2.1 Governing equations

The governing equations in dimensionless form for the fluid motion outside the drop, in the presence of inertia, are given by:

where $\boldsymbol{u}$ is the velocity field and $p$ is the pressure, with $Re\ll 1$ . The inertial velocity field in the drop exterior, which will be used in the subsequent heat transfer analysis, was derived by Raja, Subramanian & Koch (Reference Raja, Subramanian and Koch2010). To $O(Re)$ , one may use a regular perturbation analysis, and the exterior velocity field around the spherical drop may be written as:

For a planar linear flow, the Stokes field $\boldsymbol{u}^{(0)}$ is well known (Leal Reference Leal2007). The complete expressions for the components of the $O(Re)$ velocity field, $\boldsymbol{u}^{(1)}$ , are given in Raja et al. (Reference Raja, Subramanian and Koch2010). Although only valid for distances from the drop smaller than the inertial screening length of $O(Re^{-1/2})$ , it evidently suffices for purposes of the large- $Pe$ boundary layer analysis which only involves the exterior field in the immediate vicinity of the drop. Consideration of the velocity disturbance field on length scales larger than $O(Re^{-1/2})$ requires a matched asymptotic expansions approach (Subramanian et al. Reference Subramanian, Koch, Zhang and Yang2011b ). Here, the limit of small but finite $Re$ is examined when in addition, $RePe\gg 1$ ; this latter limit implies that convection due to the finite- $Re$ spiralling flow dominates diffusion everywhere except in a thin thermal boundary layer near the drop surface and in a thermal wake. The wake does not contribute to the transport at leading order, however.

The scalar transport is governed by the convection–diffusion equation, written here in non-dimensional form:

where $\unicode[STIX]{x1D6E9}=(T-T_{\infty })/(T_{0}-T_{\infty })$ , where $T$ is the temperature field. This is subject to the boundary conditions:

Here, we have neglected variations of transport properties, including those of the density and specific heat, with temperature, yielding a forced convection problem. It is worth noting that, in the absence of a body force (such as gravity), this neglect does not immediately alter the streamline topology. In other words, weakly temperature-dependent transport coefficients should preserve the closed streamline topology, although quantitative details such as the critical viscosity ratio demarcating the open and closed streamline regimes will, of course, differ. The condition of isothermality at the drop surface implies neglect of the interior phase resistance, and we have commented in some detail on its validity in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018). An important point is the likelihood of inertia-induced Lagrangian chaos for the interior flow. The resulting efficient chaotic mixing within the drop implies that the transport for large $Pe$ likely occurs through both interior and exterior boundary layers, and the $Nu$ expression derived below (see (2.94) and (2.95)), for the case of a dominant exterior phase resistance, may be readily generalized to the case of comparable phase resistances by multiplication with $(k_{i}/k)^{1/2}/[(k_{i}/k)^{1/2}+1]$ for heat transfer ( $k_{i}$ being the drop phase thermal conductivity), and with $(D_{i}/D)^{1/2}/[(D_{i}/D)^{1/2}H+1]$ for the mass transfer scenario ( $D_{i}$ being the drop phase mass diffusivity and $H$ the distribution coefficient).

The assumption of a spherical drop is also implicit in the surface boundary condition above. Although the available experimental (Torza et al. Reference Torza, Henry, Cox and Mason1971) and computational evidence (Kennedy, Pozrikidis & Skalak Reference Kennedy, Pozrikidis and Skalak1994) suggest that the effects of finite deformation will not qualitatively alter the nature of the $Nu$ -function (see Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018), the analytical expressions for the $O(Ca)$ velocity field obtained originally by Barthes-Biesel & Acrivos (Reference Barthes-Biesel and Acrivos1973), and in a recent effort (Greco Reference Greco2002), suggest otherwise. We further comment on the role of drop deformation in the conclusions section. Having set up the governing equations, we now describe the coordinate system used in the analysis.

2.2 The $(C,\unicode[STIX]{x1D70F})$ coordinate system in the closed streamline regime

As shown in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018), the surface streamlines for a drop in a planar linear flow are generalized Jeffery orbits. In the closed streamline regime, as expected, the surface streamlines are true orbits with a real-valued effective aspect ratio, albeit one that is not a geometrically determined quantity (as was the case for the original axisymmetric particle; see Leal & Hinch Reference Leal and Hinch1971), but instead depends on the parameters $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D706}$ . This insight allowed one to derive a flow-aligned, non-orthogonal coordinate system, henceforth termed the $(C,\unicode[STIX]{x1D70F})$ coordinate system, to describe the surface streamlines. From integrating the streamline equations for an ambient planar linear flow, one obtains the defining relations for the coordinates $C$ and $\unicode[STIX]{x1D70F}$ as:

where $\unicode[STIX]{x1D70F}=(-2(1-\unicode[STIX]{x1D6FC})s/\unicode[STIX]{x1D6FE}+1/\unicode[STIX]{x1D6FE})$ and $\unicode[STIX]{x1D719}_{1}$ is the azimuthal angle measured from the $x_{2}$ axis (see figure 2 a). Here, $\unicode[STIX]{x1D6FE}$ is the effective aspect ratio mentioned above, and that depends on $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D706}$ as:

where $\unicode[STIX]{x1D6FD}=(1-\unicode[STIX]{x1D6FC})/(1+\unicode[STIX]{x1D6FC})$ and $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$ .

As also seen in figure 2(a), the lines of constant $C$ and $\unicode[STIX]{x1D70F}$ are in general not orthogonal. The $C$ coordinate defines the orbit (closed streamline) one is on, while $\unicode[STIX]{x1D70F}$ gives the phase along the particular orbit. The metrics of this coordinate system in the $C$ and $\unicode[STIX]{x1D70F}$ directions are given by $h=g_{CC}=\unicode[STIX]{x1D703}_{C}$ and $k=g_{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D70F}}=(\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}^{2}+\sin ^{2}\unicode[STIX]{x1D703}\unicode[STIX]{x1D719}_{1\unicode[STIX]{x1D70F}}^{2})^{1/2}$ , respectively, and the skewness angle between the coordinate unit vectors is given by $\sin \unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x1D719}_{1\unicode[STIX]{x1D70F}}\sin \unicode[STIX]{x1D703}(\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}^{2}+\sin ^{2}\unicode[STIX]{x1D703}\unicode[STIX]{x1D719}_{1\unicode[STIX]{x1D70F}}^{2})^{-1/2}$ (Leal & Hinch Reference Leal and Hinch1971). Here, $\unicode[STIX]{x1D703}_{C},\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ etc., denote derivatives with respect to the subscripted variable, $g_{CC}$ and $g_{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D70F}}$ being the metric tensor components as per conventional notation (Aris Reference Aris2012). The contravariant unit vectors (those along the coordinate lines) in terms of the conventional spherical coordinates unit vectors are given by:

where $\hat{\unicode[STIX]{x1D753}}_{1}=\cos \unicode[STIX]{x1D719}_{1}\hat{\boldsymbol{x}}_{1}-\sin \unicode[STIX]{x1D719}_{1}\hat{\boldsymbol{x}}_{2}$ and is the unit vector in the $\hat{\unicode[STIX]{x1D753}}_{1}$ direction where $\unicode[STIX]{x1D719}_{1}$ is measured in a clockwise sense from $x_{2}$ axis. Note that $\hat{\boldsymbol{C}}$ is still along the meridian, but $\hat{\unicode[STIX]{x1D749}}$ is along the surface streamline, rather than along the azimuth, which leads to the non-orthogonality (figure 2 a). As shown in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018), a similar characterization of surface streamlines exists in the open streamline regime with $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ , but the effective aspect ratio is now purely imaginary, as may be seen from (2.9).

2.3 The approach to analysing inertial convection for $Re\ll 1$

Recasting the finite-inertia transport problem in terms of the $(C,\unicode[STIX]{x1D70F})$ coordinate system leads to a significant simplification of the problem, making an analytical solution possible. The surface streamlines are Jeffery orbits for $Re=0$ (see figure 3 a), and the near-surface closed streamlines (relevant to the convection within the thermal boundary layer) approximately so. The effect of finite inertia, by way of finite- $Re$ spiralling streamlines, is to cause a net convective drift across Jeffery (or constant- $C$ ) orbits (figure 3 b). A parallel can be drawn with the case of a solid particle where the inertial drift arises from the finite- $Re$ streamlines spiralling across the near-field circular Stokesian streamlines (figure 3 c; see Subramanian & Koch Reference Subramanian and Koch2006a ,Reference Subramanian and Koch b ). Other sources of nonlinearity have a similar effect; for instance, viscoelasticity at $O(De)$ is known to lead to a drift across closed Stokesian streamlines, $De$ here being the Deborah number, and the driving force being the divergence of the polymeric stress (see Subramanian & Koch Reference Subramanian and Koch2007).

