Fluctuating flow of an elastico-viscous fluid along an infinite porous plate with an applied magnetic field

The problem of oscillating free stream flow of an elastico-viscous, incompressible, and electrically conducting fluid along an infinite plate with suction varying periodically with time, is considered in the presence of a transverse magnetic field. The effect of the elasticity of the fluid, the magnetic fluid, and the fluctuation of suction velocity on the velocity and the skin friction is examined.     

Torsional oscillations of an infinite plate in an elastico-viscous fluid

Summary The equations of motion of an infinite plate performing torsional oscillations in Walters elastico-viscous liquid B have been solved by expanding the velocity profile in powers of the amplitude of oscillation of the plate. The first order solution consists of a transverse velocity and the second-order solution gives a radial-axial flow composed of a steady part and a fluctuating part. The steady part of the radial flow does not vanish outside the boundary layer and hence the equations are solved by another approximate method for the steady part of the flow. The effects of the non-Newtonian term is to increase the non-dimensional boundary layer to start with and subsequently to decrease it and to increase the shearing stress at the plate. The steady radial and the steady axial velocities fall short of the inelastic flow in the beginning but later on their values lie above.     

Effects of couple stresses on the oscillatory flow past an infinite plate with constant suction

Summary An analysis of a two dimensional oscillatory flow past an infinite porous plate with contant suction is carried out on taking into account the couple stresses. Here the free stream velocity oscillates about a nonzero constant mean. Approximate solutions are derived to coupled linear equations, and the expressions for the mean velocity, the transient velocity, the mean skinfriction, the amplitude and the phase of skin-friction are obtained. The solutions are followed by discussion. the effects of variations of α(ν_r/ν), β(Iν/γ) and λ, the frequency are graphically represented and physically interpreted. It is observed that the reverse type of flow does not occur in the presence of the couple stresses.

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Sommario In questo lavoro è svolta un'analisi di un flusso oscillatorio bidimensionale sopra una piastra porosa, infinita, con aspirazione costante, tenendo conto delle coppie di sforzo. La velocità della corrente libera oscilla attorno ad un valore medio costante diverso da zero. Si deducono le soluzioni approssimate per le equazioni lineari accoppiate e si ottengono le espressioni per la velocità media, la velocità transitoria, l'attrito superficiale, l'ampiezza e la fase. Si discutono le soluzioni. Si rappresentano graficamente e si interpretano fisicamente gli effetti delle variazioni di α(ν_r/ν), β(Iν/γ) e λ. Si osserva che in presenza delle coppie di sforzo nel fluido non si ha il tipo inverso di flusso.

     

Viscous dissipative fluid flow past a semi‐infinite vertical plate with variable surface temperature

An analysis of the effect of viscous dissipative heat on two‐dimensional viscous incompressible fluid flow past a semi‐infinite vertical plate with variable surface temperature is carried out. The dimensionless governing equations are unsteady, two‐dimensional, coupled, and non‐linear governing equations. A most accurate, unconditionally stable and fast converging implicit finite‐difference scheme is used to solve the non‐dimensional governing equations. Velocity and temperature of the flow have been presented graphically for various parameters occurring in the problem. The local and average skin friction and Nusselt number are also shown graphically. It is observed that greater viscous dissipative heat causes a rise in the temperature. Copyright © 2007 John Wiley & Sons, Ltd.     

