Abstract
We consider a non-standard eigenvalue problem arising in stability studies of 3-layer immiscible porous media and Hele-Shaw flows which contain the viscous profile of the middle layer as a coefficient in the eigenvalue problem. We characterize the eigenvalues and eigenfunctions of this eigenvalue problem. We then apply this characterization to an exponential viscous profile and numerically compute the associated eigenvalues and eigenfunctions. We provide an explicit sequence of numbers that give upper and lower bounds on the eigenvalues. We also discuss the limiting cases when either the length of the middle layer approaches zero or the exponential viscous profile approaches a constant viscosity profile.
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Gin, C., Daripa, P. A Study of a Non-Standard Eigenvalue Problem and its Application to Three-Layer Immiscible Porous Media and Hele-Shaw Flows with Exponential Viscous Profile. J. Math. Fluid Mech. 17, 155–181 (2015). https://doi.org/10.1007/s00021-014-0196-z
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DOI: https://doi.org/10.1007/s00021-014-0196-z