Well-Posedness of the Prandtl Boundary Layer Equations for the Upper Convected Maxwell Fluid

2013 ◽  
Vol 27 (3-4) ◽  
pp. 981-988 ◽  
Author(s):  
Michael Renardy
Keyword(s):  
Boundary Layer ◽  
Maxwell Fluid ◽  
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.

Numerical Study of Non-Newtonian Maxwell Fluid in the Region of Oblique Stagnation Point Flow over a Stretching Sheet

2015 ◽  
Vol 32 (2) ◽  
pp. 175-184 ◽  
Author(s):  
T. Javed ◽  
A. Ghaffari
Keyword(s):  
Boundary Layer ◽  
Stagnation Point ◽  
Shooting Method ◽  
Stretching Sheet ◽  
Numerical Study ◽  
Maxwell Fluid ◽  
Governing Equations ◽  
Boundary Layer Equations ◽  
Parameter Ratio ◽  
Two Dimensional Flow

AbstractIn this article, a numerical study is carried out for the steady two-dimensional flow of an incompressible Maxwell fluid in the region of oblique stagnation point over a stretching sheet. The governing equations are transformed to dimensionless boundary layer equations. After some manipulation a system of ordinary differential equations is obtained, which is solved by using parallel shooting method. A comparison with the previous studies is made to show the accuracy of our results. The effects of involving parameters are discussed in detail and the streamlines are drawn to predict the flow pattern of the fluid. It is observed that increasing velocities ratio parameter (ratio of straining to stretching velocity) helps to decrease the boundary layer thickness. Furthermore, the velocity of fluid increases by increasing the obliqueness parameter.

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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