Linear Elastic Solutions for Slotted Plates

Two-dimensional problems of finite-length blunted cracks cut into infinite plates subject to remote tractions are solved using complex variable theory. The slot geometry is composed of two flat surfaces connected by rounded ends. This special geometrical shape was derived by Riabouchinsky in the study of two-dimensional ideal fluid flow around parallel plates. The simpler antiplane slotted plate problem is addressed initially for this geometry. From this exact solution, the equivalent of a Westergaard stress potential is found and applied to the two other principal modes of fracture, which are plane elasticity problems. For a plate subject to uniform radial tension at infinity, an analytical solution is obtained that will reduce to the familiar mode I singular crack solution as the separation between the parallel faces of the slot becomes zero. For finite-width mode I slots, the rounded ends have tensile tractions which terminate at the adjoining flat surfaces of the slot, which remain traction-free. In this respect, the finite-width mode I slot problem resembles a Barenblatt cohesive zone model of a plane crack or a Dugdale plastic strip model of a plane crack, although the tractions will vary in magnitude along the slot ends rather than remaining uniform as in the former type of crack problems. Similarly, in the case of the finite-width mode II slot problem, the rounded ends of the slot have shear tractions, while the flat surfaces remain load-free. A distinguishing feature of the mode II slot solution over the mode I slot problem is that the maximum in-plane shear stress is constant along the rounded ends of the slot. Because of this, those particular regions of the boundary can represent incipient plastic yield based on either the Mises or Tresca yield condition under plane strain loading conditions. In this way, the problem resembles the plastic strip models of Dugdale, Cherepanov, Bilby-Cottrell-Swinden, and others. Notably, the mode III slot problem also has a constant maximum shear stress along the curved portions of the slot, while the entire slot boundary remains traction-free, unlike the mode II slot problem. Consequently, the mode III slot problem represents both a generalization of the standard mode III crack problem geometry, while simultaneously satisfying the boundary conditions of a plastic strip model.     

Some Steady MHD Flows of the Second Order Fluid

The exact analytic solutions of two problems of a second order fluid in presence of a uniform transverse magnetic field are investigated. The governing equation is of fourth order ordinary differential equation and is solved using perturbation method. In the first problem we discuss the flow of a second order fluid due to non-coaxial rotations of a porous disk and a fluid at infinity. In second problem the flow of a second order conducting fluid between two infinite plates rotating about the same axis is investigated, with suction or blowing along the axial direction. For second order conducting fluid it is observed that asymptotic solution exists for the velocity both in the case of suction and blowing.     

Drag reduction of a nonnewtonian fluid by fluid injection on a moving wall

Summary A boundary layer problem of a nonnewtonian fluid flow with fluid injection on a semi-infinite flat plate whose surface moves with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. Concluding similarity equations are solved numerically to show the dependence of the problem to the velocity ratio λ of the plate to uniform flow and to the injection velocity parameter C. The critical values of λ and C for each nonnewtonian power-law index n are obtained, and their significance in drag reduction is discussed. Received 26 August 1997; accepted for publication 21 October 1998     

Rarefied flow in a channel and a periodic system of channels

the steady two-dimensional isothermal rarefied flow in a channel formed by two parallel flat plates of finite length is studied on the basis of the numerical solution of a linearized kinetic problem. The channel may either be isolated or constitute a cell of a periodic cascade consisting of zero-thickness plates arranged one above the other. As the channel length increases, the flow in it approaches the asymptotic one-dimensional Poiseuille flow. It is shown that the asymptotic dependence of the gas flow rate on the low Knudsen number corresponding to an infinitely long channel is already attained for a channel of length equal to several channel widths, if the flow rate is referred to the pressure gradient at the middle of the channel rather than to the mean pressure difference at the channel ends. The effect of the boundary conditions imposed on the channel entrance is investigated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 166–175, May–June, 2000. The study was carried out with the support of the Russian Foundation for Basic Research (project No. 98-01-00443).     

Unsteady flow of a fourth-grade fluid due to an oscillating plate

The governing non-linear high-order, sixth-order in space and third-order in time, differential equation is constructed for the unsteady flow of an incompressible conducting fourth-grade fluid in a semi-infinite domain. The unsteady flow is induced by a periodically oscillating two-dimensional infinite porous plate with suction/blowing, located in a uniform magnetic field. It is shown that by augmenting additional boundary conditions at infinity based on asymptotic structures and transforming the semi-infinite physical space to a bounded computational domain by means of a coordinate transformation, it is possible to obtain numerical solutions of the non-linear magnetohydrodynamic equation. In particular, due to the unsymmetry of the boundary conditions, in numerical simulations non-central difference schemes are constructed and employed to approximate the emerging higher-order spatial derivatives. Effects of material parameters, uniform suction or blowing past the porous plate, exerted magnetic field and oscillation frequency of the plate on the time-dependent flow, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviour of the fourth-grade non-Newtonian fluid is also compared with those of the Newtonian fluid.     

