Thermocapillary migration of a planar droplet at small and large Marangoni numbers: effects of interfacial rheology

Abstract

In this paper, roles of interfacial rheology on thermocapillary migration of a planar droplet at small and large Marangoni numbers are analyzed. Under quasi-steady-state assumption, the time-independent momentum and energy equations of thermocapillary droplet migration with boundary conditions are determined. An exact solution of the steady thermocapillary migration of the deformed droplet at small Marangoni numbers is obtained. It is found that the deformed droplet has an oblate shape. The deviation from the circular section depends on the Weber number and the migration speed. The surface shear viscosity, the dilatational viscosity and the surface internal energy parameter affect the deformation of the droplet through reducing the migration speed. The validity of the steady thermocapillary droplet migration at small Marangoni numbers is confirmed by determining the conservative overall integral energy equations. At large Marangoni numbers, the non-conservative overall integral energy equations imply that thermocapillary droplet migration is always an unsteady process.

Appendix: Steady momentum and energy equations derived from the laboratory coordinate system

Using the coordinate and variable transformations from the laboratory coordinate system ($\bar{x},\bar{y}$) to a coordinate system (x, y) moving with the droplet velocity $V_{\infty }$, respectively, described below

$$\begin{aligned} {\bar{\mathbf{r}}} = \mathbf{r} + V_{\infty } t \mathbf{j} \end{aligned}$$

(32)

and

$$\begin{aligned} \begin{array}{lll} {\bar{\mathbf{v}}}({\bar{\mathbf{r}}},t) = {\mathbf{v}}({\mathbf{r}}) + V_{\infty } \mathbf{j}, &{} \bar{p}({\bar{\mathbf{r}}},t) = p({\mathbf{r}}) + p_{\infty }, &{} {\bar{T}}({\bar{\mathbf{r}}},t )- {\bar{T}}_0 = T({\mathbf{r}}) + GV_{\infty }t,\\ {\bar{\mathbf{v'}}} ({\bar{\mathbf{r}}},t) = {\mathbf{v'}}({\mathbf{r}}) + V_{\infty } \mathbf{j}, &{} \bar{p'}({\bar{\mathbf{r}}},t) = p'({\mathbf{r}}) + P'_{0}(t), &{} \bar{T'}({\bar{\mathbf{r}}},t )- {\bar{T}}_0 = T'({\mathbf{r}}) + GV_{\infty } t, \end{array} \end{aligned}$$

(33)

we have

$$\begin{aligned} \begin{aligned} {\bar{\nabla }}|_t&= \frac{\partial }{\partial \bar{x}}|_t \mathbf{i} + \frac{\partial }{\partial \bar{y}}|_t \mathbf{j} =\frac{\partial }{\partial x}|_t \mathbf{i} + \frac{\partial }{\partial y}|_t \mathbf{j} = \nabla |_t, \\ {\bar{\Delta }}|_t&= \frac{\partial ^2}{\partial \bar{x}^2}|_t + \frac{\partial ^2}{\partial \bar{y}^2}|_t =\frac{\partial ^2}{\partial x^2}|_t + \frac{\partial ^2}{\partial y^2}|_t = \Delta |_t. \end{aligned} \end{aligned}$$

(34)

For momentum equation of the continuous phase fluid in Eq. (1), we can derive its unsteady, convection and viscous terms as follows:

