Response of heat transfer from a moving flat plate in a parabolic flow

This paper deals with the solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations and the study of the response of heat transfer when there is a parabolic flow over a moving flat plate. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse thermal boundary layer equations. A general analysis is made from which we deduce (i) Solutions of two-dimensional thermal boundary layer on a moving flat plate, (ii) Solutions of thermal boundary layer on a yawed flat plate, (iii) Solutions of thermal boundary layer when there is a parabolic flow over a moving flat plate by giving different values to β and Cx. Solutions are developed for large and small times and curves are drawn representing the variations of heat transfer from the plate with time for all the cases. The limiting time is also calculated.     

Primary elements in formal modules

The stability of an elastic plate in a supersonic gas flow is considered in the presence of a boundary layer formed on the surface of the plate. The problem is solved in two statements. In the first statement, the plate is of large but finite length, and a coupled-mode type of flutter is examined (the effect of the boundary layer on another, single-mode, type of flutter has been studied earlier). In the second statement, the plate is assumed to be infinite, and the character of its instability (absolute or convective) is analyzed. In both cases, the instability is determined by a branch point of the roots of the dispersion equation, and the mathematical analysis is the same. It is proved that instability in a uniform gas flow is weakened by a boundary layer but cannot be suppressed completely, while in the case of a stable plate in a uniform flow the boundary layer leads to the destabilization of the plate.     

THE EFFECT OF BOUNDARY SHAPE ON BOUNDARY LAYER OF P-MODEL PLATE PROBLEMS WITH HARD SIMPLY SUPPORT

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Nearly parallel Blasius flow with slip

The steady boundary layer flow past a moving horizontal flat plate with a slip effect at the plate in a free stream with constant speed, slightly different from the plate speed is studied. An analytic perturbation solution of order two is obtained for the velocity. With respect to the parallel flow both the boundary layer and the inverted boundary layer characters of the flow are plotted and discussed. It is observed that under high slip, the flow becomes a nearly parallel flow with an increased speed.     

The uniform heating and bending of a two-layer plate

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic layer considered (a soft plate) is in ideal contact with a rigid isotropic thin elastically deformed layer. The ends of the plate are load-free. A boundary layer of the soft plate (a thin contact layer) is introduced, which enables the boundary conditions on the ends of the plate to be formulated in such a way that the problem has a bounded smooth solution [1]. The two-layer plate, generally speaking, is bounded along the axis perpendicular to the axes directed along the length and thickness of the plate. The resultant force and the resultant moment, applied to the end transverse sections, are equal to zero. The exact solution of the temperature problem is sought using the equations of the theory of elasticity. The plane problem of the bending of a two-layer plate acted upon by a uniformly distributed pressure applied to the side surface of an anisotropic layer is solved by a similar method. The ends of the rigid isotropic layer are clamped.     

Existence of the stationary solution of a Rayleigh-type equation

A fluid flow along a semi-infinite plate with small periodic irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure: a thin boundary layer (“lower deck”) and a classical Prandtl boundary layer (“upper deck”). The aim of this paper is to prove the existence and uniqueness of the stationary solution of a Rayleigh-type equation, which describes oscillations of the vertical velocity component in the classical boundary layer.     

The asymmetrical mixed temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic elastic layer (soft plate) investigated is in ideal contact with an absolutely rigid layer, deformable only by thermal expansion. The generalized plane temperature problem reduces to determining the stress-strain state of the soft anisotropic layer investigated using the equations of the mixed problem of elasticity theory. At the ends of the boundary layer of the soft plate (a thin contact layer), no conditions are imposed. On the remaining part of the ends of the soft plate, the boundary conditions correspond to a free boundary. The problem has a bounded smooth solution. Unlike the approach described earlier [1], it is proposed to seek an accurate solution in the form of ordinary Fourier series with respect to a single longitudinal coordinate. Solutions in polynomials are also used. It is shown that the existence of these solutions in polynomials enables the convergence of the Fourier series to be improved considerably.     

Influence of Inertia of a Compliant Surface on Viscous Instability of an Incompressible Boundary Layer

Computational Mathematics and Mathematical Physics - An incompressible boundary layer on a compliant plate is considered. The influence exerted by the inertia of the plate on the stability of the...     

Monte Carlo simulation of boundary layer transition

A statistical method for simulating a boundary layer transition flow is proposed as based on experimental data on the kinematics and dynamics of turbulent spots (Emmons spots) on a flat plate placed in an incompressible fluid. The method determines intermittency with allowance for overlapping spots, which makes it possible to determine the forces on the plate surface and the flow field near the transition region if the mean streamwise velocity field in a developed turbulent boundary layer is known as a function of the Reynolds number. In contrast to multiparameter transition models, this approach avoids the use of nonphysical parameter values.     

合成展开法求解圆薄板大挠度问题

本文用合成展开摄动法,把外场解和内层解结合起来,求解圆薄板大挠度问题.本文把Hencky的薄膜解当作外场解的一级近似解,并求出了外场解的二级近似解.利用边界内层坐标,求得了相应的各级内层解,即边界层解.本文采用最大位移和板厚之比的倒数作为小参数,所得结果大大改进了1948年作者所得的结果.     

Minimum error solutions of boundary layer equations

The minimum error solutions of boundary layer equations in the least square sense have been studied by employing the Euler-Lagrange equations. To test the method a class of problems,i.e., boundary layer on a flat plate, Hiemenz flow, boundary layer on a moving sheet and boundary layer in non-Newtonian fluids have been studied. The comparison of the results with approximate methods, like Karman-Pohlhuasen, local potential and other variational methods, shows that the present predictions are invariably better.     