Numerical predictions for unsteady viscous flow past an array of cylinders

The unsteady two-dimensional flow around an array of circular cylinders submerged in a uniform onset flow is analysed. The fluid is taken to be viscous and incompressible. The array of cylinders consists of two horizontal rows extending to infinity in the upstream and downstream directions. The centre-to-centre distance between adjacent cylinders is fixed at three diameters, and the rows are staggered. Advantage is taken of spatially periodic boundary conditions in the flow direction. This reduces the computational domain to a rectangular region surrounding a single circular cylinder. Two cases, for Reynolds numbers of 1000 and 10,000, are presented.     

Unsteady Motion of a Viscous Liquid between Rotating Parallel Walls in the Presence of a Crossflow

The paper considers the unsteady flow of a viscous incompressible fluid inside an infinitely long slot with uniform injection or suction of the fluid through the porous walls of the slot. The plates with the fluid are rotated rigidly with constant angular velocity. The unsteady flow is induced by nontorsional vibrations of the upper plate. The flowvelocity field and the tangential stress vectors exerted by the fluid on the upper and lower walls of the slot are determined. In this case, one can find an exact solution of the threedimensional nonstationary Navier–Stokes equations. No restrictions are imposed on the motion pattern of the plate.     

Hall effects on hydromagnetic flow on an oscillating porous plate

In this paper, an analysis is made on the unsteady flow of an incompressible electrically conducting viscous fluid bounded by an infinite porous flat plate. The plate executes harmonic oscillations at a frequency n in its own plane. A uniform magnetic field Ho is imposed perpendicular to the direction of the flow. It is found that the solution also exists for blowing at the plate. The temperature distribution is also obtained by taking viscous and Joule dissipation into account. The mean wall temperature θo（O） decreases with the increase in the Hall parameter m. It is found that no temperature distribution exists for the blowing at the plate.     

End effects of nominally two-dimensional thin flat plates

Differences in the structure and dynamics of nominally two-dimensional turbulent wakes are investigated experimentally for a thin flat plate, normal to a uniform flow, with two different end conditions: with and without end plates. Both cases are characterized by Karman-like vortex shedding with broadband low frequency unsteadiness. Both wakes evidence a low frequency flapping motion in addition to the slowly drifting base flow common to cylinder wakes. For the case without end plates, an interaction between the drift motion and the vortex formation process is associated with a much stronger modulation of the quasiperiodic vortex shedding amplitude when compared to the case with end plates where a flapping motion is more strongly expressed. These dynamics underlie structural differences in the mean wake and Reynolds stress fields.     

Fully Developed Mixed Convection Flow Between Inclined Parallel Plates Filled with a Porous Medium

A fully developed mixed convection flow between inclined parallel flat plates filled with a porous medium is considered through which there is a constant flow rate and with heat being supplied to the fluid by the same uniform heat flux on each plate. The equations governing this flow are made non-dimensional and are seen to depend on two dimensionless parameters, a mixed convection parameter λ and the Péclet number Pe, as well as the inclination γ of the plates to the horizontal. The velocity and temperature profiles are obtained in terms of λ, Pe and γ when the channel is inclined in an upwards direction as well as for horizontal channels. The limiting cases of small and large λ and small Pe are considered with boundary-layer structures being seen to develop on the plates for large values of λ.     

Mathematical modeling of fractional derivatives for magnetohydrodynamic fluid flow between two parallel plates by the radial basis function method

Investigations into the magnetohydrodynamics of viscous fluids have become more important in recent years, owing to their practical significance and numerous applications in astro-physical and geo-physical phenomena. In this paper, the radial base function was utilized to answer fractional equation associated with fluid flow passing through two parallel flat plates with a magnetic field. The magnetohydrodynamics coupled stress fluid flows between two parallel plates, with the bottom plate being stationary and the top plate moving at a persistent velocity. We compared the radial basis function approach to the numerical method (fourth-order Range-Kutta) in order to verify its validity. The findings demonstrated that the discrepancy between these two techniques is quite negligible, indicating that this method is very reliable. The impact of the magnetic field parameter and Reynolds number on the velocity distribution perpendicular to the fluid flow direction is illustrated. Eventually, the velocity parameter is compared for diverse conditions α, Reynolds and position (y), the maximum of which occurs at α = 0.4. Also, the maximum velocity values occur in α=0.4 and Re=1000 and the concavity of the graph is less for α=0.8.     

