2.2.1 Combined analysis

The interior and exterior bounding functions are not independent of each other. For instance, for a given $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ , $\hat{F}_{2}(r)$ cannot have a minimum in $r\in [0,1]$ , while $F_{2}(r)$ has a maximum in $r\in (1,\infty )$ . In other words, there is a transition between monotonicity and non-monotonicity (in opposite senses) for $\hat{F}_{2}(r)$ and $F_{2}(r)$ at the same critical value of $\unicode[STIX]{x1D706}$ . With regard to the earlier discussion, this implies that there is, in fact, only one fixed circle in the $(x_{1},x_{3})$ plane, of radius $[3/5+(2/5)\unicode[STIX]{x1D6FD}(1+\unicode[STIX]{x1D706})]^{1/2}$ , that appears either in the interior or exterior depending on the particular $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D706}$ ; figure 4(a,b) therefore correspond to different $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ pairs. An additional implication is that it must be possible to combine our analysis of the interior and exterior streamlines into one which is uniformly valid in the entire domain. To achieve this, one rewrites the streamline label $E$ in the exterior coordinate as $G=E+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}g(1)/(1+\unicode[STIX]{x1D706})$ , making the bounding functions continuous across the drop surface ( $r=1$ ). This allows $G$ to be the single streamline label for both the interior and exterior flows ( $G=\hat{E}$ for $r<1$ and $G=E+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}g(1)/(1+\unicode[STIX]{x1D706})$ for $r>1$ ).

Based on the behaviour of the combined (upper and lower) bounding functions, one finds two critical $\unicode[STIX]{x1D6FC}$ -dependent viscosity ratios $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ and $\unicode[STIX]{x1D706}_{c}^{\prime }(\unicode[STIX]{x1D6FC})$ , in turn leading to three distinct regimes for the streamline topologies in the drop interior and exterior. $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ is the value at which the minimum of $\hat{F}_{2}(r)$ and the maximum of $F_{2}(r)$ coincide at $r=1$ (figure 5 b). Thus, $\unicode[STIX]{x1D706}_{c}$ can be calculated by setting $r_{0}^{1}=\hat{r}_{0}^{2}=1$ , yielding:

(2.18) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})=2\unicode[STIX]{x1D6FC}/(1-\unicode[STIX]{x1D6FC}).\end{eqnarray}$$

For $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ , as mentioned earlier, $\hat{F}_{2}(r)$ has a minimum in $r\in [0,1]$ , while $F_{2}(r)$ is monotonically decreasing in $r\in [1,\infty )$ (figure 5 ai), while for $\unicode[STIX]{x1D706}_{c}<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}^{\prime }$ , $\hat{F}_{2}(r)$ is monotonically decreasing while $F_{2}(r)$ has a maximum for $r\in [\!1,\infty \!)$ (figure 5 c). In a similar fashion, at $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}^{\prime }$ , the maximum of $\hat{F}_{1}(r)$ lies at $r=0$ and for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}^{\prime }(\unicode[STIX]{x1D6FC})$ , the function $\hat{F}_{1}(r)$ is monotonically decreasing in $r\in [0,1]$ with a finite slope at $r=0$ (figure 5 di). This second critical viscosity ratio can therefore be calculated by solving ${\hat{r}_{0}}^{1}=0$ , giving:

(2.19) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{c}^{\prime }=(5\unicode[STIX]{x1D6FC}+1)/(2(1-\unicode[STIX]{x1D6FC})).\end{eqnarray}$$

Let $G_{1}$ denote the maximum value of the function $\hat{F}_{1}(r)$ at $r_{1}$ , $G_{2}$ the minimum value of the function $\hat{F}_{2}(r)$ at $r_{2}$ , and $G_{0}$ the maximum value of the shifted function $F_{2}(r)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}g(1)/(1+\unicode[STIX]{x1D706})$ attained at $r_{0}^{1}$ . In table 1, given in appendix A, the streamline topology for the three regimes (i) $0\leqslant \unicode[STIX]{x1D706}\leqslant \unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ , (ii) $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})<\unicode[STIX]{x1D706}\leqslant \unicode[STIX]{x1D706}_{c}^{\prime }(\unicode[STIX]{x1D6FC})$ and (iii) $\unicode[STIX]{x1D706}_{c}^{\prime }(\unicode[STIX]{x1D6FC})<\unicode[STIX]{x1D706}<\infty$ , based on the relative magnitudes of these extrema.

The transition in the streamline topology from one regime to the other, with increasing $\unicode[STIX]{x1D706}$ is shown in figure 5(a–d). For a fixed $\unicode[STIX]{x1D6FC}$ , as one increases $\unicode[STIX]{x1D706}$ from below $\unicode[STIX]{x1D706}_{c}$ , the first (and smaller) fixed circle in the drop interior at $\{{\hat{r}_{0}}^{1},{\hat{\unicode[STIX]{x1D703}}_{0}}^{1},{\hat{\unicode[STIX]{x1D719}}_{0}}^{1}\}$ decreases in radius, while the second (and larger) fixed circle at $\{{\hat{r}_{0}}^{2},{\hat{\unicode[STIX]{x1D703}}_{0}}^{2},{\hat{\unicode[STIX]{x1D719}}_{0}}^{2}\}$ increases in radius (see figure 5 a). At $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ , this second fixed circle merges with the fixed points at $\{{\hat{r}_{0}}^{3},{\hat{\unicode[STIX]{x1D703}}_{0}}^{3},{\hat{\unicode[STIX]{x1D719}}_{0}}^{3}\}$ and $\{{\hat{r}_{0}}^{4},{\hat{\unicode[STIX]{x1D703}}_{0}}^{4},{\hat{\unicode[STIX]{x1D719}}_{0}}^{4}\}$ , which themselves move towards each other (along $r=1$ ) for $\unicode[STIX]{x1D706}$ approaching $\unicode[STIX]{x1D706}_{c}$ , to form a fixed circle exactly on the drop surface (see figure 5 b). For $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$ , this fixed circle moves outside the drop and is now given by $\{r_{0}^{1},\unicode[STIX]{x1D703}_{0}^{1},\unicode[STIX]{x1D719}_{0}^{1}\}$ (see figure 5 c). Importantly, for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$ there now exists an annular region of closed streamlines surrounding the drop whose maximum radial extent is given by $r_{0}^{1}$ (see figure 5 c,d). Thus, $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ is the critical viscosity curve which separates the open- and closed-streamline regimes in the exterior field (see figure 1). These two regimes lead to fundamentally different mechanisms for the transport of heat or mass from drops at large $Pe$ . In the rest of the paper we solve the problem of transport from drops in the open-streamline regime where the effects of inertia do not affect the transport at leading order, except perhaps close to $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ (see figure 1). As one increases $\unicode[STIX]{x1D706}$ further, the fixed circle in the drop interior continues to become smaller while the exterior fixed circle grows in size. At $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}^{\prime }(\unicode[STIX]{x1D6FC})$ , the interior fixed circle shrinks to a point at the origin, leaving the origin as the only fixed point in the drop interior (although not relevant to the exterior transport problem analysed here, this transition in the interior streamline topology involving the disappearance of saddles for sufficiently high $\unicode[STIX]{x1D706}$ is likely to be important in chaotic advection leading to drop mixing (see Bryden & Brenner (Reference Bryden and Brenner1999))); the implications of chaos in the interior flow, for the transport rate at large $Pe$ , are further discussed in § 4. Note that the axisymmetric closed streamline region is infinite in volume for simple shear flow ( $\unicode[STIX]{x1D6FC}=0$ ) for any $\unicode[STIX]{x1D706}$ since the maximal extent of the region $r_{0}^{1}\rightarrow \infty$ for $\unicode[STIX]{x1D6FC}=0$ (with the ordinate decreasing as $r^{-3/2}$ , as a for a rigid sphere). For any $\unicode[STIX]{x1D6FC}\neq 0$ , this region grows in size for increasing $\unicode[STIX]{x1D706}$ , but asymptotes to a finite extent even as $\unicode[STIX]{x1D706}\rightarrow \infty$ .

3.2.1 The $(C,\unicode[STIX]{x1D70F})$ coordinate system

We first analyse the streamlines on the drop surface, given that they determine the flow within the thermal boundary layer. Note that the ’open-streamline’ terminology refers to the nature of the exterior streamlines for $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ . The surface streamlines in this regime cannot obviously be open. As will be seen, they begin and end at stagnation points of the surface tangential velocity field (figure 6 c). Since the drop is spherical and stationary, the radial velocity at the surface is identically zero, and the surface streamlines are determined by the velocity components in the $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$ directions, obtained by setting $r=1$ in (2.5) and (2.6):

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\unicode[STIX]{x1D703}}|_{r=1}=-{\displaystyle \frac{(1+\unicode[STIX]{x1D6FC})}{2(1+\unicode[STIX]{x1D706})}}\sin 2\unicode[STIX]{x1D703}\sin 2\unicode[STIX]{x1D719}_{1}={\displaystyle \frac{\text{d}\unicode[STIX]{x1D703}}{\text{d}s}}, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\unicode[STIX]{x1D719}}|_{r=1}={\displaystyle \frac{(1+\unicode[STIX]{x1D6FC})}{(1+\unicode[STIX]{x1D706})}}\sin \unicode[STIX]{x1D703}[\cos 2\unicode[STIX]{x1D719}_{1}+\unicode[STIX]{x1D6FD}(1+\unicode[STIX]{x1D706})]=-\sin \unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D719}_{1}}{\text{d}s}}, & \displaystyle\end{eqnarray}$$

where we have used the transformation $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D719}$ , so that $\unicode[STIX]{x1D719}_{1}$ is measured from the $x_{2}$ axis (see figure 2 a). Also, $s$ is the dimensionless time (measured in units of $\dot{\unicode[STIX]{x1D6FE}}^{-1}$ ) and $\unicode[STIX]{x1D6FD}=(1-\unicode[STIX]{x1D6FC})/(1+\unicode[STIX]{x1D6FC})$ . The above equations may be rewritten as:

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D703}}{\text{d}s}}=-{\displaystyle \frac{(1-\unicode[STIX]{x1D6FC})(\unicode[STIX]{x1D6FE}^{2}-1)}{2(\unicode[STIX]{x1D6FE}^{2}+1)}}\sin 2\unicode[STIX]{x1D703}\sin 2\unicode[STIX]{x1D719}_{1}, & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D719}_{1}}{\text{d}s}}=-{\displaystyle \frac{2(1-\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D6FE}^{2}+1}}[\unicode[STIX]{x1D6FE}^{2}\cos ^{2}\unicode[STIX]{x1D719}_{1}+\sin ^{2}\unicode[STIX]{x1D719}_{1}], & \displaystyle\end{eqnarray}$$

where

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\left[{\displaystyle \frac{(\unicode[STIX]{x1D6FD}(1+\unicode[STIX]{x1D706})+1)}{(\unicode[STIX]{x1D6FD}(1+\unicode[STIX]{x1D706})-1)}}\right]^{1/2}=\left[{\displaystyle \frac{\unicode[STIX]{x1D706}_{c}+\unicode[STIX]{x1D6FC}~\unicode[STIX]{x1D706}}{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c})}}\right]^{1/2}.\end{eqnarray}$$

Comparing (3.6) and (3.7) to equations 2(a) and 2(b) in Leal & Hinch (Reference Leal and Hinch1971) (see also Hinch & Leal (Reference Hinch and Leal1972)), one notes that these have the exact same form as the equations for the Jeffery orbits of an axisymmetric particle in a planar linear flow. For the original Jeffery orbits in a simple shear flow (Leal & Hinch Reference Leal and Hinch1971), $\unicode[STIX]{x1D6FE}$ was a geometric quantity – being the true aspect ratio for spheroids, and an effective aspect ratio for other geometries (Trevelyan & Mason Reference Trevelyan and Mason1951; Cox Reference Cox1971). Generalization of the Jeffery orbits for planar linear flows results in an effective aspect ratio which is no longer a purely geometric quantity, being a function of both the actual aspect ratio and flow type (Hinch & Leal Reference Hinch and Leal1972; Marath & Subramanian Reference Marath and Subramanian2018). For the drop surface streamlines considered here there is no geometric quantity, and the effective aspect ratio is determined by the flow type ( $\unicode[STIX]{x1D6FC}$ ) and the viscosity ratio ( $\unicode[STIX]{x1D706}$ ).

