Approximated Solution of a Nonlinear Singular Equation of Prandtl's Type

A nonlinear strongly singular integral equation, which can be reduced to a nonlinear singular integro-differential equation of Prandtl's type, is considered. A collocation method for solution is treated and the convergence of the approximated solution to the unique solution of the nonlinear integral equation is proved.     

Prandtl's Mixing Length Model - Revisited

The paper is concerned with a modification of Prandtl's mixing length model of Reynolds stresses in fully developed turbulent channel flows. Here it is a well established fact that Prandtl's model falls short to describe the Reynolds stresses correctly very close to the wall. Furthermore, the resulting solution of the time averaged velocity fails to describe the channel flow correctly from the wall to its center. To overcome these shortcomings, the only characteristic mixing length in Prandtl's model is replaced by separate mixing lengths for velocity fluctuations parallel to the wall and normal to the wall, respectively. The modified model describes the mean velocity, all Reynolds stresses, and the functional dependence between Reynolds number based on the mean velocity and the one based on the friction velocity. For all Reynolds numbers these results - and corresponding results for the production terms of Reynolds stresses and the energy balance of the mean flow - agree quantitatively with experimental data and with data obtained by numerical simulations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate solution of a class of singular integral equations of second kind

A simple method based on polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind. For a hypersingular integral equation of the second kind, this method avoids the complex function-theoretic method and produces the known exact solution to Prandtl's integral equation as a special case. For a particular singular integro-differential equation of the second kind, this also produces an approximate solution which compares favourably with numerical results obtained by various Galerkin methods. The convergence of the method for both the equations is also established.     

Heat transfer at high energy devices with prescribed cooling flow

We study the heat transfer from a high‐energy electric device into a surrounding cooling flow. We analyse several simplifications of the model to allow an easier numerical treatment. First, the flow variables velocity and pressure are assumed to be independent from the temperature which allows a reduction to Prandtl's boundary layer model and leads to a coupled nonlinear transmission problem for the temperature distribution. Second, a further simplification using a Kirchhoff transform leads to a coupled Laplace equation with nonlinear boundary conditions. We analyse existence and uniqueness of both the continuous and discrete systems. Finally, we provide some numerical results for a simple two‐dimensional model problem. Copyright © 2006 John Wiley & Sons, Ltd.     

Zaremba's problem for elliptic equations in the neighborhood of a singular point or in the infinity

In the present paper the behavior of solutions of the mixed Zaremba's problem in the neighborhood of a boundary point and at infinity is studied. In part I of this paper[4] the concept of Wiener's generalized solution of Zaremba's problem was introduced and the so called Growth Lemma for the class of domains, satisfying isoperimetric condition, was proven. In part II regularity criterion for joining points of Neumann's and Dirichlet's boundary conditions is formulated. Generalized solution in unlimited domains as a limit of Zaremba's problem's solutions in a sequence of limited domains is introduced and a regularity condition allowed to obtain an analogue of Phragmen-Lindeloeff theorem for the solutions of Zaremba's problem. Main results of the present paper are formulated in terms of divergence of Wiener's type series.     

Blowup of solutions of the unsteady Prandtl's equation

We prove that for certain classes of compactly supported C^∞ initial data, smooth solutions of the unsteady Prandtl's equation blow up in finite time. © 1997 John Wiley & Sons, Inc.     

Singularly perturbed problems in partial differential equations: a survey

This survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980–2000. In this review we have covered singularly perturbed partial differential equations. About ODEs the survey has already been done by us [see M.K. Kadalbajoo, K.C. Patidar, Appl. Math. Comput. 130 (2002) 457–510].     

Solving a nonlinear inverse Robin problem through a linear Cauchy problem

ABSTRACT

Considered in this paper is an inverse Robin problem governed by a steady-state diffusion equation. By the Robin inverse problem, one wants to recover the unknown Robin coefficient on an inaccessible boundary from Cauchy data measured on the accessible boundary. In this paper, instead of reconstructing the Robin coefficient directly, we compute first the Cauchy data on the inaccessible boundary which is a linear inverse problem, and then compute the Robin coefficient through Newton's law. For the Cauchy problem, a parameter-dependent coupled complex boundary method (CCBM) is applied. The CCBM has its own merits, and this is particularly true when it is applied to the Cauchy problem. With the introduction of a positive parameter, we can prove the regularized solution is uniformly bounded with respect to the regularization parameter which is a very good property because the solution can now be reconstructed for a rather small value of the regularization parameter. For the problem of computing the Robin coefficient from the recovered Cauchy data, a least square output Tikhonov regularization method is applied to Newton's law to obtain a stable approximate Robin coefficient. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