The convection–diffusion equation expressed in the $(r,C,\unicode[STIX]{x1D70F})$ coordinate system is (Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018):

Here, the $(C,\unicode[STIX]{x1D70F})$ components of the velocity field, using (2.10) and (2.11), can be written in terms of the $(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})$ components, as:

with $u_{C}$ , of course, being identically zero at leading order since the unit vector $\hat{\unicode[STIX]{x1D749}}$ is tangential to the Stokes surface streamlines at all points (see Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018).

Since it is the transport in a thin thermal boundary layer adjacent to the drop that is of interest, it is the near-field form of the velocity field which enters the analysis. The near-field forms of the Stokes velocity components, to the relevant order, are given by:

and the $O(Re)$ components have the forms:

Here, $y=r-1$ and is expected to be of the order of the boundary layer thickness at large $Pe$ . The expressions for $h_{r}^{(1)}$ , $h_{C}^{(1)}$ and $h_{\unicode[STIX]{x1D70F}}^{(1)}$ , which are functions of $C$ and $\unicode[STIX]{x1D70F}$ with an additional parametric dependence on $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D706}$ , are quite lengthy, and are given in appendix B. It is important to note that the radial velocity is $O(y)$ , while the tangential velocities are $O(1)$ near the surface. Further, the combination $u_{\unicode[STIX]{x1D70F}}^{(0)}/k$ is independent of both $C$ and $\unicode[STIX]{x1D70F}$ . While the latter is analogous to $u_{\unicode[STIX]{x1D719}}/\sin \unicode[STIX]{x1D703}$ for a rotating solid particle which is independent of both $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D719}$  (in spherical polar coordinates), being equal to the angular velocity (Subramanian & Koch Reference Subramanian and Koch2006b ), the no-slip boundary condition implies that $u_{r}$ and $u_{\unicode[STIX]{x1D703}}$ for a solid particle are $O(y^{2})$ and $O(y)$ , respectively. The differing near-field scalings for the drop and solid particle leads to crucial differences in the large- $Pe$ boundary layer analysis below.

From figure 3(b) one sees that the inertial streamlines are tightly wound spirals for small $Re$ , with each turn closely resembling an inertialess Jeffery orbit, and successive turns separated by $O(Re)$ . The rapid convection due to the Stokes velocity along orbits compared to the slow $O(Re)$ drift across orbits implies that, for purposes of the heat transfer analysis, the inertial convection may be interpreted in a Jeffery-orbit-averaged sense, being given by the $\unicode[STIX]{x1D70F}$ average of the $O(Re)$ velocity field along a Jeffery orbit. This is analogous to the much simpler $\unicode[STIX]{x1D719}$ -average used for a solid particle by Subramanian & Koch (Reference Subramanian and Koch2006b ). Using this estimate for the inertial convection, one may obtain the scale of the boundary layer thickness in the limit $RePe\gg 1$ . The time scale associated with the $O(Re\dot{\unicode[STIX]{x1D6FE}}a)$ $\unicode[STIX]{x1D70F}$ -averaged tangential convection due to inertia is $O[a/(Rea\dot{\unicode[STIX]{x1D6FE}})]=O(\dot{\unicode[STIX]{x1D6FE}}Re^{-1})$ , while the diffusive time scale is $O(y^{2}/D)$ , where $y$ is a scale for the boundary layer thickness set up by the inertial convection. Equating these time scales, one obtains $y\sim O(RePe)^{-1/2}$ ; in turn, this leads to $Nu\propto O(RePe)^{1/2}$ for $RePe\gg 1$ . A Jeffery-orbit-averaged drift interpretation in the above sense has recently been used to characterize the inertial orientation dynamics of anisotropic particles in linear shearing flows (Dabade, Marath & Subramanian Reference Dabade, Marath and Subramanian2016; Marath & Subramanian Reference Marath and Subramanian2018).

As we argue in appendix A, for large $Pe$ and $Re=0$ , a shear-enhanced diffusion rapidly isothermalizes the closed streamlines on a time scale of $O(Pe^{1/3}\dot{\unicode[STIX]{x1D6FE}}^{-1})$ , starting from an arbitrary initial temperature field; diffusion acting alone would need a much longer time of $O(a^{2}/D)=O(Pe\dot{\unicode[STIX]{x1D6FE}}^{-1})$ . The role of such an isothermalizing mechanism has also been discussed earlier in the context of both passive (Rhines & Young Reference Rhines and Young1983) and active scalars, an example of the latter being vorticity (Bassom & Gilbert Reference Bassom and Gilbert1998, Reference Bassom and Gilbert1999); in the latter case, shear-enhanced diffusion leads to the streamlines becoming coincident with iso-vorticity contours. For sufficient small $Re$ , each turn of a finite- $Re$ spiralling streamline must therefore become nearly isothermal due to shear-enhanced diffusion. The $\unicode[STIX]{x1D70F}$ -averaging process in the earlier paragraph is effectively along the Stokesian isotherms, and helps isolate the temperature variation caused by the inertial drift. For the solid particle, these isotherms are at a constant radial distance at leading order, and the averaging process that leads to the inertial drift needs only be along the azimuth (see Subramanian & Koch Reference Subramanian and Koch2006b ). On the other hand, although the projections of the near-field Stokesian isotherms onto the drop surface are Jeffery orbits (at a unit distance from the drop centre), the $\unicode[STIX]{x1D70F}$ -averaging process must nevertheless account for the variations in the radial distance at leading order. This is because, unlike a solid particle, the radial distance of a near-field streamline for a drop changes by an amount of order the boundary layer thickness ( $O(y)\sim O(RePe)^{-1/2}$ ), and over which the dimensionless temperature field ( $\unicode[STIX]{x1D6E9}$ ) varies by order unity.

Figure 4.

Projections of streamsurfaces of the external flow for the case of a drop in a linear flow with $\unicode[STIX]{x1D6FC}=0.2$ , $\unicode[STIX]{x1D706}=2$ . (a) Projections of the constant- $D$ surfaces on the ( $x_{1},x_{3}$ ) plane. These surfaces are surfaces of revolution about the $x_{3}$ axis as depicted by the curved arrows. (b) Projection of the near-field constant- $E$ surfaces on the ( $x_{1},x_{2}$ ) plane. These are surfaces of revolution about the $x_{2}$ axis as depicted by the curved arrows.

Thus, in the case of a drop, one needs to define a new radial coordinate which takes into account the radial displacements of the inertialess closed streamlines, displacements that, for the streamlines in the thermal boundary layer, are of the same order as the $O(RePe)^{-1/2}$ boundary layer thickness. Instead of spheres, as was the case for the solid particle and the drop in the open streamline regime (Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018), the streamsurfaces of the exterior Stokes flow serve as natural candidates for the constant coordinate surfaces, in the radial direction, in the closed streamline regime. These surfaces were originally identified by Cox et al. (Reference Cox, Zia and Mason1968) for a solid sphere in a simple shear flow, and were analysed in some detail for the case of a drop in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018). The streamsurfaces form two one-parameter families. The ones exterior to the drop, denoted as the constant- $D$ and $E$ surfaces therein, are surfaces of revolution about the $x_{3}$ and $x_{2}$ axes, respectively. It turns out that the drop surface (with or without the plane of symmetry $x_{3}=0$ ) is itself a limiting streamsurface, corresponding to $D=0,E=0$ , for both families. Apart from indicating a degeneracy (near tangency) of the two families in the limit $r\rightarrow 1$ , this also leads to two possible choices for the (generalized) radial coordinate to be used in the boundary layer analysis. We choose the constant- $E$ surfaces (figure 4 b) to formulate the boundary layer analysis below. This is because these intersect transversally with the plane of symmetry, the curves of intersection being the in-plane Stokesian streamlines. In contrast, the constant- $D$ surfaces eventually become parallel to the plane of symmetry, approaching it at infinity for any $D>0$ (figure 4 a). Thus, although the near-field constant- $D$ surfaces would have their normal in the radial direction over most of the drop, they would eventually approach a tangential orientation sufficiently close to the plane of symmetry (figure 4 a), and this leads to a spurious dominance of tangential diffusion in this region.

