Dynamics of a silicone oil drop submerged in a stratified ethanol-water bath

We analyze numerically the Marangoni flow around an immiscible droplet submerged in a stably stratified mixture of ethanol and water. The linear stability analysis shows that the base flow undergoes a supercritical Hopf bifurcation that leads to oscillations. The theoretical prediction for the critical droplet radius is consistent with previous experimental results. Ethanol diffusion in water is critical in the flow stability for both low and high droplet viscosity. Direct numerical simulations of the nonlinear oscillatory flow show that the frequency of those oscillations approximately equals that of the critical eigenmode. The nonlinear convective term of the ethanol diffusive-convective transport equation fixes the amplitude of the droplet oscillations. The viscous dissipation associated with the Marangoni flow inside the droplet considerably reduces the oscillation.

Figure 1

A sketch of the levitating drop. Deeper orange indicates higher water concentration. The ring around the droplet represents the kinematic boundary layer set by the Marangoni flow indicated by the black arrows. The ethanol concentration inside this layer increases and is homogenized by the Marangoni convection. The blue arrows represent the diffusion of ethanol from the surrounding mixture toward the boundary layer.

Figure 2

Sketch of the computational domain. The black semicircumference represents the outer boundary of the numerical domain, which solidly moves with the droplet.

Figure 3

Droplet vertical position $h\left(t\right)$ as a function of time for 100-cSt silicone oil droplets with different radii.

Figure 4

Ethanol mass fraction $Y_e$ (left) and streamlines (right) for $\{R_d=39\mu \mathrm{m}, {\mu }^{\left(i\right)}=5\mathrm{cSt}\}$ and $\{R_d=80\mu \mathrm{m}, {\mu }^{\left(i\right)}=100\mathrm{cSt}\}$. The maximum velocity is 0.423 mm/s and 0.321 mm/s for ${\mu }^{\left(i\right)}=5$ and 100 cSt, respectively.

Figure 5

Spectrum of eigenvalues for $\{R_d=39\mu \mathrm{m}, {\mu }^{\left(i\right)}=5\mathrm{cSt}\}$ (black squares) and $\{R_d=80\mu \mathrm{m}, {\mu }^{\left(i\right)}=100\mathrm{cSt}\}$ (red circles). The figure shows the eigenvalues with ${\omega }_i>-0.05{\mathrm{s}}^{-1}$ and ${\omega }_r>0$. The arrows indicate the eigenvalue of the critical eigenmode.

Figure 6

Real part of the eigenfunction of the ethanol mass fraction $Y_e$ (left) and velocity field (right) for $\{R_d=39\mu \mathrm{m}, {\mu }^{\left(i\right)}=5\mathrm{cSt}\}$ and $\{R_d=80\mu \mathrm{m}, {\mu }^{\left(i\right)}=100\mathrm{cSt}\}$.

Figure 7

Critical droplet radius $R_d$ as a function of the droplet viscosity.

Figure 8

Critical droplet radius $R_d$ (solid symbols) and maximum velocity $v_{\text{max}}$ (open symbols) as a function of the diffusion coefficient $D_e$ for $Y=0$. The droplet viscosity is 5 cSt.

Figure 9

Critical droplet radius $R_d$ (solid symbols) and maximum velocity $v_{\text{max}}$ (open symbols) as a function of the diffusion coefficient $D_e$ for $Y=0$. The droplet viscosity is 100 cSt.

Figure 10

Amplitude of the droplet oscillation as a function of the droplet radius for ${\mu }^{\left(i\right)}=5$ (solid symbols) and 100 cSt (open symbols). Here, $h_{\text{max}}$ and $h_{\text{min}}$ are the maximum and minimum values of $h\left(t\right)$, respectively, and $R_d^{*}$ is the critical droplet radius.

Figure 11

Droplet vertical position $h$, velocity at the droplet's equator $v_{\text{max}}$, ethanol concentration $Y_e$, diffusion coefficient $D_e$, outer viscosity ${\mu }^{\left(o\right)}$, density gradient $d{\rho }^{\left(o\right)}/dz$, and concentration gradient $d{Y}_e/dz$ as a function of time for ${\mu }^{\left(i\right)}=5$ cSt and $R_d=45\mu \mathrm{m}$. The quantities $Y_e, D_e, {\mu }^{\left(o\right)}, d{\rho }^{\left(o\right)}/dz$, and $d{Y}_e/dz$ are evaluated at the droplet's equator.

Figure 12

Droplet vertical position $h$, velocity at the droplet's equator $v_{\text{max}}$, ethanol concentration $Y_e$, diffusion coefficient $D_e$, outer viscosity ${\mu }^{\left(o\right)}$, density gradient $d{\rho }^{\left(o\right)}/dz$, and concentration gradient $d{Y}_e/dz$ as a function of time for ${\mu }^{\left(i\right)}=100$ cSt and $R_d=80\mu \mathrm{m}$. The quantities $Y_e, D_e, {\mu }^{\left(o\right)}, d{\rho }^{\left(o\right)}/dz$, and $d{Y}_e/dz$ are evaluated at the droplet's equator.

Figure 13

FFT (Power Spectrum Density, PSD) of the experimental values of $h\left(t\right)$ over the time interval [69.76,335.86] s for ${\mu }^{\left(i\right)}=5$ cSt and $R_d=45\mu \mathrm{m}$. The dashed vertical line indicates the frequency of the dominant eigenmode, $\omega =0.1552$ rad/s.

Figure 14

Computational subdomains and grids for the original and mapped variables. The green (magenta) lines represent the oil (water-ethanol) mesh in the real space (right panel) and in the computational domain (left panel).