ratio termed as critical viscosity ratio $\lambda _{c}^{S}$. Equating $\mu _{eff}^{S}-1$ to zero, we obtain the critical viscosity ratio from (5.7) as,

(5.8)\begin{align} \lambda_{c}^{S}&={-}(3 \alpha^{10}+125 \alpha^7-336 \alpha^5+200 \alpha^3+8 -5 (49 \alpha^{20}+14 \alpha^{17}-1175 \alpha^{14}+2352 \alpha^{12}\nonumber\\ &\quad -1288 \alpha^{10}-16 \alpha^7+64)^{1/2})/(8 (4 \alpha^{10}-25 \alpha^7+42 \alpha^5-25 \alpha^3+4)), \end{align}

beyond which a reduction in viscosity will be observed.

5.2. Extensional viscosity

By subjecting the dilute dispersion of compound particles to a uniaxial extensional flow, we can calculate the extensional viscosity (or Trouton's viscosity) $\mu _{eff}^{T}$ as the ratio of the normal stress difference to the imposed shear rate,

(5.9)\begin{equation} \mu_{eff}^{T}=\frac{\sigma_{33}-\sigma_{11}}{E_{33}}=\frac{\sigma_{33}-\sigma_{22}}{E_{33}}. \end{equation}

Hence, the extensional viscosity of a dilute dispersion of compound particles can be obtained from (5.4) as

(5.10)\begin{equation} \mu_{eff}^{T}= \left(3+3\phi\left(-\frac{c_{1}}{2 }+Ca\frac{3c_{5}}{4 } \right)\right)+{O}(\phi^{2}, \phi^{2} Ca, \phi Ca^2 ). \end{equation}

The above expression (5.10) can be deduced from the work of Santra et al. (Reference Santra, Panigrahi, Das and Chakraborty2020b) on compound drops by taking the appropriate limit.

The dependencies of the Trouton viscosity of a dilute dispersion of compound particles on the size ratio and viscosity ratio of individual compound particles are shown in figure 12. Again, instead of viscosity, it is the enhancement factor $(\mu _{eff}^{T}-3\mu _{eff})/\phi$ that is plotted against $\lambda$. For a Newtonian fluid, the Trouton viscosity is equal to $3\mu _{eff}$ and the enhancement factor is exactly zero. This limit is indicated by the flat line in figure 12. Dispersions of compound particles show deviations from this Newtonian limit – larger deviations (and thus larger $\mu _{eff}^{T}$) are observed for smaller values of $\alpha$ and $\lambda$. As $\lambda$ increases, the enhancement factor for the Trouton viscosity asymptotes to a constant, independent of $\alpha$. On the other hand, for a given $\lambda$, increase in $\alpha$ reduces the Trouton viscosity. The decrease in the Trouton viscosity with an increase in the size ratio or with an increase in the viscosity ratio occurs for the same reason as the decrease in the effective viscosity discussed in the previous section.

Figure 12. The variation of the Trouton viscosity vs the viscosity ratio $\lambda$ for various size ratios and $Ca=0.1$.

These observations can also be endowed by analysing (5.10). The Newtonian limit can be recovered for the special case, when $Ca = 0$ so that the confining interface is exactly spherical, and (5.10) reduces to give the extensional viscosity $\mu _{eff}^{T} = 3\mu _{eff}$. In general, compound particles show deviations from this Newtonian behaviour since $Ca \neq 0$ and the confining drop is deformed. Of course, as mentioned in the previous sections, the extent of the deviation depends upon the specific values of $\alpha$ and $\lambda$. For large values of $\lambda$ the enhancement factor $(\mu _{eff}^{T}-3\mu _{eff})/\phi$ asymptotes to ${36 Ca}/{35}$ a constant, which is independent of $\alpha$ and consistent with the observations in figure 12. In the limit of $\alpha \to 1$, but retaining ${O}(\alpha - 1)$ terms, the extensional viscosity of a suspension of particles that are coated with a thin fluid film of inner fluid can be calculated as,

(5.11)\begin{equation} \frac{\mu_{eff}^{T}-3\mu_{eff}}{\phi} \simeq Ca \left( -\frac{675 (\lambda-4)}{896 (\alpha-1)}+\frac{225 (45 \lambda^2-130 \lambda+136)}{3584}+{O}(\alpha-1) \right). \end{equation}

It is clear that the Trouton viscosity increases with a decrease in $\alpha$. In the opposite limit, the case of simple drops without encapsulated particles, i.e. as $\alpha \rightarrow \infty$, (5.10) provides the enhancement factor $(\mu _{eff}^{T}-3\mu _{eff})/\phi = 9 Ca (64 \lambda ^3+732 \lambda ^2+1179 \lambda +475)/ (560 (\lambda +1)^3)$. This result matches with the result of Ramachandran & Leal (Reference Ramachandran and Leal2012) with the slip coefficient being zero and with Mandal et al. (Reference Mandal, Das and Chakraborty2017) when no surfactants are present.