In summary, one needs to move from an $(r,C,\unicode[STIX]{x1D70F})$ coordinate system to an $(E,C,\unicode[STIX]{x1D70F})$ system. This transformation need not be done exactly, since only the constant- $E$ surfaces in the thermal boundary layer matter, and these deviate from sphericity only by $O(RePe)^{-1/2}$ for large $RePe$ . We therefore derive the approximate near-field forms of the constant- $D$ and $E$ surfaces; the latter will be used in the large- $Pe$ analysis. The near-field Stokes velocity components in spherical coordinates are given by:

where $s$ denotes the time measured in units of inverse shear rate; recall that $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D719}$ .

Dividing (2.23) by (2.24), one obtains:

Upon integrating this, the near-field form of the constant- $D$ surfaces, the parameter denoted here by $\unicode[STIX]{x1D713}$ , are given by:

Based on the known functional form of the constant- $D$ surfaces (Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018), $\unicode[STIX]{x1D713}$ and $D$ can be shown to be related as:

Similarly, from (2.25) and (2.24), one obtains:

Upon rearranging, this becomes:

Further, dividing (2.30) by (2.26), we get:

Integrating from $y_{m}$ to $y$ and substituting for $\cos \unicode[STIX]{x1D703}$ from (2.27), we have:

where $y_{m}$ , which emerges as a constant of integration in the above analysis, is the new coordinate which characterizes the near-field inertialess streamlines. Imposing $\tan ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D719}_{1}=0$ at $y=y_{m}$ , one obtains after some algebra:

The constant- $y_{m}$ surfaces are the near-field forms of the constant- $E$ surfaces whose exact functional forms are given in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018), and have been derived earlier by Powell (Reference Powell1983). Writing (2.33) in terms of the $(C,\unicode[STIX]{x1D70F})$ coordinate system, the expression for the new radial coordinate is:

where:

A near-field streamline is thus defined by a constant value of $y_{m}$ and $C$ , $y_{m}$ being a rescaled version of the original radial coordinate $y$ , equalling it when $\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x1D70F}=\unicode[STIX]{x03C0}/2$ . The scaling prefactor in (2.35) is a function of $C$ and $\unicode[STIX]{x1D70F}$ , and arises from the incompressibility constraint applied to an infinitesimal cylindrical fluid element, with its base on the drop. Specifically, the prefactor describes the change in height of such an element as a function of $\unicode[STIX]{x1D70F}$ as it moves along a Jeffery orbit (see figure 5). In the context of the boundary layer analysis that follows, it is worth reiterating that it is the constant- $y_{m}$ surfaces that are the near-field isotherms in the limit $Re=0$ , $Pe\rightarrow \infty$ , just as the constant- $y$ surfaces (spheres) were isotherms for a solid particle (Subramanian & Koch Reference Subramanian and Koch2006b ). The large- $Pe$ analysis below is implemented in the $(y_{m},C,\unicode[STIX]{x1D70F})$ coordinate system.

Figure 5.

A schematic of a near-field Stokes streamline, corresponding to a constant- $y_{m}$ constant- $C$ curve, showing the varying radial distance from the drop surface. To satisfy the continuity constraint a fluid element moving along such a streamline has a height that is modulated by $M(C,\unicode[STIX]{x1D70F})$ .

2.4 The boundary layer analysis for $Re\ll 1$ , $RePe\gg 1$

Having characterized the radial coordinate required for the analysis, we now transform the convection–diffusion equation from the $(y,C,\unicode[STIX]{x1D70F})$ coordinate system to a $(y_{m},C,\unicode[STIX]{x1D70F})$ coordinate system. Using the chain rule for differentiation, one may write:

The physical significance of (2.37) and (2.38) can be understood from figure 5. As one moves across Jeffery orbits (constant- $C$ curves) at a given $y$ , one ends up moving across constant- $y_{m}$ surfaces, and the consequent change in temperature corresponds to the first term in (2.37). A similar argument applies for the first term in (2.38). These terms therefore appear as additional ‘radial’ velocity contributions in the transformed convection–diffusion equation written below. For the derivative with respect to $y$ , however, the terms involving changes to the $C$ and $\unicode[STIX]{x1D70F}$ coordinates with respect to $y$ have been omitted. Such terms will, in general, be non-zero since the $(C,\unicode[STIX]{x1D70F})$ coordinate system is defined exactly at the drop surface, and the near-field closed streamlines are not exactly Jeffery orbits. For purposes of the boundary layer analysis, however, the perturbations to the $(C,\unicode[STIX]{x1D70F})$ coordinates are only $O(RePe)^{-1/2}$ .

Using the above expressions in (2.12), the convection–diffusion equation in the $(y_{m},C,\unicode[STIX]{x1D70F})$ coordinate system is given by:

where each of the velocity components in (2.39) includes a Stokes and an $O(Re)$ contribution. We have anticipated the dominance of radial diffusion in the thermal boundary layer. The term $\left(\unicode[STIX]{x2202}y_{m}/\unicode[STIX]{x2202}y\right)^{2}$ captures the variation of the effective diffusivity which is larger in regions where the constant- $y_{m}$ surfaces (isotherms at zero $Re$ ) are squeezed close together ( $\unicode[STIX]{x2202}y_{m}/\unicode[STIX]{x2202}y>1$ ) and smaller when they are further apart ( $\unicode[STIX]{x2202}y_{m}/\unicode[STIX]{x2202}y<1$ ).

Considering only the Stokes contributions to the convection terms, to begin with, one obtains:

where $u_{C}^{(0)}=0$ . Near the surface of the drop, using (2.16) and (2.19) the above equation becomes:

Further, since:

the rate of convection in the $y_{m}$ direction vanishes, This is simply a mathematical statement of the fact that the Stokes velocity field, by definition, cannot lead to convection across constant- $y_{m}$ surfaces, which are the invariant Stokes streamsurfaces. Thus, at leading order, in the limit of $Pe\gg 1$ , one has:

which implies that the leading-order temperature field is independent of $\unicode[STIX]{x1D70F}$ at steady state. The transformation to the new radial coordinate has helped one directly see the isothermality of the closed Stokesian streamlines.

Expand the temperature field, for small but finite $Re$ , as:

where $F(Re,Pe)$ is a small parameter, and will turn out to be proportional to the boundary layer thickness. Writing the convection–diffusion equation, with the velocity field now written down to $O(Re)$ , and making use of (2.42) and (2.43), one obtains:

Since $u_{\unicode[STIX]{x1D70F}}^{(0)}/k$ is a constant, one may average the above equation over the Stokesian surface streamlines. This is equivalent to an integration over $\unicode[STIX]{x1D70F}$ from $0$ to $2\unicode[STIX]{x03C0}$ , which eliminates the term involving $\unicode[STIX]{x1D6E9}^{(1)}$ , and one obtains the $\unicode[STIX]{x1D70F}$ -averaged equation that governs the leading-order boundary layer temperature field $\unicode[STIX]{x1D6E9}^{(0)}$ :

Expanding the inertial velocity components in the near field using (2.20)–(2.22), one obtains:

In (2.47), $RePe$ naturally emerges as the parameter that governs the relative magnitudes of convection and diffusion. The $\unicode[STIX]{x1D70F}$ average used points to a crucial difference between the particle and drop cases. For the rotating particle, the uniform angular velocity implies that the inertial drift is uniformly averaged over $\unicode[STIX]{x1D719}$ . For a drop, the weighting is not uniform owing to the non-trivial relation between  $\unicode[STIX]{x1D70F}$ and  $\unicode[STIX]{x1D719}$ (see (2.7)). Interpreted as an average over $\unicode[STIX]{x1D719}$ , the parts of the orbit where the fluid elements are moving slower have a greater contribution to the $\unicode[STIX]{x1D70F}$ -averaged convection in (2.47) since inertia, acting over a longer time, leads to a greater drift in this regions; and vice versa.