The above insight allows one to define a new coordinate system to describe the surface streamlines. Proceeding as in Leal & Hinch (Reference Leal and Hinch1971), and integrating (3.7), one obtains:

(3.9) $$\begin{eqnarray}\tan \unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D6FE}\tan {\displaystyle \frac{-2(1-\unicode[STIX]{x1D6FC})s}{\unicode[STIX]{x1D6FE}+1/\unicode[STIX]{x1D6FE}}}.\end{eqnarray}$$

Introducing a new variable $\unicode[STIX]{x1D70F}=(-2(1-\unicode[STIX]{x1D6FC})s)/(\unicode[STIX]{x1D6FE}+1/\unicode[STIX]{x1D6FE})$ , we have the equation:

(3.10) $$\begin{eqnarray}\tan \unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D6FE}\tan \unicode[STIX]{x1D70F}.\end{eqnarray}$$

Similarly, we can integrate (3.6), after substituting for $\unicode[STIX]{x1D719}_{1}$ in terms of $\unicode[STIX]{x1D70F}$ , and one obtains:

(3.11) $$\begin{eqnarray}\tan \unicode[STIX]{x1D703}=C[\cos ^{2}\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D6FE}^{2}\sin ^{2}\unicode[STIX]{x1D70F}]^{1/2}.\end{eqnarray}$$

The relations (3.10) and (3.11) form the basis of the $(C,\unicode[STIX]{x1D70F})$ coordinate system on the drop surface. As seen in figure 6(a), the coordinate lines $C$ and $\unicode[STIX]{x1D70F}$ are in general not orthogonal to one another. The $C$ coordinate acts as a surface-streamline label (referred to as an ‘orbit constant’ in the original context; see Leal & Hinch (Reference Leal and Hinch1971)) while the $\unicode[STIX]{x1D70F}$ coordinate gives the phase along that trajectory (orbit). The diagonal coordinate metrics in the $C$ and $\unicode[STIX]{x1D70F}$ directions denoted respectively by $h$ and $k$ , and the skewness angle $\unicode[STIX]{x1D6FC}_{1}$ ( $\unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x03C0}/2$ for an orthogonal coordinate system) are given by:

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle h=g_{CC}=\unicode[STIX]{x1D703}_{C}, & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle k=g_{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D70F}}=(\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}^{2}+\sin ^{2}\unicode[STIX]{x1D703}\unicode[STIX]{x1D719}_{1\unicode[STIX]{x1D70F}}^{2})^{1/2}, & \displaystyle\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle g_{C\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D703}_{C}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}, & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \sin \unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x1D719}_{1\unicode[STIX]{x1D70F}}\sin \unicode[STIX]{x1D703}(\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}^{2}+\sin ^{2}\unicode[STIX]{x1D703}\unicode[STIX]{x1D719}_{1\unicode[STIX]{x1D70F}}^{2})^{-1/2}, & \displaystyle\end{eqnarray}$$

where the subscripts denote derivatives with respect to the particular variable ( $g_{CC}$ , $g_{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D70F}}$ and $g_{C\unicode[STIX]{x1D70F}}$ being metric tensor components as per the conventional notation; see Aris (Reference Aris2012)). The contravariant unit vectors of the coordinate system in terms of the conventional spherical coordinates unit vectors are given by:

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{C}}=\hat{\unicode[STIX]{x1D73D}}, & \displaystyle\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\unicode[STIX]{x1D749}}=\cos \unicode[STIX]{x1D6FC}_{1}\hat{\unicode[STIX]{x1D73D}}+\sin \unicode[STIX]{x1D6FC}_{1}\hat{\unicode[STIX]{x1D753}}_{1}, & \displaystyle\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D753}}_{1}=\cos \unicode[STIX]{x1D719}_{1}\hat{\boldsymbol{x}}_{1}-\sin \unicode[STIX]{x1D719}_{1}\hat{\boldsymbol{x}}_{2}$ is the unit vector along the azimuth with $\unicode[STIX]{x1D719}_{1}$ being measured in a clockwise sense from the $x_{2}$ axis. Note that $\hat{\boldsymbol{C}}$ is still along the meridional direction, but $\hat{\unicode[STIX]{x1D749}}$ is along the surface streamline, rather than along the azimuth, which leads to the non-orthogonality. The surface velocity field is given by $\boldsymbol{u}=u_{C}\hat{\boldsymbol{C}}+u_{\unicode[STIX]{x1D70F}}\hat{\unicode[STIX]{x1D749}}$ , where:

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle u_{C}=u_{\unicode[STIX]{x1D703}}-{\displaystyle \frac{u_{\unicode[STIX]{x1D719}_{1}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D719}_{1\unicode[STIX]{x1D70F}}\sin \unicode[STIX]{x1D703}}}, & \displaystyle\end{eqnarray}$$

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\unicode[STIX]{x1D70F}}={\displaystyle \frac{ku_{\unicode[STIX]{x1D719}_{1}}}{\unicode[STIX]{x1D719}_{1\unicode[STIX]{x1D70F}}\sin \unicode[STIX]{x1D703}}}. & \displaystyle\end{eqnarray}$$

Using (3.18) along with (3.4) and (3.5), one finds $u_{C}=0$ . This is, of course, because the $\unicode[STIX]{x1D70F}$ coordinate is aligned with the Stokes surface streamlines. This reduction of $\boldsymbol{u}$ to a one-dimensional field aids a solution of the heat transfer problem both in the open streamline regime analysed here, and the inertially enhanced transport in the closed-streamline regime analysed in a companion paper (Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018).

Importantly, the $(C,\unicode[STIX]{x1D70F})$ coordinate system above enables a natural means of distinguishing between the open and closed surface Stokesian streamline topologies in planar linear flows. When the surface streamlines are closed, they must be characterized by a real-valued effective aspect ratio $\unicode[STIX]{x1D6FE}$ (as for the original axisymmetric particle). This is easily seen from (3.8) where, for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$ , $\unicode[STIX]{x1D6FE}^{2}>0$ . On the other hand, for $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ , corresponding to the open-streamline regime analysed here, $\unicode[STIX]{x1D6FE}^{2}<0$ , leading to a purely imaginary aspect ratio. In this regime, all streamlines begin or end at fixed (stagnation) points either on the symmetry plane or on the vorticity axis, with the indices of these fixed points adding up to the Euler characteristic of the sphere (see figures 6 b and 6 c). In what follows, the focus is on the case $\unicode[STIX]{x1D6FE}^{2}<0$ .

In the open-streamline regime it is convenient to rewrite the purely imaginary effective aspect ratio $\unicode[STIX]{x1D6FE}=\text{i}\hat{\unicode[STIX]{x1D6FE}}$ , where $\hat{\unicode[STIX]{x1D6FE}}^{2}=[(\unicode[STIX]{x1D706}_{c}+\unicode[STIX]{x1D6FC}~\unicode[STIX]{x1D706})/\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D706}_{c}-\unicode[STIX]{x1D706})]$ and $\hat{\unicode[STIX]{x1D6FE}}$ is real. Since $\unicode[STIX]{x1D719}_{1}$ corresponds to a physical angle, and is therefore real valued, we have by rearranging (3.10) $\unicode[STIX]{x1D70F}=-\text{i}\tanh ^{-1}(\tan \unicode[STIX]{x1D719}_{1}/\hat{\unicode[STIX]{x1D6FE}})$ . Using the well-known form of the inverse hyperbolic tangent in terms of logarithms, one obtains:

(3.20) $$\begin{eqnarray}\unicode[STIX]{x1D70F}={\displaystyle \frac{\text{i}}{2}}\log \left({\displaystyle \frac{\hat{\unicode[STIX]{x1D6FE}}-\tan \unicode[STIX]{x1D719}_{1}}{\hat{\unicode[STIX]{x1D6FE}}+\tan \unicode[STIX]{x1D719}_{1}}}\right).\end{eqnarray}$$

Similarly, we can rearrange (3.11) to get:

(3.21) $$\begin{eqnarray}C={\displaystyle \frac{\tan \unicode[STIX]{x1D703}(\hat{\unicode[STIX]{x1D6FE}}^{2}-\tan ^{2}\unicode[STIX]{x1D719}_{1})^{1/2}}{|\text{sec}\,\unicode[STIX]{x1D719}_{1}|\hat{\unicode[STIX]{x1D6FE}}}}.\end{eqnarray}$$

The branch points of the logarithm in (3.20) are at $\tan \unicode[STIX]{x1D719}_{1}=\pm \hat{\unicode[STIX]{x1D6FE}}$ . For $\unicode[STIX]{x1D719}_{1}\in [-\unicode[STIX]{x03C0},\unicode[STIX]{x03C0}]$ , this corresponds to four distinct values of $\unicode[STIX]{x1D719}_{1}$ given by $\unicode[STIX]{x1D719}_{1}^{(1)}=\tan ^{-1}\hat{\unicode[STIX]{x1D6FE}}$ , $\unicode[STIX]{x1D719}_{1}^{(2)}=\unicode[STIX]{x03C0}-\unicode[STIX]{x1D719}_{1}^{(1)}$ , $\unicode[STIX]{x1D719}_{1}^{(3)}=-\unicode[STIX]{x1D719}_{1}^{(2)}$ and $\unicode[STIX]{x1D719}_{1}^{(4)}=-\unicode[STIX]{x1D719}_{1}^{(1)}$ (figure 6 b), where we consider the principal value of the inverse tangent. These values of $\unicode[STIX]{x1D719}_{1}$ correspond precisely to the four stagnation points on the drop surface in the equatorial plane (see § 2 with regard to the discussion on streamline topology). The $\unicode[STIX]{x1D719}_{1}=\text{const.}$ lines corresponding to these four stagnation points separate the drop surface into four regions (I–IV), where the pairs (I, III) and (II, IV) are equivalent, merely separated by $\unicode[STIX]{x03C0}$ in the azimuthal coordinate (see figure 6). This is due to the underlying symmetry (to a $\unicode[STIX]{x03C0}$ rotation) of the one-parameter family of linear flows considered here. Therefore, it is sufficient to confine our analysis to two of these regions, and we consider regions I and II.