Hence, it is clear that a larger Trouton viscosity is observed for smaller values of $\alpha$ and $\lambda$. This behaviour is qualitatively similar to that of the effective shear viscosity. However, comparison with the case of $Ca = 0$ suggests that deviations from the Newtonian limit ($\mu _{eff}^T \neq 3\mu _{eff}$) arise in the dispersion of compound particles due to the deformation of the confining interface. Consequently, a time-dependent increase in the deformation of the confining drop (as discussed in § 4.1) can give rise to a time-dependent increase in the extensional viscosity. Another feature to notice is that, (5.11) shows that the enhancement factor for $\mu _{eff}^{T}$ of a dilute dispersion of compound particles can be made negative. This is similar to the case of shear viscosity and indicates that the extensional viscosity of a dilute suspension of spheres can be reduced by providing a thin film coating of a less viscous fluid on the particles.

5.3. Normal stress differences

In § 4, we have seen that an imposed flow acts to deform the confining spherical drop while the interfacial tension acts to revert it back, yielding an anisotropic shape of the confining drop. Therefore, the force distribution around the compound particle will also be anisotropic, inducing normal stress differences in a dispersion of compound particles. Normal stress differences, a characteristic of viscoelastic behaviour of a fluid, can also be determined by analysing the volume-averaged stress (5.4) as in the last two sections.

Subjecting the dilute dispersion of compound particles to a simple shear flow $\boldsymbol{u}^{imp}= x_{2} \boldsymbol {i}_{1}$, (5.4) renders the first and second normal stress differences $N_{1}$ and $N_{2}$ as

(5.12)\begin{gather} N_{1}=\sigma_{11}-\sigma_{22}=\phi Ca \left(\frac{3c_{8}}{2}\right)+{O}(\phi^{2}, \phi^{2}Ca, \phi Ca^2 ) , \end{gather}

(5.13)\begin{gather}N_{2}=\sigma_{22}-\sigma_{33}=\phi Ca \left(\frac{3(c_{5}-c_{8})}{4}\right)+{O}(\phi^{2}, \phi^{2}Ca, \phi Ca^2 ). \end{gather}

The dependency of the first and second normal stress differences on the viscosity and size ratio of compound particles is shown in figure 13(a). In all cases, $N_1$ is positive and $N_2$ is negative. Moreover, it may be noted that both $N_{1}$ and $-N_{2}$ decrease with an increase in the viscosity ratio for a given $\alpha$. The asymptotic values of the normal stresses as $\lambda \rightarrow \infty$ can be deduced from (5.12) and (5.13) respectively as $N_{1}/\phi Ca=32/5$ and $-N_{2}/\phi Ca = 20/7$. For a given viscosity ratio, both $N_1$ and $-N_2$ decrease with an increase in the size ratio. The two data sets are put together in figure 13(b), which shows the ratio of normal stress differences, $-N_{2}/N_{1}$ as a function of $\lambda$ for various $\alpha$. This ratio increases with an increase in $\lambda$ or $\alpha$. Irrespective of $\alpha$ and $\lambda$, the value of $\mod {-N_2/N_1}$ is less than $1$. This behaviour can be understood by analysing the extent of deformation of the confining drop in different planes. We have seen that the deformation of the confining drop is larger in the flow–gradient plane compared to that in the vorticity–gradient plane. Therefore, the $N_1$ which is based on the stresses in the flow–gradient plane is always larger than the $N_2$ which is based on the stresses in the vorticity–gradient plane. The maximum value of $-N_{2}/N_{1} = 0.4464$ is approached at large $\lambda$, independent of the value of $\alpha$.

Figure 13. Effect of the viscosity ratio $\lambda$ on the normal stress differences for different size ratios; (a) $N_{1}/\phi Ca$ (solid lines) and $N_{2}/\phi Ca$ (dash-dotted lines), (b) $-N_{2}/N_{1}$.