It is worth mentioning that the simplicity of the leading-order equation, (2.43), belies the non-trivial approach towards $\unicode[STIX]{x1D70F}$ -independence via a shear-enhanced diffusion mechanism. In dimensional terms, equation (2.43) holds true for times much longer than $O(Pe^{-2/3})a^{2}/D$ , and this time scale is shorter than the time of $O([(RePe)^{-1/2}]^{2})a^{2}/D$ for inertia to set up a steady temperature field in the thermal boundary layer, provided $Re\ll Pe^{-1/3}$ . Thus, for $Re\ll Pe^{-1/3}$ , the evolution towards the steady state boundary layer temperature field satisfying (2.47) would proceed through an intermediate asymptotic stage when the Jeffery-orbit-like turns of the near-surface streamlines become nearly isothermal due to shear-enhanced diffusion, but with the temperature variation across the turns still being closely related to the initial field. When $Pe^{-1/3}\ll Re\ll 1$ , there is no separation of time scales between the (Stokesian) isothermalization and inertial convection. Since the tight spiralling of the inertial streamlines continues to hold, the $\unicode[STIX]{x1D70F}$ -averaging above may still be carried out to determine the boundary layer temperature field. For a solid particle, the tight spiralling of the inertial streamlines holds even when $Re$ is of order unity provided $Pe\gg 1$ , owing to the near-surface convecting velocity becoming weak (of $O(Pe)^{-1/3}$ ) due to the no-slip boundary condition; this implies that a $\unicode[STIX]{x1D719}$ -averaged analysis may be carried out even when $Re$ is of order unity (Yang et al. Reference Yang, Zhang, Koch and Yin2011). The $\unicode[STIX]{x1D70F}$ -averaging approach is, however, no longer valid when the drop $Re$ is of order unity.

From (2.47), it is clear that the scale for the boundary layer thickness is $O(Re^{-1/2}Pe^{-1/2})$ . Indeed, on postulating a rescaling for the modified radial coordinate $y_{m}$ of the form $Y_{m}=m(Re,Pe)y_{m}$ , where $Y_{m}$ is $O(1)$ , the leading-order balance gives $m(Re,Pe)=Re^{1/2}Pe^{1/2}$ . In terms of this rescaled coordinate, the $\unicode[STIX]{x1D70F}$ -averaged convection–diffusion equation is given by:

where $Y_{m}$ and $Y$ are still related by (2.34). Using (2.33) the above equation can be rewritten as:

which is of the form:

where

The detailed closed-form expression for ${\mathcal{B}}(C)$ is given in appendix C, and that for ${\mathcal{A}}(C)$ may be obtained using a relation derived below (see (2.71)).

Note that the original three-dimensional non-axisymmetric heat transfer problem at small but finite $Re$ has been reduced, through the use of an appropriate coordinate system and phase averaging over the inertialess closed streamlines, to one resembling an axisymmetric problem, where the tangential convection is along a single coordinate ( $C$ ). The physical interpretations of the three factors ${\mathcal{A}}(C)$ , ${\mathcal{B}}(C)$ and ${\mathcal{D}}(C)$ follow naturally. ${\mathcal{A}}(C)$ represents the convective flux across the constant- $y_{m}$ surfaces due to the $\unicode[STIX]{x1D70F}$ -averaged inertial velocity field. ${\mathcal{B}}(C)$ represents the $\unicode[STIX]{x1D70F}$ -averaged convective flux tangential to the drop surface. The thermal wake must therefore correspond to the $C$ - location (implying that the wake is a Jeffery orbit to the order considered) at which this tangential convection vanishes. ${\mathcal{D}}(C)$ is the ‘orbit-dependent diffusivity’ which is the original constant molecular diffusivity ( $1/Pe$ in non-dimensional terms) modulated by the spacing of the near-field Stokesian streamsurfaces.

One now introduces a similarity variable of the form $\unicode[STIX]{x1D702}=Y_{m}/g(C)$ , where $g(C)$ characterizes the dependence of the $\unicode[STIX]{x1D70F}$ -averaged boundary layer thickness as a function of position on the drop surface. Unlike the open streamline regime examined in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018), the boundary layer thickness on account of $\unicode[STIX]{x1D70F}$ -averaging is only a function of $C$ . Transforming (2.50), the non-dimensional temperature field satisfies:

with the boundary layer thickness governed by:

where $f=g^{2}/2$ . The boundary conditions for the non-dimensional temperature are:

Solving (2.55) and using (2.57) and (2.58):

In order to solve for the boundary layer thickness profile from (2.56), without tedious algebraic manipulation, and thence calculate $Nu$ , it helps to relate the $\unicode[STIX]{x1D70F}$ -averaged radial ( ${\mathcal{A}}(C)$ ) and tangential ( ${\mathcal{B}}(C)$ ) velocities. Such a relation is immediate in simpler problems, but here one starts from the continuity equation for the Stokes velocity field, written in $(y,C,\unicode[STIX]{x1D70F})$ coordinates:

where $u_{C}^{(0)}=0$ , as before, by definition. In terms of the velocity field near the drop surface, the remaining two terms take the form:

Noting that $h_{\unicode[STIX]{x1D70F}}^{(0)}$ is independent of both $C$ and $\unicode[STIX]{x1D70F}$ and that the combination of metric factors $hk\sin \unicode[STIX]{x1D6FC}_{1}$ (the area of an elementary parallelogram) is independent of the radial coordinate at leading order, one can write the above equation as:

From (2.33) and (2.42), the radial Stokes velocity measure, $h_{r}^{(0)}$ , may be written as:

where $M(C,\unicode[STIX]{x1D70F})$ is given by (2.35). Substituting for $h_{r}^{(0)}$ in (2.62), one has the relation:

This is the statement of volume conservation applied to a cylindrical fluid element as it moves along an inertialess Stokes orbit, with its base on the surface of the spherical drop (see figure 5). The volume of such an infinitesimal fluid element is given by $\unicode[STIX]{x0394}V=hk\sin \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x0394}C\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}y_{m}/M(C,\unicode[STIX]{x1D70F})$ and the above equation is then the same as the condition that $\unicode[STIX]{x2202}(\unicode[STIX]{x0394}V)/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}|_{y_{m},C}=0$ .

Next, the continuity equation, at $O(Re)$ expressed in $(y,C,\unicode[STIX]{x1D70F})$ coordinates is given by:

Rewriting this equation for the near-field inertial velocity, one has:

Expanding the last term and using the relation (2.64), one obtains:

Note that the left-hand side of the above equation contains the first and third terms in the integrand in the expression for ${\mathcal{A}}(C)$ (see (2.51)). Upon substitution, one gets:

The first term in the above expression is an exact differential, and hence zero when integrated from $0$ to $2\unicode[STIX]{x03C0}$ . Upon rearranging the expression further, one has:

Since the term:

which is independent of $\unicode[STIX]{x1D70F}$ and can therefore be pulled out of the integral, the remaining terms can be expressed in terms of ${\mathcal{B}}(C)$ , giving the final relation between ${\mathcal{A}}(C)$ and ${\mathcal{B}}(C)$ as:

The above expression is a statement of the $\unicode[STIX]{x1D70F}$ -averaged continuity equation for the $O(Re)$ velocity field. The $\unicode[STIX]{x1D70F}$ averaging trivially excludes any convection in the $\unicode[STIX]{x1D70F}$ direction, even at $O(Re)$ , which leaves only the convection in the radial (across constant- $y_{m}$ surfaces) and $C$ directions (across Jeffery orbits) denoted by ${\mathcal{A}}(C)$ and ${\mathcal{B}}(C)$ , respectively. The continuity equation dictates that these two must be related and the relation is given by (2.71).

2.5 Solution to the boundary layer thickness equation

Substituting (2.71) in (2.56), the equation for the boundary layer thickness becomes:

which can be rewritten to give:

Integrating this between the limits $C_{inlet}$ and $C$ , one obtains:

with the boundary layer thickness, as a function of $C$ , being given by:

where

Here, $C_{inlet}$ is the location of the fluid inlet (at the drop surface) due to the $O(Re)$ velocity field. The tangential velocity component, proportional to ${\mathcal{B}}(C)$ , vanishes as $C\rightarrow C_{inlet}$ , the convection here being solely across constant $Y_{m}$ surfaces (and towards $Y_{m}=0$ ). The condition used in obtaining (2.75) and (2.76) is that the boundary layer thickness is finite at $C=C_{inlet}$ .

