2.1. Steady component of flow

The equation for the steady component of the velocity within the drop is determined by setting the time derivative equal to zero in (2.14),

(2.34) \begin{align} \frac {\mbox{d}^4 \bar {g}^{(i)}}{\mbox{d} r^{4}} - \frac {12}{r^{2}} \frac {\mbox{d}^2 \bar {g}^{(i)}}{\mbox{d} r^{2}} + \frac {24}{r^{3}} \frac {\mbox{d} \bar {g}^{(i)}}{\mbox{d} r} + \bar {F} = 0. \end{align}

Since there is no Lorentz force outside the drop, the scaled equation for $\bar {g}^{(o)}$ is

(2.35) \begin{align} \frac {\mbox{d}^4 \bar {g}^{(o)}}{\mbox{d} r^{4}} - \frac {12}{r^{2}} \frac {\mbox{d}^2 \bar {g}^{(o)}}{\mbox{d} r^{2}} + \frac {24}{r^{3}} \frac {\mbox{d} \bar {g}^{(o)}}{\mbox{d} r} = 0. \end{align}

Equation (2.35) for the flow outside the drop is solved subject to the boundary conditions (2.28) for $r \rightarrow \infty$ and (2.24) for $r=1$ to obtain the solution

(2.36) \begin{align} \bar {g}^{(o)} = C \left ( 1-\frac {1}{r^{2}} \right ), \end{align}

where $C$ is a constant of integration. The velocity components are

(2.37) \begin{align} \bar {v}_r & = \frac {C (1-r^2)(3 \cos {(\theta )}^2-1)}{r^4},\end{align}

(2.38) \begin{align} \bar {v}_\theta & = \mbox{} \frac {2 C \sin {(\theta )} \cos {(\theta )}}{r^4}.\end{align}

The constant $C$ is determined from the condition for the tangential velocity (2.15) at the surface,

(2.39) \begin{align} C = \tfrac {1}{2} \left . \bar {V}_\theta \right |_{r=1}. \end{align}

The tangential stress boundary condition, (2.26), is divided by $(\mbox{d} \bar {g}^{(o)}/\mbox{d} r)$ , which is equal to $(\mbox{d} \bar {g}^{(i)}/\mbox{d} r)$ from the tangential velocity boundary condition (2.25),

(2.40) \begin{align} \frac {\mbox{d}^2 \bar {g}^{(i)}}{\mbox{d} r^2} \left ( \frac {\mbox{d} \bar {g}^{(i)}}{\mbox{d} r} \right )^{-1} - 2 = \eta _r \left ( \frac {\mbox{d}^2 \bar {g}^{(o)}}{\mbox{d} r^{2}} \left ( \frac {\mbox{d} \bar {g}^{(o)}}{\mbox{d} r} \right )^{-1} -2 \right ) \: \: \mbox{ at } \: \: r^{\dagger } = 1. \end{align}

Substituting the solution (2.36) for $\bar {g}^{(o)}$ in (2.40), the tangential velocity boundary condition reduces to

(2.41) \begin{align} \frac {\mbox{d}^2 \bar {g}^{(i)}}{\mbox{d} r^2} \left ( \frac {\mbox{d} \bar {g}^{(i)}}{\mbox{d} r} \right )^{-1} = 2 - 5 \eta _r \: \: \mbox{ at } \: \: r^{\dagger } = 1. \end{align}

Equation (2.34) is solved for $\bar {g}^{(i)}$ subject to the boundary conditions (2.27) for $r \rightarrow 0$ and the conditions (2.24) and (2.41) at $r=1$ .

The normal stress at the surface is given by (2.20) and (2.22). Here, $\bar {F}_\theta = 0$ because the steady component of the body force is in the radial direction. The time derivative on the right-hand side of (2.22) is also zero. Inside the drop, the solution of (2.34) is substituted into (2.20) and (2.22) to obtain

(2.42) \begin{align} \bar {\sigma }_{rr}^{(i)} = \left ( \frac {1}{6} \frac {\partial ^3 \bar {g}^{(i)}}{\partial r^{3}} - \frac {3}{r^{2}} \frac {\partial \bar {g}^{(i)}}{\partial r} + \frac {6 \bar {g}^{(i)}}{r^{3}} \right ) (3 \cos {(\theta )}^2 - 1) - 2 {Ca}^{-1}. \end{align}

Here, ${Ca} = (\mu _0 H_0^2 R/\gamma )$ is the capillary number, and the last term on the right-hand side in (2.42) is the result of the non-dimensionalisation of the last term on the right-hand side in (2.20). At the surface $r=1$ , $\bar {g}^{(i)} = 0$ , and the normal stress is

(2.43) \begin{align} \left . \bar {\sigma }_{rr}^{(i)} \right |_{r=1} = \left ( \frac {1}{6} \frac {\partial ^3 \bar {g}^{(i)}}{\partial r^{3}} - 3 \frac {\partial \bar {g}^{(i)}}{\partial r} \right ) (3 \cos {(\theta )}^2 - 1) - 2 {Ca}^{-1}. \end{align}

The normal stress outside the drop is determined using the solutions (2.38) and (2.39) for $\bar {g}^{(o)}$ ,

(2.44) \begin{align} \bar {\sigma }_{rr}^{(o)} = \eta _r \left . \bar {V}_\theta \right |_{r=1} \left ( \frac {3}{r^3} - \frac {4}{ r^5} \right ) (3 \cos {(\theta )}^2 - 1). \end{align}

At the surface $r=1$ , the normal stress is

(2.45) \begin{align} \left . \bar {\sigma }_{rr}^{(o)} \right |_{r=1} = \mbox{} - \eta _r \left . \bar {V}_\theta \right |_{r=1} (3 \cos {(\theta )}^2 - 1). \end{align}

The normal stress balance at the drop surface $r=1$ is

(2.46) \begin{align} \left . \bar {\sigma }_{rr}^{(i)} \right |_{r=1} - \left . \bar {\sigma }_{rr}^{(o)} \right |_{r=1} & = - 2 {Ca}^{-1} - 2 {Ca}^{-1} \boldsymbol{\nabla }_s^{2} \bar {h}(\theta ) \nonumber \\ & = - 2 {Ca}^{-1} + 6 {Ca}^{-1} \bar {h}_0 (3 \cos {(\theta )}^2-1), \end{align}

where $\boldsymbol{\nabla }_s$ is the Laplacian along the spherical surface $r=1$ . The expressions (2.43) and (2.45) are substituted into (2.46). The first term on the right-hand side in (2.46) cancels the last term on the right-hand side in (2.43) for $\bar {\sigma }_{rr}^{(i)}$ , and the resulting equation is divided by $(3 \cos {(\theta )}^2 - 1)$ to determine $\bar {h}_0$ . Since the left-hand side of (2.46) is independent of the capillary number, it is evident that $\bar {h}_0$ is proportional to the capillary number in the limit ${Ca} \ll 1$ considered here.

2.2. Oscillatory component of flow

The oscillatory component of the forcing in (2.29) and the function $g$ in (2.30) are substituted into the momentum balance (2.14). The coefficient of $\exp {(2 \imath t)}$ in the resulting equation is

(2.47) \begin{align} \frac {\mbox{d}^4 \tilde {g}^{(i)}}{\mbox{d} r^{4}} - \frac {12}{r^{2}} \frac {\mbox{d}^2 \tilde {g}^{(i)}}{\mbox{d} r^{2}} + \frac {24}{r^{3}} \frac {\mbox{d} \tilde {g}^{(i)}}{\mbox{d} r} - \tilde {\alpha }^2 \left ( \frac {\mbox{d}^2 \tilde {g}^{(i)}}{\mbox{d} r^{2}} - \frac {6 \tilde {g}^{(i)}}{r^2} \right ) + \tilde {F} = 0, \end{align}

where the complex parameter $\tilde {\alpha }$ is proportional to the square root of the Reynolds number, ${Re}_\omega = (\rho _i \omega R^2/\eta _i)$ , based on the frequency of oscillations and the density and viscosity of the fluid within the drop,

(2.48) \begin{align} \tilde {\alpha } = \sqrt {2 \imath \rho _i \omega R^2/\eta _i} = \sqrt {2 \imath {Re}_\omega }. \end{align}

There is no forcing for the flow outside the drop, and the equivalent of (2.47) is

(2.49) \begin{align} \frac {\mbox{d}^4 \tilde {g}^{(o)}}{\mbox{d} r^{4}} - \frac {12}{r^{2}} \frac {\mbox{d}^2 \tilde {g}^{(o)}}{\mbox{d} r^{2}} + \frac {24}{r^{3}} \frac {\mbox{d} \tilde {g}^{(o)}}{\mbox{d} r} - \tilde {\alpha }_o^2 \left ( \frac {\mbox{d}^2 \tilde {g}^{(o)}}{\mbox{d} r^{2}} - \frac {6 \tilde {g}^{(o)}}{r^2} \right ) = 0, \end{align}

where

(2.50) \begin{align} \tilde {\alpha }_o = \sqrt {2 \imath \rho _o \omega R^2/\eta _o} = \sqrt {2 \imath {Re}_\omega ^{(o)}}. \end{align}

Equation (2.49) for $\tilde {g}^{(o)}$ can be solved analytically,

(2.51) \begin{align} \tilde {g}^{(o)} & = \frac {D_1}{r^2} + D_2 r^3 + D_3 \exp {(\tilde {\alpha }_o r)} \left ( 1 - 3 (\tilde {\alpha }_o r)^{-1} + 3 (\tilde {\alpha }_o r)^{-2} \right ) \nonumber \\ & \quad + D_4 \exp {({-} \tilde {\alpha }_o r)} \left ( 1 + 3 (\tilde {\alpha }_o r)^{-1} + 3 (\tilde {\alpha }_o r)^{-2} \right ). \end{align}

The constants $D_2$ and $D_3$ are zero due to the condition (2.28) that the velocity tends to zero for $r \rightarrow \infty$ . A relation between $D_1$ and $D_4$ is obtained from the zero normal velocity condition (2.24) at $r=1$ , and the equation for the stream function is

(2.52) \begin{align} \tilde {g}^{(o)} & = \frac {6 D}{\tilde {\alpha }_o^2} \left ( \frac {1}{r^2} - \exp {({-} \tilde {\alpha }_o (r - 1))} \left (\frac {1 + 3 (\tilde {\alpha }_o r)^{-1} + 3 (\tilde {\alpha }_o r)^{-2}}{1 + 3 \tilde {\alpha }_o^{-1} + 3 \tilde {\alpha }_o^{-2}} \right ) \right ). \end{align}

