Numerical study of mixed convection flow of a micropolar fluid towards permeable vertical plate with convective boundary condition

In this article, the mixed convective flow of a micropolar fluid along a permeable vertical plate under the convective boundary condition is analyzed. The scaling group of transformations is applied to get the similarity representation of the system of partial differential equations of the problem and then the resulting equations are solved by using Spectral Quasi-Linearisation Method. This study reveals that the dual solutions exists for certain values of mixed convection parameter. The outcomes are analyzed with dual solutions in detail. Effects of micropolar parameter, Biot number and suction/injection parameters on different flow profiles are discussed and depicted graphically.     

A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition

This paper considers the classical problem of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid. It is well known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, no such solution has been attempted for the convective surface boundary condition. The paper demonstrates that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x^?1/2. Numerical solutions of the resulting similarity energy equation are provided for representative Prandtl numbers of 0.1, 0.72, and 10 and a range of values of the parameter characterizing the hot fluid convection process. For the case of constant heat transfer coefficient, the same data provide local similarity solutions.     

Free convection flow about a cone under mixed thermal boundary conditions and a magnetic field

A similarity analysis was performed to investigate the laminar free-convection boundary-layer flow in the presence of a transverse magnetic field over a vertical down-pointing cone with mixed thermal boundary conditions. Boundary layer velocity and temperature profiles were determined numerically for various values of the magnetic parameter and the Prandtl number. The results show that the magnetic field suppresses the velocity profiles and increases the skin friction. The temperature profiles were expanded with increasing values of the magnetic parameter resulting in higher surface temperatures. A transformation relating the similarity solutions of the boundary-layer velocity and temperature profiles associated with different values of the mixed thermal boundary condition parameter was obtained.     

ASYMPTOTIC BEHAVIOR OF SOLUTION BRANCHES OF NONLOCAL BOUNDARY VALUE PROBLEMS

In this article, by employing an oscillatory condition on the nonlinear term,a result is proved for the existence of connected component of solutions set of a nonlocal boundary value problem, which bifurcates from infinity and asymptotically oscillates over an interval of parameter values. An interesting and immediate consequence of such oscillation property of the connected component is the existence of infinitely many solutions of the nonlinear problem for all parameter values in that interval.     

Numerical study of porous channel flow in a rotating system by a homotopy method

Problems in lubrication, transpiration cooling, dialysis, gaseous diffusion processes, subterranean rivers fed by porous bedrock, and flow under the polar icecap are examples of porous channel flow in a rotating system. Using similarity equations the Navier-Stokes equations reduce to a 10th order two-point boundary value problem. For certain parameter ranges this problem is very difficult and is not amenable to simple shooting techniques or classical locally convergent iterative methods. A recently developed globally convergent homotopy algorithm is applied to the boundary value problem, and numerical results are reported for a wide range of parameter values.     

On the Solution Structure of Nonlinear Hill's Equation I,Global Results

The solution structure in W^2,p(?) × ? of nonlinear Hill's equation is discussed in full detail. In a recent article, the existence of unbounded solution components was shown for values of the branching parameter in the gaps of the continuous spectrum of the linearized problem. This result is the starting point of further investigations concerning the existence region of the corresponding solution components. In particular, new phenomena such as asymptotic bifurcation from infinity at a specific parameter value inside of each gap of the spectrum can be shown, if the nonlinearity satisfies a growth condition. The main assumption is the concentration of the nonlinearity to a compact interval which allows the reduction to an equivalent nonlinear Sturm - Liouville problem with parameter dependent boundary conditions, if the parameter does not belong to the continuous spectrum. Extending this problem to all real parameter values makes it possible to get information about the existence region of the solution components with the help of a priori bounds for solutions of the Sturm-Liouville problem.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Micro/nano sliding plate problem with Navier boundary condition

For Newtonian flow through micro or nano sized channels, the no-slip boundary condition does not apply and must be replaced by a condition which more properly reflects surface roughness. Here we adopt the so-called Navier boundary condition for the sliding plate problem, which is one of the fundamental problems of fluid mechanics. When the no-slip boundary condition is used in the study of the motion of a viscous Newtonian fluid near the intersection of fixed and moving rigid plane boundaries, singular pressure and stress profiles are obtained, leading to a non-integrable force on each boundary. Here we examine the effects of replacing the no-slip boundary condition by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. The Navier boundary condition, possesses a single parameter to account for the slip, the slip length ℓ, and two solutions are obtained; one integral transform solution and a similarity solution which is valid away from the corner. For the former the tangential stress on each boundary is obtained as a solution of a set of coupled integral equations. The particular case solved is right-angled corner flow and equal slip lengths on each boundary. It is found that when the slip length is non-zero the force on each boundary is finite. It is also found that for a suffciently large distance from the corner the tangential stress on each boundary is equal to that of the classical solution. The similarity solution involves two restrictions, either a right-angled corner flow or a dependence on the two slip lengths for each boundary. When the tangential stress on each boundary is calculated from the similarity solution, it is found that the similarity solution makes no additional contribution to the tangential stress of that of the classical solution, thus in agreement with the findings of the integral transform solution. Values of the radial component of velocity along the line θ = π /4 for increasing distance from the corner for the similarity and integral transform solutions are compared, confirming their agreement for sufficiently large distances from the corner. (Received: November 9, 2005)     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

Numerical Solution of the Inverse Nonequilibrium Sorption Problem with a Time-Dependent Boundary Condition

We consider the quasi-linear problem of nonequilibrium sorption dynamics with external-diffusion kinetics and a boundary condition that contains the time derivative of a solution component. A numerical method is proposed for describing the inverse problem to recover the nonlinear parameter of the system of differential equations—the inverse of the sorption isotherm. Convergence of the difference scheme for the direct problem is proved. Numerical solutions of both the direct and the inverse problem are obtained for various parameter values.     