The profile of the thermal boundary layer thickness, and in particular, the locations of the inlet and wake, provides another important distinction between drops and solid particles. For a solid particle, the inertial drift always occurs from the poles towards the plane of symmetry (Subramanian & Koch Reference Subramanian and Koch2006a ,Reference Subramanian and Koch b ), a fact readily explained by the centrifugal forces pushing fluid out at the plane of symmetry. From the heat transfer perspective, the thermal wake is thus always at this plane. In the $\unicode[STIX]{x1D719}$ -averaged analysis given in Subramanian & Koch (Reference Subramanian and Koch2006a ,Reference Subramanian and Koch b ), the signature of this wake is a divergence of the $\unicode[STIX]{x1D719}$ -averaged boundary layer thickness at $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ . For the drop, the location of the wake arising from the inertial convection may be calculated using the fact that the tangential flux (at the surface) goes to zero, and in addition, the fluid drifts radially outwards at this location. Thus, $C_{wake}$ is determined by the equation

with the constraint $h_{r}^{(1)}>0$ , $h_{r}^{(1)}$ being defined in (2.21). Solving (2.77) using symbolic manipulation (Mathematica, Wolfram Research Inc. 2010) one finds, as for a particle, that the wake is always at the plane of symmetry for $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{bif}$ where $\unicode[STIX]{x1D6FC}_{bif}\approx 0.35$ (the single wake regime; see figure 7). However, for $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{bif}$ , the wake is at the plane of symmetry only for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{bif}(\unicode[STIX]{x1D6FC})\,({>}\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC}))$ . It bifurcates and lifts off this plane for smaller $\unicode[STIX]{x1D706}$ , being located at an intermediate $C$ -orbit; there are a pair of such orbits on either side of the plane of symmetry, corresponding to $C=\pm C_{wake}$ , in this bifurcated wake regime (see figure 8). As shown in figure 6(a), the bifurcated wake regime in the ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane is the finite region bounded by the curves $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ and $\unicode[STIX]{x1D706}_{bif}(\unicode[STIX]{x1D6FC})$ . For a fixed $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{bif}$ , as $\unicode[STIX]{x1D706}$ increases from zero, one is first in the open streamline regime where the effect of inertia is perturbative. At $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ , the wake is at the poles ( $C=0$ ) and the $\unicode[STIX]{x1D70F}$ -averaged inertial convection has a uniaxial extensional character, being directed from the equator to the poles. As $\unicode[STIX]{x1D706}$ increases further, the wake location moves towards the equator, separating distinct regions where the averaged inertial convection is directed towards the poles ( $C_{wake}<C<\infty$ ) and towards the equator ( $0<C<C_{wake}$ ); see figure 8(b). At $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{bif}(\unicode[STIX]{x1D6FC})$ , the wake just merges with the plane of symmetry, and remains there for all greater values of $\unicode[STIX]{x1D706}$ , with the $\unicode[STIX]{x1D70F}$ -averaged spiralling now being bi-axial in character over the entire drop; see figure 7(b). Figure 6(b) depicts the migration of the thermal wake, from the pole to the equatorial plane, with increasing $\unicode[STIX]{x1D706}$ in the interval $(\unicode[STIX]{x1D706}_{c},\unicode[STIX]{x1D706}_{bif})$ , for $\unicode[STIX]{x1D6FC}=0.1$ . The two distinct wake regimes for a drop arise from a non-trivial reversal in the direction of the inertial force over certain regions of the drop surface. Unlike a solid particle, at small $Re$ , where the near-field velocity is dominated by solid-body rotation (the extensional component being weaker by $O(RePe)^{-1/3}$ ), for a drop, the extensional component of the disturbance flow remains of comparable importance even at the surface, as is evident from the non-planar nature of surface streamlines. Non-planar trajectories leading to a complicated behaviour of the inertial force, including direction reversal, have been found in other scenarios (Subramanian & Brady Reference Subramanian and Brady2006; Marath, Dwivedi & Subramanian Reference Marath, Dwivedi and Subramanian2017; Marath & Subramanian Reference Marath and Subramanian2018). Interestingly, even for a solid particle, (Yang et al. Reference Yang, Zhang, Koch and Yin2011) the wake lifts off the equatorial plane at a finite $Re$ , this being broadly consistent with the inertial slowing down of the particle rotation (Mikulencak & Morris Reference Mikulencak and Morris2004), and the resulting weakening of inertial forces causing a drift towards the equator. Regardless of the particular wake regime, a fluid parcel approaches the drop via an incoming convective channel, spirals towards the wake (towards, or both towards and away from, the plane of symmetry depending on the particular regime), and eventually exits through an outgoing convective channel.

Figure 6.

(a) The single and bifurcated wake regimes in the ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane. The bifurcated wake regime is the shaded region enclosed between the curves $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ and $\unicode[STIX]{x1D706}_{bif}(\unicode[STIX]{x1D6FC})$ . The two curves intersect at $\unicode[STIX]{x1D6FC}_{bif}\approx 0.35$ ; for $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{bif}$ , there is only the single wake regime. (b) The variation of the orbit constant at which the wake occurs with viscosity ratio for $\unicode[STIX]{x1D6FC}\approx 0.1$  (dashed vertical line in (a)). The wake location shifts from being at the poles at $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ to being at the equatorial plane for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{bif}$ .

Table 1.

The inlet and wake values of the orbit constant $C$ for different regions in the ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane.

Figure 7.

Inertial surface streamlines and the boundary layer profile in the single wake regime: $\unicode[STIX]{x1D6FC}=0.2$ , $\unicode[STIX]{x1D706}=2$ ( $\unicode[STIX]{x1D706}_{bif}=0.547$ ),  $Re=0.1$ . (a) Three-dimensional view, (b) projection onto the ( $x_{2},x_{3}$ ) plane (gradient–vorticity plane). The black arrows show the direction of rapid convection in the $\unicode[STIX]{x1D70F}$ -direction and red arrows depict the much slower inertial drift across constant- $C$ orbits. The boundary layer growth is schematically represented as the shaded portion, and implies the existence of a single wake at the equatorial plane ( $C_{wake}\rightarrow \infty$ ). The red dot in (a) shows the starting point of the streamline.

Figure 8.

Inertial surface streamlines and the boundary layer profile in the bifurcated wake regime: $\unicode[STIX]{x1D6FC}=0.2$ , $\unicode[STIX]{x1D706}=0.52$ ( $\unicode[STIX]{x1D706}_{bif}=0.547$ ),  $Re=0.1$ . (a) Three-dimensional view, (b) projection onto the ( $x_{2},x_{3}$ ) plane (gradient–vorticity plane). The black arrows show the direction of rapid convection in the $\unicode[STIX]{x1D70F}$ -direction and red arrows depict the much slower inertial drift across constant- $C$ orbits. The boundary layer growth is schematically represented as the shaded portion. Note the bifurcated wake corresponding to the locations $C=\pm C_{wake}$ . The red dots in (a) show starting points for the streamlines.

Figure 9.

(a) A contour plot of ${\mathcal{H}}_{c}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ defined by ${\mathcal{H}}_{c}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})=Nu/(Pe)^{1/2}$ , for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$ (closed streamline regime). $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ and $\unicode[STIX]{x1D706}_{bif}(\unicode[STIX]{x1D6FC})$ are depicted by the blue dotted and green dot-dashed curves, respectively. (b) A three-dimensional view of the $Nu$ -prefactor surface on the ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane. (c) Slices of ${\mathcal{H}}_{c}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ as a function of $\unicode[STIX]{x1D706}$ for different $\unicode[STIX]{x1D6FC}$ values. In (a), (b) and (c) the divergence of the prefactor with increasing $\unicode[STIX]{x1D6FE}$ ( $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}$ ) is evident. Note the minima in ${\mathcal{H}}_{c}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ at $\unicode[STIX]{x1D706}\approx \unicode[STIX]{x1D706}_{bif}$ for $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{bif}$ due to a broadened wake. The contours in (a) are restricted to $\unicode[STIX]{x1D6FE}<10$ in order to highlight the variations away from the rapid changes near $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ .

In light of the above discussion, the value of $C_{inlet}$ in (2.75) is determined by the particular wake regime in the ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane. $C_{inlet}=0$ (poles) for the single wake regime, whereas $C_{inlet}$ can be both $0$ and $\infty$ in the bifurcated regime. The values of $C_{inlet}$ and $C_{wake}$ in each of these regimes are summarized in table 1.

2.6 Nusselt number calculation

Using the expression for the boundary layer thickness in (2.75), one proceeds to calculate the dimensionless heat transfer defined as:

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

The large- $Pe$ transport rate, at leading order, is the areal integral of the inverse boundary layer thickness modulated by the spacing between the Stokesian isotherms.

3.1 Breakdown of the large- $Pe$ asymptotic analyses for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{\pm }$

To begin with, based on the expressions already derived here and in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018), we obtain precise estimates for the intervals of viscosity ratios where the Jeffery-orbit-averaged analysis for the closed streamline regime, and the non-orthogonal coordinate-based boundary layer analysis in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018) for the open streamline regime, cease to remain valid. A more detailed description of the $Nu$ -variation, as one traverses these intervals (in the process, moving across the critical viscosity curve) in the ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane, is given later.