The boundary condition (2.26) is simplified using the above solution for $\tilde {g}^{(o)}$ . If we divide the right- and left-hand sides of (2.26) by $\left . (\mbox{d} \tilde {g}^{(o)}/\mbox{d} r) \right |_{r=1}$ , which is equal to $\left . (\mbox{d} \tilde {g}^{(i)}/\mbox{d} r) \right |_{r=1}$ (boundary condition (2.25)), we obtain

(2.53) \begin{align} \frac {\mbox{d}^2 \tilde {g}^{(i)}}{\mbox{d} r^2} \left (\frac {\mbox{d} \tilde {g}^{(i)}}{\mbox{d} r} \right )^{-1} & = 2 (1 - \eta _r) + \eta _r \frac {\mbox{d}^2 \tilde {g}^{(o)}}{\mbox{d} r^2} \left (\frac {\mbox{d} \tilde {g}^{(o)}}{\mbox{d} r} \right )^{-1} \: \: \mbox{at} \: \: r=1. \end{align}

The right-hand side of the above boundary condition is evaluated using the solution (2.52),

(2.54) \begin{align} \frac {\mbox{d}^2 \tilde {g}^{(i)}}{\mbox{d} r^2} \left (\frac {\mbox{d} \tilde {g}^{(i)}}{\mbox{d} r} \right )^{-1} & = 2 (1 - \eta _r) - \eta _r \left ( \frac {3 + 3 \tilde {\alpha }_o + \tilde {\alpha }_o^2}{1 + \tilde {\alpha }_o} \right ) \: \: \mbox{at} \: \: r=1. \end{align}

Equation (2.47) is solved for $\tilde {g}^{(i)}$ subject to the boundary conditions (2.24) and (2.54) at $r=1$ and the conditions (2.27) for $r \rightarrow 0$ .

The shear and normal stresses are simplified using the same procedure as that used in (2.42)–(2.46) for a steady flow, but with the important difference that the inertial term on the momentum equation is $(\tilde {\alpha }^2/r) (\partial \tilde {g}/ \partial r) \sin {(\theta )}^2 \cos {(\theta )}$ in dimensionless form. The oscillatory component of the pressure is expressed as

(2.55) \begin{align} \tilde {p}^{(i)} = \tilde {P}^{(i)}(r) (3 \cos {(\theta )}^2 - 1 ). \end{align}

This is substituted into the $\theta$ momentum balance to obtain the oscillating component of (2.19),

(2.56) \begin{align} \frac {\tilde {\alpha }^2}{r} \frac {\mbox{d} \tilde {g}^{(i)}}{\mbox{d} r} = \frac {6 \tilde {P}^{(i)}}{r} + \frac {1}{r} \frac {\mbox{d}^3 \tilde {g}^{(i)}}{\mbox{d} r^{3}} - \frac {6}{r^3} \frac {\mbox{d} \tilde {g}^{(i)}}{\mbox{d} r} + \frac {12 \tilde {g}^{(i)}}{r^{4}} + \tilde {F}_\theta (r). \end{align}

The oscillatory components (coefficients of $\exp {(2 \imath t)}$ ) in the expressions for the tangential and normal stresses are

(2.57) \begin{align} \tilde {\sigma }^{(i)}_{rr} & = - \tilde {p}^{(i)} + 2 \frac {\partial \tilde {v}^{(i)}_r}{\partial r} = \tilde {\Sigma }^{(i)}_{rr} (3 \cos {(\theta )}^2 - 1), \\[-12pt] \nonumber \end{align}

(2.58) \begin{align} \tilde {\sigma }^{(i)}_{\theta r} & = \left ( r \frac {\partial }{\partial r} \left ( \frac {\tilde {v}^{(i)}_\theta }{r} \right ) + \frac {1}{r} \frac {\partial \tilde {v}^{(i)}_r}{\partial \theta } \right ) = \tilde {\Sigma }^{(i)}_{\theta r} \sin {(\theta )} \cos {(\theta )}, \\[1pt] \nonumber \end{align}

where, $\tilde {\Sigma }^{(i)}_{rr}$ and $\tilde {\Sigma }^{(i)}_{\theta r}$ are

(2.59) \begin{align} \tilde {\Sigma }^{(i)}_{rr} & = - \tilde {P}^{(i)} + \left ( \frac {4 \tilde {g}^{(i)}}{r^{3}} - \frac {2}{r^{2}} \frac {\mbox{d} \tilde {g}^{(i)}}{\partial r} \right ), \nonumber \\ & = \frac {1}{6} \frac {\mbox{d}^3 \tilde {g}^{(i)}}{\mbox{d} r^{3}} - \frac {3}{r^{2}} \frac {\mbox{d} \tilde {g}^{(i)}}{\mbox{d} r} + \frac {6 \tilde {g}^{(i)}}{r^{3}} + \frac {r \tilde {F}_{\theta }}{6} - \frac {\tilde {\alpha }^2}{6} \frac {\mbox{d} \tilde {g}^{(i)}}{\mbox{d} r},\end{align}

(2.60) \begin{align} \tilde {\Sigma }^{(i)}_{\theta r} & = \left ( \frac {1}{r} \frac {\mbox{d}^2 \tilde {g}^{(i)}}{\mbox{d} r^{2}} - \frac {2}{r^{2}} \frac {\mbox{d} \tilde {g}^{(i)}}{\mbox{d} r} + \frac {6 \tilde {g}^{(i)}}{r^{3}} \right ).\end{align}

The stresses outside the drop are calculated from $\tilde {g}^{(o)}$ (2.52), in a similar manner. The pressure is expressed as

(2.61) \begin{align} \tilde {p}^{(o)} = \tilde {P}^{(o)} \big(3 \cos {(\theta )}^2-1 \big). \end{align}

The pressure is determined from the $\theta$ momentum equation, resulting in an equation similar to (2.56) but without the forcing term,

(2.62) \begin{align} \frac {\eta _r \tilde {\alpha }_o^2}{r} \frac {\mbox{d} \tilde {g}^{(o)}}{\mbox{d} r} = \frac {6 \tilde {P}^{(o)}}{r} + \eta _r \left ( \frac {1}{r} \frac {\mbox{d}^3 \tilde {g}^{(o)}}{\mbox{d} r^{3}} - \frac {6}{r^3} \frac {\mbox{d} \tilde {g}^{(o)}}{\mbox{d} r} + \frac {12 \tilde {g}^{(o)}}{r^{4}} \right ). \end{align}

Here, the factors $\eta _r$ on the left- and right-hand sides of the equation result from the use of the viscosity of the inner fluid, $\eta _i$ , for non-dimensionalisation. The shear and normal stresses, equivalent to (2.57)–(2.60), are

(2.63) \begin{align} \tilde {\sigma }^{(o)}_{rr} & = - \tilde {p}^{(o)} + 2 \eta _r \frac {\partial \tilde {v}^{(o)}_r}{\partial r} = \tilde {\Sigma }^{(o)}_{rr} (3 \cos {(\theta )}^2 - 1 ),\end{align}

(2.64) \begin{align} \tilde {\sigma }^{(o)}_{\theta r} & = \eta _r \left ( r \frac {\partial }{\partial r} \left ( \frac {\tilde {v}^{(o)}_\theta }{r} \right ) + \frac {1}{r} \frac {\partial \tilde {v}^{(o)}_r}{\partial \theta } \right ) = \tilde {\Sigma }^{(o)}_{\theta r} \sin {(\theta )} \cos {(\theta )},\end{align}

where, $\tilde {\Sigma }^{(o)}_{rr}$ and $\tilde {\Sigma }^{(o)}_{\theta r}$ are

(2.65) \begin{align} \tilde {\Sigma }^{(o)}_{rr} & = \eta _r \left ( \frac {1}{6} \frac {\mbox{d}^3 \tilde {g}^{(o)}}{\mbox{d} r^{3}} - \frac {3}{r^{2}} \frac {\mbox{d} \tilde {g}^{(o)}}{\mbox{d} r} + \frac {6 \tilde {g}^{(o)}}{r^{3}} \right ) - \frac {\eta _r \tilde {\alpha }_o^2}{6} \frac {\mbox{d} \tilde {g}^{(o)}}{\mbox{d} r}, \end{align}

(2.66) \begin{align} \tilde {\Sigma }^{(o)}_{\theta r} & = \eta _r \left ( \frac {1}{r} \frac {\mbox{d}^2 \tilde {g}^{(o)}}{\mbox{d} r^{2}} - \frac {2}{r^{2}} \frac {\mbox{d} \tilde {g}^{(o)}}{\mbox{d} r} + \frac {6 \tilde {g}^{(o)}}{r^{3}} \right ). \end{align}

The oscillatory response of the surface deformation is defined by the parameter $\tilde {h}_0$ (2.31)–(2.33). This is determined from the oscillatory component of the normal stress equation, which is the analogue of (2.46) for the steady displacement,

(2.67) \begin{align} \left . \tilde {\sigma }^{(i)}_{rr} \right |_{r=1} - \left . \tilde {\sigma }^{(o)}_{rr} \right |_{r=1} = 6 {Ca}^{-1} \tilde {h}_0 (3 \cos {(\theta )}^2-1 ), \end{align}

where (2.57) and (2.63) are substituted for $\tilde {\sigma }^{(i)}_{rr}$ and $\tilde {\sigma }^{(o)}_{rr}$ .

