Radial source flows in porous media: Linear stability analysis of axial and helical perturbations in miscible displacements

Abstract

Linear stability results are presented for axial and helical perturbation waves in radial porous media displacements involving miscible fluids of constant density. A numerical eigenvalue problem is formulated and solved in order to evaluate the relevant dispersion relations as functions of the Peclet number and the viscosity ratio. In contrast to the constant algebraic growth rates of purely azimuthal perturbations [C. T. Tan and G. M. Homsy, Phys. Fluids 30, 1239 (1987)], axial perturbations are seen to grow with a time-dependent growth rate. As a result, there exists a critical time up to which the most dangerous axial wavenumbers are larger, and beyond which the most dangerous azimuthal wavenumbers have higher values. This raises the possibility that early on, the smaller flow scales appear in the axial direction, whereas the later flow stages are dominated by smaller azimuthal features. By rescaling the axial wavenumber, the explicit appearance of time can be eliminated. The maximum growth rate of axial perturbations, as well as their most dangerous and cutoff wavenumbers, are seen to increase with the Peclet number and the viscosity ratio. The most dangerous wavenumber is observed to shift towards the lower end of the spectrum as the Peclet number increases. With increasing viscosity contrast, it first moves towards the lower part of the spectrum, only to shift towards the higher end later on. In the limit of large Pe, asymptotic solutions are obtained for the growth of axial disturbances. Numerical solutions of the full eigenvalue problem generally show good agreement with these asymptotic solutions for large Peclet numbers. Over the entire range of wave vector directions between the purely axial and azimuthal extrema, helical waves display an approximately constant maximum growth rate. The wavenumber of maximum growth as well the maximum growth rate of helical waves can be evaluated from the corresponding purely azimuthal and axial problems. This suggests that in three-dimensional flows the nature of the initial conditions plays an important role.