Limit oscillatory cycles in the single mode flutter of a plate

The development of the single mode flutter of an elastic plate in a supersonic gas flow is investigated in a non-linear formulation. In the case of a small depression in the instability zone, there is a unique limit cycle corresponding to a unique growing mode. Several new non-resonant limit cycles arise when a second increasing mode appears and the domains of their existence and stability are found. Limit cycles with an internal resonance, in which there is energy exchange between the modes, can exist for the same parameters. Relations between the amplitudes of the limit cycles and the parameters of the problem are obtained that enable one to estimate the risk of the onset of flutter.     

Flow Past a Swept Wing with a Compliant Surface: Stabilizing the Attachment-Line Boundary Layer

Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi-rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment-line boundary layer flow, which is an exact similarity solution of the Navier–Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow-field is three-dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two-dimensional Tollmien–Schlichting modes in the classical boundary-layer flow past a compliant surface.     

Hydromagnetic rotating flow of a fourth‐order fluid past a porous plate

The rotating flow in the presence of a magnetic field is a problem belonging to hydromagnetics and deserves to be more widely studied than it has been to date. In the non‐linear regime the literature is scarce. We develop the governing equations for the unsteady hydromagnetic rotating flow of a fourth‐order fluid past a porous plate. The steady flow is governed by a boundary value problem in which the order of differential equations is more than the number of available boundary conditions. It is shown that by augmenting the boundary conditions based on asymptotic structures at infinity it is possible to obtain numerical solutions of the nonlinear hydromagnetic equations. Effects of uniform suction or blowing past the porous plate, exerted magnetic field and rotation on the flow phenomena, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviours of the Newtonian fluid and second‐, third‐ and fourth‐order non‐Newtonian fluids are compared for the special flow problem, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

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.     

Crocco variable formulation for uniform shear flow over a stretching surface with transpiration: Multiple solutions and stability

The simultaneous effects of transpiration through and tangential movement of a semi-infinite flat plate on the self-similar boundary layer flow driven by uniform shear in the far field is considered. Difficulties with standard shooting techniques are overcome using Crocco variables which also serve to better elucidate the solution structure. The stabilities of dual, triple and even quadruple steady flow solutions encountered in different ranges of plate stretching and wall stress are determined using a linear temporal stability analysis for the self-similar flow.      

On the heat transfer from a finite plate in channel flow for vanishing Prandtl number

Summary The temperature field produced by a finite, hot plate at zero incidence in uniform channel flow is solved exactly for the limiting case of zero prandtl number by means of the Wiener-Hopf technique. The heat transfer on the plate is found to agree with the corresponding boundary layer result over most of the plate for Péclét numbers as low as ten. Extensions to similar Ossen-flow problems are indicated.     

Unsteady MHD flow of a non-Newtonian fluid on a porous plate

This paper deals with the study of the MHD flow of non-Newtonian fluid on a porous plate. Two exact solutions for non-torsionally generated unsteady hydromagnetic flow of an electrically conducting second order incompressible fluid bounded by an infinite non-conducting porous plate subjected to a uniform suction or blowing have been analyzed. The governing partial differential equation for the flow has been established. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. The effect of presence of the material constants of the second order fluid on the velocity field is discussed.     

On the analytic solution of nonlinear flow problem involving Oldroyd 8-constant fluid

The problem dealing with the steady flow of an Oldroyd 8-constant fluid over a suddenly moved plate is considered. The fluid is electrically conducting and a uniform magnetic field is applied in the transverse direction. An analytical solution of the nonlinear boundary value problem is obtained using homotopy analysis method (HAM). The behavior of the material constants and the magnetic field is seen on the velocity distribution. It is noted that the boundary layer thickness decreases by increasing the magnetic parameter.     

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.     

Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

比热容非常值对高超声速平板边界层稳定性的影响

在高超声速条件下,边界层中气体的温度可能很高,以致气体的比热容不再是常数而与温度有关．这时边界层中的流动稳定性如何是值得研究的问题．采用线性稳定性理论,考虑比热容与温度有关时高超声速可压缩平板边界层的稳定性,并与假定比热容为常值的情况作比较,发现对第一模态和第二模态波的中性曲线、最大增长率都有影响．因此,在高超声速情况下,比热容随温度变化是研究边界层稳定性时必须考虑的一个因素．     

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.     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

A moving-wall boundary layer flow of a slightly rarefied gas free stream over a moving flat plate

In the current work, the boundary layer flow of a slightly rarefied gas free stream over a moving flat plate is presented and solved numerically. The first-order slip boundary condition is adopted in the derivation. The dimensionless velocity and shear stress profiles are plotted and discussed. A theoretical derivation of the estimated solution domain is developed, which will give a very close estimation to the exact solution domain obtained numerically. The influences of velocity slip at the wall on the velocity and shear stress are also addressed.     

The Trailing Edge Problem For Mixed Convection Flow Past a Horizontal Plate

The influence of buoyancy onto the boundary‐layer flow past a horizontal plate aligned parallel to a uniform free stream is characterized by the buoyancy parameter K = Gr/Re^5/2 where Gr and Re are the Grashof and Reynolds number, respectively. An asymptotiy analysis of the complete flow field including potential flow, boundary layer, wake and interaction region is given for small buoyancy parameters and large Reynolds numbers in the distinguished limit KRe^1/4 = O(1). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Absolute Instability of Incompressible Boundary Layer over a Compliant Plate

An incompressible boundary layer on a compliant plate is considered. The influence exerted by the tensile stress and bending stiffness of the plate on the stability of the boundary layer is investigated in the limit of high Reynolds numbers on the basis of the triple-deck theory. It is shown that upstream-propagating growing waves can be generated in a certain range of parameters characterizing the plate properties. As a result, the flow becomes absolutely unstable in the conventional sense.     

Hydromagnetic effects on the three-dimensional flow past a porous plate

Hydromagnetic effects on the three-dimensional flow of an electrically conducting viscous incompressible fluid past a porous plate with periodic suction has been analysed. The uniform flow is subjected to a transversely applied magnetic field. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. Approximate solutions for the components of velocity field and the skin-frictions due to them are obtained and discussed with the help of a graph and tables.     

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.