3.1. Flow outside the drop

The velocity components outside the drop are given by (2.37) and (2.38), with the constant $C$ given by (2.39). This is a biaxial extensional flow, where there is extension along the equatorial $x{-}y$ plane and compression along the $z$ axis. Consider a flow driven by the rate of deformation tensor of the form

(3.1) \begin{align} \bar {G} = \left ( \begin{array}{ccc} \hat {\boldsymbol{e}}_x & \hat {\boldsymbol{e}}_y & \hat {\boldsymbol{e}}_z \end{array} \right ) \left ( \begin{array}{ccc} \tfrac {1}{2} & 0 & 0 \\ 0 & \tfrac {1}{2} & 0 \\ 0 & 0 & -1 \end{array} \right ) \left ( \begin{array}{c} \hat {\boldsymbol{e}}_x \\ \hat {\boldsymbol{e}}_y \\ \hat {\boldsymbol{e}}_z \end{array} \right ). \end{align}

In Stokes flow, the general expression for the velocity and pressure fields in indicial notation are

(3.2) \begin{align} \bar {v}_i & = C_1 G_{ij} \Phi ^{(1)}_j + C_2 G_{jk} \Phi ^{(3)}_{ijk} + \frac {p x_i}{2 \eta },\end{align}

(3.3) \begin{align} \bar {p} & = 2 \eta C_3 G_{jk} \Phi ^{(2)}_{jk},\end{align}

where each index represents a vector direction, and a repeated index implies a dot product. Here, $C_1$ , $C_2$ and $C_3$ are constants, $\Phi ^{(1)}_i$ , $\Phi ^{(2)}_{ij}$ and $\Phi ^{(3)}_{ijk}$ are the solutions of the Laplace equation in spherical coordinates,

(3.4) \begin{align} \Phi _i^{(1)} & = \frac {\partial }{\partial x_i} \left ( \frac {1}{r} \right ) = \mbox{} - \frac {x_i}{r^3}, \: \: \Phi _{ij}^{(2)} = \frac {\partial }{\partial x_i} \frac {\partial }{\partial x_j} \left ( \frac {1}{r} \right ) = \mbox{} \frac {3 x_i x_j}{r^5} - \frac {\delta _{ij}}{r^3}, \nonumber \\ \Phi _{ijk}^{(3)} & = \frac {\partial }{\partial x_i} \frac {\partial }{\partial x_j} \frac {\partial }{\partial x_k} \left ( \frac {1}{r} \right ) = \frac {3 \delta _{ij} x_k}{r^5} + \frac {3 \delta _{jk} x_i}{r^5} + \frac {3 \delta _{ik} x_j}{r^5} - \frac {15 x_i x_j x_k}{r^7}. \end{align}

Here, $\delta _{ij}$ is the Kronecker delta. The velocity field is incompressible only if $C_1 = 0$ in (3.2). The zero normal velocity boundary condition at the surface implies that

(3.5) \begin{align} v_i n_i = v_i x_i = 0 \: \: \mbox{at} \: \: r=1, \end{align}

where $\boldsymbol{n} = \boldsymbol{x}$ is the unit vector at the surface $r=1$ . The zero normal velocity condition provides a relation between the constants $C_2$ and $C_3$ ,

(3.6) \begin{align} C_2 G_{jk} x_i \Phi ^{(3)}_{ijk} + C_3 x_i^2 G_{jk} \Phi _{jk}^{(2)} = 0 \: \: \mbox{at} \: \: r=1. \end{align}

This is solved to obtain $C_3 = 3 C_2$ , and the velocity field is a function of one constant $C_2$ .

The radial and azimuthal components of the velocity and the pressure can now be calculated from the expressions (3.2) and (3.3) with $C_1 = 0$ and $C_3 = 3 C_2$ ,

(3.7) \begin{align} \bar {v}_r & = \frac {9 C_2 (1-r^2 ) (3 \cos {(\theta )}^2 - 1 )}{2 r^4}, \end{align}

(3.8) \begin{align} \bar {v}_\theta & = \frac {9 C_2 \sin {(\theta )} \cos {(\theta )}}{r^4},\end{align}

(3.9) \begin{align} \bar {p} & = - \frac {9 C_2 \eta _r (3 \cos {(\theta )}^2 - 1 )}{r^3}.\end{align}

Comparing (2.38), (2.39) and (3.8), we find that

(3.10) \begin{align} C_2 = \frac {\left . \bar {V}_\theta \right |_{r=1}}{9}. \end{align}

The velocity vectors outside the drop are shown in figure 2. There is a biaxial extensional flow with compression along the axial direction and extension in the equatorial plane. This is generated by an axisymmetric point force dipole aligned along the axial direction.

Figure 2.

The velocity vectors (arrows) for the flow outside the drop due to an oscillating magnetic field. The applied magnetic field is along the $z$ axis, and $x$ is along any direction in the equatorial plane. The solid quarter-circle is the boundary of the drop. The flow is in the azimuthal direction ( $\bar {v}_r = 0$ ) on the blue dashed line.

The ‘particle stress’ for a suspension of particles in the Stokes regime is defined as

(3.11) \begin{align} \bar {\sigma }_{ij}^{p} = \frac {1}{V} \int _S \mbox{d} S \: \left . \bar {\sigma }_{ik} n_k x_j \right |_{r=1}, \end{align}

where $V$ is the volume of the particle, $S$ is the surface and $\boldsymbol{n}$ is the outwards unit normal. For biaxial extension, the particle stresses are related, $\bar {\sigma }_{xx}^{(p)} = \bar {\sigma }_{yy}^{(p)} = - 2 \bar {\sigma }_{zz}^{(p)}$ , and it is sufficient to determine any one component. The component $\bar {\sigma }_{zz}^{(p)}$ is

(3.12) \begin{align} \bar {\sigma }_{zz}^{(p)} = \frac {1}{V} \int _S \mbox{d} S \: (\cos {(\theta )} \bar {\sigma }_{rr} - \sin {(\theta )} \bar {\sigma }_{\theta r}) \cos {(\theta )}, \end{align}

where $\bar {\sigma }_{rr}$ and $\bar {\sigma }_{\theta r}$ are given in (2.20) and (2.21). Substituting the expressions (3.7) and (3.8) for the velocity components and (3.10) for the constant $C_2$ , we find that

(3.13) \begin{align} \bar {\sigma }_{zz}^{(p)} = \frac {6 \eta _r \left . \bar {V}_\theta \right |_{r=1}}{5}. \end{align}

There is a factor $\eta _r$ in the above expression because the viscosity to be used in (2.20) and (2.21) is $\eta _o$ for the fluid outside the drop, whereas the velocities are scaled by the viscosity $\eta _i$ of the inner fluid in table 1.

3.2. Asymptotic results for $\beta \ll 1$

The approximation (B13) for $\bar {F}$ is substituted into (2.34), and solved analytically to obtain

(3.14) \begin{align} \bar {g}^{(i)} = M_1 r^5 + M_2 r^3 + M_3 + \frac {M_4}{r^2} - \frac {\beta ^4 r^7}{10\,080}, \end{align}

where $M_1, M_2, M_3$ and $M_4$ are constants of integration. The boundary condition (2.27) for $r \rightarrow 0$ requires $M_3=M_4=0$ . The normal velocity condition (2.24) and the tangential stress balance condition (2.41) are used to determine the constants $M_1$ and $M_2$ , and the final expression for $\bar {g}^{(i)}$ is

(3.15) \begin{align} \bar {g}^{(i)} = \frac {\beta ^4 r^3 (1-r^2 ) (5 r^2 (1+\eta _r ) - 5 \eta _r - 9 )}{50\,400(1 + \eta _r)}. \end{align}

The tangential velocity measure $\bar {V}_\theta$ (2.16) and stress measure $\bar {\Sigma }_{\theta r}$ (2.23) at the surface $r=1$ are

(3.16) \begin{align} \left . \bar {V}_\theta \right |_{r=1} & = \left . \frac {\bar {v}_\theta }{\sin {(\theta )} \cos {(\theta )}} \right |_{r=1} = \frac {\beta ^4}{6300 (1 + \eta _r)}, \nonumber \\ \left . \bar {\Sigma }_{\theta r} \right |_{r=1} & = \left . \frac {\bar {\sigma }_{\theta r}}{\sin {(\theta )} \cos {(\theta )}} \right |_{r=1} = \mbox{} - \frac {\beta ^4 \eta _r}{1260 (1 + \eta _r)}. \end{align}

The parameter $\bar {h}_0$ , which characterises the steady shape deformation is determined from (2.46), where the normal stresses inside and outside the drop are given by (2.43) and (2.45), respectively,

(3.17) \begin{align} {Ca}^{-1} \bar {h}_0 = \mbox{} - \frac {\beta ^4 (17 + 18 \eta _r)}{75\,600 (1 + \eta _r)}. \end{align}

The parameter $\bar {h}_0$ is negative in the limit $\beta \ll 1$ , and it increases proportional to $\beta ^4$ . The implications of this are discussed in § 3.5.

3.3. Asymptotic results for $\beta \gg 1$

In this limit, the approximation (B14) for $\bar {F}$ is substituted into (2.34). The solution for $\bar {g}^{(i)}$ is expressed as the sum of the general solution $\bar {g}^{(i)}_g$ and the particular solution $\bar {g}^{(i)}_p$ . The general solution of the homogeneous equation which satisfies the condition (2.27) for $r \rightarrow 0$ is

(3.18) \begin{align} \bar {g}^{(i)}_g = N_1 r^3 + N_2 r^5, \end{align}

where $N_1$ and $N_2$ are constants to be determined from the boundary conditions at the interface. The approximation for the particular solution $\bar {g}^{(i)}_p$ in the limit $\beta \gg 1$ is determined as follows. Since the variation of $\bar {F}$ is restricted to a region of thickness $\beta ^{-1}$ close to the wall, the scaled distance from the wall can be defined as $\xi =\beta (1-r)$ . When expressed in terms of the scaled variable $\xi$ , (2.34) is

(3.19) \begin{align} \beta ^4 \frac {\mbox{d}^4 \bar {g}^{(i)}_p}{\mbox{d} \xi ^{4}} - \frac {12 \beta ^2}{(1 - \xi /\beta )^2} \frac {\mbox{d}^2 \bar {g}^{(i)}_p}{\mbox{d} \xi ^2} - \frac {24 \beta }{(1 - \xi /\beta )^3} \frac {\mbox{d} \bar {g}^{(i)}_p}{\mbox{d} \xi } + \left ( \frac {9 \beta \exp {\big[{-} \sqrt {2} \xi \big]}}{4 \sqrt {2} (1-\xi /\beta )^2} \right ) = 0. \end{align}

Here, we have used the approximation (B14) for $\bar {F}$ . The first and last terms on the right-hand side are dominant terms in the limit $\beta \gg 1$ . The following solution is obtained by retaining only these two terms and approximating $(1 - \xi /\beta ) \approx 1$ :

(3.20) \begin{align} \bar {g}^{(i)}_p = \mbox{} - \frac {9 \exp {\big[{-} \sqrt {2} \xi \big]}}{16 \sqrt {2} \beta ^3} = - \frac {9 \exp {\big[{-} \sqrt {2} \beta (1-r) \big]}}{16 \sqrt {2} \beta ^3}. \end{align}

The solution for $\bar {g}^{(i)}$ is the sum of (3.18) for $\bar {g}^{(i)}_g$ and (3.20) for $\bar {g}^{(i)}_p$ . In the resulting equation, one of the constants is determined from the condition (2.24) that the normal velocity is zero at $r=1$ . The resulting solution has one unknown constant,