It is instructive to first examine the inertialess flow topology for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{+}$ , corresponding to $\unicode[STIX]{x1D6FE}\rightarrow \infty$ for the closed streamline regime (note that, for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{-}$ , corresponding to an approach from the open streamline side, $\unicode[STIX]{x1D6FE}=i\hat{\unicode[STIX]{x1D6FE}}$ with $\hat{\unicode[STIX]{x1D6FE}}\rightarrow \infty$ ; the scaling arguments below for the Stokes streamlines remain unchanged, however). The Stokes surface streamlines are Jeffery orbits, and $\unicode[STIX]{x1D6FE}\rightarrow \infty$ corresponds to the nearly meridional orbits of a slender rod. Such a rod changes orientation quickly when not aligned with the flow axis, while rotating very slowly in the nearly aligned phase. For $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}$ , as shown in figure 12(a), the surface streamlines are indeed nearly meridional. A given fluid element moves rapidly along a surface streamline away from the flow axis, while spending a large time in the vicinity of the flow–vorticity plane. This is consistent with the scalings for the near-field $\unicode[STIX]{x1D70F}$ –velocity component, and the coordinate metrics. Away from the flow–vorticity plane (the non-aligned phase), $u_{\unicode[STIX]{x1D70F}}^{(0)}|_{r\rightarrow 1}=k\,\text{d}\unicode[STIX]{x1D70F}/\text{d}t\sim O(1)$ , corresponding to an angular velocity of $O(\dot{\unicode[STIX]{x1D6FE}}^{-1})$ . Now, $\unicode[STIX]{x1D70F}$ changes at a uniform rate of $O(\unicode[STIX]{x1D6FE}^{-1})$ , so $\text{d}\unicode[STIX]{x1D70F}/\text{d}t\sim O(\unicode[STIX]{x1D6FE}^{-1})$ ; with $k\sim O(\unicode[STIX]{x1D6FE})$ , this gives $\text{d}t\sim O(1)$ for the non-aligned phase. On the other hand, close to the flow–vorticity plane (aligned phase), one has $u_{\unicode[STIX]{x1D70F}}^{(0)}|_{r\rightarrow 1}\sim O(1/\unicode[STIX]{x1D6FE}^{2})$ . Again, $\text{d}\unicode[STIX]{x1D70F}/\text{d}t\sim O(\unicode[STIX]{x1D6FE}^{-1})$ but $k\sim O(1/\unicode[STIX]{x1D6FE})$ , leading to $\text{d}t\sim O(\unicode[STIX]{x1D6FE})$ for the aligned phase. Thus, in dimensional terms, a fluid element spends a time of $O(\unicode[STIX]{x1D6FE}\dot{\unicode[STIX]{x1D6FE}}^{-1})$ near the flow vorticity plane and a time of $O(\dot{\unicode[STIX]{x1D6FE}}^{-1})$ over the rest of its orbit.

For small but finite $Re$ , and with $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{+}$ , the $O(Re)$ inertial drift acts over a large $O(\unicode[STIX]{x1D6FE}\dot{\unicode[STIX]{x1D6FE}}^{-1})$ period corresponding to the aligned phase of the original meridional Stokes streamlines, leading to an $O(\unicode[STIX]{x1D6FE}Re)$ angular drift. When $\unicode[STIX]{x1D6FE}$ is $O(Re^{-1})$ or larger, this aligned-phase inertial drift is of order unity; away from the flow–vorticity plane, the Stokes velocity is again dominant with the drift only being $O(Re)$ . These drift scalings may also be seen based on the scaling for the velocity field $u_{C}$ and the relevant metric $h$ , in the $C$ direction. For the non-aligned phase, using $\text{d}C/\text{d}t=u_{C}^{(1)}/h$ , $h\sim O(1/\unicode[STIX]{x1D6FE})$ , so that with $\text{d}t\sim O(1)$ , $\unicode[STIX]{x0394}C\sim O(Re)$ , where $\unicode[STIX]{x0394}C$ is the change in orbit constant. On the other hand, for the aligned phase, $u_{C}^{(1)},h\sim O(1)$ , with $\text{d}t\sim O(\unicode[STIX]{x1D6FE})$ , so $\unicode[STIX]{x0394}C\sim O(\unicode[STIX]{x1D6FE}Re)$ . For fixed $Re$ , it is the $O(\unicode[STIX]{x1D6FE})$ scaling of the change in orbit constant that manifests as an $O(\unicode[STIX]{x1D6FE}^{1/2})$ divergence of the $Nu$ -prefactor above. For $\unicode[STIX]{x1D6FE}\sim O(Re^{-1})$ , a single inertial streamline consists of well-separated meridional portions that rapidly traverse the drop surface away from the flow–vorticity plane, with the separation between successive meridional portions arising from the aforementioned large inertial drift (across several Stokes surface streamlines) during the aligned phase. Examples of such streamlines, in both the single ( $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{bif}$ ) and bifurcated wake ( $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{bif}$ ) regimes, are shown in figure 12. The aligned-phase drift across several Stokesian orbits is apparent since the finite- $Re$ streamline has drifted across a sizeable fraction of the drop surface in relatively few turns. Clearly, for $\unicode[STIX]{x1D6FE}Re\sim O(1)$ , the near-surface inertial streamlines are no longer tight spirals as assumed in the analysis presented earlier in § 2.4; thus, the notion of a Jeffery-orbit-averaged inertial drift is no longer applicable (see Marath & Subramanian Reference Marath and Subramanian2018 for a treatment of the analogue of this regime for the orientation dynamics of anisotropic particles). Using (2.9), this implies that the expressions for the $Nu$ -prefactor, given by (2.94) and (2.95), are no longer valid when $\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c}\leqslant 2Re^{2}(1+\unicode[STIX]{x1D6FC})/(1-\unicode[STIX]{x1D6FC})$ .

Figure 12.

(a) The geometry of inertialess Stokesian streamlines on the drop surface for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}$ ); here, $\unicode[STIX]{x1D6FC}=0.5$ and $\unicode[STIX]{x1D706}=2.001$ ( $\unicode[STIX]{x1D706}_{c}=2$ ). The streamlines have a nearly meridional character converging to and diverging away from the flow–vorticity plane. (b,c) Inertial surface streamlines in the intermediate regime ( $\unicode[STIX]{x1D6FC}=0.2$ , $\unicode[STIX]{x1D706}_{c}=0.5$ , $\unicode[STIX]{x1D706}_{bif}=0.547$ , $Re=0.1$ ) for (b) single wake ( $\unicode[STIX]{x1D706}=0.6$ ) and (c) bifurcated wake ( $\unicode[STIX]{x1D706}=0.501$ ) regimes. The large drift during a given turn is apparent from the small number of turns taken to drift across a large portion of the drop surface. Note the different drift direction in the two regimes even though the inertialess streamlines in both cases have the same character (see (a)).

Now we consider the breakdown of the open streamline analysis. From Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018), the analytical prediction for the Nusselt number in this regime may be written as:

in terms of the inverse boundary layer thicknesses over regions (I and II) on the drop corresponding to distinct streamline topologies. The boundary layer thicknesses $g_{I}$ and $g_{II}$ approach a common limiting form at the inlet given by