On the choice of the coupling parameter in boundary integral formulations of the exterior acoustic problem

Most commonly used boundary integral formulations of the exterior problem in time-harmonic acoustics, which are valid for all wave-numbers, involve a non-zero coupling parameter. In this paper we analyse the qualitative behaviour of the direct boundary integral formulation due to Burton and Miller. In particular we include a regularised formulation for the Neumann problem. By deriving the eigensystems of the operators we are able to obtain ′almost optimal′ values for the coupling parameter, as a function of the wave-number, so as to minimize the condition number of various formulations     

Variational approach to a non-local identification problem related to non-linear ion transport

A variational approach to a non-linear non-local identification problem related to the non-linear transport equation is studied. Introducing a similarity transformation, the problem is formulated as an identification problem for a non-linear differential equation of second order with an additional non-local condition. For the solution of the forward problem stability in H¹-norm with respect to the identification parameter is obtained. Using this result the existence of a solution to the identification problem is proved. Some results of computational experiments are given. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Adaptive testing of multiple hypotheses for stochastic processes

The multiple hypotheses testing problem is studied when the family of underlying probability distributions involves a nuisance parameter. A necessary and sufficient condition for the existence of an adaptive decision procedure is obtained when the nuisance parameter takes values in a compact topological space and observations (possibly in continuous time) have log-likelihood ratios obeying the large deviatons principle. By definition, an adaptive procedure is asymptotically fully efficient for each value of the nuisance parameter and does not depend on it. The mentioned adaptation condition is illustrated by Gaussian and Markov processes     

Application of M-PML absorbing boundary conditions to the numerical simulation of wave propagation in anisotropic media. Part II: Stability

This paper deals with detailed properties of M-PML (Multiaxial Perfectly Matched Layer) absorbing boundary conditions, which are used when the computational domain is limited. These conditions are stable for any type of anisotropy with the correct choice of a stabilization parameter. In the first part of this paper, [3], the authors show that the reflectivity is a linear function of the stabilization parameter. Based on this study, the problem of finding an optimal stabilization parameter that provides stability and minimal reflection is formulated. A necessary M-PML stability condition that allows limiting the lower values of the stabilization parameter is obtained. It is shown that this criterion is not sufficient.     

The Pattern of Multiple Rings from Morphogenesis in Development

Under certain conditions the problem of morphogenesis in development and the problem of morphology in block copolymers may be reduced to one geometric problem. In two dimensions two new types of solutions are found. The first type of solution is a disconnected set of many components, each of which is close to a ring. The sizes and locations of the rings are precisely determined from the parameters and the domain shape of the problem. The solution of the second type has a coexistence pattern. Each component of the solution is either close to a ring or to a round disc. The first-type solutions are stable for certain parameter values but unstable for other values; the second-type solutions are always unstable. In both cases one establishes the equal area condition: the components in a solution all have asymptotically the same area.     

Asymptotic study of turbulent Poiseuille flow with wall transpiration

An incompressible, pressure–driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary–value problem for a second–order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. A limiting transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one–parameter family of curves. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and a priori bounds for steady stagnation flow toward a stretching cylinder

We investigate the nonlinear boundary value problem (BVP) that is derived from a similarity transformation of the Navier-Stokes equations governing fluid flow toward a stretching permeable cylinder. Existence of a solution is proven for all values of the Reynolds number and for both suction and injection, and uniqueness results are obtained in the case of a monotonic solution. A priori bounds on the skin friction coefficient are also obtained. These bounds achieve any desired order of accuracy as the injection parameter tends to negative infinity.     

The effect of a parameter in a boundary condition on the stability of difference schemes

A nonlocal difference scheme for the heat conduction equation with a real parameter γ > 1 or γ < 0 in the boundary condition is considered. Thus, the analysis of this problem is generalized to all real values of the parameter in the boundary condition. The spectrum of the spatial operator of the scheme is studied. The results obtained are used to formulate a stability criterion for the difference scheme. The case γ = ?1 proves to be special, because the system of eigenfunctions of the operator does not form a basis in the space of grid functions.     

Homogenization of the boundary value problem for the laplace operator in a perforated domain with a rapidly oscillating nonhomogeneous Robin-type condition on the boundary of holes in the critical case

The asymptotic behavior of solutions to a boundary value problem in a domain periodically perforated by small holes with a rapidly oscillating nonhomogeneous Robin-type condition on their boundaries is investigated in the case of critical parameter values.     

Error structure and identification condition in maximum likelihood nonmetric multidimensional scaling

The author addresses two previously unresolved issues in maximum likelihood estimation (MLE) for multidimensional scaling (MDS). First, a theoretically consistent error model for nonmetric MLDMS is proposed. In particular, theoretical arguments are given that the disturbance should be multiplicative with distance when a stochastic choice model is used on rank-ordered similarity data. This assumption implies that the systematic component of similarity in the rank order is a logarithmic function of distances between stimuli. Second, a problem with identification condition of the maximum likelihood estimators is raised. The author provides a set of constraints that guarantees the identification in MLE, and produces more desirable asymptotic confidence regions that are parameter independent. An example using perception of business schools illustrates these ideas and demonstrates the computational tractability of the MLE approach to MDS.