(3.21) \begin{align} \bar {g}^{(i)} = N \big(r^5 - r^3 \big) + \frac {9 \big(r^3 - \exp { \big[{-} \sqrt {2} \beta (1-r)\big]} \big)}{16 \sqrt {2} \beta ^3}. \end{align}

The constant $N$ is determined using the boundary condition (2.41), and the final solution for $\bar {g}^{(i)}$ is

(3.22) \begin{align} \bar {g}^{(i)} & = \frac {9 \big(r^3 - \exp {\big[{-} \sqrt {2} \beta (1-r) \big]} \big)}{16 \sqrt {2} \beta ^3} - \frac {9 r^3 (1-r^2 )}{80 \sqrt {2} \beta (1 + \eta _r)} \nonumber \\ & \quad + \frac {9(2 - 5 \eta _r) r^3 (1-r^2 )}{160 \beta ^2 (1+\eta _r)} + \frac {27 \eta _r r^3 (1-r^2 )}{32 \sqrt {2} \beta ^3 (1+\eta _r)}. \end{align}

The third and fourth terms on the right-hand side in the above equation are included, because they become dominant for $\eta _r \gg 1$ . Using the approximation (3.22) for $\bar {g}^{(i)}$ , the measures for the tangential velocity and stress at the interface, (2.16) and (2.23), are

(3.23) \begin{align} \left . \bar {V}_\theta \right |_{r=1} & = \left . \frac {\bar {v}_\theta }{\sin {(\theta )} \cos {(\theta )}} \right |_{r=1} = \frac {9}{40 \sqrt {2} \beta (1 + \eta _r)}, \nonumber \\ \left . \bar {\Sigma }_{\theta r} \right |_{r=1} & = \left . \frac {\bar {\sigma }_{\theta r}}{\sin {(\theta )} \cos {(\theta )}} \right |_{r=1} = \mbox{} - \frac {9 \eta _r}{8 \sqrt {2} \beta (1 + \eta _r)}. \end{align}

The drop deformation in the limit of low capillary number is characterised by the parameter $\bar {h}_0$ , which is determined from (2.46), (2.43) and (2.45). In the limit $\beta \gg 1$ , $\tilde {g}^{(i)} \sim \beta ^{-1}$ (3.22). The pressure is evaluated from (2.56), in which the dominant term is the third derivative of $\tilde {g}^{(i)}$ on the right-hand side which is $O(1)$ due to the first term on the right-hand side of (3.22). Due to this term, $\bar {\sigma }_{rr}^{(i)}$ is $O(1)$ . The stress in the outer fluid $\bar {\sigma }_{rr}^{(o)}$ is $O(\beta ^{-1})$ , because the flow is driven by the tangential velocity and stress at the interface which is proportional to $\beta ^{-1}$ . Therefore, the parameter $\bar {h}_0$ is calculated from the relation (2.46),

(3.24) \begin{align} {Ca}^{-1} \bar {h}_0 = \mbox{} \frac {\left . \bar {\Sigma }_{rr}^{(i)} \right |_{r=1}}{6} = \mbox{} \frac {1}{36} \left . \frac {\mbox{d}^3 \tilde {g}^{(i)}}{\mbox{d} r^3} \right |_{r=1} = \mbox{} - \frac {1}{32}. \end{align}

Here, we have retained only the $O(1)$ contribution to $(\mbox{d}^3 \tilde {g}^{(i)}/\mbox{d} r^3)$ on the right-hand side.

3.4. Numerical results for $\beta \sim 1$

The fourth-order differential equation (2.34) for $\bar {g}^{(i)}$ , with the function $\bar {F}$ specified by (B12), is solved numerically as follows. A third-order polynomial expansion of $\bar {F}$ about $r=0$ is used to determine the function $\bar {g}^{(i)}$ analytically at $r=0.02$ that satisfies the conditions (2.27) at $r=0$ . The solution contains two constants, since (2.34) is a fourth-order equation. For initial guesses of these two constants, numerical integration using a fourth-order Runge–Kutta scheme is used to determine the value of $\bar {g}^{(i)}$ and its derivatives at $r=1$ . A two-dimensional Newton–Raphson iteration procedure is used to converge to the values of the constants that satisfy the conditions (2.24) and (2.41).

The stream function is calculated from $\bar {g}^{(i)}$ using (2.11), and the velocities within the drop are determined from the relations (2.15). The vector plots for the circulation within the drop are shown in figure 3 for $\beta =1$ and for three different viscosity ratios. The vectors are shown in the meridional $x{-}z$ plane, where $z$ is the axis in the direction of the magnetic field, and $x$ is any direction on the equatorial plane perpendicular to the magnetic field. The velocity field is axisymmetric about the $z$ axis. The velocity field below the equatorial plane is a mirror image of that above the equatorial plane. Therefore, the velocity field is shown only for $z\gt 0$ in figure 3.

Figure 3.

The velocity vectors (arrows) and the streamlines (dotted lines) for the flow in a drop due to an oscillating magnetic field for different values of the viscosity ratio $\eta _r$ . The applied magnetic field is along the $z$ axis, and $x$ is along any direction in the equatorial plane. The solid quarter-circle is the boundary of the drop. The flow is radial ( $\bar {v}_\theta = 0$ ) along the dashed red quarter circle. The flow is in the azimuthal direction ( $\bar {v}_r = 0$ ) on the blue dashed line.

An axisymmetric circulation is observed in each hemisphere, where the flow is inwards at the equatorial plane, and outwards along the axis. The circulation is from the axis to the equatorial plane close to the surface, and in the opposite direction near the centre. There is a spherical surface, shown by the dashed red lines in figure 3, where $\bar {v}_\theta = 0$ , and the flow is in the radial direction. The velocity is the azimuthal direction, and $\bar {v}_r = 0$ , along the dashed blue lines. The intersection of these two is the centre of circulation where the velocity is zero. The inward velocity at the equatorial plane can be understood from the direction of the body force density. Equations (A31)–(A33) show that the electric field is in the $- \hat {\boldsymbol{e}}_\phi$ direction along the surface. Equation (A24) shows that the magnetic field near the surface has contributions in the $\hat {\boldsymbol{e}}_r$ and $- \hat {\boldsymbol{e}}_\theta$ direction for $\beta \ll 1$ , while (A29) shows that the magnetic field is primarily in the $- \hat {\boldsymbol{e}}_\theta$ direction for $\beta \gg 1$ . Consequently, the force density, which is the cross product of the electric and magnetic fields, is in the $- \hat {\boldsymbol{e}}_r$ direction at the equator, resulting in an inwards flow. The electric field and the force density decrease to zero along the axis, but mass conservation results in an outwards flow. This results in the circulation pattern observed in figure 3.

As the ratio of viscosities $\eta _r$ is increased, figure 3 shows that the flow pattern does not change qualitatively. However, the tangential velocity at the surface decreases as $\eta _r$ increases. The limit $\eta _r \rightarrow \infty$ is equivalent to the zero tangential velocity condition at the surface. However, figure 3 shows that internal circulation persists even when the tangential velocity at the surface is zero.

The velocity measures $\bar {V}_r = \bar {v}_r/(3 \cos {(\theta )}^2-1)$ and $\bar {V}_{\theta } = \bar {v}_\theta /(\cos {(\theta )} \sin {(\theta )})$ , (2.15)–(2.16), are shown as a function of $r$ in figure 4. The radial velocity, shown by the red curves, is zero at the centre and the surface of the drop, and has a maximum in between. The velocity in the azimuthal direction decreases from zero near the centre, reaches a minimum and then increases and becomes positive. The dashed red line in figure 3 shows the location at which the azimuthal velocity passes through zero. The tangential velocity has zero slope at the surface for $\eta _r \rightarrow 0$ due to the zero-stress condition at the surface, and it decreases to zero for $\eta _r \rightarrow \infty$ consistent with the no-slip condition.

Figure 4.

The velocity measures $\bar {V}_r = \bar {v}_r/(3 \cos {(\theta )}^2-1)$ ((2.16), red line) and $\bar {V}_{\theta } = \bar {v}_\theta /(\cos {(\theta )} \sin {(\theta )})$ ((2.16), blue line) for $\eta _r \rightarrow 0$ (a,d,g), $\eta _r=1$ (b,e,h) and $\eta _r \rightarrow \infty$ (c,f,i), and for $\beta = 1$ (a–c), $\beta = 3$ (d–f) and $\beta = 10$ (g–i). The dashed black lines in (a)–(c) are obtained from the $\beta \ll 1$ asymptotic solution (3.15), and the dashed black lines in the (g)–(i) are obtained from the $\beta \gg 1$ asymptotic solution (3.22).

The $\beta \ll 1$ asymptotic results, derived from (3.15), are shown by the dashed black lines in figure 4(a–c) for $\beta = 1$ . The numerical and asymptotic results are found to be in quantitative agreement. The $\beta \gg 1$ asymptotic results, derived from (3.22), are shown by the dashed black lines in figure 4(g–i). These asymptotic results qualitatively capture the variation in the velocity components, but there are quantitative differences. For $\beta = 30$ (not shown here), we find quantitative agreement to within 2 % between the asymptotic and numerical results.

The tangential velocity and stress measures at the surface, $\left . \bar {V}_\theta \right |_{r=1}$ and $\left . \bar {\Sigma }_{\theta r} \right |_{r=1}$ , are shown as a function of $\beta$ in figure 5. From the asymptotic expressions (3.16) and (3.23), $\left . \bar {V}_\theta \right |_{r=1} \propto (1+\eta _r)^{-1}$ and $\left . \Sigma _{\theta r} \right |_{r=1} \propto \eta _r/ (1+\eta _r)$ in both limits for $\beta \ll 1$ and $\beta \gg 1$ . Therefore, the products $(1 + \eta _r) \left . \bar {V}_\theta \right |_{r=1}$ and $\mbox{} - (1 + \eta _r) \left . \bar {\Sigma }_{\theta r} / \eta _r \right |_{r=1}$ are shown as a function of $\beta$ in figure 5.

Figure 5.

The measure (a) $(1 + \eta _r) \bar {V}_\theta$ of the tangential velocity and (b) $\mbox{} - ((1 + \eta _r) /\eta _r) \bar {\Sigma }_{\theta r}$ of the tangential stress at the surface $r=1$ as a function of $\beta$ for viscosity ratio $\eta _r \rightarrow 0$ ( $\circ$ ), $\eta _r = 1.0$ ( $\triangle$ ), $\eta _r \rightarrow \infty$ ( $\nabla$ ). The definitions of $\bar {V}_\theta$ and $\bar {\Sigma }_{\theta r}$ are given in (2.16) and (2.23), respectively. The dotted red line on the left is the $\beta \ll 1$ asymptotic result (3.16). The dotted red line on the right is the $\beta \gg 1$ asymptotic result (3.23).