MHD free convection flow of visco-elastic fluid past an infinite vertical porous plate

 This work provides a comprehensive theoretical analysis of a two-dimensional unsteady free convection flow of an incompressible, visco-elastic fluid past an infinite vertical porous plate. Solutions for the zero order perturbation velocity profile, the first order perturbation velocity and temperature profiles in closed form are obtained with the help of Laplace transform technique. The numerical solutions are carried out for the Prandtl number 0.1, 0.72, 1.0, 1.5 and 2.0 which are appropriate for different types of liquid metals and for different values of magnetic field parameter, M. Received on 1 September 1999     

Soret and Dufour effects on unsteady MHD flow past an infinite vertical porous plate with thermal radiation

The objective of the present study is to investigate the effect of flow parameters on the free convection and mass transfer of an unsteady magnetohydrodynamic flow of an electrically conducting, viscous, and incompressible fluid past an infinite vertical porous plate under oscillatory suction velocity and thermal radiation. The Dufour （diffusion thermo） and Soret （thermal diffusion） effects are taken into account. The problem is solved numerically using the finite element method for the velocity, the temperature, and the concentration field. The expression for the skin friction, the rate of heat and mass transfer is obtained. The results are presented numerically through graphs and tables for the externally cooled plate （Gr 〉 0） and the externally heated plate （Gr 〈 0） to observe the effects of various parameters encountered in the equations.     

On hydromagnetic fluctuating flow past an infinite porous plate in slip flow regime

The effect of a uniform external magnetic field on the laminar, incompressible rarefied gas flow along an infinite porous flat plate is studied under the following conditions: 1) there is uniform suction, 2) the external flow velocity varies periodically with time in magnitude but not in direction, 3) the magnetic Reynolds number is small and 4) the current occurs under slip flow boundary conditions. Expressions for the velocity and temperature fields in the boundary layer are obtained. The response of skin friction, and heat transfer to the fluctuating stream is studied for variations in the rarefaction parameter h ₁, the magnetic field parameter M, and the frequency of the fluctuating stream.Nomenclature c _p specific heat of the gas - f ₁ Maxwells reflection coefficient - f ₂ thermal accommodation coefficient - G as defined in (36) - h ₁ rarefaction parameter (L ₁ v ₀/) - h ₂ nondimensional temperature jump coefficient (L ₂ v ₀/) - H amplitude of the skin friction - k thermal conductivity - K _n Knudsen number - L mean free path - L ₁ (2–f ₁/f ₁) L - L ₂ - M magnetic field parameter ( ₀ B ₀ ² /v ₀ ² ) - m 1/2[1+(1+4M+4i)^1/2], m _r+im _i - n ₁ 1/2[1+(1+4M)^1/2] - q heat flux - R suction Reynolds number - T temperature - x, y coordinates along and perpendicular to the plates - u, v velocity components along x, y-directions - density - kinematic viscosity - ₀ electrical conductivity - Prandtl number - frequency of the fluctuating stream - nondimensional frequency parameter (/v ₀ ² ) - nondimensional distance from wall (v ₀ y/) - phase lead - U _{0 0} mean velocity in the boundary layer - U _{0 1}, U _{0 2} amplitude of the velocity fluctuation in the boundary layer - specific heat ratio     

Effect of suction upon unsteady laminar free convection flow on a vertical infinite flat plate

Summary Unsteady laminar free convection flow past a vertical infinite flat plate subjected to suction is considered. Exact solutions of momentum and energy equations are obtained in two cases: (1) When the plate temperature is proportional to some power of time and (2) when the heat flux at the plate is proportional to some power of time. It is assumed that the suction velocity varies as (time)^–1/2. Expressions for the temperature and velocity profiles are obtained in closed forms in both the cases. Effect of suction on velocity, temperature, skin friction and rate of head transfer is studied for Prandtl numbers 0.02, 0.1, 0.72, 1 and 10.     

An existence theorem for the flow of a non-newtonian fluid past an infinite porous plate

Non-Newtonian fluid mechanics affords an excellent opportunity for studying many of the mathematical methods which have been developed to analyse non-linear problems in mechanics. The flow of an incompressible fluid of grade three past an infinite porous flat plate, subject to suction at the plate, is governed by a non-linear differential equation that is particularly well suited to demonstrate the power and usefulness of three such techniques. We establish an existence theorem using shooting methods. Next, we investigate the problem using a perturbation analysis. It is not clear that the perturbation solution converges and thus may not be the appropriate solution for a certain range of a material constant (which is not the perturbation parameter). Finally, we employ a numerical method which is particularly suited to the problem in question.     