The logarithm in (3.20) is real for $\tan ^{2}\unicode[STIX]{x1D719}_{1}<\hat{\unicode[STIX]{x1D6FE}}^{2}$ , which corresponds to $-\unicode[STIX]{x1D719}_{1}^{(1)}<\unicode[STIX]{x1D719}_{1}<\unicode[STIX]{x1D719}_{1}^{(1)}$ and $\unicode[STIX]{x1D719}_{1}^{(2)}<\unicode[STIX]{x1D719}_{1}<\unicode[STIX]{x1D719}_{1}^{(3)}$ , making $\unicode[STIX]{x1D70F}$ purely imaginary in these two regions (I and III). Also, from (3.21) the surface-streamline label $C$ is purely real in these regions. The other two regions, II and IV, lie in the range $\unicode[STIX]{x1D719}_{1}^{(1)}<\unicode[STIX]{x1D719}_{1}<\unicode[STIX]{x1D719}_{1}^{(2)}$ and $\unicode[STIX]{x1D719}_{1}^{(3)}<\unicode[STIX]{x1D719}_{1}<\unicode[STIX]{x1D719}_{1}^{(4)}$ , respectively. In these regions, the combination $(\hat{\unicode[STIX]{x1D6FE}}-\tan \unicode[STIX]{x1D719}_{1})/(\hat{\unicode[STIX]{x1D6FE}}+\tan \unicode[STIX]{x1D719}_{1})<0$ , implying the logarithm is complex valued. Thus, we rewrite the equation for $\unicode[STIX]{x1D70F}$ as:

(3.22) $$\begin{eqnarray}\unicode[STIX]{x1D70F}={\displaystyle \frac{\text{i}}{2}}\log \left[{\displaystyle \frac{\tan \unicode[STIX]{x1D719}_{1}-\hat{\unicode[STIX]{x1D6FE}}}{\tan \unicode[STIX]{x1D719}_{1}+\hat{\unicode[STIX]{x1D6FE}}}}\exp (\text{i}\unicode[STIX]{x03C0})\right],\end{eqnarray}$$

where $(\tan \unicode[STIX]{x1D719}_{1}-\hat{\unicode[STIX]{x1D6FE}})/(\tan \unicode[STIX]{x1D719}_{1}+\hat{\unicode[STIX]{x1D6FE}})>0$ . Expanding this, we have:

(3.23) $$\begin{eqnarray}\unicode[STIX]{x1D70F}={\displaystyle \frac{\text{i}}{2}}\log \left[{\displaystyle \frac{\tan \unicode[STIX]{x1D719}_{1}-\hat{\unicode[STIX]{x1D6FE}}}{\tan \unicode[STIX]{x1D719}_{1}+\hat{\unicode[STIX]{x1D6FE}}}}\right]-{\displaystyle \frac{\unicode[STIX]{x03C0}}{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}+\unicode[STIX]{x03C0}/2$ is now purely imaginary. This motivates the introduction of a new variable $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}+(\unicode[STIX]{x03C0}/2)$ , where $\unicode[STIX]{x1D70F}^{\prime }$ is purely imaginary. From (3.21), $C$ in regions II and IV is purely imaginary.

In summary, we have:

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle (C,\unicode[STIX]{x1D70F})\equiv (C,\text{i}\hat{\unicode[STIX]{x1D70F}})\quad \text{in regions I and III}, & \displaystyle\end{eqnarray}$$

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle (C,\unicode[STIX]{x1D70F})\equiv (\text{i}{\hat{C}},\text{i}\hat{\unicode[STIX]{x1D70F}}-\unicode[STIX]{x03C0}/2)\quad \text{in regions II and IV}. & \displaystyle\end{eqnarray}$$

with ( ${\hat{C}},\hat{\unicode[STIX]{x1D70F}}$ ) being real valued. In § 3.2.2, we use (3.24) and (3.25) to formulate the large- $Pe$ boundary-layer analysis.

From figure 6 one sees that almost all surface streamlines begin and end at stagnation points on the drop surface and are therefore entirely confined to one of the four regions above. Focussing our attention on regions I and II, the stagnation point, $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D719}_{1}^{(1)}$ , is an inlet since fluid outside moves radially inwards towards the drop. This location corresponds to the phase variable taking the limiting value $\hat{\unicode[STIX]{x1D70F}}=-\infty$ in both regions. The $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D719}_{1}^{(2)}$ and $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D719}_{1}^{(4)}$ locations are outlets, and correspond to $\hat{\unicode[STIX]{x1D70F}}=\infty$ . These inlet and outlet values of $\hat{\unicode[STIX]{x1D70F}}$ are regardless of the particular value of $C$ or ${\hat{C}}$ . Thus, along any surface streamline, as one moves from the inlet to the outlet location (from $\unicode[STIX]{x1D719}_{1}^{(1)}$ to $\unicode[STIX]{x1D719}_{1}^{(4)}$ in region I, and from $\unicode[STIX]{x1D719}_{1}^{(1)}$ to $\unicode[STIX]{x1D719}_{1}^{(2)}$ in region II), $\hat{\unicode[STIX]{x1D70F}}$ varies through all finite values from $-\infty$ to $+\infty$ (except for $C={\hat{C}}=0$ ; see below). The surface-streamline labels $C$ and ${\hat{C}}$ may be mapped as follows. The limiting trajectories $C,{\hat{C}}\rightarrow \infty$ correspond to circular arcs in the plane $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ , with $\hat{\unicode[STIX]{x1D70F}}$ again increasing from $-\infty$ to $\infty$ along each arc. For finite $C$ and ${\hat{C}}$ , the surface streamlines are non-planar curves, the minimum value of $\unicode[STIX]{x1D703}$ accessed decreasing from $\unicode[STIX]{x03C0}/2$ towards $0$ with decreasing $C$ (or ${\hat{C}}$ ). The limiting trajectories for $C=0$ and ${\hat{C}}=0$ , in regions I and II respectively, correspond to constant- $\unicode[STIX]{x1D719}_{1}$ lines; that is, they are portions of meridians that connect the inlet and outlet stagnation points to the point $\unicode[STIX]{x1D703}=0$ . In these cases, unlike the earlier streamlines with non-zero $C$ (or ${\hat{C}}$ ), $\hat{\unicode[STIX]{x1D70F}}=-\infty$ along one half of the trajectory (from the inlet to $\unicode[STIX]{x1D703}=0$ ), and $\hat{\unicode[STIX]{x1D70F}}=\infty$ along the other half (from $\unicode[STIX]{x1D703}=0$ to the outlet). The fluid velocity is nevertheless finite along these trajectories, and so is the thermal boundary-layer thickness. As a result, these limiting trajectories do not require any special consideration in the boundary-layer analysis below (said differently, the eventual $Nu$ integral does not exhibit a divergence for $C,{\hat{C}}\rightarrow 0$ ).

3.2.2 Boundary-layer analysis

The convection–diffusion equation in the $(r,C,\unicode[STIX]{x1D70F})$ coordinate system has the form:

(3.26) $$\begin{eqnarray}u_{r}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}r}}+{\displaystyle \frac{u_{C}}{h}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}C}}+{\displaystyle \frac{u_{\unicode[STIX]{x1D70F}}}{k}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}={\displaystyle \frac{1}{Pe}}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6E9},\end{eqnarray}$$

where $u_{r}$ is the velocity along the radial direction; $u_{C}$ and $u_{\unicode[STIX]{x1D70F}}$ are given by (3.18) and (3.19). Anticipating the dominance of radial diffusion in the thermal boundary layer, this equation simplifies to:

(3.27) $$\begin{eqnarray}u_{r}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}r}}+{\displaystyle \frac{u_{C}}{h}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}C}}+{\displaystyle \frac{u_{\unicode[STIX]{x1D70F}}}{k}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}={\displaystyle \frac{1}{Pe}}{\displaystyle \frac{1}{r^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}}\left(r^{2}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}r}}\right).\end{eqnarray}$$

Since the thickness of this boundary layer is small (scaling as $O(aPe^{-1/2})$ for a drop) compared to the drop radius for large $Pe$ , only the near-surface velocity field is relevant to the leading-order boundary-layer analysis. Defining the boundary-layer coordinate, $y=r-1$ and expanding the velocity field (expressed in $(C,\unicode[STIX]{x1D70F})$ coordinates) near the drop surface, one gets from (2.4) and (3.19), respectively:

(3.28) $$\begin{eqnarray}\displaystyle u_{r} & = & \displaystyle \left[-{\displaystyle \frac{3C^{2}(1+\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6FE}\sin 2\unicode[STIX]{x1D70F}}{(1+\unicode[STIX]{x1D706})(1+C^{2}(\cos ^{2}\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D6FE}^{2}\sin ^{2}\unicode[STIX]{x1D70F}))}}\right]y+O(y^{2}),\end{eqnarray}$$

(3.29) $$\begin{eqnarray}\displaystyle & = & \displaystyle h_{r}(C,\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})y+O(y^{2}),\end{eqnarray}$$

(3.30) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{u_{C}}{h}} & = & \displaystyle h_{C}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})+O(y)\end{eqnarray}$$

(3.31) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{u_{\unicode[STIX]{x1D70F}}}{k}} & = & \displaystyle {\displaystyle \frac{-2\unicode[STIX]{x1D6FE}(1+\unicode[STIX]{x1D6FC})}{(1+\unicode[STIX]{x1D706})(\unicode[STIX]{x1D6FE}^{2}-1)}}+O(y),\end{eqnarray}$$

(3.32) $$\begin{eqnarray}\displaystyle & = & \displaystyle h_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})+O(y),\end{eqnarray}$$

where the leading-order term $h_{C}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})=0$ since $u_{C}=0$ , by definition, at the drop surface (3.18). Note that the dependence of the coefficients $h_{r}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ and $h_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ on $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D706}$ has a specific form: a complicated non-separable dependence via the effective aspect ratio $\unicode[STIX]{x1D6FE}$ , and an explicit separable dependence via the factor $(1+\unicode[STIX]{x1D6FC})/(1+\unicode[STIX]{x1D706})$ . We will see later that one may include the dependence on $(1+\unicode[STIX]{x1D6FC})/(1+\unicode[STIX]{x1D706})$ into the Péclet number, so that the residual $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ dependence of the heat transfer appears solely as a function of $\unicode[STIX]{x1D6FE}$ .

Defining the boundary-layer coordinate as $Y=Pe^{m}y$ , where $m$ is the exponent characterizing the boundary-layer thickness, and with $Y$ as $O(1)$ , one obtains at leading order:

(3.33) $$\begin{eqnarray}h_{r}(C,\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})Y{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}Y}}+h_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}){\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}={\displaystyle \frac{Pe^{2m}}{Pe}}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}Y^{2}}}.\end{eqnarray}$$

The leading-order balance implies $m=1/2$ giving the expected scaling as $y\sim Pe^{-1/2}Y$ . The use of a non-orthogonal $(C,\unicode[STIX]{x1D70F})$ coordinate system has also led to a major simplification with the tangential rate of convection, $h_{\unicode[STIX]{x1D70F}}$ , being independent of both $C$ and $\unicode[STIX]{x1D70F}$ (see (3.19)). The $(C,\unicode[STIX]{x1D70F})$ coordinate system has thus allowed one to establish an exact parallel between the analysis for a drop, and that for a (rotating) solid particle. In the latter case, the tangential (azimuthal) rate of convection, given by $u_{\unicode[STIX]{x1D719}}/(r\sin \unicode[STIX]{x1D703})$ , is trivially a constant, being equal to the angular velocity of the rotating particle. Thus, $\unicode[STIX]{x1D70F}$ for the drop here plays the role of $\unicode[STIX]{x1D719}$ for the particle (Subramanian & Koch Reference Subramanian and Koch2006a ,Reference Subramanian and Koch b , Reference Subramanian and Koch2007).