The limiting case of $\alpha \rightarrow \infty$ which corresponds to a dilute suspension of drops that does not contain encapsulated particles, is also shown in figure 13(a). The mathematical expressions that correspond to these curves can be obtained from (5.12) and (5.13) respectively, as $N_{1}/\phi Ca = (16 \lambda +19)^2 / (40 (\lambda +1)^2)$ and $N_{2}/\phi Ca = -(800 \lambda ^3+1926 \lambda ^2+1623 \lambda +551 )/(280 (\lambda +1)^3 )$, and they are consistent with the works of Ramachandran & Leal (Reference Ramachandran and Leal2012) and Mandal et al. (Reference Mandal, Das and Chakraborty2017). On the other hand, in the limit $\alpha \rightarrow 1$, which corresponds to solid particles coated with a thin film of inner fluid, we can determine the normal stress differences as

(5.14)\begin{equation} \frac{N_{1}}{\phi Ca}=\frac{45}{32 (\alpha-1)^2} -\frac{15 (15 \lambda-34)}{64 (\alpha-1)}+\frac{ (3375 \lambda^2-5460 \lambda+4412)}{512}+{O}(\alpha-1), \end{equation}

and

(5.15)\begin{align} \frac{N_{2}}{\phi Ca}={-}\frac{45}{64 (\alpha-1)^2}+\frac{15 (45 \lambda-89)}{448 (\alpha-1)} -\frac{16875 \lambda^2-18720 \lambda+10484}{7168}+{O}(\alpha-1). \end{align}

The apparent large increase in $N_1$ and $-N_2$ as $\alpha \rightarrow 1$ represents the result of increased deformation of the confining interface in this limit. These expressions hold correct only when $Ca < Ca_{crit}$, i.e. when the thin film coated on the particle is stable (without breakup) in the imposed flow.

Hence, as observed for the extensional viscosity, both $N_1$ and $N_2$ are also dependent upon extent of deformation of the confining drop. Consequently, the normal stress differences for a dilute dispersion of compound particles will exhibit capillary-number-dependent and time-dependent behaviour following the extent of deformation of the confining interface.

5.4. Small-amplitude oscillatory shear flow

Finally, we characterise the linear viscoelastic behaviour of a dilute dispersion of compound particles by subjecting the dispersion to a SAOS. In general, the rheological response of the dispersion may be linear or nonlinear, and the transition from linear to nonlinear response can be observed by increasing the amplitude of the shear rate at a fixed frequency. In the following, we restrict our analysis to the linear regime, where the analytical calculations are possible.

Consider an imposed oscillatory shear flow of the form

(5.16)\begin{equation} \boldsymbol{u}^{imp}= \exp{(\textrm{i}\omega t)} x_{2} \boldsymbol{i}_{1}, \end{equation}

where $\omega$ is the frequency of oscillation. In the linear regime, shear stress is related to the complex modulus ($G^*$) (Ramachandran & Leal Reference Ramachandran and Leal2012) via,

(5.17)\begin{equation} \langle \sigma_{12}\rangle= \frac{G^{*}}{\omega} 2E_{12}, \end{equation}

where $G^{*}=G^{\prime }+ \textrm {i} \omega G^{\prime \prime }$, $G^{\prime }$ is the elastic or storage modulus and $G^{\prime \prime }$ is the viscous or loss modulus. Comparing (5.17) and the volume-averaged stress, (5.4), we find that

(5.18)\begin{equation} G^{*}=\omega\left(1-\phi \frac{c_1}{2}\right). \end{equation}

The constant $c_1$ in (5.18), associated with the velocity field, is dependent on the drop shape parameter $b_1$.

Previously, in § 4.1, we described the time evolution $b_1$, and thus the shape of the confining interface in an imposed simple shear flow. Here, we need to analyse the time evolution of the shape parameter $b_{1}$ in an imposed oscillatory shear flow, which is obtained from the leading-order kinematic boundary condition, (2.11), as

(5.19)\begin{equation} \frac{\partial b_{1}}{\partial t}+\textrm{i} \omega b_{1}={-}\frac{b_{1}-b_{1}^{*}}{t_{c}}. \end{equation}

In the long time limit, where the unsteady term can be neglected, so that we can obtain a frequency-dependent shape parameter,

(5.20)\begin{equation} b_{1}=\frac{b_{1}^{*}}{1+\textrm{i} \omega t_{c}}. \end{equation}

Following the procedure developed in Ramachandran & Leal (Reference Ramachandran and Leal2012), we evaluate $c_{1}$ in terms of $b_{1}$ using (5.20) and substitute it in (5.18), to obtain the expression for the complex modulus ($G^*$) as,