Figure 5 shows that $(1 + \eta _r) \left . \bar {V}_\theta \right |_{r=1}$ and $\mbox{} - ((1 + \eta _r)/\eta _r) \left . \bar {\Sigma }_{\theta r} \right |_{r=1}$ collapse on to a universal curve independent of $\eta _r$ . The $\beta \ll 1$ asymptotic expressions in (3.16), shown by the dotted red lines on the left, are in quantitative agreement with the numerical results for $\beta \lesssim 1$ . The $\beta \gg 1$ asymptotic expressions in (3.23), shown by the dotted red lines on the right, accurately predict the numerical results for $\beta \gtrsim 30$ . The velocity and stress measures pass through a maximum at $\beta \approx 6.1$ . From (3.16) and (3.23), it is also observed that

(3.25) \begin{align} \mbox{} - ((1 + \eta _r)/\eta _r) \left . \bar {\Sigma }_{\theta r} \right |_{r=1} = 5 (1 + \eta _r) \left . \bar {V}_\theta \right |_{r=1}, \end{align}

for $\beta \ll 1$ and $\beta \gg 1$ . In figure 5, the scale in the ordinate in figure 5(b) is five times that in figure 5(a). The identical nature of the curves in figures 5(a) and 5(b) shows that (3.25) is quantitatively accurate for $\beta \sim 1$ also.

3.5. Drop deformation

The deformation of the drop due to the steady flow is examined in the limit of low capillary number; it is shown in § 5 that this limit is applicable for practical situations. The deformation is characterised by the parameter $\bar {h}_0$ defined in (2.31)–(2.33), which is determined from the normal stress balance (2.46) in the low capillary number limit. This parameter is negative, indicating that a spherical drop deforms into an oblate spheroid which is extended at the equator and compressed at the poles. The deformed shape is shown for $\bar {h}_0 = - 0.05$ and $- 0.1$ in figure 6(a). The shape is axisymmetric about the $z$ axis.

Figure 6.

(a) The shape of the conducting drop for the spherical drop (solid line), $\bar {h}_0=-0.05$ (dashed line) and $\bar {h}_0=-0.1$ (dotted line), and (b) the parameter $-{Ca}^{-1} \bar {h}_0$ in (2.46) as a function of $\beta$ for $\eta _r \rightarrow 0$ ( $\circ$ ), $\eta _r = 1$ ( $\triangle$ ) and $\eta _r \rightarrow \infty$ ( $\nabla$ ). The dotted red lines on the left in (b) show (3.17) in the limit $\beta \ll 1$ , and the dotted red line on the right is $- {Ca}^{-1} \bar {h}_0 = {1}/{32}$ (3.24).

From the normal stress balance (2.46) at the interface, it is evident that $\bar {h}_0 \propto {Ca}$ , since the left-hand side is independent of ${Ca}$ . The parameter $- {Ca}^{-1} \bar {h}_0$ is plotted as a function of $\beta$ in figure 6. This figure shows that $-{Ca}^{-1} \bar {h}_0$ increases proportional to $\beta ^4$ for $\beta \ll 1$ , and it approaches a constant value for $\beta \gg 1$ . The dashed red lines on the left show the asymptotic results (3.17) for different values of $\eta _r$ . There are three red lines for $\eta _r \rightarrow 0$ , $\eta _r = 1$ and $\eta _r \rightarrow \infty$ . However, these are indistinguishable because the numerical values are very close. There is excellent agreement between the asymptotic and numerical results for $\beta \lesssim 1$ . The dashed red line on the right is the $\beta \gg 1$ asymptotic result, (3.24). This is in quantitative agreement with the numerical results for $\eta _r \gtrsim 10$ .

3.6. Joule heating

An issue of importance is whether sufficient heat can be generated by Joule heating due to the eddy currents to increase the temperature of the surrounding fluid. Attention is restricted to the total rate of heat generation in the entire drop. At steady state, this has to be the heat transfer rate from the drop to the surrounding fluid. Based on this, the difference between the drop and the ambient fluid is calculated in § 5.

The rate of generation of heat energy per unit volume is expressed in dimensional form as

(3.26) \begin{align} q^\dagger = \frac {\boldsymbol{J}^\dagger \boldsymbol{\cdot } \boldsymbol{J}^\dagger }{\kappa }. \end{align}

If we substitute $\boldsymbol{J}^\dagger = {1}/{2} (\kern1.5pt \tilde {\boldsymbol{\kern-1.5pt J}}^\dagger \exp {(\imath t)} + (\kern1.5pt \tilde {\boldsymbol{\kern-1.5pt J}}^{\dagger } \exp {(\imath t)})^\ast )$ , the steady part of the energy generation per unit volume is

(3.27) \begin{align} \bar {q}^\dagger = \frac {\kern1.5pt \tilde {\boldsymbol{\kern-1.5pt J}}^\dagger \boldsymbol{\cdot } \kern1.5pt \tilde {\boldsymbol{\kern-1.5pt J}}^{\dagger \ast }}{2 \kappa } = \frac {\kappa \kern1.5pt \tilde {\boldsymbol{\kern-1.5pt E}}^\dagger \boldsymbol{\cdot } \kern1.5pt \tilde {\boldsymbol{\kern-1.5pt E}}^{\dagger \ast }}{2}. \end{align}

The non-dimensional energy generation per unit volume is defined as $\bar {q} = (\bar {q}^\dagger /(H_0^2/ R^2 \kappa ))$ . Equation (3.27) is non-dimensionalised,

(3.28) \begin{align} \bar {q} = \frac {\kern1.5pt \tilde {\boldsymbol{\kern-1.5pt E}} \boldsymbol{\cdot } \kern1.5pt \tilde {\boldsymbol{\kern-1.5pt E}}^{\ast }}{2}. \end{align}

Substituting (A31) for $\kern1.5pt \tilde {\boldsymbol{\kern-1.5pt E}}$ , the heat generation rate per unit volume is

(3.29) \begin{align} \bar {q} & = \frac {9 \sin {(\theta )}^2}{8 \sin {(\tilde {\beta })} \sin {(\tilde {\beta }^\ast )}} \frac {\mbox{d} \tilde {\chi }}{ \mbox{d} r} \frac {\mbox{d} \tilde {\chi }^\ast }{\mbox{d} r} \nonumber \\& = \frac {9 \sin {(\theta )}^2}{8 \big(\cosh {\big(\sqrt {2} \beta \big)} - \cos {\big(\sqrt {2} \beta \big)}\big)} \left ( \frac {\beta ^2 \big(\cosh {\big(\sqrt {2} \beta r \big)} + \cos {\big(\sqrt {2} \beta r \big)}\big)}{r^2} \right . \nonumber \\& \quad \left . \mbox{} - \frac {\sqrt {2} \beta \big(\sinh {\big(\sqrt {2} \beta r \big)}+\sin {\big(\sqrt {2} \beta r \big)}\big)}{r^3}\! +\! \frac {\big(\cosh {\big(\sqrt {2} \beta r \big)} - \cos {\big(\sqrt {2} \beta r \big)} \big)}{r^4} \!\right ). \end{align}

The total heat generated per unit time is

(3.30) \begin{align} \bar {Q} & = 2 \pi \int _{0}^{\pi } \sin {(\theta )} \mbox{d} \theta \int _0^1 r^2 \mbox{d} r \: \bar {q} \nonumber \\ & = 3 \pi \left ( \frac {\beta \big(\sinh {\big(\sqrt {2} \beta \big)}+\sin {\big(\sqrt {2} \beta \big)}\big)}{\sqrt {2} \big(\cosh {\big(\sqrt {2} \beta \big)} - \cos {\big(\sqrt {2} \beta \big)}\big)} - 1 \right ). \end{align}

Here, the heat generation rate $\bar {Q}$ is non-dimensionalised by $(H_0^2 R/ \kappa ))$ . The following limiting values of the scaled heat generation rate are derived in the limits $\beta \ll 1$ and $\beta \gg 1$ :

(3.31) \begin{align} \bar {Q} & = \frac {\beta ^4 \pi }{15} \: \: \: \mbox{ for } \: \: \: \beta \ll 1,\end{align}

(3.32) \begin{align} & = \frac {3 \pi \beta }{\sqrt {2}} \: \: \: \mbox{ for } \: \: \: \beta \gg 1.\end{align}

The non-dimensional heat generation rate $\bar {Q}$ is shown as a function of $\beta$ in figure 7. The $\beta \ll 1$ asymptotic expression (3.31) is in quantitative agreement with the numerical solution for $\beta \lesssim 2$ , while the $\beta \gg 1$ asymptotic expression (3.32) accurately predicts the heat generation rate for $\beta \gtrsim 20$ .

Figure 7.

The non-dimensional heat generation rate per unit time $\bar {Q}$ due to Joule heating as a function of $\beta$ . The dotted red line on the left is the asymptotic expression (3.31) for $\beta \ll 1$ , and the dotted red line on the right is the asymptotic expression (3.32) for $\beta \gg 1$ .

