Boundary-Layer Similarity Near the Edge of a Rotating Disk

1975 ◽  
Vol 42 (3) ◽  
pp. 584-590 ◽  
Author(s):  
R. J. Bodonyi ◽  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Flow Field ◽  
Boundary Value Problem ◽  
Asymptotic Expansions ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Rotating Disk ◽  
Angular Speed ◽  
Rotating Fluid

Numerical solutions of the similarity equations governing the flow near the edge of a finite rotating disk are found to be possible only for −2.06626 ≤ α ≤ 1, where α is the ratio of the disk’s angular speed to that of the rigidly rotating fluid far from the disk. Furthermore, for α ≤ −1 the solutions of the boundary-value problem are not unique, and along one of the solution branches a singular structure of the flow field is approached as α → −1. Using the method of matched asymptotic expansions an approximate solution is found along the singular branch which explains some of the problems encountered in finding numerical solutions.

On rotationally symmetric flow above an infinite rotating disk

1975 ◽  
Vol 67 (4) ◽  
pp. 657-666 ◽  
Author(s):  
R. J. Bodonyi
Keyword(s):  
Transition Layer ◽  
Numerical Solutions ◽  
Outer Boundary ◽  
Rotating Disk ◽  
Angular Speed ◽  
Wall Region ◽  
Rotating Fluid ◽  
Symmetric Flow ◽  
Rotationally Symmetric ◽  
Rotationally Symmetric Flow

The similarity equations for rotationally symmetric flow above an infinite counter–rotating disk are investigated both numerically and analytically. Numerical solutions are found when α, the ratio of the disk's angular speed to that of the rigidly rotating fluid far from it, is greater than −0.68795. It is deduced that there exists a critical value αcr, of α above which finite solutions are possible. The value of α and the limiting structure as α → αcrare found using the method of matched asymptotic expansions. The flow structure is found to consist of a thin viscous wall region above which lies a thick inviscid layer and yet another viscous transition layer. Furthermore, this structure is not unique: there can be any number of thick inviscid layers, each separated from the next by a viscous transition layer, before the outer boundary conditions on the solution are satisfied. However, comparison with the numerical solutions indicates that a single inviscid layer is preferred.

Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity

2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Maximum Norm ◽  
Lateral Part ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

Nearly equilibrium dissociating boundary-layer flows by the method of matched asymptotic expansions

1966 ◽  
Vol 26 (4) ◽  
pp. 793-806 ◽  
Author(s):  
George R. Inger
Keyword(s):  
Boundary Layer ◽  
Asymptotic Expansions ◽  
High Speed ◽  
Numerical Solutions ◽  
The Body ◽  
Matched Asymptotic Expansions ◽  
Form Analysis ◽  
Perturbation Problem ◽  
Valid Solution ◽  
Non Equilibrium

The approach to equilibrium in a non-equilibrium-dissociating boundary-layer flow along a catalytic or non-catalytic surface is treated from the standpoint of a singular perturbation problem, using the method of matched asymptotic expansions. Based on a linearized reaction rate model for a diatomic gas which facilitates closed-form analysis, a uniformly valid solution for the near equilibrium behaviour is obtained as the composite of appropriate outer and inner solutions. It is shown that, under near equilibrium conditions, the primary non-equilibrium effects are buried in a thin sublayer near the body surface that is described by the inner solution. Applications of the theory are made to the calculation of heat transfer and atom concentrations for blunt body stagnation point and high-speed flat-plate flows; the results are in qualitative agreement with the near equilibrium behaviour predicted by numerical solutions.

scholarly journals Boundary Layers in a Two-Point Boundary Value Problem with a Caputo Fractional Derivative

2015 ◽  
Vol 15 (1) ◽  
pp. 79-95 ◽  
Author(s):  
Martin Stynes ◽  
José Luis Gracia
Keyword(s):  
Boundary Layer ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Order Term ◽  
Boundary Value ◽  
Point Boundary ◽  
Leading Term ◽  
Special Case

AbstractA two-point boundary value problem is considered on the interval $[0,1]$, where the leading term in the differential operator is a Caputo fractional derivative of order δ with $1<\delta <2$. Writing u for the solution of the problem, it is known that typically $u^{\prime \prime }(x)$ blows up as $x\rightarrow 0$. A numerical example demonstrates the possibility of a further phenomenon that imposes difficulties on numerical methods: u may exhibit a boundary layer at x = 1 when δ is near 1. The conditions on the data of the problem under which this layer appears are investigated by first solving the constant-coefficient case using Laplace transforms, determining precisely when a layer is present in this special case, then using this information to enlighten our examination of the general variable-coefficient case (in particular, in the construction of a barrier function for u). This analysis proves that usually no boundary layer can occur in the solution u at x = 0, and that the quantity $M = \max _{x\in [0,1]}b(x)$, where b is the coefficient of the first-order term in the differential operator, is critical: when $M<1$, no boundary layer is present when δ is near 1, but when M ≥ 1 then a boundary layer at x = 1 is possible. Numerical results illustrate the sharpness of most of our results.

scholarly journals ABOUT THE APPROXIMATE SOLUTION OF THE USUAL AND GENERALIZED HILBERT BOUNDARY VALUE PROBLEMS FOR ANALYTICAL FUNCTIONS

2000 ◽  
Vol 5 (1) ◽  
pp. 119-126
Author(s):  
V. R. Kristalinskii
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Cauchy Type ◽  
Analytical Functions ◽  
Method Of Solution ◽  
Cauchy Type Integrals ◽  
Hilbert Boundary Value Problem

In this article the methods for obtaining the approximate solution of usual and generalized Hilbert boundary value problems are proposed. The method of solution of usual Hilbert boundary value problem is based on the theorem about the representation of the kernel of the corresponding integral equation by τ = t and on the earlier proposed method for the computation of the Cauchy‐type integrals. The method for approximate solution of the generalized boundary value problem is based on the method for computation of singular integral of the formproposed by the author. All methods are implemented with the Mathcad and Maple.

scholarly journals On a method for approximate solution of a mixed boundary value problem for an elliptic equation

2021 ◽  
Vol 23 (1) ◽  
pp. 58-71
Author(s):  
Mahmut E. Fairuzov ◽  
Fedor V. Lubyshev
Keyword(s):  
Boundary Condition ◽  
Approximate Solution ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Variable Coefficients ◽  
Natural Boundary Condition ◽  
Mixed Boundary Value Problem ◽  
Integration Region

A mixed boundary value problem for an elliptic equation of divergent type with variable coefficients is considered. It is assumed that the integration region is a rectangle, and the boundary of the integration region is the union of two disjoint pieces. The Dirichlet boundary condition is set on the first piece, and the Neumann boundary condition is set on the other one. The given problem is a problem with a discontinuous boundary condition. Such problems with mixed conditions at the boundary are most often encountered in practice in process modeling, and the methods for solving them are of considerable interest. This work is related to the paper [1] and complements it. It is focused on the approbation of the results established in [1] on the convergence of approximations of the original mixed boundary value problem with the main boundary condition of the third boundary value problem already with the natural boundary condition. On the basis of the results obtained in this paper and in [1], computational experiments on the approximate solution of model mixed boundary value problems are carried out.

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