Effect of viscosity and shear flow on the nonlinear two fluid interfacial structures

Abstract

A nonlinear formulation is presented to deal with the combined action of Rayleigh-Taylor and Kelvin-Helmholtz instabilities as well as combined Ricthmyer-Meshkov and Kelvin-Helmholtz instabilities at the two fluid interface under the influence of viscosity and consequent shear flow. Using Layzer's model, the development of the interfacial structures like bubbles is investigated analytically and numerically. It is found that the growth and normal velocity of the structures are dependent on the relative velocity shear and the kinematic coefficient of viscosity of both the fluids. Both the bubble growth and growth rate are reduced significantly for fluids of higher viscosity coefficient with small velocity shear difference. It is also observed that, for viscous fluids, the transverse velocity of the perturbed interface becomes slower under certain conditions.

Influence of surface tension on two fluids shearing instability

“…[17]. In absence of surface tension, the asymptotic values coincide with the results obtained in our previous work [14].…”

“…According to the extended Layzer model [10,11,14,20], the velocity potentials describing the motion for the upper (heavier) and lower (lighter) fluids are assumed to be given by…”

“…In our previous works [14,[20][21][22][23], we have considered η 1 (t) = 0 due to the absence of velocity shear parallel to the unperturbed interface. However, in presence of streaming motion of the fluids, the tip of the bubble moves parallel to unperturbed interface with velocity η1 (t).…”

“…We have restricted our study near the peak of the perturbed structure where |k(x−η 1 (t))| ≪ 1. Thus, we can neglect the terms of O(|x−η 1 | i ) (i ≥ 3) [14]. With this point of view, we have…”

“…Layzer [10] described the formation of the structure using an expansion near the tip of the bubble or the spike up to second order in the transverse coordinates in two dimensional motion and this approach was extended in Ref. [14] for Kelvin-Helmholtz instability. It is well known [15] that the surface tension reduces the linear Rayleigh-Taylor Growth rate.…”