4.1. Flow outside the drop

The velocity fields outside the drop are calculated from (2.52) for $\tilde {g}^{(o)}$ , and the definitions (2.11) for the stream function and (2.5) for the velocity. The solution (2.52) satisfies the condition (2.24) ( $\tilde {v}_r = 0$ ) at $r=1$ , and the tangential velocity condition (2.25) is used to express the constant $D$ in terms of $\left . \tilde {V}_\theta \right |_{r=1}$ ,

(4.1) \begin{align} \tilde {v}_r = \tilde {V}_r \big(3 \cos {(\theta )}^2 - 1 \big), \: \: \: \tilde {v}_\theta = \tilde {V}_\theta \sin {(\theta )} \cos {(\theta )}, \end{align}

where

(4.2) \begin{align} \frac {\tilde {V}_r}{\tilde {V}_\theta |_{r=1}} & = \frac {\big[ \big(3 + 3 \tilde {\alpha }_o r + \tilde {\alpha }_o^2 r^2 \big) \exp {({-} \tilde {\alpha }_o (r-1))} - \big(3 + 3 \tilde {\alpha }_o + \tilde {\alpha }_o^2 \big) \big]}{r^4 \tilde {\alpha }_o^2 (1 + \tilde {\alpha }_o)},\end{align}

(4.3) \begin{align} \frac {\tilde {V}_\theta }{\tilde {V}_\theta |_{r=1}} & = \frac {\big[\big(6 + 6 \tilde {\alpha }_o r + 3 \tilde {\alpha }_o^2 r^2 + \tilde {\alpha }_o^3 r^3 \big) \exp {({-} \tilde {\alpha }_o (r-1))} - 2 \big(3 + 3 \tilde {\alpha }_o + \tilde {\alpha }_o^2 \big) \big]}{ r^4 \tilde {\alpha }_o^2 (1 + \tilde {\alpha }_o)}.\end{align}

The real and imaginary parts of the ratios $(\tilde {V}_r/\tilde {V}_\theta |_{r=1})$ and $(\tilde {V}_\theta /\tilde {V}_\theta |_{r=1})$ are shown as a function of $r-1$ by the red and blue lines in figure 8, respectively. Three different values of the Reynolds number, ${Re}_\omega = 1, 30$ and $1000$ are considered. Also shown by the black lines are the real parts of $(\tilde {V}_r/\tilde {V}_\theta |_{r=1})$ and $(\tilde {V}_\theta /\tilde {V}_\theta |_{r=1})$ for a steady flow with ${Re}_\omega = 0$ ; these are derived from (2.37)–(2.39), and the imaginary parts are zero in this case. It is evident that the magnitudes of the velocity and the extent of the flow both decrease as ${Re}_\omega$ is increased. For a steady flow, (2.37)–(2.38) show that the magnitude of the velocity decreases as $r^{-2}$ for $r \gg 1$ . In contrast, (4.2)–(4.3) show that $\tilde {V}_r$ and $\tilde {V}_\theta$ contain one term that decreases proportional to $r^{-4}$ , while the other term decreases exponentially over distance $\tilde {\alpha }_o^{-1} \sim 1/\sqrt {{Re}_\omega }$ . Thus, the disturbance due to the oscillatory component of the velocity field is much shorter in range in comparison with that due to the steady component. Sufficiently far from the drop, the dominant contribution is due to the steady component.

Figure 8.

The velocity scaled velocity measures $(\tilde {V}_r/\tilde {V}_\theta |_{r=1})$ (a) and $(\tilde {V}_\theta /\tilde {V}_\theta |_{r=1})$ (b) as a function of the distance from the surface, $r-1$ for ${Re}_\omega = 1$ ( $\circ$ ), ${Re}_\omega = 30$ ( $\triangle$ ), ${Re}_\omega = 1000$ ( $\nabla$ ). The red lines are real parts, and the blue lines are the imaginary parts. The black line is the result (2.37)–(2.38) for ${Re}_\omega = 0$ .

4.2. Limit $\beta \ll 1$

In this limit, the approximation (B25) for $\tilde {F}$ is substituted into (2.47). The analytical solution for the resulting equation consists of a general solution and a particular solution which is a polynomial in $r$ ,

(4.4) \begin{align} \tilde {g}^{(i)} & = \frac {K_1}{r^2} + K_2 r^3 + K_3 \big ( \sinh {(\tilde {\alpha } r)} (1 + 3 (\tilde {\alpha } r)^{-2}) - 3 (\tilde {\alpha } r)^{-1} \cosh {(\tilde {\alpha } r)} \big ) \nonumber \\ & \quad + K_4 \big ( \cosh {(\tilde {\alpha } r)} (1 + 3 (\tilde {\alpha } r)^{-2}) - 3 (\tilde {\alpha } r)^{-1} \sinh {(\tilde {\alpha } r)} \big ) \nonumber \\ & \quad + \left (\frac {\beta ^4 r^5}{280 \tilde {\alpha }^2} \right ), \end{align}

where $K_1, K_2, K_3$ and $K_4$ are constants of integration. The first four terms in the above solution are identical to (2.51) for $\tilde {g}^{(o)}$ , but are expressed in a slightly different manner to facilitate the application of boundary conditions. The boundary conditions (2.27) for $r \rightarrow 0$ require that $K_1 = K_4 = 0$ . A relation between the constants $K_2$ and $K_3$ is obtained from the zero normal velocity boundary condition $\tilde {g}^{(i)} = 0$ at $r=1$ (2.24),

(4.5) \begin{align} \tilde {g}^{(i)} & = K \left [ \sinh {(\tilde {\alpha } r)} (1 + 3 (\tilde {\alpha } r)^{-2} ) - r^{3} \sinh {(\tilde {\alpha })} (1 + 3 \tilde {\alpha }^{-2}) \right . \nonumber \\ & \quad \left . \mbox{} - 3 (\tilde {\alpha } r)^{-1} \cosh {(\tilde {\alpha } r)} + 3 r^{3} \tilde {\alpha }^{-1} \cosh {(\tilde {\alpha })} \right ] + \beta ^4 \left ( \frac {r^5-r^3}{280 \tilde {\alpha }^2} \right ). \end{align}

The constant $K$ is determined from the tangential stress boundary condition (2.54).

Since the last term on the right-hand side in (4.5) is proportional to $\beta ^4$ , and the right-hand side of boundary condition at the surface (2.54) does not depend on $\beta$ , the constant $K$ is also proportional to $\beta ^4$ . Consequently, the functions $\tilde {g}^{(i)}$ , the components of the velocity and the stress are all proportional to $\beta ^4$ . Therefore, the results are expressed as ratios of the dynamical quantities and $\beta ^4$ .

Vector plots for the real and imaginary parts of velocity vectors $\tilde {\boldsymbol{v}}$ are shown in figures 9(a–c) and 9(d–f) for three different values of ${Re}_\omega$ . The real and imaginary parts of $\tilde {V}_r$ and $\tilde {V}_\theta$ , defined in (2.16), are shown as a function of $r$ in figure 9(g–i). For ${Re}_\omega = 1$ , the real part of $\tilde {v}$ is much larger than the imaginary part, and the real part is similar to that for a steady flow in figure 3 for $\eta _r = 1$ . This is expected at low Reynolds number, where viscous effects are dominant and the flow is quasisteady. The term proportional to $\tilde {\alpha }^2 = 2 \imath {Re}_\omega$ is neglected in (2.47), and $\beta \ll 1$ the asymptotic expression (B25) for $\tilde {F}$ is real. Due to this, the solution for the stream function and the velocity components are also real. The flow field responds instantaneously to the forcing at the same time instant. Therefore, the real part of the velocity field is similar to that for steady forcing. Moreover, the imaginary part of $\tilde {\boldsymbol{v}}$ , which is related to the out-of-phase component of the velocity field, is small compared with the real part of $\tilde {\boldsymbol{v}}$ related to the in-phase component. Since the flow is quasisteady, the variation of $\textrm{Re}(\tilde {V}_r)$ and $\mbox{Re}(\tilde {V}_\theta )$ in figure 9(g) is numerically close to that of $\bar {V}_r$ and $\bar {V}_\theta$ in figure 4 for $\beta = 1$ and $\eta _r = 1$ .

Figure 9.

Vector plots of the real part (a–c) and imaginary part (d–f) of the velocity vectors and (g–i) the velocity measures $\tilde {V}_r = \tilde {v}_r/(3 \cos {(\theta )}^2-1)$ ((2.16), $\circ$ ) and $\tilde {V}_{\theta } = \tilde {v}_\theta /(\sin {(\theta )} \cos {(\theta )})$ ((2.16), $\triangle$ ) in the limit $\beta \ll 1$ , all scaled by $\beta ^4$ , for $\eta _r = 1$ and for three different values of ${Re}_\omega$ . In (g)–(i), the real parts are the red lines and the imaginary parts are the blue lines.

For ${Re}_\omega = 1000$ , figure 9 shows that the imaginary part of the velocity vector is much larger than the real part. In the bulk of the drop, the term proportional to $\tilde {\alpha }^2 = 2 \imath {Re}_\omega$ in (2.47) is dominant. Since $\tilde {F}$ is real for $\beta \ll 1$ (see (B25)), the leading-order solution for $\tilde {g}^{(i)}$ and $\tilde {\boldsymbol{v}}$ are imaginary for ${Re}_\omega \gg 1$ within the drop. However, there is a sharp increase in the real part close to the surface due to viscous effects. If the viscous terms are neglected in (2.47), a second-order differential equation results, and it is not possible to satisfy all the boundary conditions. The boundary layer approximation is invoked, where inertial and viscous effects are comparable in a layer of thickness ${Re}_\omega ^{-1/2}$ . The resulting sharp variations in the $\tilde {V}_r$ and $\tilde {V}_\theta$ close to the surface are captured in figure 9(i).

The oscillatory parts of the tangential velocity measure $\tilde {V}_\theta$ (2.16) and stress measure $\tilde {\Sigma }_{\theta r}$ (2.23) at the surface $r=1$ can be evaluated analytically in the limits ${Re}_\omega \ll 1$ and ${Re}_\omega \gg 1$ . In the limit ${Re}_\omega \ll 1$ , the term proportional to $\tilde {\alpha }^2$ in (2.47) is neglected, and the equation becomes identical to that for a steady flow. The expressions for the tangential velocity and stress measures are identical to the relations (3.16),

(4.6) \begin{align} \left . \tilde {V}_\theta \right |_{r=1} & = \left . \frac {\tilde {v}_\theta }{\sin {(\theta )} \cos {(\theta )}} \right |_{r=1} = \frac {\beta ^4}{6300 (1 + \eta _r)}, \nonumber \\ \left . \tilde {\Sigma }_{\theta r} \right |_{r=1} & = \left . \frac {\tilde {\sigma }_{\theta r}}{\cos {(\theta )} \sin {(\theta )}} \right |_{r=1} = - \frac {\beta ^4 \eta _r}{1260 (1 + \eta _r)}. \end{align}

In the limit ${Re}_\omega \gg 1$ , the velocity and stress measures are

(4.7) \begin{align} \left . \tilde {V}_\theta \right |_{r=1} & = - \frac {\imath \beta ^4}{280 {Re}_\omega \left (1+ \eta _r \sqrt {{Re}_\omega ^{(o)}/ {Re}_\omega } \right )}, \nonumber \\ \left . \tilde {\Sigma }_{\theta r} \right |_{r=1} & = \frac {\beta ^4 ({-}1+\imath ) \eta _r \sqrt {{Re}_\omega ^{(o)}/ {Re}_\omega }}{280 \sqrt {{Re}_\omega } \left (1+ \eta _r \sqrt {{Re}_\omega ^{(o)}/ {Re}_\omega } \right )}. \end{align}

Here, we find that the shear stress at the surface decreases proportional to ${Re}_\omega ^{-1/2}$ while the surface velocity decreases proportional to ${Re}_\omega ^{-1}$ .

