Theoretical analysis on the onset of buoyancy-driven instability of horizontal interfaces between miscible fluids in a Hele–Shaw cell

Abstract

The stability of the initially sharp interface between two miscible fluids saturated in a Hele–Shaw cell is analyzed. Based on the 2-dimensional Navier–Stokes–Brinkman (NSB) equation and its modification, new linear stability equations are derived in the similar (τ, ζ)-domain, and solved analytically and numerically with and without a quasi-steady state approximation (QSSA). Through the exact eigenanalysis without the QSSA, it is found that the system is unconditionally stable and the most unstable initial disturbance exists. The initial value problem analysis is conducted in the global (τ, z)-domain and also the similar (τ, ζ)-domain. It is interesting that for the present system, the linear stability characteristics obtained using the various methods support to each other. Using the linear stability results as a starting point, fully-nonlinear analysis is also carried out. The present linear and nonlinear studies suggest that the Brinkman's correction plays an important role in the onset and development of disturbance fields.