Numerical study of flow-induced oscillations of two rigid plates elastically hinged at the two ends of a stationary plate in a cross-flow

The flow-induced oscillation (FIO) of bluff bodies is commonly encountered in the fluid structure interaction (FSI) problems. In this study, we use an unstructured moving grid strategy and simulate the FIO of two rigid plates, which are elastically hinged at the two ends of a fixed flat plate in a cross-flow. We use a hybrid finite-element-volume (FEV) method in an arbitrary Lagrangian–Eulerian (ALE) framework to study FIO of the two hinged plates. The current simulations are carried out for wide ranges of flow Reynolds number (50–175), spring stiffness coefficient, and the two hinged plates' moment of inertia magnitudes. The influences of these parameters are investigated on the magnitudes of maximum deflection angle, the amplitude of oscillation, the total lift and drag coefficients, and so on. The study is also carried out in the transition period to describe the in-phase and out-of-phase angular oscillations occurring for the two elastically hinged plates with respect to each other. After the transition period, the two hinged plates eventually arrive to a similar periodic oscillation; however, with some phase lags. We find that the achieved phase lag is equal to the phase lag between the two pairs of flow vortices, which are alternatively shed into the flow from the upper and lower hinged plates. Similar to past FIO problems, the current model also exhibits two important lock-in and phase-switch FSI phenomena; however, in angular directions. There is a phase jump of approximately 170° between the aerodynamic lift coefficient and angular oscillations of hinged plates, which nearly occurs in the middle of lock-in region. Indeed, our literature review shows that this is the first time to report the phase-switch phenomenon in angular oscillations of three-element bluff bodies in a FSI problem.     

Effect of ion slip on the time-varying Hartmann flow of a non-Newtonian viscoelastic fluid with heat transfer

Ion slip in a time-varying Hartmann flow of a conducting incompressible non-Newtonian viscoelastic fluid between two parallel horizontal insulating porous plates is studied with allowance for heat transfer. A uniform and constant pressure gradient is applied in the axial direction. An external uniform magnetic field and uniform suction and injection through the surface of the plates are applied in the normal direction. The two plates are maintained at different but constant temperatures; the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained with the use of finite differences, and the effect of various physical parameters on both the velocity and temperature fields is discussed.     

Self-similar problem of separated ideal-fluid flow over an expanding plate

The research carried out in [1–8] is developed by considering the self-similar problem of the unsteady separated flow over a plate expanding from a point with the constant velocity D of a plane-parallel stream of ideal fluid with velocity V_∞. At infinity the flow is uniform, steady and normal to the surface of the plate. A wide range of values of the parameter α=V_∞/D is investigated. On the value of α there depends, in particular, the direction of shedding of the vortex sheets (VS) which, in accordance with the Joukowsky-Chaplygin condition, occur in separated flow over a plate. A comparison is made with the results obtained when the sheets are replaced by vortex filaments (VF). In accordance with [9] the choice of the intensity of the VF ensures, like the introduction of VS, the finiteness of the flow velocity at the edges of the plate. Within the framework of the unsteady analogy and the law of plane sections the problem of the flow over a delta wing at an angle of attack reduces to the unsteady flow over an expanding plate investigated. In addition to [3, 9], this question was also examined in [10–15]. In [11–15] and in [3] the analysis is based on VS and in [9, 10] on VF. Special attention is paid to the topology of the flow, in particular, to the structure of the so-called conical streamlines and their points of convergence and divergence (this was done in [3] for a special, nonlinear law of expansion of the plate and a variable free-stream velocity). The results obtained for the models with VS and VF are compared over a broad range of values of α, not only with respect to the integral characteristics, as in [12], but also with respect to the flow patterns. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 62–69, September–October, 1988.     

Comprehensive correlations for laminar mixed convection on vertical and horizontal flat plates

This paper presents comprehensive correlation equations of the local Nusselt numbers and surface shear stresses for laminar forced convection, natural convection, and mixed convection on vertical and horizontal flat plates which are maintained with uniform wall temperature or uniform surface heat flux. The correlation for pure forced convection and pure natural convection are very accurate for any Prandtl number between 0.001 and infinity. The correlations for mixed convection coincide very well with the numerical results over the entire regimes of mixed convection intensity and Prandtl number for the eight cases of the two plates with distinct thermal boundary conditions and buoyancy-assisting and-opposing flow conditions.     

