The Taylor-Saffman Instability Criteria

Therefore, as previously stated, we assume that we have a pair of superposed Newtonian fluids. We denote the vertical upward coordinate as z, with x and y being parallel to the plane of the horizontal interface that separates the two fluids. We fix the fluid interface as being initially at z = 0. There is no advantage for this particular problem in carrying out the analysis in terms of dimensionless variables. Hence we simply retain dimensional equations throughout. For simplicity, we therefore drop the use of primes to denote dimensional variables. All variables in what follows are dimensional. 

We should note that this description of the system as being two fluids separated by a flat interface already has inherent in it the spatial averaging to a scale of resolution that is much larger than the individual pore level of description. The volume-averaged velocity in each fluid is determined by Darcy s law. As noted earlier, we assume that the fluids (and the interface between them) move with a uniform velocity V in the positive z direction. It is therefore convenient to consider the problem with respect to a moving reference frame that is fixed at the unperturbed fluid interface, i.e., we introduce z, which is related to the original laboratory frame of reference as 

From this perspective of the moving coordinate z, the two fluids at infinity appear to be stationary. 

We assume that the flat interface between the two fluids, now designated as z = 0, is perturbed with an arbitrary infinitesimal perturbation of shape. As usual, for a linear stability analysis, we consider only a single Fourier mode in each of the x and y directions, with the wave number (or wavelength) as a parameter in the stability analysis. Hence we consider a perturbation of the form 

In the present analysis, we seek to determine the fate of a perturbation to the interface shape when there is a mean, uniform velocity V in the upward direction normal to the interface. Hence, in the laboratory reference frame 

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