For $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{-}$ , corresponding to $\hat{\unicode[STIX]{x1D6FE}}\rightarrow \infty$ , the inlet boundary layer thickness above is $O(\hat{\unicode[STIX]{x1D6FE}}/Pe)^{1/2}$  (for $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}\sim O(1)$ ). This may again be explained based on Jeffery-orbit scalings for slender rods; measured in units of $a$ , $(\hat{\unicode[STIX]{x1D6FE}}/Pe)^{1/2}$ is the diffusion length corresponding to the $O(\hat{\unicode[STIX]{x1D6FE}}\dot{\unicode[STIX]{x1D6FE}}^{-1})$ duration of the aligned phase of nearly meridional orbits found above. The inlet region, with a tangential velocity of $O(1/\hat{\unicode[STIX]{x1D6FE}}^{2})$ , has an angular extent of $O(1/\hat{\unicode[STIX]{x1D6FE}})$ . Thus, the boundary layer thickness is $O(\hat{\unicode[STIX]{x1D6FE}}/Pe)^{1/2}$ only over a small region with an angular extent of $O(1/\hat{\unicode[STIX]{x1D6FE}})$ , and reduces to an asymptotically smaller value of $O(Pe^{-1/2})$ over the remainder of the drop. The boundary layer assumption therefore breaks down in this inlet region when the transverse and longitudinal length scales are of the same order, that is, when $O(\hat{\unicode[STIX]{x1D6FE}}/Pe)^{1/2}\sim O(1/\hat{\unicode[STIX]{x1D6FE}})$ , or $\hat{\unicode[STIX]{x1D6FE}}\sim O(Pe^{1/3})$ . Using the definition of $\hat{\unicode[STIX]{x1D6FE}}$ in terms of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D706}$ , the $Nu$ -prediction given in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018) is no longer valid when $\unicode[STIX]{x1D706}_{c}-\unicode[STIX]{x1D706}\leqslant 2Pe^{-2/3}(1+\unicode[STIX]{x1D6FC})/(1-\unicode[STIX]{x1D6FC})$ . For smaller $\unicode[STIX]{x1D706}_{c}-\unicode[STIX]{x1D706}$ , tangential diffusion in the inlet region will significantly modify the temperature distribution entering the ‘main’ $O(Pe^{-1/2})$ boundary layer that develops beyond the inlet. Since $\hat{\unicode[STIX]{x1D6FE}}^{-1}$ is also the angular scale of the separation between the inlet and outlet (wake) regions (these correspond to the locations $\unicode[STIX]{x1D719}_{1}^{(1)-(4)}$ in the plane of symmetry; see figure 6 in Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018), these regions are no longer asymptotically separated when $\unicode[STIX]{x1D706}_{c}-\unicode[STIX]{x1D706}\sim O(Pe^{-2/3})$ , and the resulting ‘short-circuiting’ of heat between the inlet and wake fluids will lead to additional deviations from the original theoretical prediction. Note that the boundary layer assumptions are still satisfied for the $O(Pe^{-1/2})$ boundary layer over the remainder of the drop. Although, it is the inverse of this boundary layer thickness that determines the rate of transport, for a fixed $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ , $Nu$ will no longer scale as $O(Pe^{1/2})$ when $\unicode[STIX]{x1D706}_{c}-\unicode[STIX]{x1D706}<O(Pe^{-2/3})$ on the open streamline side, since the prefactor which determines the temperature distribution entering the boundary layer is now a function of $\hat{\unicode[STIX]{x1D6FE}}Pe^{-1/3}$ .

To summarize then, the analytical predictions for the closed streamline regime given in earlier sections, and the prediction for the open streamline regime given in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018) are no longer valid when $-Pe^{-2/3}(1+\unicode[STIX]{x1D6FC})/(1-\unicode[STIX]{x1D6FC})<\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c}<Re^{2}(1+\unicode[STIX]{x1D6FC})/(1-\unicode[STIX]{x1D6FC})$ . Note that the lower bound for the $\unicode[STIX]{x1D706}$ -interval, on the open streamline side, has been derived assuming the absence of inertial effects in accordance with the analysis of Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018). However, including the effects of weak inertia should not affect this estimate since an inertial-drift-induced rearrangement of the nearly isothermal streamlines at the inlet, as happens in the interval $Re^{2}\gg \unicode[STIX]{x1D706}_{c}-\unicode[STIX]{x1D706}\gg Pe^{-2/3}$ , should not alter the temperature distribution over the remainder of the drop.

3.2 The $Nu$ -behaviour in the transition regime ( $\unicode[STIX]{x1D706}\approx \unicode[STIX]{x1D706}_{c}$ )

We first examine the inertialess scaling behaviour of $Nu$ in the vicinity of $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ . As already seen in the earlier sub-section, the prediction of the large- $Pe$ boundary layer analysis in the open streamline regime must break down once $\unicode[STIX]{x1D706}_{c}-\unicode[STIX]{x1D706}$ becomes $O(Pe^{-2/3})$ . For $\unicode[STIX]{x1D706}$ values closer to $\unicode[STIX]{x1D706}_{c}$ , the $Nu$ -prefactor is a function of $\hat{\unicode[STIX]{x1D6FE}}Pe^{-1/3}$ , implying a deviation from the $Pe^{1/2}$ scaling in the open streamline regime. This is the case even as one crosses over the critical curve, and until the dimensions of the closed streamline envelope become comparable to the boundary layer thickness. To understand the $Nu$ -scalings on the closed streamline side, one therefore needs the geometry of the separatrix surface for $\unicode[STIX]{x1D6FE}\rightarrow \infty$ .

Figure 13.

A schematic of the extent of the annular region of closed (spiralling) streamlines adjacent to the drop surface in the limit of large $\unicode[STIX]{x1D6FE}$ . Note the asymptotically different thickness of $O(1/\unicode[STIX]{x1D6FE}^{2})$ near the flow–vorticity plane, compared to the $O(1/\unicode[STIX]{x1D6FE}^{5})$ thickness over the remainder of the drop.

The circle of fixed points, corresponding to the intersection of the separatrix surface with the flow–vorticity plane, denotes its maximum radial extent (see Krishnamurthy & Subramanian Reference Krishnamurthy and Subramanian2018); the radius of this circle, for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}$ , has the form $y_{m}=r_{0}-1\approx ((\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c})(1-\unicode[STIX]{x1D6FC})^{2}/10\unicode[STIX]{x1D6FC}(1+\unicode[STIX]{x1D6FC}))$ , or alternatively, $y_{m}\approx (1-\unicode[STIX]{x1D6FC}/5\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6FE}^{-2}$ in terms of the effective aspect ratio. From (2.33), this scaling is seen to be valid within an angle of $O(1/\unicode[STIX]{x1D6FE})$ bracketing the flow–vorticity plane (that is, $\unicode[STIX]{x1D719},\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D719}_{1}\sim O(1/\unicode[STIX]{x1D6FE})$ ). The radial extent over the remainder of the drop can again be inferred from (2.33) or (2.34). One finds $r_{0}-1\approx y_{m}/\unicode[STIX]{x1D6FE}^{3}$ for $\unicode[STIX]{x1D70F},\unicode[STIX]{x1D719}\sim O(1)$ , and since $y_{m}=1/\unicode[STIX]{x1D6FE}^{2}$ , $r_{0}-1\sim 1/\unicode[STIX]{x1D6FE}^{5}$ . Thus, the thin separatrix envelope is pronouncedly anisotropic for large $\unicode[STIX]{x1D6FE}$ , with a thickness of $O(1/\unicode[STIX]{x1D6FE}^{2})$ in a small $O(1/\unicode[STIX]{x1D6FE})$ angular region enclosing the flow–vorticity plane, and a much smaller thickness of $O(1/\unicode[STIX]{x1D6FE}^{5})$ over the remainder of the drop; see figure 13. For arbitrary $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FE})$ , the diffusion-limited plateau attained by $Nu$ , for $Pe\rightarrow \infty$ with $Re=0$ , requires a detailed numerical calculation along the lines of Yu-Fang & Acrivos (Reference Yu-Fang and Acrivos1968) and Poe & Acrivos (Reference Poe and Acrivos1976) of the diffusion equation in a streamline-aligned coordinate system (rather intriguingly, such a calculation, for the elliptic linear flows not considered here, yields $Nu=0$ ! In other words, the constant of order unity that $Nu$ must asymptote to is identically zero, in turn suggesting a non-uniformity of the diffusion-limited approximation in an unbounded domain; it is likely that $Nu\sim O(1/\ln Pe)$ for elliptic linear flows (see Poe & Acrivos Reference Poe and Acrivos1976)). For $\unicode[STIX]{x1D706}\approx \unicode[STIX]{x1D706}_{c}$ , however, the temperature may be assumed to vary linearly across the asymptotically thin closed streamline region, from $T_{0}$ at the drop surface to $T_{\infty }$ at the separatrix surface. In this limit, the geometry of the separatrix surface alone (rather than the detailed configuration of the closed streamlines within) can be used to determine the diffusion-limited $Nu$ -plateau. One starts from the following approximate expression for $Nu$ :

which is the areal average of the inverse thickness of the separatrix surface, $y_{sep}^{-1}$ , over the unit sphere. Writing the metric factors in terms of $C$ and $\unicode[STIX]{x1D70F}$ , and substituting for $y_{sep}$ from (2.33), with $y_{m}$ denoting the radius of the fixed circle above, one obtains:

which, on integration, gives:

As expected, it is the $O(1/\unicode[STIX]{x1D6FE}^{5})$ portion of the separatrix envelope which controls the scaling of the diffusion-limited asymptote at leading order; the larger $O(\unicode[STIX]{x1D6FE}^{-2})$ thickness, being restricted to an $O(1/\unicode[STIX]{x1D6FE})$ angular region, only has an $O(\unicode[STIX]{x1D6FE})$ contribution towards this estimate.