From here on, the analysis is standard (Leal Reference Leal2007). Defining a similarity variable $\unicode[STIX]{x1D702}=Y/g(C,\unicode[STIX]{x1D70F})$ , where $g(C,\unicode[STIX]{x1D70F})$ characterizes the dependence of the boundary-layer thickness with respect to position on the drop surface, the non-dimensional temperature is found to satisfy:

(3.34) $$\begin{eqnarray}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6E9}}{\text{d}\unicode[STIX]{x1D702}^{2}}}+2\unicode[STIX]{x1D702}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6E9}}{\text{d}\unicode[STIX]{x1D702}}}=0,\end{eqnarray}$$

where

(3.35) $$\begin{eqnarray}h_{\unicode[STIX]{x1D70F}}g{\displaystyle \frac{\unicode[STIX]{x2202}g}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}-h_{r}g^{2}=2,\end{eqnarray}$$

and:

(3.36) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E9}=1\quad \text{at }\unicode[STIX]{x1D702}=0, & \displaystyle\end{eqnarray}$$

(3.37) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E9}\rightarrow 0\quad \text{as }\unicode[STIX]{x1D702}\rightarrow \infty . & \displaystyle\end{eqnarray}$$

Solving (3.34), and using (4.3) and (4.4), the temperature profile in terms of the similarity variable is given as:

(3.38) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D702})=1-{\displaystyle \frac{2}{\sqrt{\unicode[STIX]{x03C0}}}}\int _{0}^{\unicode[STIX]{x1D702}}\exp (-t^{2})\,\text{d}t.\end{eqnarray}$$

Next, we solve the equation for the boundary-layer thickness given by (3.35). Introducing a new variable $f=g^{2}/2$ , we get:

(3.39) $$\begin{eqnarray}h_{\unicode[STIX]{x1D70F}}{\displaystyle \frac{\text{d}f}{\text{d}\unicode[STIX]{x1D70F}}}-2h_{r}f=2.\end{eqnarray}$$

Substituting for $h_{r}$ and $h_{\unicode[STIX]{x1D70F}}$ from (3.28) and (3.19), the equation has the form:

(3.40) $$\begin{eqnarray}{\displaystyle \frac{\text{d}f}{\text{d}\unicode[STIX]{x1D70F}}}-{\displaystyle \frac{3C^{2}(\unicode[STIX]{x1D6FE}^{2}-1)\sin 2\unicode[STIX]{x1D70F}}{[1+C^{2}(\cos ^{2}\unicode[STIX]{x1D70F}_{1}+\unicode[STIX]{x1D6FE}^{2}\sin ^{2}\unicode[STIX]{x1D70F}_{1})]}}f=-{\displaystyle \frac{(1+\unicode[STIX]{x1D706})(\unicode[STIX]{x1D6FE}^{2}-1)}{(1+\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6FE}}}.\end{eqnarray}$$

The integrating factor for (3.40) is easily shown to be given by $[1+C^{2}(\cos ^{2}\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D6FE}^{2}\sin ^{2}\unicode[STIX]{x1D70F})]^{-3}$ , and using this, one obtains the following expression for the boundary-layer thickness:

(3.41) $$\begin{eqnarray}\displaystyle g(C,\unicode[STIX]{x1D70F}) & = & \displaystyle \left[-{\displaystyle \frac{2(1+\unicode[STIX]{x1D706})(\unicode[STIX]{x1D6FE}^{2}-1)[1+C^{2}(\cos ^{2}\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D6FE}^{2}\sin ^{2}\unicode[STIX]{x1D70F})]^{3}}{(1+\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6FE}}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.\times \int _{\unicode[STIX]{x1D70F}_{inlet}}^{\unicode[STIX]{x1D70F}}[1+C^{2}(\cos ^{2}\unicode[STIX]{x1D70F}_{1}+\unicode[STIX]{x1D6FE}^{2}\sin ^{2}\unicode[STIX]{x1D70F}_{1})]^{-3}\text{d}\unicode[STIX]{x1D70F}_{1}\right]^{1/2},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{inlet}$ is determined from the constraint that the boundary-layer thickness be finite at the inlet stagnation points ( $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D719}_{1}^{(1)}$ and $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D719}_{1}^{(3)}$ ); at the inlets, the local flow is a linear extension convecting heat towards the stagnation point, balancing the outward diffusion. From the point of view of the later integration, it is useful to note that $g(C,\unicode[STIX]{x1D70F})$ above has the form:

(3.42) $$\begin{eqnarray}\displaystyle & \displaystyle g(C,\unicode[STIX]{x1D70F})=\left[-{\displaystyle \frac{2(1+\unicode[STIX]{x1D706})(\unicode[STIX]{x1D6FE}^{2}-1)}{(1+\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6FE}}}Q(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE}){\mathcal{I}}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE})\right]^{1/2}, & \displaystyle\end{eqnarray}$$

(3.43) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{I}}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE})=\int _{\unicode[STIX]{x1D70F}_{inlet}}^{\unicode[STIX]{x1D70F}}Q(C,\unicode[STIX]{x1D70F}_{1};\unicode[STIX]{x1D6FE})^{-1}\text{d}\unicode[STIX]{x1D70F}_{1}, & \displaystyle\end{eqnarray}$$

where $Q(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE})=[1+C^{2}(\cos ^{2}\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D6FE}^{2}\sin ^{2}\unicode[STIX]{x1D70F})]^{3}$ .

We now write down the forms of the boundary-layer thicknesses in the two distinct regions identified above on the drop surface in the open streamline regime (regions (I, III) and (II, IV) in figure 6 c). Using the forms of $C$ and $\unicode[STIX]{x1D70F}$ specific to regions (I, III), as given by (3.24), the integral appearing in the equation for the boundary-layer thickness (3.43) has the form:

(3.44) $$\begin{eqnarray}\displaystyle {\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})=\int _{-\infty }^{\hat{\unicode[STIX]{x1D70F}}}Q_{I}(C,\hat{\unicode[STIX]{x1D70F}}_{1};\hat{\unicode[STIX]{x1D6FE}})^{-1}\,\text{d}\hat{\unicode[STIX]{x1D70F}}_{1}, & & \displaystyle\end{eqnarray}$$

where $Q_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})=[1+C^{2}(\cosh ^{2}\hat{\unicode[STIX]{x1D70F}}+\hat{\unicode[STIX]{x1D6FE}}^{2}\sinh ^{2}\hat{\unicode[STIX]{x1D70F}})]^{3}$ . Using this, the boundary-layer thickness expressions in region I(and III) are given by:

(3.45) $$\begin{eqnarray}\displaystyle g_{I}(C,\hat{\unicode[STIX]{x1D70F}})=\left[{\displaystyle \frac{2(1+\unicode[STIX]{x1D706})(\hat{\unicode[STIX]{x1D6FE}^{2}}+1)}{(1+\unicode[STIX]{x1D6FC})\hat{\unicode[STIX]{x1D6FE}}}}Q_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}}){\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})\right]^{1/2}. & & \displaystyle\end{eqnarray}$$

It is possible to perform the integration in (3.45) analytically, and ${\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})$ in regions (I, III) is given by:

(3.46) $$\begin{eqnarray}\displaystyle {\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}}) & = & \displaystyle {\displaystyle \frac{1}{(C^{2}\hat{\unicode[STIX]{x1D6FE}}^{2}-1)^{3}A^{5}}}\left[{\displaystyle \frac{3A^{4}-2A^{2}+3}{8}}\{\tan ^{-1}(\tanh \hat{\unicode[STIX]{x1D70F}}/A)-\tan ^{-1}(-1/A)\}\right.\nonumber\\ \displaystyle & & \displaystyle +\,{\displaystyle \frac{A(1-A^{4})\tanh \hat{\unicode[STIX]{x1D70F}}}{2(A^{2}+\tanh ^{2}\hat{\unicode[STIX]{x1D70F}})}}+{\displaystyle \frac{A(1+A^{2})^{2}(A^{2}-\tanh ^{2}\hat{\unicode[STIX]{x1D70F}})\tanh \hat{\unicode[STIX]{x1D70F}}}{8(A^{2}+\tanh ^{2}\hat{\unicode[STIX]{x1D70F}})^{2}}}\nonumber\\ \displaystyle & & \displaystyle \left.-\,{\displaystyle \frac{3A(A^{2}-1)}{8}}\right],\end{eqnarray}$$

where $A=[(1+C^{2})/(C^{2}\hat{\unicode[STIX]{x1D6FE}}^{2}-1)]^{1/2}$ . That the boundary-layer thickness is finite at the inlet stagnation point $\unicode[STIX]{x1D719}^{(1)}$ may now be seen. As mentioned earlier, the inlet corresponds to $\hat{\unicode[STIX]{x1D70F}}\rightarrow -\infty$ with $C$ arbitrary. Applying the condition $\hat{\unicode[STIX]{x1D70F}}\rightarrow -\infty$ to (3.45), and noting that $Q(C,\hat{\unicode[STIX]{x1D70F}})\sim \text{e}^{-6\hat{\unicode[STIX]{x1D70F}}}$ and ${\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}})\sim \text{e}^{6\hat{\unicode[STIX]{x1D70F}}}$ in this limit, one obtains:

(3.47) $$\begin{eqnarray}g_{I}(\hat{\unicode[STIX]{x1D70F}}\rightarrow -\infty )=\left[{\displaystyle \frac{(1+\unicode[STIX]{x1D706})(1+\hat{\unicode[STIX]{x1D6FE}}^{2})}{3(1+\unicode[STIX]{x1D6FC})\hat{\unicode[STIX]{x1D6FE}}}}\right]^{1/2}.\end{eqnarray}$$

The outlet of the surface trajectories corresponds to $\hat{\unicode[STIX]{x1D70F}}\rightarrow \infty$ , again with $C$ arbitrary. For $\hat{\unicode[STIX]{x1D70F}}\rightarrow \infty$ , $Q(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})\sim \text{e}^{6\hat{\unicode[STIX]{x1D70F}}}$ and from (3.46) we see that ${\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})$ is a function of $C$ and $\hat{\unicode[STIX]{x1D6FE}}$ alone. Thus, the boundary-layer thickness is divergent in the limit $\hat{\unicode[STIX]{x1D70F}}\rightarrow \infty$ , indicating the presence of a thermal wake.