The asymptotic expressions (4.6) and (4.7) show that the velocity measures are independent of ${Re}_\omega$ for ${Re}_\omega \ll 1$ , but they depend on both ${Re}_\omega$ and ${Re}_\omega ^{(o)}$ for ${Re}_\omega \gg 1$ . For the special case ${Re}_\omega = {Re}_\omega ^{(o)}$ , the velocity and stress measures $((1+\eta _r) \tilde {V}_\theta /\beta ^4)$ and $((1+\eta _r) \tilde {\Sigma }_{\theta r}/\beta ^4 \eta _r)$ are functions of ${Re}_\omega$ . These are shown in figures 10 and 11. As predicted by (4.6) and (4.7), the magnitudes of $\tilde {V}_\theta |_{r=1}$ and $\tilde {\Sigma }_{\theta r} |_{r=1}$ tend to values independent of ${Re}_\omega$ and $\eta _r$ for ${Re}_\omega \ll 1$ , and the real parts are much larger than the imaginary parts. For ${Re}_\omega \gg 1$ , the velocity measure in figure 10(a) decreases proportional to ${Re}_\omega ^{-1}$ , the stress measure in figure 11(a) decreases proportional to ${Re}_\omega ^{-1/2}$ , and both these are independent of $\eta _r$ for ${Re}_\omega ^{(o)} = {Re}_\omega$ . The velocity measure is imaginary for ${Re}_\omega \gg 1$ in figure 10(b), while the stress measure has real and imaginary parts consistent with (4.7). There is some variation in both the measures with $\eta _r$ for intermediate values of ${Re}_\omega$ ; while the magnitude of the velocity measure decreases monotonically for all values of $\eta _r$ , the magnitude of the stress measure exhibits a maximum at an intermediate value of ${Re}_\omega$ .

Figure 10.

The measure (a) $((1 + \eta _r) |\tilde {V}_\theta | / \beta ^4) |_{r=1}$ and (b) $(\textrm{Re}(\tilde {V}_\theta )/|\tilde {V}_\theta |)|_{r=1}$ (red) and $(\mbox{Im}(\tilde {V}_\theta )/|\tilde {V}_\theta |)|_{r=1}$ (blue) as a function of ${Re}_\omega$ for viscosity ratio $\eta _r \rightarrow 0$ ( $\circ$ ), $\eta _r = 1.0$ ( $\triangle$ ), $\eta _r \rightarrow \infty$ ( $\nabla$ ). The definition of $\tilde {V}_\theta$ is given in (2.16). The dotted red line on the left in (a) is the ${Re}_\omega \ll 1$ asymptotic result (4.6). The dotted red line on the right in (a) is the ${Re}_\omega \gg 1$ asymptotic result (4.7).

Figure 11.

The measure (a) $\mbox{} - ((1 + \eta _r) | \tilde {\Sigma }_{\theta r} |_{r=1}/\eta _r \beta ^4)$ and (b) $(\textrm{Re}(\tilde {\Sigma }_{\theta r})/|\tilde {\Sigma }_{\theta r}|) |_{r=1}$ (red) and $ (\mbox{Im}(\tilde {\Sigma }_{\theta r})/|\tilde {\Sigma }_{\theta r}|) |_{r=1}$ (blue) as a function of ${Re}_\omega$ in the limit $\beta \ll 1$ for viscosity ratio $\eta _r \rightarrow 0$ ( $\circ$ ), $\eta _r = 1.0$ ( $\triangle$ ), $\eta _r \rightarrow \infty$ ( $\nabla$ ). The definition of $\tilde {\Sigma }_{\theta r}$ is given in (2.23). The dotted red line on the left in (a) is the ${Re}_\omega \ll 1$ asymptotic result (4.6). The dotted red line on the right in (a) is the ${Re}_\omega \gg 1$ asymptotic result (4.7).

The oscillatory drop deformation is characterised by the parameter $\tilde {h}_0$ defined in (2.31)–(2.33). This parameter is related to the normal stress difference across the drop by (2.67), where the pressure is evaluated from the $\theta$ momentum balance (2.56). Here, the forcing term $\tilde {F}_\theta$ turns out to be the dominant term in the momentum balance equation. This forcing term, evaluated in Appendix B, (B23), is $O(\beta ^2)$ in the limit $\beta \ll 1$ . In contrast, $\tilde {g}^{(i)} \propto \beta ^4$ (see discussion after (4.5)). Therefore, the dominant contribution to the pressure in the drop is due to the forcing $\tilde {F}_\theta$ , and the $O(\beta ^2)$ contribution to the pressure is evaluated by neglecting all terms except the pressure and $\tilde {F}_\theta$ in (2.56),

(4.8) \begin{align} \tilde {P} = \frac {\imath \beta ^2 r^2}{24}. \end{align}

This is substituted into the stress balance across the interface, (2.67), to obtain

(4.9) \begin{align} {Ca}^{-1} \tilde {h}_0 = \mbox{} - \frac {\imath \beta ^2}{144}. \end{align}

It should be emphasised that $\tilde {h}_0 \sim \beta ^2$ in the limit $\beta \ll 1$ , in contrast to the stream function and the velocity field which are $O(\beta ^4)$ . Thus, the oscillation of the drop is primarily due to pressure fluctuations which acts perpendicular to the surface, and the $O(\beta ^2)$ contribution to the flow velocity in the drop is zero. The oscillation amplitude does not depend on ${Re}_\omega$ in the leading approximation in the limit $\beta \ll 1$ .

4.3. Limit $\beta \gg 1$

The approximation (B26) is substituted into the (2.47) for $\tilde {g}^{(i)}$ . The solution is the sum of a general solution of the homogeneous equation, $\tilde {g}^{(i)}_g$ , which has the same form as the first term on the right-hand side in (4.5), and the particular solution. Since the force density (B26) is $O(1)$ in a region of thickness $\beta ^{-1}$ at the surface, the scaled variable is defined as $\xi = \beta (1-r)$ . The equation (2.47) for $\tilde {g}^{(i)}_p$ is

(4.10) \begin{align} 0 & = \beta ^4 \frac {\mbox{d}^4 \tilde {g}^{(i)}_p}{\mbox{d} \xi ^{4}} - \frac {12 \beta ^2}{(1 - \xi /\beta )^2} \frac {\mbox{d}^2 \bar {g}^{(i)}_p}{\mbox{d} \xi ^2} - \frac {24 \beta }{(1 - \xi /\beta )^3} \frac {\mbox{d} \tilde {g}^{(i)}_p}{\mbox{d} \xi } + \tilde {\alpha }^2 \left ( \beta ^2 \frac {\mbox{d}^2 \tilde {g}^{(i)}_p}{\mbox{d} \xi ^2} \right . \nonumber \\ & \quad \left . \mbox{} - \frac {6 \tilde {g}^{(i)}_p}{( 1 - \xi /\beta )^2} \right ) - \left ( \frac {9 \beta (1 + \imath ) \exp {[{-} \sqrt {2} (1 + \imath ) \xi ]}}{4 \sqrt {2} (1-\xi /\beta )^2} \right ). \end{align}

In the limit $\beta \gg 1$ , the first and last terms on the right-hand side are the largest terms in the expansion. In the denominator of the last term on the right, $(\xi /\beta )^2$ is neglected in comparison with $1$ , and the equation is integrated to obtain

(4.11) \begin{align} \tilde {g}^{(i)}_p & = - \frac {9 (1 + \imath ) \exp {\big({-} \sqrt {2} (1+\imath ) \xi \big )}}{64 \sqrt {2} \beta ^3} \nonumber \\ & = - \frac {9 (1 + \imath ) \exp {\big({-} \sqrt {2} (1+\imath ) \beta (1-r) \big)}}{64 \sqrt {2} \beta ^3}. \end{align}

The sum of the general and particular solutions is

(4.12) \begin{align} \tilde {g}^{(i)} & = L \left[ \sinh {(\tilde {\alpha } r)} \big(1 + 3 (\tilde {\alpha } r)^{-2} \big) - 3 (\tilde {\alpha } r)^{-1} \cosh {(\tilde {\alpha } r)} \right. \nonumber \\ & \left. \quad - r^{3} (\sinh {(\tilde {\alpha })} \big(1 + 3 \tilde {\alpha }^{-2} \big) - 3 \tilde {\alpha }^{-1} \cosh {(\tilde {\alpha })}) \right] \nonumber \\ & \quad - \frac {9 (1 + \imath ) \big[\exp {\big({-} \sqrt {2} (1+\imath ) \beta (1-r) \big)} - r^{3} \big]}{64 \sqrt {2} \beta ^3}. \end{align}

This solution satisfies the conditions (2.27) at $r=0$ and the zero normal velocity condition (2.24) at $r=1$ .

The constant $L$ is determined from the stress boundary condition, (2.54). The magnitude of $L$ in the limit $\beta \gg 1$ is estimated as follows. The term on the right-hand side in (2.54) is independent of $\beta$ . On the left-hand side, the second derivative $(\mbox{d}^2 \tilde {g}^{(i)}/\mbox{d} r^2)$ in the numerator contains a contribution from the particular solution that is $O(\beta ^{-1})$ , and an $O(1)$ contribution from the general solution that is multiplied by $L$ . The first derivative in the denominator on the left-hand side of (2.54) contains a contribution from the particular solution that is proportional to $O(\beta ^{-2})$ , and an $O(1)$ contribution from the general solution that is multiplied by $L$ . From the balance between the left- and right-hand sides, it is inferred that $L \sim \beta ^{-1}$ in the limit $\beta \gg 1$ . Similarly, the coefficient $D$ in the expression (2.51) for $\tilde {g}^{(o)}$ is also proportional to $\beta ^{-1}$ from the tangential stress balance condition. From this, the velocity and stress measures (2.16) and (2.23), are all proportional to $\beta ^{-1}$ .

The solution for $\tilde {g}^{(i)}$ , (4.12), is substituted into the boundary condition (2.54), and $r$ is set equal to $1$ at the surface. The constant $L$ is expressed as $L_0 \beta ^{-1}$ , and only the $O(\beta ^{-1})$ terms are retained in the resulting boundary condition to obtain $L_0$ . This is then substituted into equation the solution (2.27) to determine the $O(\beta ^{-1})$ contribution to $\tilde {g}^{(i)}$ and its first and second derivatives. These are then used to determine the velocity fields.

