The Eigenvalue Problem and Saint-Venant Decay Rate for a Nonhomogeneous Semi-Infinite Strip in Plane Strain

2013 ◽  
Vol 706-708 ◽  
pp. 1822-1826
Author(s):  
Qing Yang ◽  
Kai Zhang ◽  
Bai Lin Zheng ◽  
Jian Xin Zhu
Keyword(s):  
Numerical Calculation ◽  
Eigenvalue Problem ◽  
Decay Rate ◽  
Plane Strain ◽  
Plane Problem ◽  
Significant Influence ◽  
Positive Eigenvalue ◽  
Infinite Strip ◽  
Homogeneous Plane

The eigenvalue problem referring to a nonhomogeneous semi-infinite strip in plane strain is investigated here, by using the analogous methodology proposed by Papkovich and Fadle in homogeneous plane problem. Two types of nonhomogeneity are considered: (i) the modulus varies with the thickness coordinate exponentially, (ii) it varies with the length coordinate exponentially. The eigenvalues for these two cases are obtained by numerical calculation. By considering the smallest positive eigenvalue, the Saint-Venant Decay rate of the problem is estimated, which indicates material nonhomogeneity can have a significant influence on the Saint-Venant decay rate.

The eigenvalue problem and Saint-Venant decay rate for a nonhomogeneous semi-infinite strip

2014 ◽  
Vol 27 (6) ◽  
pp. 588-596 ◽  
Author(s):  
Qing Yang ◽  
Bailin Zheng ◽  
Kai Zhang ◽  
Jianxin Zhu
Keyword(s):  
Eigenvalue Problem ◽  
Decay Rate ◽  
Infinite Strip

The Freestream Matching Condition for Stagnation Point Turbulent Flows: An Alternative Formulation

1996 ◽  
Vol 63 (1) ◽  
pp. 95-100 ◽  
Author(s):  
R. Abid ◽  
C. G. Speziale
Keyword(s):  
Plane Strain ◽  
Stagnation Point ◽  
Turbulent Flows ◽  
Ad Hoc ◽  
Matching Condition ◽  
Alternative Formulation ◽  
Mean Velocity ◽  
Turbulent Dissipation ◽  
Alternative Means ◽  
Homogeneous Plane

The problem of plane stagnation point flow with freestream turbulence is examined from a basic theoretical standpoint. It is argued that the singularity which arises in the standard K–ε model results from the use of an inconsistent freestream boundary condition. The inconsistency lies in the implementation of a production-equals-dissipation equilibrium hypothesis in conjunction with a freestream mean velocity field that corresponds to homogeneous plane strain—a turbulent flow for which the standard K–ε model does not predict such a simple equilibrium. The ad hoc adjustment that has been made in the constants of the ε-transport equation to eliminate this singularity is shown to be inconsistent for homogeneous plane-strain turbulence as well as other benchmark turbulent flows. An alternative means to eliminate this singularity—without compromising model predictions in more basic turbulent flows—is proposed based on the incorporation of nonequilibrium vortex stretching effects in the turbulent dissipation rate equation.

Fast scalar decay in a shear flow: modes and pseudomodes

2007 ◽  
Vol 572 ◽  
pp. 219-229 ◽  
Author(s):  
J. VANNESTE ◽  
J. G. BYATT-SMITH
Keyword(s):  
Shear Flow ◽  
Eigenvalue Problem ◽  
Decay Rate ◽  
Shear Flows ◽  
Passive Scalar ◽  
Decay Rates ◽  
Constant Speed ◽  
Fast Decay ◽  
Small Noise ◽  
Stream Direction

The decay of a passive scalar in a sinusoidal shear flow translating in the cross-stream direction at a constant speed u is studied in the limit of small diffusivity κ. The decay rate, obtained by solving an eigenvalue problem, is found to tend to a non-zero constant as κ→0 when u is of order κ1/2. This result, establishing that fast decay is possible in shear flows, is fragile however: because of the existence of pseudomodes, the addition of a small noise leads to decay rates that decrease to zero with κ as κ2/5.

Existence of the first eigenvalue of the eigenvalue problem for the Laplace-Beltrami operator on the unit sphere

2018 ◽  
Vol 55 (3) ◽  
pp. 374-382
Author(s):  
Mariusz Bodzioch ◽  
Mikhail Borsuk ◽  
Sebastian Jankowski
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Behaviour ◽  
Unit Sphere ◽  
Beltrami Operator ◽  
Positive Eigenvalue ◽  
Conical Point ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Asymptotic Behaviour Of Solutions ◽  
Oblique Derivative Problems

In this paper we formulate and prove that there exists the first positive eigenvalue of the eigenvalue problem with oblique derivative for the Laplace-Beltrami operator on the unit sphere. The firrst eigenvalue plays a major role in studying the asymptotic behaviour of solutions of oblique derivative problems in cone-like domains. Our work is motivated by the fact that the precise solutions decreasing rate near the boundary conical point is dependent on the first eigenvalue.

Consolidation under embankment-type loads

1976 ◽  
Vol 13 (1) ◽  
pp. 72-77 ◽  
Author(s):  
N. Babu Shanker ◽  
K. S. Sarma ◽  
M. Venkataratnam
Keyword(s):  
Plane Strain ◽  
Integral Transformation ◽  
Infinite Strip ◽  
Clay Layer ◽  
Transformation Technique ◽  
Loading Pattern ◽  
Distributed Load ◽  
Uniformly Distributed Load ◽  
Repeated Integral ◽  
Line Loads

Plane strain problems of consolidation (or poro-elasticity) can be solved using the two displacement functions defined by McNamee and Gibson with the help of a repeated integral transformation technique. The problem of a semi-infinite clay layer whose surface is subjected to an embankment-type of normal trapezoidal pressure applied along an infinite strip is treated here. The general loading pattern selected easily degenerates into a rectangular (uniformly distributed) load for which NcNamee and Gibson gave the solutions, to the triangular loads and also to the line loads. Not only the settlements, but also the pore pressures have been evaluated under these types of loads when the surface is either pervious or impervious.The nondimensional solutions presented are useful to highway and embankment engineers. There is also an example of the use of these solutions.

scholarly journals An integro-differential inequality related to the smallest positive eigenvalue of p(x)-Laplacian Dirichlet problem

2016 ◽  
Vol 15 (1) ◽  
pp. 27-36
Author(s):  
Damian Wiśniewski ◽  
Mariusz Bodzioch
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Unit Sphere ◽  
Differential Inequality ◽  
Beltrami Operator ◽  
Positive Eigenvalue ◽  
Differential Inequalities

AbstractWe consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.

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