An exact solution for the flow of a non-newtonian fluid past an infinite porous plate

Summary The flow of an incompressible fluid of second grade past an infinite porous plate subject to either suction or blowing at the plate is studied. It is found that existence of solutions is tied in with the sign of material moduli and in marked contrast to the Classical Newtonian, fluid solutions can be exhibited for the blowing problem.

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Sommario Si studia la corrente di un fluido incomprimibile di secondo grado che lambisce una lastra porosa da cui è succhiato o soffiato. Si trova che l'esistenza delle soluzioni è legata al segno dei moduli del materiale e, in netto contrasto col fluido newtoniano classico, si possono trovare soluzioni per il problema del soffiamento.

     

A new exact solution for the flow of a Maxwell fluid past an infinite plate

A new exact solution corresponding to the flow of a Maxwell fluid over a suddenly moved flat plate is determined. This solution is in all accordance with a previous one and for λ→0 it goes to the well-known solution for Navier-Stokes fluids.     

Oscillating plate temperature effects on a flow past an infinite vertical porous plate with constant suction and embedded in a porous medium

 An approximate solution to the problem of flow of a viscous incompressible dissipative fluid past an infinite vertical porous plate embedded in a porous medium is presented here. The plate temperature is assumed to be oscillating about a constant mean temperature. Mean velocity and mean temperature, the transient velocity and temperature profiles are shown graphically. The mean skin-friction and the mean rate of heat transfer are also shown graphically. The expressions for the amplitude and the phase of the skin-friction and the rate of heat transfer are derived and their numerical values are listed in Tables. The effects of different parameters governing the unsteady flow are discussed. Received on 23 November 1998     

Numerical solutions for the flow of a fluid of grade three past an infinite porous plate

This paper presents a numerical study of the flow of an incompressible fluid of grade three past an infinite porous flat plate, subject to suction at the plate. This flow is governed by a non-linear differential equation that is particularly well suited to demonstrate the power and usefulness of different numerical techniques. In this work, the numerical solutions are obtained using a Runge-Kutta method of fourth order. The accuracy of the method for this problem is demonstrated.     

Free convection effects on the oscillatory flow of an electrically conducting rarefied gas past an infinite, vertical plate with constant suction

An analysis of a two-dimensional, unsteady flow of an electrically conducting, viscous, incompressible rarefied gas past an infinite vertical porous plate is carried out under the following assumptions: (i) the suction velocity normal to the plate is constant (ii) the free stream velocity oscillates in time about a constant mean (iii) the plate temperature is constant (iv) the difference between the temperature of the plate and the free stream is moderately large causing the free convection currents (v) first order velocity-slip and the temperature jump boundary conditions (vi) transverse magnetic field (vii) induced magnetic field is negligible.Approximate solutions to the coupled, non-linear equations governing the flow are derived for the mean velocity, mean temperature, mean-skin-friction, mean rate of heat transfer, transient velocity and temperature, fluctuating parts of the velocity profiles, the amplitude and the phase of the skin-friction and the rate of heat-transfer. They are shown graphically followed by a discussion. The effects of ±G (Grashof number), ±E (Eckert number), M (Magnetic field parameter), h ₁ (rarefaction parameter), h ₂ (temperature jump coefficient), (frequency) are discussed for heating (G<0) or cooling (G>0) of the plate by the free convection currents.Nomenclature |B| amplitude of skin-friction - B ₀ applied magnetic field - c _p specified heat at constant pressure - E Eckert number - f ₁ Maxwell's reflection coefficient - f ₂ thermal accommodation coefficient - g _x acceleration due to gravity - G Grashof number - h ₁ rarefaction parameter (L ₁ v ₀/) - h ₂ non-dimensional temperature jump coefficient (L ₂ v ₀/) - k thermal conductivity - K _n Knudsen number - L mean free path - L ₁ (2–f ₁)L/f ₁ - L ₂ - l ₁ characteristic length - M magnetic field parameter - M _r, M _i fluctuating parts of velocity - m - P Prandtl number - p pressure - q rate of heat transfer - q _m mean rate of heat transfer - |Q| amplitude of rate of heat transfer - R suction Reynolds number - T temperature of fluid - T _w temperature of the plate - T temperature of the fluid in free stream - t time - t dimensionless time - U free stream velocity - U dimensionless free stream velocity - U mean of U(t) - u, v velocity components in x, y directions - u dimensionless velocity in x direction - u ₀ mean velocity - u ₁ fluctuating part of velocity - v ₀ suction velocity - x, y coordinate system - x, y dimensionless coordinates - frequency of the free stream oscillations - dimensionless frequency - dimensionless temperature - ₁ fluctuating part of temperature - phase angle of skin-friction - phase angle of rate of heat transfer - density of the fluid in the boundary layer - density of the fluid in the free stream - viscosity - kinematic viscosity - electrical conductivity of the fluid - small positive constant - skin-friction - _m mean skin-friction - specific heat ratio - ₁ coefficient of volume expansion     