(5.21)\begin{align} G^{\prime}+\textrm{i} \omega G^{\prime \prime} &=\omega \left(1-\frac{\phi }{2}\right.(5(( (40- 100 \alpha^3 + 100 \alpha^7 - 40 \alpha^{10} )\lambda - 4 (\alpha-1)^4 (4 + 16 \alpha\nonumber\\ &\quad + 40 \alpha^2 + 55 \alpha^3 + 40 \alpha^4 + 16 \alpha^5 + 4 \alpha^6 ) \lambda^2)+ \textrm{i} \omega ({-}48 - 200 \alpha^3 + 336 \alpha^5\nonumber\\ &\quad - 225 \alpha^7 - 38 \alpha^{10} + (16 + 400 \alpha^3 - 672 \alpha^5 + 250 \alpha^7 + 6 \alpha^{10} )\lambda\nonumber\\ &\quad + 8 ({-}1 + \alpha)^4 (4 + 16 \alpha + 40 \alpha^2 + 55 \alpha^3 + 40 \alpha^4 + 16 \alpha^5 + 4 \alpha^6 ) \lambda^2 ) ) /\nonumber\\ &\quad ( ({-}40 + 100 \alpha^3 - 100 \alpha^7 + 40 \alpha^{10} ) \lambda +10 ({-}1 + \alpha)^4 (4 + 16 \alpha + 40 \alpha^2\nonumber\\ & \quad + 55 \alpha^3+ 40 \alpha^4 + 16 \alpha^5 + 4 \alpha^6 ) \lambda^2 + \textrm{i} \omega (48 + 200 \alpha^3 - 336 \alpha^5 + 225 \alpha^7\nonumber\\ & \quad + 38 \alpha^{10}+ (- 96+ 100 \alpha^3 - 168 \alpha^5 +75 \alpha^7+ 89 \alpha^{10} ) \lambda +12 ({-}1 + \alpha)^4\nonumber\\ &\quad \left. \vphantom{\frac{\phi }{2}} \times (4 + 16 \alpha + 40 \alpha^2 + 55 \alpha^3 + 40 \alpha^4 + 16 \alpha^5 + 4 \alpha^6 ) \lambda^2 ) ) )\right). \end{align}

In the limit of $\alpha \rightarrow \infty$, which corresponds to a drop without a suspended particle, the complex modulus is given as

(5.22)\begin{equation} G^{\prime}+\textrm{i} \omega G^{\prime \prime}=\omega \left(1+ 5\phi \frac{4 \lambda (5+ 2 \lambda )+ \textrm{i} \omega (19 - 3\lambda - 16 \lambda^2 )}{ 40 \lambda(1 +\lambda ) + \textrm{i} \omega ( 38 + 89 \lambda +48 \lambda^2 ) } \right), \end{equation}

and this matches with the expressions derived in Ramachandran & Leal (Reference Ramachandran and Leal2012).

Figure 14 shows the variation of the storage ($G^{\prime }$) and loss ($G^{\prime \prime }$) moduli with frequency ($\omega$) for different size and viscosity ratios. The value of the storage modulus $G^{\prime }$, which varies as $\omega ^2$, is close to zero at low frequencies, but it increases with an increase in frequency, eventually reaching a plateau at high frequencies. On the other hand, the loss modulus $G^{\prime \prime }$ varies as $\omega$ at low and high frequencies. Figure 14 shows that viscous modulus, represented as $G^{\prime \prime }-\omega$ decreases with increase in frequency. It may be noticed that, irrespective of the size or viscosity ratio, the viscous modulus ($G^{\prime \prime }$) dominates the elastic modulus ($G^{\prime }$) at low frequencies and vice versa at high frequencies. In other words, at low frequencies, the viscous response dominates the elastic response and the dispersion behaves like a viscoelastic fluid. At high frequencies, the elastic response dominates the viscous response and thus the dispersion behaves like a viscoelastic solid. This difference in behaviour arises because, at low frequencies, the confining interface relaxes fast enough compared to the imposed time scale, i.e. $t_c < \omega ^{-1}$, leading to a fluid-like behaviour of the dilute dispersion. But at high frequencies, the interface does not relax fast enough to follow up with changes in the imposed flow that the dispersion of compound particles will show a history dependence. It can also be observed that, when $\alpha$ is sufficiently small, the viscous modulus dominates the elastic modulus irrespective of frequency. Therefore, a dilute suspension of spheres coated with a thin film always behaves like a viscoelastic fluid, provided $Ca\ll 1$. This analysis is valid only if (i) the time scale of the imposed flow ($1/\omega$) is much larger than the momentum diffusion time scale ($\rho a^2/\mu$) and (ii) the characteristic time scale of the droplet ($t_c$) is much smaller than the characteristic time scale of the imposed flow ($\omega ^{-1}$, $G^{-1}$). Mathematically, the above conditions are, $\rho a^2/\mu \ll \omega ^{-1}$, $t_c\ll \omega ^{-1}$ and $t_c \ll G^{-1}$. The first inequality is maintained by the choice of Reynolds number, while the second and third inequalities are maintained by carefully selecting the values of $\omega$, and $G^{-1}$.

Figure 14. The normalised components of the complex moduli $G^{\prime }/\phi$ (solid lines) and $(G^{\prime \prime }-\omega )/\phi \omega$ (dash-dotted lines) vs the frequency $\omega$ (see (5.21)). Effects of (a) the size ratio $\alpha$ (for $\lambda =1$) and (b) the viscosity ratio $\lambda$ (for $\alpha =2$).