A. Rogerson et E. Meiburg, SHEAR STABILIZATION OF MISCIBLE DISPLACEMENT PROCESSES IN POROUS-MEDIA, Physics of fluids. A, Fluid dynamics, 5(6), 1993, pp. 1344-1355
The interface region between two fluids of different densities and vis
cosities in a porous medium in which gravity is directed at various an
gles to the interface is analyzed. Under these conditions, base states
exist that involve both tangential and normal velocity components. Th
ese base states support traveling waves. In the presence of a normal d
isplacement velocity, the amplitude of these waves grows according to
the viscous fingering instability. For the immiscible case, it can eas
ily be shown that the growth rate is not affected by the tangential ve
locities, while surface tension results in the usual stabilization. Fo
r the case of two miscible fluids, the stability of the base states us
ing the quasi-steady-state approximation is investigated. The resultin
g equations are solved analytically for time t=0 and a criterion for i
nstability is formulated. The stability of the flow for times t>0 is i
nvestigated numerically using a spectral collocation method. It is fou
nd that the interaction of pressure forces and viscous forces is modif
ied by tangential shear as compared to the classical problem, resultin
g in a stabilizing effect of the tangential shear. The key to understa
nding the physical mechanism behind this stabilization lies in the vor
ticity equation. While the classical problem gives rise to a dipole st
ructure of the vorticity field, tangential shear leads to a quadrupole
structure of the perturbation vorticity field, which is less unstable
. This quadrupole structure is due to the finite thickness of the tang
ential base state velocity profile, i.e., the finite thickness of the
dispersively spreading front, and hence cannot emerge on the sharp fro
nt maintained in immiscible displacements.