Shape oscillations of a viscoelastic droplet suspended in a viscoelastic host liquid

A study on the small-amplitude oscillation of a viscoelastic droplet suspended in an immiscible viscoelastic host liquid is carried out. The viscoelasticity of the inner and outer liquids is described by Jeffreys constitutive equation. The analytical characteristic equation is derived and the complex frequency is solved numerically. The effect of the outer host liquid on the oscillation of the droplet is examined for the fundamental mode $n=2$. It is found that the damping rate and the frequency of oscillation of the droplet are decreased as the host liquid gets denser in the inviscid case. The boundaries between periodic oscillation and aperiodic decay in the parametric plane of the Ohnesorge number and the relative stress relaxation time are captured. The viscosity of the outer liquid makes the supercritical and subcritical bifurcation phenomena disappear and leads to periodic oscillations of the droplet all the time. Moreover, the viscosity of the outer liquid exhibits a dual effect on the decay of the amplitude of oscillation. The elasticity of the outer liquid affects the oscillation of the droplet monotonically: the stress relaxation time decreases the damping rate and the frequency of oscillation of the droplet, whereas the strain retardation time increases them limitedly.

Figure 1

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the Ohnesorge number of the inner liquid Oh for different values of the density ratio ${\rho }_r$. The case of relatively large elasticity of the inner liquid, ${\lambda }_{1r,i}=30, {\lambda }_{2r,i}=3$.

Figure 2

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the Ohnesorge number of the inner liquid Oh for different values of the density ratio ${\rho }_r$. The case of relatively small elasticity of the inner liquid, ${\lambda }_{1r,i}=0.018, {\lambda }_{2r,i}=0.0018$.

Figure 3

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the relative stress relaxation time of the inner liquid ${\lambda }_{1r,i}$ for different values of the density ratio ${\rho }_r$. The case of relatively large viscosity of the inner liquid, $\text{Oh}=10$.

Figure 4

The boundaries in the ${\lambda }_{1r,i}-\text{Oh}$ plane.

Figure 5

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the relative stress relaxation time of the inner liquid ${\lambda }_{1r,i}$ for different values of the density ratio ${\rho }_r$. The case of relatively small viscosity of the inner liquid, $\text{Oh}=0.1$.

Figure 6

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the Ohnesorge number of the inner liquid Oh for different values of the Ohnesorge number of the outer liquid $C$. The case of relatively large elasticity of the inner liquid, ${\lambda }_{1r,i}=30, {\lambda }_{2r,i}=3$. ${\rho }_r=1$.

Figure 7

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the Ohnesorge number of the inner liquid Oh for different values of the Ohnesorge number of the outer liquid $C$. The case of relatively small elasticity of the inner liquid, ${\lambda }_{1r,i}=0.018, {\lambda }_{2r,i}=0.0018$. ${\rho }_r=1$.

Figure 8

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the relative stress relaxation time of the inner liquid ${\lambda }_{1r,i}$ for different values of the Ohnesorge number of the outer liquid $C$. The case of relatively large viscosity of the inner liquid, $\text{Oh}=10$. ${\rho }_r=1$.

Figure 9

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the relative stress relaxation time of the inner liquid ${\lambda }_{1r,i}$ for different values of the Ohnesorge number of the outer liquid $C$. The case of relatively small viscosity of the inner liquid, $\text{Oh}=0.1$. ${\rho }_r=1$.

Figure 10

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the Ohnesorge number of the inner liquid Oh for different values of the relative stress relaxation time of the outer liquid ${\lambda }_{1r,o}$. The case of relatively large elasticity of the inner liquid, ${\lambda }_{1r,i}=30, {\lambda }_{2r,i}=3$. ${\rho }_r=1, C=1$.

Figure 11

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the Ohnesorge number of the inner liquid Oh for different values of the relative stress relaxation time of the outer liquid ${\lambda }_{1r,o}$. The case of relatively small elasticity of the inner liquid, ${\lambda }_{1r,i}=0.018, {\lambda }_{2r,i}=0.0018$. ${\rho }_r=1, C=1$.

Figure 12

(a) The damping rate $\text{Re}\left(\omega \right)$ and (b) the angular frequency $\text{Im}\left(\omega \right)$ versus the Ohnesorge number of the inner liquid Oh for different values of the relative strain retardation time of the outer liquid ${\lambda }_{2r,o}/{\lambda }_{1r,o}$. The case of relatively large elasticity of the inner liquid, ${\lambda }_{1r,i}=30, {\lambda }_{2r,i}=3$. ${\rho }_r=1, C=1, {\lambda }_{1r,o}=1$.