Now, in regions (II, IV), the $C$ and $\unicode[STIX]{x1D70F}$ coordinates have the form given by (3.25). Using these transformations the integral appearing in the boundary-layer thickness equation in region II (and IV) has the form:

(3.48) $$\begin{eqnarray}{\mathcal{I}}_{II}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})=\int _{-\infty }^{\hat{\unicode[STIX]{x1D70F}}}Q_{II}({\hat{C}},\hat{\unicode[STIX]{x1D70F}}_{1};\hat{\unicode[STIX]{x1D6FE}})^{-1}\,\text{d}\hat{\unicode[STIX]{x1D70F}}_{1},\end{eqnarray}$$

where $Q_{II}({\hat{C}},\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})=[1+{\hat{C}}^{2}(\hat{\unicode[STIX]{x1D6FE}}^{2}\cosh ^{2}\hat{\unicode[STIX]{x1D70F}}+\sinh ^{2}\hat{\unicode[STIX]{x1D70F}})]^{3}$ . Using (3.48), the boundary-layer thickness expressions in region II (and IV) as:

(3.49) $$\begin{eqnarray}\displaystyle g_{II}(C,\hat{\unicode[STIX]{x1D70F}})=\left[{\displaystyle \frac{2(1+\unicode[STIX]{x1D706})(\hat{\unicode[STIX]{x1D6FE}^{2}}+1)}{(1+\unicode[STIX]{x1D6FC})\hat{\unicode[STIX]{x1D6FE}}}}Q_{II}({\hat{C}},\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}}){\mathcal{I}}_{II}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})\right]^{1/2}. & & \displaystyle\end{eqnarray}$$

Again, performing the integration with respect to $\hat{\unicode[STIX]{x1D70F}}$ in (3.48) analytically, ${\mathcal{I}}_{II}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})$ in region II (and IV) takes the form:

(3.50) $$\begin{eqnarray}\displaystyle {\mathcal{I}}_{II}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}}) & = & \displaystyle {\displaystyle \frac{1}{({\hat{C}}^{2}-1)^{3}\hat{A}^{5}}}\left[{\displaystyle \frac{3\hat{A}^{4}-2\hat{A}^{2}+3}{8}}\{\tan ^{-1}(1/\hat{A}\tanh \hat{\unicode[STIX]{x1D70F}})-\tan ^{-1}(-1/\hat{A})\}\right.\nonumber\\ \displaystyle & & \displaystyle +\,{\displaystyle \frac{\hat{A}(1-\hat{A}^{4})\tanh \hat{\unicode[STIX]{x1D70F}}}{2(\hat{A}^{2}\tanh ^{2}\hat{\unicode[STIX]{x1D70F}}+1)}}+{\displaystyle \frac{\hat{A}(1+\hat{A}^{2})^{2}(\hat{A}^{2}\tanh ^{2}\hat{\unicode[STIX]{x1D70F}}-1)\tanh \hat{\unicode[STIX]{x1D70F}}}{8(\hat{A}^{2}\tanh ^{2}\hat{\unicode[STIX]{x1D70F}}+1)^{2}}}\nonumber\\ \displaystyle & & \displaystyle \left.-{\displaystyle \frac{3\hat{A}(\hat{A}^{2}-1)}{8}}\right],\end{eqnarray}$$

where $\hat{A}=[(1+{\hat{C}}^{2}\hat{\unicode[STIX]{x1D6FE}}^{2})/({\hat{C}}^{2}-1)]^{1/2}$ . As expected, since regions (I, III) and (II, IV) share a common inlet and outlet (see figure 6), the boundary-layer thickness given by (3.49) has the finite value given by (3.47) at the inlet ( $\hat{\unicode[STIX]{x1D70F}}\rightarrow -\infty$ ). It is again divergent at the outlet, indicative of a thermal wake.

3.2.3 Nusselt number calculation

Having calculated the boundary-layer thickness as a function of $(C,\unicode[STIX]{x1D70F})$ on the drop surface, the dimensionless rate of heat transfer, the Nusselt number, is given by:

(3.51) $$\begin{eqnarray}Nu=-{\displaystyle \frac{1}{4\unicode[STIX]{x03C0}}}\int _{S}\left.{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}y}}\right|_{y=0}\,\text{d}S=-{\displaystyle \frac{Pe^{1/2}}{4\unicode[STIX]{x03C0}}}\int _{S}\left.{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}Y}}\right|_{Y=0}\,\text{d}S,\end{eqnarray}$$

where $S$ denotes the surface area of the drop. Rewriting this in terms of the similarity variable $\unicode[STIX]{x1D702}$ , we get:

(3.52) $$\begin{eqnarray}Nu=-{\displaystyle \frac{1}{4\unicode[STIX]{x03C0}}}\int _{S}\left.{\displaystyle \frac{1}{g}}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6E9}}{\text{d}\unicode[STIX]{x1D702}}}\right|_{\unicode[STIX]{x1D702}=0}\,\text{d}S={\displaystyle \frac{Pe^{1/2}}{2\unicode[STIX]{x03C0}^{3/2}}}\int _{S}{\displaystyle \frac{\text{d}S}{g(C,\unicode[STIX]{x1D70F})}}.\end{eqnarray}$$

From (3.52) $Nu$ is just proportional to the areal average of the inverse boundary-layer thickness. In $(C,\unicode[STIX]{x1D70F})$ coordinates, the differential areal element on the unit sphere is given by $\text{d}S=hk\sin \unicode[STIX]{x1D6FC}_{1}\,\text{d}C\text{d}\unicode[STIX]{x1D70F}$ , $\sin \unicode[STIX]{x1D6FC}_{1}$ accounting for the non-orthogonality. Since the boundary-layer thickness has different forms in regions (I, III) and (II, IV), $Nu$ is written as a sum of two separate integrals as:

(3.53) $$\begin{eqnarray}\displaystyle Nu & = & \displaystyle (4){\displaystyle \frac{Pe^{1/2}}{2\unicode[STIX]{x03C0}^{3/2}}}\left[\int _{S_{I}}{\displaystyle \frac{\text{d}S_{I}}{g_{I}}}+\int _{S_{II}}{\displaystyle \frac{\text{d}S_{II}}{g_{II}}}\right],\nonumber\\ \displaystyle & = & \displaystyle Nu_{I}+Nu_{II},\end{eqnarray}$$

where the prefactor of four is included, since the drop surface is divided into eight regions, four which are equivalent to region I and four to region II (figure 6 b depicts only one hemisphere); the integrals $S_{I}$ , $S_{II}$ denote integration over regions I and II, respectively. It is possible to perform the integration with respect to the $\hat{\unicode[STIX]{x1D70F}}$ variable analytically (this is typically the case, as seen from simpler examples; for instance, the case of uniform flow past a drop, see Leal (Reference Leal2007)). Considering the Nusselt number contribution from regions (I, III), and substituting for the boundary-layer thickness from (3.45), one obtains:

(3.54) $$\begin{eqnarray}\displaystyle Nu_{I} & = & \displaystyle {\displaystyle \frac{2Pe^{1/2}}{\unicode[STIX]{x03C0}^{3/2}}}\int _{S_{I}}\left[{\displaystyle \frac{(1+\unicode[STIX]{x1D6FC})\hat{\unicode[STIX]{x1D6FE}}}{2(1+\unicode[STIX]{x1D706})(\hat{\unicode[STIX]{x1D6FE}^{2}}+1)Q_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}}){\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})}}\right]^{1/2}\text{d}S_{I},\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{(2(1+\unicode[STIX]{x1D6FC})\hat{\unicode[STIX]{x1D6FE}})^{1/2}Pe^{1/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\unicode[STIX]{x1D706})^{1/2}(1+\hat{\unicode[STIX]{x1D6FE}}^{2})^{1/2}}}\int _{0}^{\infty }\int _{-\infty }^{\infty }\left[{\displaystyle \frac{1}{Q_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}}){\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})}}\right]^{1/2}hk\sin \unicode[STIX]{x1D6FC}_{1}\,\text{d}C\,\text{d}\hat{\unicode[STIX]{x1D70F}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Using (3.12), (3.13) and (3.15), the combination $hk\sin \unicode[STIX]{x1D6FC}_{1}$ in terms of $C$ and $\unicode[STIX]{x1D70F}$ is given by

(3.55) $$\begin{eqnarray}hk\sin \unicode[STIX]{x1D6FC}_{1}={\displaystyle \frac{C\hat{\unicode[STIX]{x1D6FE}}}{[1+C^{2}(\cosh ^{2}\hat{\unicode[STIX]{x1D70F}}+\hat{\unicode[STIX]{x1D6FE}}^{2}\sinh ^{2}\hat{\unicode[STIX]{x1D70F}})]^{3/2}}}={\displaystyle \frac{C\hat{\unicode[STIX]{x1D6FE}}}{Q_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})^{1/2}}}.\end{eqnarray}$$

Substituting this expression in (3.54):

(3.56) $$\begin{eqnarray}\displaystyle Nu_{I}={\displaystyle \frac{[2(1+\unicode[STIX]{x1D6FC})\hat{\unicode[STIX]{x1D6FE}}]^{1/2}Pe^{1/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\unicode[STIX]{x1D706})^{1/2}(1+\hat{\unicode[STIX]{x1D6FE}}^{2})^{1/2}}}\int _{0}^{\infty }\int _{-\infty }^{\infty }{\displaystyle \frac{1}{[{\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}})]^{1/2}Q_{I}(C,\hat{\unicode[STIX]{x1D70F}})}}\,\text{d}C\,\text{d}\hat{\unicode[STIX]{x1D70F}}, & & \displaystyle\end{eqnarray}$$

where, from (3.44) that $\unicode[STIX]{x2202}{\mathcal{I}}_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})/\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D70F}}=1/Q_{I}(C,\hat{\unicode[STIX]{x1D70F}};\hat{\unicode[STIX]{x1D6FE}})$ , the integrand together with $d\hat{\unicode[STIX]{x1D70F}}$ can be written as an exact differential. Carrying out this integration, one gets:

(3.57) $$\begin{eqnarray}\displaystyle Nu_{I}={\displaystyle \frac{2^{3/2}(1+\unicode[STIX]{x1D6FC})^{1/2}\hat{\unicode[STIX]{x1D6FE}}^{3/2}Pe^{1/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\unicode[STIX]{x1D706})^{1/2}(1+\hat{\unicode[STIX]{x1D6FE}}^{2})^{1/2}}}\int _{0}^{\infty }C[{\mathcal{I}}_{I}(C,\infty ;\hat{\unicode[STIX]{x1D6FE}})]^{1/2}\,\text{d}C, & & \displaystyle\end{eqnarray}$$

since ${\mathcal{I}}_{I}(C,-\infty ;\hat{\unicode[STIX]{x1D6FE}})=0$ from (3.44). This gives $Nu$ for a drop in a hyperbolic planar linear flow in terms of an integral over the $C$ coordinate. Physically, the form for $Nu$ implies one may first calculate the heat transfer along a given surface streamline (the $\hat{\unicode[STIX]{x1D70F}}$ -integration), and then the average across streamlines (the $C$ -integration). This ordering of the integrals along and across streamlines has the consequence that the large- $Pe$ rate of heat transfer from the drop remains unchanged to a flow reversal despite the temperature field in the reversed flow bearing no obvious symmetry relation to that in the original flow. Flow reversal would mean that the $\hat{\unicode[STIX]{x1D70F}}$ -integrals above would now run from the original outlet to the inlet, and the $C$ integral would be unchanged, thus giving the same overall heat transfer rate; in fact, the heat transfer is invariant to reversal for the individual regions (I–IV) on the drop (figure 6 b), since the distribution of boundary-layer thicknesses is unaltered. Batchelor (Reference Batchelor1979) made this observation in the context of a rigid particle, and the flow-reversal invariance has been shown by way of a general theorem by Brenner (Reference Brenner1967, Reference Brenner1970).

