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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References
Boyd J.P.: Chebyshev and Fourier Spectral Methods. Dover Publications, New York (2001)
Churchill R.V.: Expansions in series of non-orthogonal functions. Bull. Am. Math. Soc. 48, 143–150 (1942)
Daripa P.: Hydrodynamic stability of multi-layer Hele-Shaw flows. J. Stat. Mech. Theory Exp. 12, 28 (2008)
Daripa, P.: Studies on stability in three-layer Hele-Shaw flows. Phys. Fluids 20 (2008)
Daripa P., Ding X.: A numerical study of instability control for the design of an optimal policy of enhanced oil recovery by tertiary dispalcement processes. Transp. Porous Media 93(3), 673–703 (2012)
Daripa P., Pasa G.: On the Growth Rate for Three-Layer Hele-Shaw Flows: Variable and Constant Viscosity Cases. Int. J. Engg. Sci 43, 877–884 (2005)
Daripa, P., Pasa, G.: A simple derivation of an upper bound in the presence of a viscosity gradient in three-layer Hele-Shaw flows. J. Stat. Mech. pp. 11 (2006). doi:10.1088/1742-5468/2006/01/P01014
Gin, C.: Stability and motion of interfaces in Hele-Shaw models of porous media flows in chemical enhanced oil recovery, Ph.D Thesis, Department of Mathematics, Texas A&M University (2015)
Gorell S.B., Homsy G.M.: A theory of the optimal policy of oil recovery by the secondary displacement process. SIAM J. Appl. Math. 43, 79–98 (1983)
Homsy G.M.: Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311 (1987)
Ince E.L.: Ordinary Differential Equations. Dover Publications, New York (1956)
Saffman P.G.: Viscous fingering in Hele–Shaw cells. J. Fluid Mech. 173, 73–94 (1986)
Saffman P.G., Taylor G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245, 312–329 (1958)
Schmid P.J., Henningson D.S.: Stability and Transition in Shear Flows. Springer, New York (2001)
Trefethen, L.N.: Spectral Methods in MATLAB, Software, Environments, Tools, Society for Industrial and Applied Mathematics, Philadelphia (2000)
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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