Boundary layer along the initial length of a flat plate at zero incidence with homogeneous suction

An investigation has been made into the two-dimensional laminar incompressible boundary layer along the initial length of a semi-infinite flat plate at zero incidence with homogeneous suction. The momentum and the kinetic energy integral equations have been numerically integrated with the aid of a singly infinite system of boundary layer velocity profiles. The results obtained are well in agreement with the known exact solution and the process of integration is simpler to be carried out by the use of a monoparametric family of velocity profiles. The method can be used to investigate the boundary layer along a flat plate with arbitrary suction starting either at the leading edge or at some point downstream.     

Analysis of the neutral layer offset of bimetal composite plate in the straightening process using boundary element subfield method

In the classical straightening theory, it supposes that the geometric central layer and stress neutral layer are a coincidence. However, there is some offset in fact. This is one of the reasons why straightening force is inaccurate in the straightening process. In this paper, the boundary element subfield method is used to analyze the three-dimension elastic–plastic deformation of a bimetal composite plate in the straightening process. At first, the boundary integral equation of a bimetal composite plate is established by the boundary element subfield method. Then, through analyzing the deformation in the rolled piece straightening, it shows that the geometric central layer does not coincide with the neutral layer. The formula of the neutral layer offset is established and the change law is discussed. At the same time, the influence of the neutral layer offset on the precision of straightening force is researched. From the numerical analysis, it shows that the error of the straightening force reaches to 5% whether considering the neutral layer offset. This demonstrates that the neutral layer offset is one of the important factors to give the straightening force inaccurately. It is ought to consider the neutral layer offset when the model of straightening force is established.     

Solutions of thermal boundary layer equations when temperature gradient at the moving flat plate in parabolic flow is prescribed

Solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations are studied when the temperature gradient at the moving flat plate in parabolic flow is prescribed. A general analysis is made and different cases are studied by giving values to β and Cx which determine the gradient and curvature of the outer flow steam lines. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse the thermal boundary layer equations. The response of temperature of the plate are studied for large and small times and curves are drawn representing the variation of temperature with time for various cases. The limiting time is also calculated.     

Effect of viscous dissipation in natural convection along a heated vertical plate

The steady laminar boundary layer flow along a vertical stationary heated plate is studied taking into account the viscous dissipation of the fluid. The results are obtained with the numerical solution of the boundary layer equations. Both upward and downward flow is considered for the isothermal and uniform flux case. It is found that the interaction between the viscous heating and the buoyancy force has a strong influence on the results.     

Effects of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference

This paper concerns with studying the steady and unsteady MHD micropolar flow and mass transfers flow with constant heat source in a rotating frame of reference in the presence chemical reaction of the first-order, taking an oscillatory plate velocity and a constant suction velocity at the plate. The plate velocity is assumed to oscillate in time with a constant frequency; it is thus assumed that the solutions of the boundary layer are the same oscillatory type. The governing dimensionless equations are solved analytically after using small perturbation approximation. The effects of the various flow parameters and thermophysical properties on the velocity and temperature fields across the boundary layer are investigated. Numerical results of velocity profiles of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. The results show that there exists completely oscillating behavior in the velocity distribution.     

On the shear boundary layer of tension plates

The present paper deals with a generic class of problems for plates subjected to loadings combining a high in-plane tension and a small lateral pressure. It develops the governing differential equations in the singular pertubation form, through the postulation of retaining only one of the Kirchhoff's assumptions, that the plate thickness in the boundary layer region is invariant. The solution by using the standard perturbation method is discussed. The postulation is justified when it is demonstrated that in the shear boundary layer the plate thickness is of higher-order smallness. The general method of solution by the standard perturbation technique is applied to an annular plate problem. Problems of different combinations of supports at the inner and the outer boundaries are solved. The case in which both edges are simply supported is presented as an illustration of the solution technique. In other cases results only are presented. The effect of support on the boundaries is also discussed. The shear effect is found to be most significant at a clamped edge. In the special geometry, it is possible to demonstrate that, when the condition on membrane force is not met as required in the general theory, thagnitude of the boundary layer changes. Specifically, the paper presents a case in which the membrhich the membrane force is zero at the inner edge.     

Dynamics of flow near a separation point

Dynamical behavior of a boundary layer separation is studied using the experimental approach. The dynamical nature of the phenomenon is demonstrated on a simple case of a flat–plate boundary layer in adverse pressure gradient. The Time–Resolved PIV technique was utilized for monitoring instantaneous structure of the separation region and its time development. Distinctive coherent structures and their dynamical behavior were identified using the Proper Orthogonal Decomposition method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The boundary layer transition at supersonic velocities

The transition to turbulent flow in a boundary layer at supersonic velocities, the study of which was started at the Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Academy of Sciences of the USSR on the initiative of V. V. Struminskii is considered. It is shown that complex investigations into this problem, including the stability of the laminar boundary layer and structure of the perturbations in the operational part of a wind tunnel at supersonic velocities, enable the mechanism of the boundary layer transition on a flat plate to be established and demonstrate the decisive effect of the spectral composition of the external flow perturbations and the blunting of the leading edge of the model that enables one to determine the role of the unit Reynolds number.     

Hydromagnetische Stabilität laminarer Grenzschicht an einer längsangeströmten ebenen Platte

Summary The purpose of the paper is to consider the stability for wavelike disturbances in the steady, twodimensional, laminar boundary layer of a magnetic field, which is applied uniformly normal to the flat plate. The results show that the critical Reynolds number (R _c ^* ) increases remarkably with the characteristic parameter (). The increase of the critical Reynolds number depends not only on the shape parameter of the velocity distribution in the boundary layer but also on the peculiarity of the velocity profile. It is also found that the boundary layer holds itself laminar all over the flat plate, when the magnetic parameterN is greater than 1.25×10^–7, then a reduction of the skin-frictin drag might be expeced.