Substituting for ${\mathcal{I}}_{I}(C,\infty ;\hat{\unicode[STIX]{x1D6FE}})$ from (3.46), the final expression for the Nusselt number in region I is given by:

(3.58) $$\begin{eqnarray}\displaystyle & \displaystyle Nu_{I}={\displaystyle \frac{2^{1/2}(1+\unicode[STIX]{x1D6FC})^{1/2}\hat{\unicode[STIX]{x1D6FE}}^{3/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\unicode[STIX]{x1D706})^{1/2}(1+\hat{\unicode[STIX]{x1D6FE}}^{2})^{1/2}}}F_{I}(\hat{\unicode[STIX]{x1D6FE}})Pe^{1/2}, & \displaystyle\end{eqnarray}$$

(3.59) $$\begin{eqnarray}\displaystyle & \displaystyle F_{I}(\hat{\unicode[STIX]{x1D6FE}})=\int _{0}^{\infty }C\left[{\displaystyle \frac{(3A^{4}-2A^{2}+3)\tan ^{-1}(1/A)-3A(A^{2}-1)}{(C^{2}\hat{\unicode[STIX]{x1D6FE}}^{2}-1)^{3}A^{5}}}\right]^{1/2}\,\text{d}C. & \displaystyle\end{eqnarray}$$

Similarly, we find the Nusselt number for regions equivalent to region II being given by:

(3.60) $$\begin{eqnarray}\displaystyle & \displaystyle Nu_{II}={\displaystyle \frac{2^{1/2}\hat{\unicode[STIX]{x1D6FE}}^{3/2}(1+\unicode[STIX]{x1D6FC})^{1/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\unicode[STIX]{x1D706})^{1/2}(1+\hat{\unicode[STIX]{x1D6FE}}^{2})^{1/2}}}F_{II}(\hat{\unicode[STIX]{x1D6FE}})Pe^{1/2}, & \displaystyle\end{eqnarray}$$

(3.61) $$\begin{eqnarray}\displaystyle & \displaystyle F_{II}(\hat{\unicode[STIX]{x1D6FE}})=\int _{0}^{\infty }{\hat{C}}\left[{\displaystyle \frac{(3\hat{A}^{4}-2\hat{A}^{2}+3)\tan ^{-1}(1/\hat{A})-3\hat{A}(\hat{A}^{2}-1)}{({\hat{C}}^{2}-1)^{3}\hat{A}^{5}}}\right]^{1/2}\text{d}{\hat{C}}. & \displaystyle\end{eqnarray}$$

Note that the $A$ values in (3.59) and (3.61) are proportional to $(C^{2}\hat{\unicode[STIX]{x1D6FE}}^{2}-1)^{-1/2}$ and $(C^{2}-1)^{-1/2}$ , respectively, so the singularities at $C=\hat{\unicode[STIX]{x1D6FE}}^{-1}$ and $1$ in (3.59) and (3.61) are only apparent. The Nusselt number is given by the sum of $Nu_{I}$ and $Nu_{II}$ :

(3.62) $$\begin{eqnarray}Nu={\displaystyle \frac{2^{1/2}\hat{\unicode[STIX]{x1D6FE}}^{3/2}(1+\unicode[STIX]{x1D6FC})^{1/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\unicode[STIX]{x1D706})^{1/2}(1+\hat{\unicode[STIX]{x1D6FE}}^{2})^{1/2}}}(F_{I}(\hat{\unicode[STIX]{x1D6FE}})+F_{II}(\hat{\unicode[STIX]{x1D6FE}}))Pe^{1/2}={\mathcal{F}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})Pe^{1/2},\end{eqnarray}$$

where the integrals in $F_{I}(\hat{\unicode[STIX]{x1D6FE}})$ and $F_{II}(\hat{\unicode[STIX]{x1D6FE}})$ are readily evaluated numerically. One may absorb the prefactor $[(1+\unicode[STIX]{x1D6FC})/(1+\unicode[STIX]{x1D706})]^{1/2}$ into the Péclet number, so as to define an effective Péclet number, $\tilde{Pe}$ , given by $\tilde{Pe}=(1+\unicode[STIX]{x1D6FC})Pe/(1+\unicode[STIX]{x1D706})$ . Although the analysis above is not valid for elliptic linear flows ( $-1<\unicode[STIX]{x1D6FC}<0$ ), one may nevertheless note that the presence of the factor $(1+\unicode[STIX]{x1D6FC})$ is because the disturbance velocity field that convects heat away must vanish in the limit of solid-body rotation. The transport for solid-body rotation evidently remains diffusion limited; this also remains true for all elliptic linear flows, although the diffusion-limited estimate is now expected to be a function of $\unicode[STIX]{x1D6FC}$ , increasing for $\unicode[STIX]{x1D6FC}\rightarrow 0$ (see Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018)). The factor $1/(1+\unicode[STIX]{x1D706})$ in $\tilde{Pe}$ denotes the increasing inefficiency of convection for higher drop viscosities. Such a term also arises for the simpler case of a uniform flow past a drop, and allows one to define the range of $\unicode[STIX]{x1D706}$ values beyond which the current analysis breaks down. It will be seen below that, for $\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}_{c}\rightarrow \infty$ with $\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c}\sim O(1)$ , $\hat{\unicode[STIX]{x1D6FE}}\rightarrow \infty$ , and the function of $\hat{\unicode[STIX]{x1D6FE}}$ in (3.62) approaches a constant. In this limit $Nu\propto \unicode[STIX]{x1D706}^{-1/2}$ , and for a $\unicode[STIX]{x1D706}$ of $O(Pe^{1/3})$ , the $Nu$ transitions to the $Pe^{1/3}$ scaling for a solid particle, implying a breakdown of the analysis owing to the near-surface shear flow (relevant for a solid particle) becoming comparable to the surface slip velocity.

The Nusselt number expressed in terms of $\tilde{Pe}$ has a prefactor which only depends on $\hat{\unicode[STIX]{x1D6FE}}$ , and is given by:

(3.63) $$\begin{eqnarray}Nu={\displaystyle \frac{2^{1/2}\hat{\unicode[STIX]{x1D6FE}}^{3/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\hat{\unicode[STIX]{x1D6FE}}^{2})^{1/2}}}(F_{I}(\hat{\unicode[STIX]{x1D6FE}})+F_{II}(\hat{\unicode[STIX]{x1D6FE}}))\tilde{Pe}^{1/2}.\end{eqnarray}$$

Figure 7.

(a) Surface plot of ${\mathcal{F}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})=Nu/Pe^{1/2}$ as a function of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D706}$ for $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ (the open-streamline regime). Constant- $\hat{\unicode[STIX]{x1D6FE}}$ contours are shown within this region on the $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ plane with $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ corresponding to $\hat{\unicode[STIX]{x1D6FE}}=\infty$ (blue dotted curve); along a given constant- $\hat{\unicode[STIX]{x1D6FE}}$ curve, ${\mathcal{F}}$ is simply proportional to $[(1+\unicode[STIX]{x1D6FC})/(1+\unicode[STIX]{x1D706})]^{1/2}$ . (b) Contour plot of ${\mathcal{F}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})=Nu/Pe^{1/2}$ . (c) Plots of $Nu/Pe^{1/2}$ as a function of $\unicode[STIX]{x1D706}$ for different values of $\unicode[STIX]{x1D6FC}$ . Each of these curves terminates when $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ . The inset in (c) shows ${\mathcal{F}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ for larger $\unicode[STIX]{x1D706}$ values.

A plot of the $Nu/Pe^{1/2}$ surface in the $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ plane, confined to the open-streamline domain, is shown in figure 7(b). One observes a decreasing trend of the $Nu$ -prefactor with both increasing $\unicode[STIX]{x1D706}$ (figure 7 c) and ambient vorticity, with the highest rates of transport corresponding to that of a bubble ( $\unicode[STIX]{x1D706}=0$ ) in planar extension ( $\unicode[STIX]{x1D6FC}=1$ ). For planar extension, in which case $\hat{\unicode[STIX]{x1D6FE}}=1$ , the dependence of $Nu$ on $\unicode[STIX]{x1D706}$ is captured solely by the $(1+\unicode[STIX]{x1D706})^{-1/2}$ term. It will be seen later in § 3.3 that this scaling of $Nu$ with the viscosity ratio is also a feature of three-dimensional extensional flows. For other values of $\unicode[STIX]{x1D6FC}$ , the dependence on $\unicode[STIX]{x1D706}$ is more complicated since $\hat{\unicode[STIX]{x1D6FE}}$ remains a function of $\unicode[STIX]{x1D706}$ for these cases. Two special cases are discussed in what follows.

3.2.4 Special cases

1. (i) Planar extensional flow ( $\unicode[STIX]{x1D6FC}=1$ ): For this flow, the equatorial stagnation points on the drop surface lie at odd multiples of $\unicode[STIX]{x03C0}/4$ , since $\hat{\unicode[STIX]{x1D6FE}}=1$ . Further, $A=\hat{A}$ and $F_{I}=F_{II}$ , thus making regions I and II equivalent, as must be the case by symmetry. The Nusselt number is now eight times the integral over a single region (now an octant of the unit sphere). From (3.62) one obtains:

   (3.64) $$\begin{eqnarray}\displaystyle Nu={\displaystyle \frac{2^{3/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\unicode[STIX]{x1D706})^{1/2}}}F_{I}(1)Pe^{1/2}, & & \displaystyle\end{eqnarray}$$
   where
   (3.65) $$\begin{eqnarray}\displaystyle F_{I}(1)=\int _{0}^{\infty }C\left[{\displaystyle \frac{(3A^{4}-2A^{2}+3)\tan ^{-1}(1/A)-3A(A^{2}-1)}{(C^{2}-1)^{3}A^{5}}}\right]^{1/2}\,\text{d}C, & & \displaystyle\end{eqnarray}$$
   and $A=(C^{2}+1/C^{2}-1)^{1/2}$ . Evaluating the $C$ -integral numerically, one finds:
   (3.66) $$\begin{eqnarray}Nu={\displaystyle \frac{1.0380}{(1+\unicode[STIX]{x1D706})^{1/2}}}Pe^{1/2}.\end{eqnarray}$$
   In terms of the effective Péclet number $\tilde{Pe}$ defined above, $Nu$ is given by:
   (3.67) $$\begin{eqnarray}Nu=1.0380\tilde{Pe}^{1/2}.\end{eqnarray}$$
   The above agrees with the result originally derived by Polyanin (Reference Polyanin1984) for transport from a drop in a planar extensional flow; the numerical prefactor in this case was different ( $0.737$ ) since the Péclet number was defined, based on the actual component of the velocity gradient tensor ((2.2) in § 2), as $Pe_{D}=\dot{\unicode[STIX]{x1D6FE}}(1+\unicode[STIX]{x1D6FC})a^{2}/D$ .