Time-optimal steering of a point mass onto the surface of a sphere at zero velocity

The problem of the time-optimal steering of a point mass onto the surface of a sphere at zero velocity, by a control force of bounded magnitude is investigated. It is assumed that the surface is penetrable and that the point may “land” on the sphere either from the outside or from the inside. An optimal control, in the open-loop and feedback form of trajectories the optimal time and the Bellman function are constructed using Pontrya'gin's maximum principle. The multidimensional boundary-value problem is reduced, by introducing self-similar variables, to the numerical solution of an algebraic equation of degree four and a transcendental equation. It is shown that the boundary-value problem degenerates when the optimal trajectory is nearly linear; a solution of the synthesis problem is constructed in the degenerate case. The efficacy of the approach proposed here is illustrated by specific examples in which families of trajectories are computed, and by an analysis of control regimes.     

Radial deformation of a nonhomogeneous spherically isotropic elastic sphere with a concentric spherical inclusion

The elasticity of a spherically isotropic medium bounded by two concentric spherical surfaces subjected to normal pressures is discussed. The material of the structure is spherically isotropic and, in addition, is continuously inhomogeneous with mechanical properties varying exponentially as the square of the radius. An exact solution of the problem in terms of Whittaker functions is presented. The St. Venant’s solution in the case of homogeneous material and Lamé’s solution in the case of homogeneous isotropic material are derived from the general solution. The problem of a solid sphere of the same medium under the external pressure is also solved as a particular case of the above problem. Finally, the displacements and stresses of a composite sphere consisting of a solid spherical body made of homogeneous material and a nonhomogeneous concentric spherical shell covering the inclusion, both of them being spherically isotropic, are obtained when the sphere is under uniform compression.     

Fluid flow in an inhomogeneous granular medium

The seepage of a compressible fluid in an inhomogeneous undeformable granular medium is investigated. It is assumed that the fluid flow in a porous space is described by the Navier–Stokes equations. It is shown that, in the case of an inhomogeneous velocity field, a tensor of additional effective stresses occurs in connection with the transfer of fluid particles in a transverse direction when flow occurs around the granules of the medium in a longitudinal direction. Using the fundamental propositions of Reynolds’ averaging theory and Prandtl's mixing path, the structure of the effective viscosity coefficient is determined and hypotheses are formulated which enable it to be assumed to be independent of the flow velocity. It is established by comparison with experimental data that the effective viscosity coefficient can exceed the viscosity coefficient of the flowing fluid by an order of magnitude. The equations of average motion are obtained, which in the case of an incompressible fluid have the form of the Navier–Stokes equations with body forces proportional to the velocity. It is established that, in addition to the well-known dimensionless flow numbers, there is a new number which characterizes the ratio of the Darcy porous drag forces to the effective viscosity forces. The proposed equations are extended to the case of the flow of an aerated fluid. The components of the angular momentum vector are used as the required functions instead of the components of the velocity vector. This enables a solving system of equations to be obtained, which, apart from the notation, is identical with the similar equations for the case of an incompressible fluid. The solution of a new problem of the fluid flow in a plane channel with permeable walls is presented using three models: Darcy's law for an incompressible and aerated fluid, and also of an aerated fluid taking the effective viscosity into account. It is established that, for the same pressure drop, the maximum flow rate corresponds to Darcy's law. Compressibility leads to its reduction, but by simultaneously taking into account the compressibility and the effective viscosity one obtains minimum values of the flow rate. The effective viscosity and aeration of the fluid has a considerable effect on the flow parameters.     

均衡及约束凸优化问题公共解的一般迭代算法

梯度投影法在解决约束凸极小化问题中起到了重要的作用. 基于Tian的一般迭代算法, 本文将梯度投影法和平均算子方法相结合, 首次提出隐式和显式的复合迭代算法, 寻求均衡问题和约束凸极小化问题的公共解. 在适当条件下, 获得了强收敛定理.     

