scholarly journals Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics

2020 ◽  
Vol 71 (4) ◽  
Author(s):  
Yuri Trakhinin
Keyword(s):  
Shock Waves ◽  
Shallow Water ◽  
Structural Stability ◽  
Vortex Sheets ◽  
Shallow Water Magnetohydrodynamics

Assessment of Structural Stability of Pure and Xylenol Orange Dye Doped Potassium Dihydroegn Phosphate Probed by Shock Waves

2021 ◽  
Author(s):  
A. Sivakumar ◽  
A. Saranraj ◽  
S. Sahaya Jude Dhas ◽  
P. Sivaprakash ◽  
S. Arumugam ◽  
...  
Keyword(s):  
Shock Waves ◽  
Structural Stability ◽  
Xylenol Orange

scholarly journals Structural stability of shock waves in 2D compressible elastodynamics

2019 ◽  
Vol 378 (3-4) ◽  
pp. 1471-1504 ◽  
Author(s):  
Alessandro Morando ◽  
Yuri Trakhinin ◽  
Paola Trebeschi
Keyword(s):  
Shock Waves ◽  
Structural Stability

Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear

2013 ◽  
Vol 723 ◽  
pp. 289-317 ◽  
Author(s):  
Andrew L. Stewart ◽  
Paul J. Dellar
Keyword(s):  
Shallow Water ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Three Dimensional ◽  
Direct Calculation ◽  
Shear Instability ◽  
Vertical Shear ◽  
Vortex Sheets ◽  
Multilayer Shallow Water Equations ◽  
Complete Coriolis Force

AbstractWe analyse the hyperbolicity of our multilayer shallow water equations that include the complete Coriolis force due to the Earth’s rotation. Shallow water theory represents flows in which the vertical shear is concentrated into vortex sheets between layers of uniform velocity. Such configurations are subject to Kelvin–Helmholtz instabilities, with arbitrarily large growth rates for sufficiently short-wavelength disturbances. These instabilities manifest themselves through a loss of hyperbolicity in the shallow water equations, rendering them ill-posed for the solution of initial value problems. We show that, in the limit of vanishingly small density difference between the two layers, our two-layer shallow water equations remain hyperbolic when the velocity difference remains below the same threshold that also ensures the hyperbolicity of the standard shallow water equations. Direct calculation of the domain of hyperbolicity becomes much less tractable for three or more layers, so we demonstrate numerically that the threshold for the velocity differences, below which the three-layer equations remain hyperbolic, is also unchanged by the inclusion of the complete Coriolis force. In all cases, the shape of the domain of hyperbolicity, which extends outside the threshold, changes considerably. The standard shallow water equations only lose hyperbolicity due to shear parallel to the direction of wave propagation, but the complete Coriolis force introduces another mechanism for loss of hyperbolicity due to shear in the perpendicular direction. We demonstrate that this additional mechanism corresponds to the onset of a transverse shear instability driven by the non-traditional components of the Coriolis force in a three-dimensional continuously stratified fluid.

Constitutive surfaces in fluid mechanics

1980 ◽  
Vol 88 (3) ◽  
pp. 517-546 ◽  
Author(s):  
M. J. Sewell ◽  
D. Porter
Keyword(s):  
Flow Stress ◽  
Shock Waves ◽  
Fluid Mechanics ◽  
Total Energy ◽  
Special Reference ◽  
Shallow Water ◽  
Gas Dynamics ◽  
Inviscid Fluid ◽  
Hydraulic Jumps ◽  
Shallow Water Theory

AbstractThe new concept of a constitutive surface is introduced into inviscid fluid mechanics, with special reference to compressible gas dynamics and to shallow water theory. The detailed shape of such surfaces is calculated, including the manner of their transition across singularities where the Mach or Froude number passes through unity. The calculation draws upon recent work describing the transition of a Legendre transformation through its singularity. For example, mass flow Q, total energy h and flow stress P are always related by part of a ‘swallowtail’ surface, regardless of the particular motion.The addition of dynamical conditions defines particle history tracks which always lie on constitutive surfaces even for unsteady flow, except that they may jump from one part to another of such a surface when shock waves or hydraulic jumps are passed through.Illustrations given include the steady flow of a general gas through a standing normal shock, general shallow water theory, and flow along a sloping-sided channel. Connections with existing literature are described.

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