2. (ii) $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{-}$ ( $\hat{\unicode[STIX]{x1D6FE}}\rightarrow \infty$ ): The changing geometry of the surface streamlines, as $\unicode[STIX]{x1D706}$ increases from $0$ to $\unicode[STIX]{x1D706}_{c}$ , is shown in figure 8. In the limit $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{-}$ , the streamlines on the drop surface have a nearly meridional character, and the regions II and IV are vanishingly small. This is because the pairs of stagnation points in the equatorial plane at $\unicode[STIX]{x1D719}_{1}^{(1)}$ , $\unicode[STIX]{x1D719}_{1}^{(2)}$ and $\unicode[STIX]{x1D719}_{1}^{(3)}$ , $\unicode[STIX]{x1D719}_{1}^{(4)}$ , are almost coincident, and now lie near the flow axis at $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x1D719}_{1}=-\unicode[STIX]{x03C0}/2$ . This implies, and it is straightforward to show, that $Nu_{II}\rightarrow 0$ as $\hat{\unicode[STIX]{x1D6FE}}\rightarrow \infty$ . We now estimate the $Nu$ in this limit, only accounting for contributions from regions I and III. Taking the limit $\hat{\unicode[STIX]{x1D6FE}}\rightarrow \infty$ in (3.58) and (3.59), and noting that $A\sim O(1/\hat{\unicode[STIX]{x1D6FE}})$ in this limit, the expression for $Nu$ simplifies to:

   (3.68) $$\begin{eqnarray}Nu_{I}={\displaystyle \frac{3^{1/2}}{\unicode[STIX]{x03C0}}}\left({\displaystyle \frac{1+\unicode[STIX]{x1D6FC}}{1+\unicode[STIX]{x1D706}}}\right)^{1/2}Pe^{1/2}\int _{0}^{\infty }{\displaystyle \frac{C^{1/2}\,\text{d}C}{(1+C^{2})^{5/4}}}.\end{eqnarray}$$
   Note that the dominant contribution to the $Nu$ from region I still comes from $O(1)$ values of $C$ (justified a posteriori by the convergence of the limiting integral in (3.68). Evaluating the integral over $C$ , one obtains:
   (3.69) $$\begin{eqnarray}Nu_{I}=0.66\left({\displaystyle \frac{1+\unicode[STIX]{x1D6FC}}{1+\unicode[STIX]{x1D706}}}\right)^{1/2}Pe^{1/2}.\end{eqnarray}$$
   The above matches with our numerical evaluation of $Nu$ near $\unicode[STIX]{x1D706}_{c}$ using the full expression (3.62). Note that the above analysis leads to a finite $Nu$ even as $\hat{\unicode[STIX]{x1D6FE}}\rightarrow \infty \,(\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}^{-})$ . The transition from this finite estimate to a value determined by inertia in the closed-streamline regime ( $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$ ) is examined in a companion effort (Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018)). It is shown therein that an analysis similar to that in Subramanian & Koch (Reference Subramanian and Koch2006a ) and Subramanian & Koch (Reference Subramanian and Koch2006b ) leads to $Nu\propto Re^{1/2}Pe^{1/2}$ with a prefactor that diverges when approaching the critical curve from the closed-streamline side ( $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{+}$ ). Thus, the leading-order $Nu$ -surface arising from a combination of the open- and closed-streamline analyses is singular all along the critical curve ( $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ ).

Figure 8.

Variation of the surface-streamline topology in the open-streamline regime for $\unicode[STIX]{x1D6FC}=0.5$ , and for different $\unicode[STIX]{x1D706}$ values: (a) $\unicode[STIX]{x1D706}=0$ , (b) $\unicode[STIX]{x1D706}=1.5$ and (c)  $\unicode[STIX]{x1D706}=1.99$ . As $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{-}$ , regions II and IV, bracketing the flow-vorticity plane, diminish, with the streamlines in regions I and III attaining a progressively meridional character.

3.3.1 $(C,\unicode[STIX]{x1D70F})$ coordinate system

The usual way of analysing the class of ambient flows under consideration, keeping in mind the biaxial extension for $\unicode[STIX]{x1D716}=1$ , would be to use a spherical coordinate system with its polar axis aligned with the principal axis corresponding to $E_{3}=-(1+\unicode[STIX]{x1D716})$ (see (3.70)). For $\unicode[STIX]{x1D716}=1$ , this would lead to the boundary-layer analysis familiar from textbook examples, with the surface streamlines being meridians (constant- $\unicode[STIX]{x1D719}$ contours), and the convection being along $\unicode[STIX]{x1D703}$ (the polar angle). The meridional streamlines resulting from the above choice of polar axis do not lend themselves to a $(C,\unicode[STIX]{x1D70F})$ characterization, however. In order to have both $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D70F}$ vary along a streamline (leading to an invertible transformation between the two) even when $\unicode[STIX]{x1D716}=1$ (see (3.77)), the polar axis needs to be chosen orthogonal to the symmetry axis of the biaxial extension, so the surface streamlines begin and end at distinct $\unicode[STIX]{x1D719}$ locations (in turn corresponding to different $\unicode[STIX]{x1D70F}$ locations). This merely involves a rearrangement of the principal axes of the rate-of-strain tensor such that the new rate-of-strain tensor is given by:

(3.71) $$\begin{eqnarray}\unicode[STIX]{x1D640}=\left[\begin{array}{@{}ccc@{}}-(1+\unicode[STIX]{x1D716}) & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & \unicode[STIX]{x1D716}\end{array}\right].\end{eqnarray}$$

The velocity fields in the drop exterior, expressed in the spherical coordinate system (with $x_{3}$ as the polar axis), are given by:

(3.72) $$\begin{eqnarray}\displaystyle & \displaystyle u_{r}=r[A(r)r^{2}+B(r)][\unicode[STIX]{x1D716}\cos ^{2}\unicode[STIX]{x1D703}-{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D716}+(2+\unicode[STIX]{x1D716})\cos 2\unicode[STIX]{x1D719})\sin ^{2}\unicode[STIX]{x1D703}], & \displaystyle\end{eqnarray}$$

(3.73) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\unicode[STIX]{x1D703}}=-{\textstyle \frac{1}{4}}B(r)r[3\unicode[STIX]{x1D716}+(2+\unicode[STIX]{x1D716})\cos 2\unicode[STIX]{x1D719}]\sin 2\unicode[STIX]{x1D703}, & \displaystyle\end{eqnarray}$$

(3.74) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\unicode[STIX]{x1D719}}=B(r)(2+\unicode[STIX]{x1D716})r\sin \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

where $A(r)$ and $B(r)$ have the same definition as in the planar linear flow case (see § 2). On the drop surface $u_{r}=0$ , while $u_{\unicode[STIX]{x1D703}}$ and $u_{\unicode[STIX]{x1D719}}$ reduce to:

(3.75) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\unicode[STIX]{x1D703}}|_{r=1}=-{\displaystyle \frac{1}{4(1+\unicode[STIX]{x1D706})}}[3\unicode[STIX]{x1D716}+(2+\unicode[STIX]{x1D716})\cos 2\unicode[STIX]{x1D719}]\sin 2\unicode[STIX]{x1D703}={\displaystyle \frac{\text{d}\unicode[STIX]{x1D703}}{\text{d}s}}, & \displaystyle\end{eqnarray}$$

(3.76) $$\begin{eqnarray}\displaystyle & \displaystyle u_{\unicode[STIX]{x1D719}}|_{r=1}={\displaystyle \frac{1}{(1+\unicode[STIX]{x1D706})}}(2+\unicode[STIX]{x1D716})\sin \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D719}=\sin \unicode[STIX]{x1D703}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D719}}{\text{d}s}}, & \displaystyle\end{eqnarray}$$

where $s$ is the dimensionless time coordinate (now measured in units of $E_{1}^{-1}$ ). Proceeding as for the planar linear flow case, we integrate (3.75) and (3.76) in order to obtain the streamline equations. From (3.76) we get:

(3.77) $$\begin{eqnarray}\displaystyle & \displaystyle \tan \unicode[STIX]{x1D719}=\exp (\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

(3.78) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}={\displaystyle \frac{(2+\unicode[STIX]{x1D716})s}{1+\unicode[STIX]{x1D706}}}. & \displaystyle\end{eqnarray}$$

Similarly, integrating (3.75), we get:

(3.79) $$\begin{eqnarray}\tan \unicode[STIX]{x1D703}=C(1+\exp 2\unicode[STIX]{x1D70F})^{1/2}\exp \left[-{\displaystyle \frac{(1+2\unicode[STIX]{x1D716})\unicode[STIX]{x1D70F}}{2+\unicode[STIX]{x1D716}}}\right],\end{eqnarray}$$

where $C$ , the constant of integration, again acts as a surface streamline label. Equations (3.77) and (3.79) constitute the $(C,\unicode[STIX]{x1D70F})$ equations for a drop in a three-dimensional extensional flow. The metric factors and skewness angle of this coordinate system are still given by (3.12), (3.13) and (3.15), respectively, while the unit vectors in the $C$ and $\unicode[STIX]{x1D70F}$ directions are given by (3.16) and (3.17), respectively. Note that $\hat{\boldsymbol{C}}$ still points along the direction of the polar angle, and this arises from the azimuthal angle in (3.77) being only a function of $\unicode[STIX]{x1D70F}$ (as for the planar linear flow case).

Figure 9.

Surface streamline plots for drop in an extensional flow (a) planar extensional flow $E_{2}=0$ , $E_{1}\neq 0$ (plane of the flow is the ( $x_{2},x_{3}$ ) plane), (b) general three-dimensional extensional flow $E_{1}\neq E_{2}$ and (c) axisymmetric (about the $x_{1}$ axis) extensional flow with $E_{1}=E_{2}$ .

A physical picture of the new $(C,\unicode[STIX]{x1D70F})$ coordinate system and its mapping to spherical coordinates follows. As before, the surface streamlines begin and terminate at stagnation points. Four of these stagnation points are in the equatorial plane ( $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ ) at $\unicode[STIX]{x1D719}=0,\unicode[STIX]{x03C0}/2,\unicode[STIX]{x03C0},3\unicode[STIX]{x03C0}/2$ while the remaining two lie on the vorticity axis at $\unicode[STIX]{x1D703}=0,\unicode[STIX]{x03C0}$ , as seen from (3.75) and (3.76). The drop surface is divided into eight equivalent octants, and for purposes of the analysis, it is sufficient to confine our attention to one of these; three of the stagnation points, at the vertices of a single octant, are shown in figure 9(b). Note that this division into eight octants is no longer true for axisymmetric extension ( $\unicode[STIX]{x1D716}=1$ ), in which case there are a pair of identical hemispheres separated by a stagnation ring lying on the $(x_{2},x_{3})$ plane. This flow may nevertheless be analysed as a limit of the general non-axisymmetric case. The other limiting flow corresponds to the planar extension already analysed above, the only difference being that the plane of this flow is now the $(x_{1},x_{3})$ plane (see figure 9 a). For the general case, the stagnation points at $\unicode[STIX]{x1D719}=0,\unicode[STIX]{x03C0}$ act as inlets while those at $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2,3\unicode[STIX]{x03C0}/2$ are outlets (see figure 9 b). The mapping of the $\unicode[STIX]{x1D70F}$ coordinate is apparent from (3.77), with $\unicode[STIX]{x1D70F}\rightarrow -\infty$ and $\infty$ at the inlet and outlet points, respectively. The streamlines are smooth curves for all non-zero $C$ . One of the limiting streamlines, $C=0$ , is discontinuous consisting of the arcs $\unicode[STIX]{x1D719}=0$ and $\unicode[STIX]{x03C0}/2$ , with $\unicode[STIX]{x1D703}$ arbitrary, meeting at right angles at $\unicode[STIX]{x1D703}=0$ . The other limiting streamline, $C\rightarrow \infty$ , is an arc lies in the equatorial plane ( $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ ). These mappings are depicted in figure 9(b).