Magnetogasdynamic flow past an infinite porous flat plate in slip flow regime

This paper presents an exact solution for the flow of a rarefied ionized gas over an infinite porous plate in the presence of a transverse magnetic field, by using the well known continuum approach. An attempt is made to bring out the salient features of the interaction between the applied magnetic field and the flow of a rarefied conducting gas. The analysis reveals that the skin friction, and the heat transfer into the plate are reduced due to gas rarefaction.     

Visco-elastic fluid flow past an infinite vertical porous plate in the presence of first-order chemical reaction

An analysis has been developed to study the unsteady free convection flow of an incompressible visco-elastic fluid on a continuously moving vertical porous plate in the presence of a first-order chemical reaction. The governing equations are solved numerically using an implicit finite difference technique. The obtained numerical solutions are compared with the analytical solutions. The velocity profiles are presented. A parametric analysis is performed to illustrate the influences of the visco-elastic parameter, the dimensionless chemical reaction parameter, and the plate moving velocity on the steady state velocity profiles, the time dependent friction coefficient, the Nusselt number, and the Sherwood number.     

Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing

An analysis is made of Hall effects on the steady shear flow of a viscous incompressible electrically conducting fluid past an infinite porous plate in the presence of a uniform transverse magnetic field. It is shown that for suction at the plate, steady shear flow solution exists only when S²<Q, where S and Q are the suction and magnetic parameters, respectively. The primary flow velocity decreases with increase in Hall parameter m. But the cross-flow velocity first increases and then decreases with increase in m. Similar results are obtained for variation of the induced magnetic field with m. It is further found that for blowing at the plate, steady shear flow solution exists only when , where S₁ is the blowing parameter.     

Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid over unsteady stretching sheet

This article studies the three-dimensional boundary layer flow of an elasticoviscous luid over a stretching surface. Velocity of the stretching sheet is assumed to be ime-dependent. Effect of mass transfer with higher order chemical reaction is further onsidered. Computations are made by the homptopy analysis method (HAM). Convergence f the obtained series solutions is explicitly analyzed. Variations of embedding arameters on the velocity and concentration are graphically discussed. Numerical computations f surface mass transfer are reported. Comparison of the present results with he numerical solutions is also given.     

Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid over unsteady stretching sheet

This article studies the three-dimensional boundary layer flow of an elasticoviscous fluid over a stretching surface. Velocity of the stretching sheet is assumed to be time-dependent. Effect of mass transfer with higher order chemical reaction is further considered. Computations are made by the homptopy analysis method (HAM). Convergence of the obtained series solutions is explicitly analyzed. Variations of embedding parameters on the velocity and concentration are graphically discussed. Numerical computations of surface mass transfer are reported. Comparison of the present results with the numerical solutions is also given.