Stability analysis of shear flows in a Hele-Shaw cell

A mathematical model describing motion of an inhomogeneous incompressible fluid in a Hele-Shaw cell is considered. Linear stability analysis of shear flow class is provided. The role of inertia, linear friction and impermeable boundaries in Kelvin-Helmholtz instability is studied. Hierarchy of simplified one-dimensional models of viscosity- and density-stratified flows is obtained in long-wave approximation. Interpretation of Saffman-Taylor instability is given in the framework of these models.

[1]  J. Azaiez,et al.  Inertial effects of miscible viscous fingering in a Hele-Shaw cell , 2015 .

[2]  Christophe Chevalier,et al.  Inertial effects on Saffman–Taylor viscous fingering , 2006, Journal of Fluid Mechanics.

[3]  O. E. Ivashnev,et al.  Effect of small parameters on the structure of the front formed by unstable viscous-fluid displacement from a Hele-Shaw cell , 2007 .

[4]  N. N. Yanenko,et al.  Systems of Quasilinear Equations and Their Applications to Gas Dynamics , 1983 .

[5]  G. Taylor,et al.  The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  H. Stone,et al.  Control of interfacial instabilities using flow geometry , 2012, Nature Physics.

[7]  J. Miranda,et al.  Influence of inertia on viscous fingering patterns: rectangular and radial flows. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[9]  R. Govindarajan,et al.  Instabilities in Viscosity-Stratified Flow , 2014 .

[10]  A. Chesnokov,et al.  Shear Hele-Shaw flows of a weakly compressible liquid , 2015 .

[11]  George M. Homsy,et al.  Viscous fingering in porous media , 1987 .

[12]  P. Drazin.  Introduction to Hydrodynamic Stability , 2002 .

[13]  A. Chesnokov,et al.  Shallow Water Equations for Shear Flows , 2011 .

[14]  Matthias Heil,et al.  Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells , 2013, Journal of Fluid Mechanics.

[15]  G. Homsy,et al.  Stability of miscible displacements in porous media: Rectilinear flow , 1986 .

[16]  M. Rabaud,et al.  Shear instability of two-fluid parallel flow in a Hele–Shaw cell , 1997 .