scholarly journals Well-posedness issues for the Prandtl boundary layer equations

2013 ◽  
pp. 1-10 ◽  
Author(s):  
David Gérard-Varet ◽  
Nader Masmoudi
Keyword(s):  
Boundary Layer ◽  
Well Posedness ◽  
Boundary Layer Equations ◽  
Prandtl Boundary Layer

Potential Systems and Nonlocal Conservation Laws of Prandtl Boundary Layer Equations on the Surface of a Sphere

2017 ◽  
Vol 72 (4) ◽  
pp. 351-357 ◽  
Author(s):  
R. Naz
Keyword(s):  
Boundary Layer ◽  
Conservation Laws ◽  
Linear Combination ◽  
Conservation Law ◽  
Local Conservation ◽  
Boundary Layer Equations ◽  
Potential System ◽  
Nonlocal Conservation Laws ◽  
Prandtl Boundary Layer ◽  
Nonlocally Related Systems

Abstract:The potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere have been investigated. The multiplier approach yields two local conservation laws for the Prandtl boundary layer equations on the surface of a sphere. Two potential variables ψ and ϕ are introduced corresponding to first and second conservation law. Moreover, another potential variable p is introduced by considering the linear combination of both conservation laws. Two level one potential systems involving a single nonlocal variable ψ or ϕ are constructed. One level two potential system involving both nonlocal variables ψ and ϕ is established. The nonlocal variable p is utilised to derive a spectral potential system. The nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere are derived by computing the local conservation laws of its potential systems. The nonlocal conservation laws are utilised to derive the further nonlocally related systems.

scholarly journals ENERGY LOSSES UNDER WAVE ACTION

1970 ◽  
Vol 1 (12) ◽  
pp. 16 ◽  
Author(s):  
P.D. Treloar ◽  
A. Brebner
Keyword(s):  
Boundary Layer ◽  
Energy Dissipation ◽  
Wave Height ◽  
Dissipation Function ◽  
Wave Action ◽  
Energy Losses ◽  
Boundary Layer Equations ◽  
Wave Height Attenuation ◽  
Prandtl Boundary Layer ◽  
Made In

Wave-height attenuation measurements were made in two identical flumes of different widths and the results used to separate bottom energy losses from sidewall energy losses These energy losses, in the form of rates of energy dissipation, were then compared with their theoretical values as calculated by solving the linearized Prandtl boundary layer equations and evaluating the Rayleigh dissipation function Using these results, an adjusted formula for the wave-height attenuation modulus was determined.

Oscillatory solutions of the Falkner-Skan equation

1985 ◽  
Vol 397 (1813) ◽  
pp. 415-418 ◽  
Keyword(s):  
Boundary Layer ◽  
Periodic Solution ◽  
Incompressible Flow ◽  
Symbolic Dynamics ◽  
Similarity Solutions ◽  
Oscillatory Solutions ◽  
Smale Horseshoe ◽  
Boundary Layer Equations ◽  
Prandtl Boundary Layer ◽  
Infinitely Many Periodic Solutions

The Falkner-Skan equation for similarity solutions of the Prandtl boundary-layer equations for incompressible flow is analysed for both positive and negative values of the parameter β . For β < — 1 branches of solutions with any number of intervals of overshoot are found analytically, and confirm recent numerical results. For β > 1 we have proved that there is a periodic solution. We conjecture that for β > 2 there are infinitely many periodic solutions and that a form of ‘symbolic dynamics’, of the kind associated with a Smale ‘horseshoe map’ can be constructed. We have shown this rigorously for β close to 2.

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