Stability Of Shear Shallow Water Flows with Free Surface

Stability of inviscid shear shallow water flows with free surface is studied in the framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is shown that all shear flows having monotonic convex velocity profiles are stable. The hydrodynamic approximations of the model corresponding to the classes of flows with piecewise linear continuous and discontinuous velocity profiles are derived and studied. It is shown that these approximations possess Hamiltonian structure and a complete system of Riemann invariants, which are found in an explicit form. Sufficient conditions for hyperbolicity of the governing equations for such multilayer flows are formulated. The generalization of the above results to the case of stratified fluid is less obvious, however, it is established that vorticity has a stabilizing effect.

[1]  Is Landau Damping Possible in a Shear Fluid Flow? , 2013 .

[2]  J. J. Stoker.  Water Waves: The Mathematical Theory with Applications , 1957 .

[3]  G. Russo,et al.  ANALYTICAL AND NUMERICAL SOLUTIONS OF THE SHALLOW WATER EQUATIONS FOR 2D ROTATIONAL FLOWS , 2004 .

[4]  Y. Manin,et al.  Conservation laws and Lax representation of Benney's long wave equations , 1979 .

[5]  E. Tabak,et al.  Stability Properties and Nonlinear Mappings of Two and Three‐Layer Stratified Flows , 2009 .

[6]  P. L. Sachdev.  Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems , 2000 .

[7]  Ronald C. Davidson,et al.  Methods in Nonlinear Plasma Theory , 1973 .

[8]  L. V. Ovsyannikov.  Two-layer “Shallow water” model , 1979 .

[9]  M. Pavlov,et al.  Reductions of Kinetic Equations to Finite Component Systems , 2012 .

[10]  Y. Manin,et al.  Equations of long waves with a free surface. II. Hamiltonian structure and higher equations , 1978 .

[11]  S. Gavrilyuk,et al.  Linear Stability of Parallel Inviscid Flows of Shallow Water and Bubbly Fluid , 2004 .

[12]  Rama Govindarajan,et al.  An Introduction to Hydrodynamic Stability , 2010 .

[13]  Vladimir E. Zakharov,et al.  Benney equations and quasiclassical approximation in the method of the inverse problem , 1980 .

[14]  N. C. Freeman.  Simple waves on shear flows: similarity solutions , 1972, Journal of Fluid Mechanics.

[15]  L. V. Bogdanov,et al.  Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type , 2003, nlin/0312013.

[16]  J. C. Burns.  Long waves in running water , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  V. Teshukov,et al.  Characteristic properties of the system of equations of a shear flow with nonmonotonic velocity profile , 1995 .

[18]  D. J. Benney.  Some Properties of Long Nonlinear Waves , 1973 .

[19]  T. H. Stix,et al.  The Theory Of Plasma Waves , 1962 .

[20]  A. Chesnokov.  On the Propagation of Long‐Wave Perturbations in a Two‐Layer Free‐Boundary Rotational Fluid , 2004 .

[21]  E. Tabak,et al.  Shear instability for stratified hydrostatic flows , 2009 .