$$\begin{aligned} \begin{aligned} \frac{\partial {\bar{\mathbf{v}}}}{\partial t}|_{{\bar{\mathbf{r}}}}&= \frac{\partial (\mathbf{v} + V_{\infty } \mathbf{j})}{\partial t}|_{{\bar{\mathbf{r}}}} = \frac{\partial \mathbf{v}}{\partial t}|_{{\bar{\mathbf{r}}}} =\frac{\partial \mathbf{v}}{\partial x}|_t \frac{\partial x}{\partial t}|_{{\bar{\mathbf{r}}}} +\frac{\partial \mathbf{v}}{\partial y}|_t \frac{\partial y}{\partial t}|_{{\bar{\mathbf{r}}}} +\frac{\partial \mathbf{v}}{\partial t}|_\mathbf{r} \frac{\partial t}{\partial t}|_{{\bar{\mathbf{r}}}} \\&= \frac{\partial \mathbf{v}}{\partial y}|_t (-V_{\infty }) +\frac{\partial \mathbf{v}}{\partial t}|_\mathbf{r} =- V_{\infty } \frac{\partial \mathbf{v}}{\partial y}, \\ {\bar{\mathbf{v}}} \cdot {\bar{\nabla }} {\bar{\mathbf{v}}}|_t&= (\mathbf{v} + V_{\infty } \mathbf{j}) \cdot {\bar{\nabla }} (\mathbf{v} +V_{\infty } \mathbf{j})|_t =(\mathbf{v} + V_{\infty } \mathbf{j}) \cdot {\bar{\nabla }} \mathbf v|_t \\&= \mathbf{v} \cdot \nabla \mathbf{v} + V_{\infty } \frac{\partial \mathbf{v}}{\partial y}, \\ {\bar{\nabla }} \bar{p}|_t&= {\bar{\nabla }} (p + p_\infty )|_t = {\bar{\nabla }} p|_t = \nabla p, \\ {\bar{\Delta }} {\bar{\mathbf{v}}}|_t&= {\bar{\Delta }} (\mathbf{v} +V_{\infty } \mathbf{j})|_t = {\bar{\Delta }} \mathbf{v}|_t =\Delta \mathbf{v}, \end{aligned} \end{aligned}$$

(35)

where $\frac{\partial x}{\partial t}|_{{\bar{\mathbf{r}}}}=\frac{\partial x}{\partial t}|_{\bar{x}}=0$, $\frac{\partial y}{\partial t}|_{{\bar{\mathbf{r}}}} =\frac{\partial y}{\partial t}|_{\bar{y}}=-V_{\infty }$ and $\frac{\partial \mathbf{v}}{\partial t}|_\mathbf{r}=0$. Then, substituting Eq. (35) into the first equation in Eq. (1), we obtain the steady momentum equation of the continuous phase fluid

$$\begin{aligned} \rho \mathbf{v} \cdot \nabla \mathbf{v}= -\nabla p +\mu \Delta \mathbf{v}. \end{aligned}$$

(36)

And for energy equation of the continuous phase fluid in Eq. (1), we can write its unsteady, convection and conductivity terms as follows:

$$\begin{aligned} \begin{aligned} \frac{\partial \bar{T}}{\partial t}|_{{\bar{\mathbf{r}}}}&= \frac{\partial T}{\partial t}|_{{\bar{\mathbf{r}}}} + G V_{\infty } =\frac{\partial T}{\partial x}|_t \frac{\partial x}{\partial t}|_{{\bar{\mathbf{r}}}} +\frac{\partial T}{\partial y}|_t \frac{\partial y}{\partial t}|_{{\bar{\mathbf{r}}}} +\frac{\partial T}{\partial t}|_\mathbf{r} \frac{\partial t}{\partial t}|_{{\bar{\mathbf{r}}}} + G V_{\infty } \\&= \frac{\partial T}{\partial y}|_t (-V_{\infty }) +\frac{\partial T}{\partial t}|_\mathbf{r} + G V_{\infty } =- V_{\infty } \frac{\partial T}{\partial y} + G V_{\infty }, \\ {\bar{\mathbf{v}}} \cdot {\bar{\nabla }} \bar{T}|_t&= (\mathbf{v} + V_{\infty } \mathbf{j}) \cdot {\bar{\nabla }} (T +GV_{\infty } t)|_t =(\mathbf{v} + V_{\infty } \mathbf{j}) \cdot {\bar{\nabla }} T|_t \\&= \mathbf{v} \cdot \nabla T + V_{\infty } \frac{\partial T}{\partial y}, \\ {\bar{\Delta }} \bar{T}|_t&= {\bar{\Delta }} (T +GV_{\infty } t)|_t = {\bar{\Delta }} T|_t =\Delta T, \end{aligned} \end{aligned}$$

(37)

where $\frac{\partial T}{\partial t}|_\mathbf{r}=0$. Then, substituting Eq. (37) into the third equation in Eq. (1), we obtain the steady energy equation of the continuous phase fluid

$$\begin{aligned} G V_{\infty } + \mathbf{v} \nabla T= \kappa \Delta T. \end{aligned}$$

(38)

Similarly, we can also transform the momentum and energy equations within the droplet as above.