Vector plots of the real and imaginary parts of $\tilde {\boldsymbol{v}}$ for $\beta \gg 1$ are shown in figures 12(a–c) and 12(d–f). For ${Re}_\omega = 1$ , the flow is expected to be quasisteady and the real and imaginary parts of the velocity vector are in phase with the forcing. Since the particular solution in (4.11) is complex with real and imaginary parts of equal magnitude, the solution for $\tilde {\boldsymbol{v}}$ has real and imaginary parts of equal magnitude and opposite sign. It should also be noted that the forcing is restricted to a thin region of thickness $\beta ^{-1}$ at the surface in the limit $\beta \gg 1$ . This forcing causes a flow throughout the drop in the limit ${Re}_\omega \ll 1$ due to the quasisteady nature of momentum diffusion.

Figure 12.

Vector plots of the real part (a–c) and imaginary part (d–f) of the velocity vectors and the velocity measures $\beta \tilde {V}_r = \beta \tilde {v}_r/(3 \cos {(\theta )}^2-1)$ ((2.16), red line) and $\beta \tilde {V}_{\theta } = \beta \tilde {v}_\theta /(\sin {(\theta )} \cos {(\theta )})$ ((2.16), $\triangle$ ) in the limit $\beta \gg 1$ , for $\eta _r = 1$ and for three different values of ${Re}_\omega$ . The real parts are shown by the red lines, and the imaginary parts are shown by the blue lines.

For ${Re}_\omega = 1000$ , figure 12(c,f,i) shows that the flow is restricted to a thin region near the surface of the drop for two reasons. Firstly, the forcing in (4.11) is restricted to a thin region of thickness $\beta ^{-1}$ at the surface. Secondly, the diffusion length for momentum, $(\eta_i/\rho_i \omega)^{1/2}$ , is much smaller than the radius of the drop for ${Re}_\omega \gg 1$ . Due to this, momentum diffusion is restricted to a thin region at the surface in a manner similar to that for the Stokes oscillatory boundary layer. This results in a sharp variation in the tangential and normal velocity at the surface observed in figure 12(i). The imaginary part of the velocity $\tilde {\boldsymbol{v}}$ is much larger than the real part in this case.

The oscillatory parts of the tangential velocity measure $\tilde {V}_\theta$ (2.16) and stress measure $\tilde {\Sigma }_{\theta r}$ (2.23) at the surface $r=1$ are evaluated analytically in the limits ${Re}_\omega \ll 1$ and ${Re}_\omega \gg 1$ . In the limit ${Re}_\omega \ll 1$ ,

(4.13) \begin{align} \left . \tilde {V}_\theta \right |_{r=1} & = \mbox{} \left . \frac {\tilde {v}_\theta }{\sin {(\theta )} \cos {(\theta )}} \right |_{r=1} = \mbox{} - \frac {9 (1 - \imath )}{80 \sqrt {2} \beta (1+\eta _r)}, \nonumber \\ \left . \tilde {\Sigma }_{\theta r} \right |_{r=1} & = \left . \frac {\tilde {\sigma }_{\theta r}}{\sin {(\theta )} \cos {(\theta )}} \right |_{r=1} = \mbox{} \frac {9 (1 - \imath )\eta _r}{16 \sqrt {2} \beta (1 + \eta _r)}. \end{align}

In the limit ${Re}_\omega \gg 1$ , the velocity and stress measures are

(4.14) \begin{align} \left . \tilde {V}_\theta \right |_{r=1} & = \mbox{} \frac {9 \imath }{16 \sqrt {2} \beta \left (1+ \eta _r \sqrt {\textit{Re}_\omega ^{(o)}/\textit {Re}_\omega } \right ) \sqrt {\textit {Re}_\omega }}, \nonumber \\ \left . \tilde {\Sigma }_{\theta r} \right |_{r=1} & = \mbox{} \frac {9 (1 - \imath ) \eta _r \sqrt {\textit{Re}_\omega ^{(o)}/\textit {Re}_\omega }}{16 \sqrt {2} \beta \left (1 + \eta _r \sqrt {\textit{Re}_\omega ^{(o)}/\textit {Re}_\omega } \right )}. \end{align}

Equation (4.13) shows that the velocity and stress measures are independent of ${Re}_\omega$ for ${Re}_\omega \ll 1$ . In the limit ${Re}_\omega \gg 1$ , the stress measure is independent of ${Re}_\omega$ , but the velocity measure decreases proportional to ${Re}_\omega ^{-1/2}$ . This is as expected for the Stokes oscillatory boundary layer driven by a fixed stress amplitude at the interface instead of a fixed velocity.

The measures $((1+\eta _r) \beta \tilde {V}_\theta )$ and $((1+\eta _r) \beta \tilde {\Sigma }_{\theta r}/\eta _r)$ are shown as a function of ${Re}_\omega$ in figures 13 and 14. Here, we have set ${Re}_\omega ^{(o)} = {Re}_\omega$ . The magnitude of the scaled velocity measure tends to a constant value that is independent of $\eta _r$ for ${Re}_\omega \ll 1$ , and it decreases proportional to ${Re}_\omega ^{-1/2}$ for ${Re}_\omega \gg 1$ . The results in figure 13 are in agreement with asymptotic results, (4.13) and (4.14). The scaled stress measure in figure 14 tends to the same constant value, independent of $\eta _r$ , for both ${Re}_\omega \ll 1$ and ${Re}_\omega \gg 1$ . The velocity and stress measures exhibit variation with $\eta _r$ over a narrow range for intermediate values of ${Re}_\omega$ .

Figure 13.

The measure (a) $((1 + \eta _r) \beta | \tilde {V}_\theta |_{r=1})$ and (b) $\textrm{Re}(\tilde {V}_\theta )/|\tilde {V}_\theta | |_{r=1}$ (red) and $\mbox{Im}(\tilde {V}_\theta )/|\tilde {V}_\theta |$ (blue) as a function of $\beta$ for viscosity ratio $\eta _r \rightarrow 0$ ( $\circ$ ), $\eta _r = 1.0$ ( $\triangle$ ), $\eta _r \rightarrow \infty$ ( $\nabla$ ). The definition of $\tilde {V}_\theta$ is given in (2.16). The dotted red line on the left in (a) is the ${Re}_\omega \ll 1$ asymptotic result (4.13). The dotted red line on the right in (a) is the ${Re}_\omega \gg 1$ asymptotic result (4.14).

Figure 14.

The measure (a) $((1 + \eta _r) |\tilde {\Sigma }_{\theta r}| \beta /\eta _r) |_{r=1}$ and (b) $(\tilde {\Sigma }_{\theta r})/|\tilde {\Sigma }_{\theta r}| |_{r=1}$ and $\mbox{Im}(\tilde {\Sigma }_{\theta r})/|\tilde {\Sigma }_{\theta r}|$ as a function of ${Re}_\omega$ in the limit $\beta \ll 1$ for viscosity ratio $\eta _r = 0$ ( $\circ$ ), $\eta _r = 1.0$ ( $\triangle$ ), $\eta _r \rightarrow \infty$ ( $\nabla$ ). The definition of $\tilde {\Sigma }_{\theta r}$ is given in (2.23). The dotted red line on the left in (a) is the ${Re}_\omega \ll 1$ asymptotic result (4.13). The dotted red line on the right is the ${Re}_\omega \gg 1$ asymptotic result (4.14).

There is a non-monotonic variation of the tangential stress measures with ${Re}_\omega$ , but not the tangential velocity measures. The stress scales as the ratio of the tangential velocity and the penetration depth of the magnetic field into the fluid. In the limit ${Re}_\omega \ll 1$ , figures 10 and 13 show that the scaled velocity approaches a constant, and the characteristic length is the drop radius, so the stress approaches a constant. In the limit ${Re}_\omega \gg 1$ , the penetration depth is proportional to ${Re}_\omega ^{-1/2}$ , which is similar to the Stokes second problem for an oscillating boundary layer. The velocity in the limit $\beta \ll 1$ in figure 10 decreases proportional to ${Re}_\omega ^{-1}$ , and therefore the stress in figure 11 decreases proportional to ${Re}_\omega ^{-1/2}$ . Similarly, in the limit $\beta \gg 1$ , the velocity in figure 13 decreases proportional to ${Re}_\omega ^{-1/2}$ for ${Re}_\omega \gg 1$ , and therefore the stress approaches a constant value in figure 14. The stress exhibits a non-monotonic behaviour for intermediate values of ${Re}_\omega$ . This is because the tangential velocity decreases more slowly than the penetration depth as ${Re}_\omega$ is increased, resulting in an initial increase in the stress. This is followed by a decrease in the stress to a constant in the ${Re}_\omega \gg 1$ asymptotic regime.

The parameter $\tilde {h}_0$ (2.31)–(2.33) that characterises the oscillatory drop deformation is evaluated from the normal stress balance across the drop, (2.67). In the expression (2.59) for the normal stress within the drop, the largest term is the first term on the right-hand side, $(1/6)(\mbox{d}^3 \tilde {g}^{(i)}/\mbox{d} r^3)$ . In the (4.12) for $\tilde {g}^{(i)}$ in the limit $\beta \gg 1$ , the first term on the right-hand side is $O(\beta ^{-1})$ because $L$ is $O(\beta ^{-1})$ (see discussion after (4.12)), while the second term on the right-hand side is $O(\beta ^{-3})$ . In the expression for the third derivative, the derivative of the first term (proportional to $L$ ) is $O(\beta ^{-1})$ , whereas the third derivative of the second term on the right-hand side in (4.12) is $O(1)$ . Therefore, the dominant contribution to the normal stress (2.59) is the $O(1)$ contribution due to the second term on the right-hand side in (4.12). The expression for the stress outside the drop depends on $\tilde {g}^{(o)}$ (2.52), and the constant $D$ in this equation is $O(\beta ^{-1})$ (see discussion after (4.12)). From the approximation (B24), the contribution due to $\tilde {F}_\theta$ on the right-hand side of (2.59) is also $O(1)$ in the limit $\beta \gg 1$ . Therefore, the largest contribution to the interfacial normal stress balance is due to the third derivative of the particular solution $\tilde {g}^{(i)}_p$ and the Lorentz force within the drop,

(4.15) \begin{align} \left . \left ( \frac {1}{6} \frac {\mbox{d}^3 \tilde {g}^{(i)}_p}{\mbox{d} r^3} + \frac {r \tilde {F}_\theta }{6} \right ) \right |_{r=1} = 6 {Ca}^{-1} \tilde {h}_0. \end{align}

Substituting the expression (4.11) for $\tilde{g}_p^{(i)}$ and (B24) for $\tilde {F}_{\theta }$ , we obtain

(4.16) \begin{align} {Ca}^{-1} \tilde {h}_0 = \mbox{} - \frac {1}{32}. \end{align}