Flow stability in shallow droplets subject to localized heating of the bottom plate

We investigate theoretically the stability of thermo-capillary convection within a droplet when subject to localized heating from below. To model the droplet, we use a mathematical model based on lubrication theory. We formulate a base-state droplet profile, and we examine its stability with respect to small-amplitude perturbations in the azimuthal direction. Such linear stability analysis reveals that the base state is stable across a wide parameter space. We carry out transient simulations in three spatial dimensions: the simulations reveal that when the heating is slightly off-centered with respect to the droplet center, vortices develop within the droplet. The vortices persist when the contact line is pinned. These findings are consistent with experimental studies of locally heated sessile droplets.

Figure 1

Schematic description of the experimentally observed flow in a locally heated sessile droplet.

Figure 2

The classical axisymmetric Marangoni current observed in prior works, in the case of uniform heating of the substrate.

Figure 3

Schematic description of the fluid mechanical problem of droplet spreading, as derived from the Navier–Stokes equations in the lubrication limit.

Figure 4

Equilibrium solutions for different heat sources. We take $\alpha =0.6$. All other parameters $(\mathrm{Ma},\mathrm{Bi},\mathrm{\Theta })$ are taken to be unity.

Figure 5

Contour plot of the equilibrium droplet volume as a function of equilibrium contact angle $\alpha$ and Marangoni number $\mathrm{Ma}$, for (a) $s=0.2$, and (b) $s=0.3$. Other physical parameters are $(\mathrm{Bi},\mathrm{\Theta })=(1,1)$. Empty regions correspond to cases where the droplet ruptures. The broken-line curves in panel (a) indicate different paths through the parameter space used in the linear stability analysis: a constant-$\alpha$ path, and a constant-volume path.

Figure 6

Plot showing the occurrence of a weak solution at a critical value of $\alpha$, for fixed $\mathrm{Ma}$. Parameter values: $\mathrm{Ma}=1$, $\alpha =0.1$, and $s=0.2$. The blue curve refers to the classical solution of the regularized boundary-value problem on $0<r<1$ (regularization parameter: $\delta ={10}^{-4}$). The regularized classical solution fails: $h$ turns negative, and the numerical solution does not converge as the regularization parameter tends to zero (in nonsingular cases, the regularization technique does converge, and $h$ remains non-negative). The red curve has been generated using BVP4C in matlab and refers to the piecewise solution [Eq. (34)], which is non-negative on $r_1<r<1$, and which is zero in the core region $0\le r\le r_1$.

Figure 7

A specimen hump solution of Eq. (32). Parameters: $\mathrm{Ma}=1$, $\alpha =0.2$, $s=0.2$. Other parameters: $\mathrm{Bi}=0$ and $\mathrm{\Theta }=0$.

Figure 8

Dispersion curves for different values of $\mathrm{Bi}$, $\mathrm{\Theta }$, and $\alpha$. Increasing $\mathrm{Bi}$, $\mathrm{\Theta }$ going across the rows, and increasing $\alpha$ going down the middle column. The value $s=0.2$ is used throughout.

Figure 9

Dispersion curves for fixed equilibrium droplet volume, where $\alpha$ is allowed to vary. Other parameters are $(\mathrm{Bi},\mathrm{\Theta })=(1,1)$. The value $s=0.2$ is used throughout.

Figure 10

Time evolution of initial states. Lines with circles: the equilibrium solution generated using the boundary-value solvers in Sec. 3. Solid lines: the equilibrium solution evolved over 100 dimensionless time units using the precursor-film method. The inset in panel (a) shows the distribution of the parameter values used in the parameter study. We also use a dimensionless precursor-film thickness $\varepsilon =0.001$. Other numerical parameters are $\mathrm{\Delta }x=\mathrm{\Delta }y=0.005$, $\mathrm{\Delta }t=0.05$, and $s=0.2$.

Figure 11

Evolution of the ring rupture process with axisymmetric heating profile $T_s\left(r\right)={\mathrm{e}}^{-r^2/0.4^2}$. Relative temperature is shown on the left and the stream function is shown on the right. The scale is maintained throughout the snapshots. The precursor-film parameters are $(\mathcal{A},\varepsilon ,m,n)=(50,0.01,2,3)$ and all other parameters $(\mathrm{Ma},\mathrm{Bi},\mathrm{\Theta })$ are taken to be unity. At $t=0$, the localized heating is turned on and the droplet is no longer at equilibrium, hence the streamlines intersecting with the droplet surface.

Figure 12

The height of the droplet at $r=0$ for the ring rupture process in Fig. 11.

Figure 13

An $xy$ slice of the $z$ vorticity ${\omega }_z$ in an off-center heated droplet using the precursor film description with different hotspot size. The precursor parameters are $(\mathcal{A},\varepsilon ,m,n)=(50,0.01,2,3)$, heating location at $(x,y)=(-0.01,0)$, and all other parameters $(\mathrm{Ma},\mathrm{Bi},\mathrm{\Theta })$ are taken to be unity.

Figure 14

Droplet characteristics for two different heating locations: $x_0=-0.2$ and $x_0=-0.3$. (a) Time evolution of ${max}_{x,y}{\omega }_z(x,y,z_0,t)$ where $z_0=max\left(h\right)/2$. (b) Time evolution of the droplet center. Aside from the heating location, all parameters are the same as Fig. 13.

Figure 15

The $z$ component of the vorticity ${\omega }_z$ with pinned circular contact line at $r=1$ and $\alpha =0.6$. (a) Off-centered heating, with hotspot location at $(x,y)=(-0.001,0)$. (b) Centered heating. Dashed line represents the level-zero contour, which divides the droplet into four circulation regions. All other parameters $(\mathrm{Ma},\mathrm{Bi},\mathrm{\Theta })$ are taken to be unity.

Figure 16

Relative surface temperature profile $\vartheta (x,y;h)$ of Fig. 15.

Figure 17

Plot showing how the shooting method is insensitive to the choice of the regularization parameter $\delta$. Parameter values $\mathrm{Ma}=\mathrm{\Theta }=\mathrm{Bi}=1$, $\alpha =0.6$, $w=0.2$.

Figure 18

Plot showing the first four eigenfunctions for $k=1$. Solid lines: $N=100$. The legend shows the eigenvalues. Open circles: $N=50$. Solid lines: $N=100$. Other parameters: $(\mathrm{Ma},\mathrm{Bi},\mathrm{\Theta })=(1,1,1)$, $\alpha =0.6$, and $w=0.2$.

Figure 19

Example of (a) a spectral grid in a Cartesian arrangement and (b) the mapping onto a polar arrangement. The pole at $r=0$ is avoided by construction, the periodic boundary condition is applied in the azimuthal direction, and the only actual boundary is the $r_1$ row corresponding to the contact line $r=1$. It is also crucial that every node has a reflection through the origin for the derivatives to be computed.

Figure 20

Convergence analysis showing the dependence of the residual $r_n$ on $N$ and $M$. Parameter values: $(\mathrm{Ma},\mathrm{Bi},\mathrm{\Theta })=(1,1,1)$, $\alpha =0.6$, and $w=0.2$. The results have been produced using matlab.