With $\unicode[STIX]{x1D706}$ increasing beyond $\unicode[STIX]{x1D706}_{c}$ , the $Nu$ scaling behaviour must change due to the boundary layer thickness becoming comparable to the (increasing) radial extent of the separatrix envelope. For the pair of thickness scales of the separatrix surface obtained above, for large $\unicode[STIX]{x1D6FE}$ , this gives $\unicode[STIX]{x1D6FE}\sim O(Pe^{1/5})$ and $\unicode[STIX]{x1D6FE}\sim O(Pe^{1/10})$ for transitions in $Nu$ -scaling. The second, in principle, corresponds to the onset of the diffusion limitation where $Nu$ saturates at the value given by (3.5) for sufficiently large $Pe$ ; in terms of $\unicode[STIX]{x1D706}$ , this corresponds to $\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c}\sim O(Pe^{-1/5})$ . For typical values of $Pe$ of $O(10^{3})$ , however, the aforementioned separatrix-based scaling estimates are of order unity, and the diffusion-limited plateau is therefore expected to be of order unity, attained when $\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c}\sim O(1)$ . Figure 14(a) sketches the scaling behaviour of $Nu$ in the vicinity of the critical curve, for $Re=0$ , as one traverses across constant- $\unicode[STIX]{x1D6FE}$  (or $\hat{\unicode[STIX]{x1D6FE}}$ ) curves on the ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane. Importantly, while for $Pe=\infty$ , the $Nu$ -curve is singular (it is $\infty$ for $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ , but is finite and independent of $Pe$ for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$ , diverging as $O(\unicode[STIX]{x1D6FE}^{5})$ or $(\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c})^{-5/2}$ for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{+}$ ), it is a smooth function of $\unicode[STIX]{x1D706}$ for any finite $Pe$ .

Figure 14.

A schematic of the Nusselt number variation as a function of $\unicode[STIX]{x1D706}$ as one crosses $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ , from the open to the closed streamline regime for $Re=0$ . $Nu\sim (\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c})^{-5/2}$ is the asymptotic form of the diffusion-limited curve for $Pe=\infty$ and $\unicode[STIX]{x1D6FE}\gg 1$ .

Figure 15.

(a) A plot of the $Nu$ -prefactor, defined by ${\mathcal{H}}_{c/o}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})=Nu/Pe^{1/2}$ , over the entire ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane, for $Re=0.1$ . $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ and $\unicode[STIX]{x1D706}_{bif}(\unicode[STIX]{x1D6FC})$ are depicted by the blue dotted curve and green dot-dashed curves, respectively. The lower and upper black dashed lines mark the boundaries of validity for the open streamline and closed streamline analysis, respectively, as given by (3.6), and are calculated using the values $Sc=10^{7}$ and $Re=0.1$ . (b) Slices of the $Nu$ prefactor as a function of $\unicode[STIX]{x1D706}$ for different $\unicode[STIX]{x1D6FC}$ . The intervals where the open and closed streamline results are valid are shown using symbols; the dashed lines, across the transition regime, are obtained via linear interpolation and are only meant as a guide for the eye. The inset shows a magnified view of the closed streamline interval for clarity.

Figure 16.

(a) A plot of the $Nu$ prefactor, defined by ${\mathcal{H}}_{c/o}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})=Nu/Pe^{1/2}$ , for the entire ( $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706}$ ) plane, for $Re=0.25$ and $Sc=10^{7}$ . (b) A magnified view highlights variations near $\unicode[STIX]{x1D6FC}=0$ and for $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{bif}$ . $\unicode[STIX]{x1D706}_{c}(\unicode[STIX]{x1D6FC})$ and $\unicode[STIX]{x1D706}_{bif}(\unicode[STIX]{x1D6FC})$ are depicted by the blue dotted curve and green dot-dashed curves, respectively. The lower and upper black dashed lines mark the boundaries of validity for the open streamline and closed streamline analysis. (c) Slices of the $Nu$ prefactor as a function of $\unicode[STIX]{x1D706}$ for different $\unicode[STIX]{x1D6FC}$  (details appear in previous figure).

Consider now the variation in $Nu$ across $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ with the inclusion of inertia, but in the limit $RePe\gg 1$ when there exists a thermal boundary layer on the closed streamline side. Based on the inertialess scaling behaviour of $Nu$ above, there emerged three transition scales for the effective aspect ratio. These were $Pe^{1/3}$ on the open streamline side, and $Pe^{1/5}$ and $Pe^{1/10}$ , respectively, on the closed streamline side; although as indicated earlier, the remaining two scales are of order unity in practice, so the associated scaling regimes are not well separated. For small but finite $Re$ , a transition aspect ratio of $O(Re^{-1})$ has already been identified that governs the nature of the spiralling inertial streamlines; $\unicode[STIX]{x1D6FE}\ll Re^{-1}$ ensures that the inertial streamlines are tightly knit spirals. For small $Re$ , one may still appeal to the notion of a separatrix envelope, although it now roughly corresponds to the region of tightly spiralling streamlines rather than closed ones; equating the small $O(RePe)^{-1/2}$ inertial boundary layer thickness to the $O(1/\unicode[STIX]{x1D6FE}^{5})$ thickness of the separatrix envelope leads to a critical aspect-ratio scale of $O(RePe)^{1/10}$ . The $Nu$ calculation detailed in earlier sections is valid when the thermal boundary layer is much thinner than the separatrix envelope, and in addition, consists of tightly spiralling inertial streamlines. This requires $\unicode[STIX]{x1D6FE}<\min [(RePe)^{1/10},Re^{-1}]$ . For typical $Pe$ values, the former scale is of order unity, and thus, the breakdown of the Jeffery-orbit-averaged analysis, for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{+}$  ( $\unicode[STIX]{x1D6FE}\rightarrow \infty$ ) will always occur on account of the thermal boundary layer becoming comparable in thickness to the separatrix envelope that rapidly thins with increasing $\unicode[STIX]{x1D6FE}$ . The nature of this breakdown will be similar to what happens when $\unicode[STIX]{x1D6FE}\sim O(Re^{-1})$ , however. When $\unicode[STIX]{x1D6FE}$ becomes large for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}$ , a significant fraction of the streamlines in the thermal boundary layer will no longer be tight spirals on account of their vicinity to the (fictitious) inertialess separatrix surface (rather than due to the meridional nature of the Stokesian streamlines). The resulting long circulation periods leads to large inertial drifts, and the Jeffery-orbit phase-averaged approach is no longer valid. This in turn implies that, although interesting, the regime where $\unicode[STIX]{x1D6FE}\sim Re^{-1}$ , and one needs to account for the localized inertial drift (across meridional Stokesian streamlines) in the vicinity of the flow–vorticity plane in order to estimate the rate of transport (see figure 12), may not be relevant in practice.

In conclusion, provided $Re\ll 1$ , the large- $Pe$ analyses in the open and closed streamline regimes are no longer valid when

The consideration of scalings associated with the separatrix envelope has allowed one to modify the bound for $\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}_{c}$ , on the closed streamline side relative to that in § 3.1; although, this bound must still be regarded as tentative, on account of the rather complicated behaviour in the vicinity of the critical viscosity curve. An unambiguous threshold might only be revealed in a detailed numerical investigation. For $\unicode[STIX]{x1D706}$ values lying in the aforementioned interval, the transport resistance will be distributed across both the open and spiralling streamline regions. This, of course, violates the assumptions made in both the earlier sections of this paper, and in Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018), where the resistance to transport has been assumed to be entirely restricted to either within or outside the separatrix envelope, respectively. In order to obtain a (crude) quantitative estimate, we use (3.6) for a specific mass transport scenario involving gas bubbles in molten glass (discussed in more detail in the conclusions section below), that conforms to the asymptotic regime examined here. For this application, $Sc$ is $O(10^{7})$ . Taking $Re=0.1$ and $Re=0.25$ , corresponding to millimetre-sized bubbles in ambient shear rates of $1s^{-1}$ and $2.5s^{-1}$ , respectively, the intervals around $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{c}$ where the two analyses must break down can be estimated. The combined results of Krishnamurthy & Subramanian (Reference Krishnamurthy and Subramanian2018) and the current work, accounting for the intervals of validity for the two $Re$ values above, are given in figures 15 and 16. It is worth noting that, depending on the particular choice of parameters, a significant portion of the bifurcated wake region might, in fact, lie in the transition regime.