Asymptotic Solution of the Boltzmann-Krook Equation for the Rayleigh Shear Flow Problem

The problem solved is that of the interaction between a laminar boundary layer on a semi-infinite flat plate and an oncoming shear flow of finite lateral dimensions bounded by uniform irrotational flow extending to infinity. The pressures along the plate and upstream of the same are deduced (to a linearized approximation) in the form of a Fourier integral based on the solution of a simpler periodic flow problem. It is found that while the assumption of an infinite, uniform shear flow gives asymptotically correct interaction pressure gradients on the plate near the leading edge, the pressure level even there (compared to upstream infinity) is strongly influenced by the boundedness of the external shear. At distances from the leading edge which are large compared to the lateral extent of the shear flow, the pressure gradients along the plate are shown to be vanishingly smaller than in the infinite shear case.

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On shear flow past flat plates

Journal of Fluid Mechanics ◽

10.1017/s0022112062001354 ◽

1962 ◽

Vol 14 (3) ◽

pp. 452-462 ◽

Cited By ~ 3

Author(s):

Richard M. Mark

Keyword(s):

Boundary Layer ◽

Approximate Solution ◽

Shear Flow ◽

Asymptotic Solution ◽

Constant Pressure ◽

Viscous Incompressible Fluid ◽

Two Dimensional ◽

Flat Plates ◽

Exact Calculations ◽

Good Agreement

The boundary layer on a semi-infinite flat plate placed in a two-dimensional, unbounded, steady, constant shear flow of a viscous incompressible fluid is examined on the basis of the constant-pressure assumption. An asymptotic solution is obtained first for large vorticity numbers. Then an approximate solution is found for arbitrary vorticity numbers that gives good agreement with exact calculations for the extreme cases of small and large vorticity numbers. The present calculations are limited to the boundary layer on the top side of the plate only.

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A shear flow problem for compressible viscous micropolar fluid: Uniqueness of a generalized solution

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.5727 ◽

2019 ◽

Vol 42 (18) ◽

pp. 6358-6368 ◽

Cited By ~ 1

Author(s):

Loredana Simčić

Keyword(s):

Shear Flow ◽

Micropolar Fluid ◽

Flow Problem ◽

Generalized Solution

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Matched asymptotics for two-dimensional planing hydrofoils with spoilers

Journal of Fluid Mechanics ◽

10.1017/s0022112097008215 ◽

1998 ◽

Vol 358 ◽

pp. 259-281 ◽

Cited By ~ 6

Author(s):

G. M. FRIDMAN

Keyword(s):

Asymptotic Solution ◽

Flat Plate ◽

Linear Theory ◽

Asymptotic Expansions ◽

Flow Problem ◽

Matched Asymptotic Expansions ◽

Asymptotic Solutions ◽

Two Dimensional ◽

Trailing Edge ◽

Matched Asymptotics

The purpose of the paper is to demonstrate the effectiveness of the matched asymptotic expansions (MAE) method for the planing flow problem. The matched asymptotics, taking into account the flow nonlinearities in those regions where they are most pronounced (i.e. in the vicinity of the edges), are shown to significantly extend the range where the linear theory gives good results. Two model problems are used: the planing flat plate with a spoiler on the trailing edge and the curved planing foil. Asymptotic solutions obtained by the MAE method are compared with those obtained using linear and exact nonlinear theories. Based on the results, the asymptotic solution to the planing problem under the gravity is proposed.

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