Magnetohydrodynamic oblique stagnation-point flow

Laminar two-dimensional stagnation flow of a viscous and electrically conducting fluid obliquely impinging on a flat plate in the presence of a uniform applied magnetic field is formulated as a similarity solution of the Navier-Stokes equations. The relative importance of this flow is measured by the dimensionless strain rate and magnetohydrodynamic parameters γ and M. The viscous problem is reduced to a coupled pair of ordinary differential equations governed by γ and M. It is found that the parameter M causes a shift in the position of the point of zero skin friction along the wall.     

Finite Difference Method for the Acoustic Radiation of an Elastic Plate Excited by a Turbulent Boundary Layer: A Spectral Domain Solution

A finite difference method is developed to study, on a two-dimensional model, the acoustic pressure radiated when a thin elastic plate, clamped at its boundaries, is excited by a turbulent boundary layer. Consider a homogeneous thin elastic plate clamped at its boundaries and extended to infinity by a plane, perfectly rigid, baffle. This plate closes a rectangular cavity. Both the cavity and the outside domain contain a perfect fluid. The fluid in the cavity is at rest. The fluid in the outside domain moves in the direction parallel to the system plate/baffle with a constant speed. A turbulent boundary layer develops at the interface baffle/plate. The wall pressure fluctuations in this boundary layer generates a vibration of the plate and an acoustic radiation in the two fluid domains. Modeling the wall pressure fluctuations spectrum in a turbulent boundary layer developed over a vibrating surface is a very complex and unresolved task. Ducan and Sirkis [1] proposed a model for the two-way interactions between a membrane and a turbulent flow of fluid. The excitation of the membrane is modeled by a potential flow randomly perturbed. This potential flow is modified by the displacement of the membrane. Howe [2] proposed a model for the turbulent wall pressure fluctuations power spectrum over an elastomeric material. The model presented in this article is based on a hypothesis of one-way interaction between the flow and the structure: the flow generates wall pressure fluctuations which are at the origin of the vibration of the plate, but the vibration of the plate does not modify the characteristics of the flow. A finite difference scheme that incorporates the vibration of the plate and the acoustic pressure inside the fluid cavity has been developed and coupled with a boundary element method that ensures the outside domain coupling. In this paper, we focus on the resolution of the coupled vibration/interior acoustic problem. We compare the results obtained with three numerical methods: (a) a finite difference representation for both the plate displacement and the acoustic pressure inside the cavity; (b) a coupled method involving a finite difference representation for the displacement of the plate and a boundary element method for the interior acoustic pressure; (c) a boundary element method for both the vibration of the plate and the interior acoustic pressure. A comparison of the numerical results obtained with two models of turbulent wall pressure fluctuations spectrums - the Corcos model [3] and the Chase model [4] - is proposed. A difference of 20 dB is found in the vibro-acoustic response of the structure. In [3], this difference is explained by calculating a wavenumber transfer function of the plate. In [6], coupled beam-cavity modes for similar geometry are calculated by the finite difference method. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mixed flow of an ideal fluid on a plane containing a recess

In the exact formulation, a study is made of the solution to the problem of the flow of an ideal incompressible fluid on a flat surface in a recess in the form of a half-cylinder in the direction at right angles to the flow. In the recess, the flow is assumed to have uniform vorticity, while in the exterior unbounded flow it is irrotational. On the separating streamline, the Bernoulli constant has a discontinuity of a given magnitude.     

Flow and heat transfer of an electrically conducting third grade fluid past an infinite plate with partial slip

The effects of partial slip on the steady flow and heat transfer of an electrically conducting, incompressible, third grade fluid past a horizontal plate subject to uniform suction and blowing is investigated. Two distinct heat transfer problems are studied. In the first case, the plate is assumed to be at a higher temperature than the fluid; and in the second case, the plate is assumed to be insulated. The momentum equation is characterized by a highly nonlinear boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. Numerical solutions for the governing nonlinear equations are obtained over the entire range of physical parameters. The effects of slip, magnetic parameter, non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. It is interesting to find that the velocity and the thermal boundary layers decrease with an increase in the slip, and as the slip increases to infinity, the flow behaves as though it were inviscid.