3.3.2 Boundary-layer analysis

The boundary-layer analysis for this case closely follows the one presented earlier for planar linear flows. The convection–diffusion equation in $(C,\unicode[STIX]{x1D70F})$ coordinates is given by (3.26), and as before, $u_{C}$ is identically zero. In the limit $Pe\gg 1$ , the surface velocity field again contributes to the leading-order heat transfer. Expanding the radial and tangential velocities expressed in $(C,\unicode[STIX]{x1D70F})$ coordinates in terms of $y=r-1$ , one obtains:

(3.80) $$\begin{eqnarray}\displaystyle u_{r} & = & \displaystyle \left\{{\displaystyle \frac{3\left[C^{2}((\exp (2\unicode[STIX]{x1D70F})-1)-\unicode[STIX]{x1D716})+\exp \left[{\displaystyle \frac{2(1+2\unicode[STIX]{x1D716})\unicode[STIX]{x1D70F}}{2+\unicode[STIX]{x1D716}}}\right]\unicode[STIX]{x1D716}\right]}{\left[\exp \left[{\displaystyle \frac{2(1+2\unicode[STIX]{x1D716})\unicode[STIX]{x1D70F}}{2+\unicode[STIX]{x1D716}}}\right]+C^{2}(1+\exp (2\unicode[STIX]{x1D70F}))\right](1+\unicode[STIX]{x1D706})}}\right\}y+O(y^{2}),\end{eqnarray}$$

(3.81) $$\begin{eqnarray}\displaystyle & = & \displaystyle h_{r}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D716},\unicode[STIX]{x1D706})y+O(y^{2}),\end{eqnarray}$$

(3.82) $$\begin{eqnarray}\displaystyle u_{\unicode[STIX]{x1D70F}}/k & = & \displaystyle {\displaystyle \frac{2+\unicode[STIX]{x1D716}}{1+\unicode[STIX]{x1D706}}}+O(y),\end{eqnarray}$$

(3.83) $$\begin{eqnarray}\displaystyle & = & \displaystyle h_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D716},\unicode[STIX]{x1D706}).\end{eqnarray}$$

Note that the convective term in the $\unicode[STIX]{x1D70F}$ direction is independent of $C$ and $\unicode[STIX]{x1D70F}$ ; again mimicking the much simpler case of a solid particle in a planar linear flow with vorticity. Substituting the above in the convection–diffusion equation and defining a scaled boundary layer coordinate we find, as earlier, $Y=Pe^{1/2}y$ . Defining a similarity variable $\unicode[STIX]{x1D702}=Y/g(C,\unicode[STIX]{x1D70F})$ , where $g(C,\unicode[STIX]{x1D70F})$ denotes the boundary-layer thickness, the non-dimensional temperature satisfies (3.34) and the boundary conditions are given by (4.3) and (4.4). The non-dimensional temperature field is therefore given by (B 13). The boundary-layer thickness satisfies (3.35) with $h_{r}$ and $h_{\unicode[STIX]{x1D70F}}$ now being given by (3.81) and (3.83). Proceeding as before, the final expression for the boundary-layer thickness is given by:

(3.84) $$\begin{eqnarray}g(C,\unicode[STIX]{x1D70F})=\left[{\displaystyle \frac{4(1+\unicode[STIX]{x1D706})}{2+\unicode[STIX]{x1D716}}}Q(C,\unicode[STIX]{x1D70F}){\mathcal{I}}(C,\unicode[STIX]{x1D70F})\right]^{1/2},\end{eqnarray}$$

where

(3.85) $$\begin{eqnarray}Q(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D716})=\exp \left[-{\displaystyle \frac{6(1+\unicode[STIX]{x1D716})\unicode[STIX]{x1D70F}}{2+\unicode[STIX]{x1D716}}}\right]\left\{\exp \left[{\displaystyle \frac{2(1+2\unicode[STIX]{x1D716})\unicode[STIX]{x1D70F}}{2+\unicode[STIX]{x1D716}}}\right]+(1+\exp (2\unicode[STIX]{x1D70F}))C^{2}\right\}^{3},\end{eqnarray}$$

and

(3.86) $$\begin{eqnarray}{\mathcal{I}}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D716})=\int _{-\infty }^{\unicode[STIX]{x1D70F}}Q(C,\unicode[STIX]{x1D70F}_{1};\unicode[STIX]{x1D716})^{-1}\,\text{d}\unicode[STIX]{x1D70F}_{1}.\end{eqnarray}$$

Interestingly, the boundary-layer thickness profile has a form similar to that in the planar linear flow case (see (3.45)). Just as in the previous case, it is straightforward to see that the boundary-layer thickness is finite at the inlet ( $\unicode[STIX]{x1D70F}\rightarrow -\infty$ ). This is because $Q(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D716})\sim \exp [-(6(1+\unicode[STIX]{x1D716})\unicode[STIX]{x1D70F})/2+\unicode[STIX]{x1D716}]$ as $\unicode[STIX]{x1D70F}\rightarrow -\infty$ and ${\mathcal{I}}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D716})\sim \exp [(6(1+\unicode[STIX]{x1D716})\unicode[STIX]{x1D70F})/2+\unicode[STIX]{x1D716}]$ , so that their product is finite. Likewise, $g(C,\unicode[STIX]{x1D70F})$ diverges for $\unicode[STIX]{x1D70F}\rightarrow \infty$ , indicative of a thermal wake.

3.3.3 Nusselt number calculation

The dimensionless rate of heat transfer is given by:

(3.87) $$\begin{eqnarray}\displaystyle Nu=(8){\displaystyle \frac{Pe^{1/2}}{2\unicode[STIX]{x03C0}^{3/2}}}\int _{0}^{\infty }\int _{-\infty }^{\infty }{\displaystyle \frac{hk\sin \unicode[STIX]{x1D6FC}_{1}}{g(C,\unicode[STIX]{x1D70F})}}\,\text{d}C\,\text{d}\unicode[STIX]{x1D70F}, & & \displaystyle\end{eqnarray}$$

where the prefactor appears since the total drop surface consists of eight equivalent octants for $\unicode[STIX]{x1D716}<1$ (figure 9 b depicts one of these). The combination $hk\sin \unicode[STIX]{x1D6FC}_{1}$ for this case is given by:

(3.88) $$\begin{eqnarray}\displaystyle hk\sin \unicode[STIX]{x1D6FC}_{1} & = & \displaystyle {\displaystyle \frac{C\exp \left[{\displaystyle \frac{3(E_{1}+E_{2})\unicode[STIX]{x1D70F}}{2E_{1}+E_{2}}}\right]}{\left\{\exp \left[{\displaystyle \frac{2(E_{1}+2E_{2})\unicode[STIX]{x1D70F}}{2E_{1}+E_{2}}}\right]+C^{2}(1+\exp (2\unicode[STIX]{x1D70F}))\right\}^{3/2}}},\end{eqnarray}$$

(3.89) $$\begin{eqnarray}\displaystyle & = & \displaystyle CQ(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D716})^{-1}.\end{eqnarray}$$

Substituting from (3.84) and (3.89) in (3.87), the integration with respect to $\unicode[STIX]{x1D70F}$ is in the form of an exact differential. Since ${\mathcal{I}}(C,-\infty ;\unicode[STIX]{x1D716})=0$ , one obtains:

(3.90) $$\begin{eqnarray}\displaystyle Nu=Pe^{1/2}{\displaystyle \frac{4(2+\unicode[STIX]{x1D716})^{1/2}}{\unicode[STIX]{x03C0}^{3/2}(1+\unicode[STIX]{x1D706})^{1/2}}}\int _{0}^{\infty }C[{\mathcal{I}}(C,\infty ;\unicode[STIX]{x1D716})]^{1/2}\,\text{d}C={\mathcal{G}}(\unicode[STIX]{x1D716},\unicode[STIX]{x1D706})Pe^{1/2}, & & \displaystyle\end{eqnarray}$$

where ${\mathcal{I}}(C,\unicode[STIX]{x1D70F};\unicode[STIX]{x1D716})$ is given by (3.86). Thus, we have the expression for the Nusselt number in terms of a two-dimensional integral which can be readily carried out numerically. The above expression can be rewritten by defining an effective Péclet number:

(3.91) $$\begin{eqnarray}\tilde{Pe}={\displaystyle \frac{(2+\unicode[STIX]{x1D716})Pe}{(1+\unicode[STIX]{x1D706})}},\end{eqnarray}$$

whence one obtains:

(3.92) $$\begin{eqnarray}\displaystyle Nu={\displaystyle \frac{4}{\unicode[STIX]{x03C0}^{3/2}}}\tilde{Pe}^{1/2}\int _{0}^{\infty }C[{\mathcal{I}}(C,\infty ;\unicode[STIX]{x1D716})]^{1/2}\,\text{d}C, & & \displaystyle\end{eqnarray}$$

with the prefactor in (3.92) only a function of $\unicode[STIX]{x1D716}$ . This prefactor is plotted in figure 10 as a function $\unicode[STIX]{x1D716}$ , for $\unicode[STIX]{x1D716}$ varying between $0$ and $1$ , where the endpoints correspond to planar extension and axisymmetric extensional flow ( $x_{1}$ being the axis of symmetry).

Figure 10.

Plot of the Nusselt number prefactor given in (3.92) as a function of $\unicode[STIX]{x1D716}$ for $0\leqslant \unicode[STIX]{x1D716}\leqslant 1$ . The endpoints correspond to planar extensional flow ( $\unicode[STIX]{x1D716}=0$ ) and axisymmetric extensional flow ( $\unicode[STIX]{x1D716}=1$ ).

3.3.4 Special cases

1. (i) Axisymmetric extensional flow $\unicode[STIX]{x1D716}=1$ : For this case, ${\mathcal{I}}(C,\unicode[STIX]{x1D70F})$ reduces to:

   (3.93) $$\begin{eqnarray}{\mathcal{I}}(C,\unicode[STIX]{x1D70F})=\int _{-\infty }^{\unicode[STIX]{x1D70F}}{\displaystyle \frac{\exp (4\unicode[STIX]{x1D70F}_{1})\,\text{d}\unicode[STIX]{x1D70F}_{1}}{[\exp (2\unicode[STIX]{x1D70F}_{1})+(1+\exp (2\unicode[STIX]{x1D70F}_{1}))C^{2}]^{3}}},\end{eqnarray}$$
   which can be integrated to yield:
   (3.94) $$\begin{eqnarray}{\mathcal{I}}(C,\unicode[STIX]{x1D70F})={\displaystyle \frac{\exp (4\unicode[STIX]{x1D70F})}{4C^{2}[C^{2}+(1+C^{2})\exp (2\unicode[STIX]{x1D70F})]^{2}}},\end{eqnarray}$$
   and
   (3.95) $$\begin{eqnarray}{\mathcal{I}}(C,\infty )={\displaystyle \frac{1}{4C^{2}(1+C^{2})^{2}}}.\end{eqnarray}$$
   The rather simple expression for ${\mathcal{I}}(C,\infty )$ allows an entirely analytical estimate of the Nusselt number. Substituting this in (3.90), and evaluating the resulting integral with respect to $C$ yields:
   (3.96) $$\begin{eqnarray}Nu=\left[{\displaystyle \frac{3}{\unicode[STIX]{x03C0}(1+\unicode[STIX]{x1D706})}}\right]^{1/2}Pe^{1/2}.\end{eqnarray}$$
   The above agrees exactly with the expression derived by Gupalo & Riazantsev (Reference Gupalo and Riazantsev1972), serving as a validation for our approach. In terms of the effective Péclet number defined earlier in the section:
   (3.97) $$\begin{eqnarray}Nu=\left({\displaystyle \frac{3}{\unicode[STIX]{x03C0}}}\right)^{1/2}\tilde{Pe}^{1/2}.\end{eqnarray}$$

2. (ii) Planar extensional flow $\unicode[STIX]{x1D716}=0$ : The Nusselt number was derived for this flow in § 3.2 (the case $\unicode[STIX]{x1D6FC}=1$ ), the only difference here being the plane of the ambient flow which is the $(x_{1},x_{3})$ plane. Evaluating the integral in (3.90) numerically, for this case, we find:

   (3.98) $$\begin{eqnarray}Nu={\displaystyle \frac{0.734}{(1+\unicode[STIX]{x1D706})^{1/2}}}Pe^{1/2}.\end{eqnarray}$$
   The equivalence with the result given in the earlier section on planar linear flows can be seen by noting that $E_{1}=\dot{\unicode[STIX]{x1D6FE}}(1+\unicode[STIX]{x1D6FC})$ .