The diffraction in a class of unbounded domains connected through a hole

In this paper, the unique solvability, Fredholm property, and the principle of limiting absorption are proved for a boundary value problem for the system of Maxwell's equations in a semi‐infinite rectangular cylinder coupled with a layer by an aperture of arbitrary shape. Conditions at infinity are taken in the form of the Sveshnikov–Werner partial radiation conditions. The method of solution employs Green's functions of the partial domains and reduction to vector pseudodifferential equations considered in appropriate vectorial Sobolev spaces. Singularities of Green's functions are separated both in the domain and on its boundary. The smoothness of solutions is established. Copyright © 2003 John Wiley & Sons, Ltd.     

Motion of a rigid body in a viscous fluid

We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary.     

The efficient frontier for spot and forward purchases: an application to electricity

A local electricity distribution company (LDC) can reduce its exposure to the inherent risks of spot-price volatility and uncertain future demand via forward contracts. Management's problem is to determine the optimal forward-contract purchase. We propose a practical three-stage approach for dealing with the problem. The first stage determines an optimal purchase by solving a cost-constrained risk-minimization problem. The second stage derives the efficient frontier of tradeoffs between expected cost and cost risk from the first-stage solution, at various bounds on the expected cost. The optimal solution is found by melding the frontier with management's risk preferences. In the third stage, the model's parameters are estimated from data typically available to an LDC and used to determine its forward-contract purchase.     

Simulation and analysis of turbulent MHD channel flow with a streamwise magnetic field

We perform large-eddy simulations of turbulent MHD channel flow with a streamwise magnetic field using a pseudo spectral method. The streamwise magnetic field leads to turbulent drag reduction due to the selective Joule damping of certain flow structures. Near the walls, the turbulent mean velocity profile retains the logarithmic layer but the von Karman constant decreases with increasing magnetic field strength. In the outer region, the flow is characterized by persistent streaky structures of large streamwise extent, which lead to a rather flat mean velocity profile. In addition, the streamwise velocity fluctuations develop a pronounced second peak upon increasing the magnetic induction as well as a second logarithmic layer that increases in steepness. We find that Prandtl's classical mixing-length model with a variable Kármán constant can describe the modified logarithmic layer reasonably accurately in a wide range of Reynolds and Hartmann numbers. However, the flow modification near the center of the channel is not properly captured by this approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Finite Difference Scheme for an Inverse Integro-Differential Problem Using Semigroup Theory: A Functional Analytic Approach

In this article, the problem of reconstructing an unknown memory kernel from an integral overdetermination in an abstract linear (of convolution type) evolution equation of parabolic type is considered. After illustrating some results of the existence and uniqueness of a solution for the differential problem, we study its approximation by Rothe's method. We prove a result of stability and another concerning the order of approximation of the solution in dependence of its regularity. The main tool is a maximal regularity result for solutions to abstract parabolic finite difference schemes. Two model problems to which the results are applicable are illustrated.     

Dynamics of a two-layer beam with layer slip

A possible type of structural damping in mechanical systems when there is a distinct dependence of the oscillation decrement on the amplitude is investigated, using the example of the solution of a model problem on the oscillations of a two-layer beam. It is assumed that layer slip only occurs along the beam axis and that the layers together in the transverse direction. The interaction between the layers is of an elastic-friction form. The equations of motion of the beam in Timoshenko's form are numerically integrated using Godunov's difference scheme.     

The free-parameter solution of the convection equations in a vertical cylinder with a volume heat source

An exact solution of the free-convection equations is constructed in the Oberbeck–Boussinesq approximation, describing the flow of a viscous heat-conducting fluid in a vertical cylinder of large radius when heated by radiation. The initial problem is reduced to an operator equation with an extremely non-linear operator, satisfying Schauder's theorem in C[0,1]. An iteration procedure is proposed for determining the independent parameter, that occurs in the solution, which enables three different values to be obtained and, correspondingly, three classes of solution of the initial problem. The linear stability of all the solutions obtained is investigated and it is shown that, for chosen values of the problem parameters, the most dangerous one is the plane wave mode and two instability mechanisms are present. The flow structure and the type of instability depend considerably on the values of the free parameter.     

A boundary value problem for the spherically symmetric motion of a pressureless gas with a temperature‐dependent viscosity

We consider an initial‐boundary value problem for the equations of spherically symmetric motion of a pressureless gas with temperature‐dependent viscosity µ(θ) and conductivity κ(θ). We prove that this problem admits a unique weak solution, assuming Belov's functional relation between µ(θ) and κ(θ) and we give the behaviour of the solution for large times. Copyright © 2009 John Wiley & Sons, Ltd.