Group method analysis of magneto-elastico-viscous flow along a semi-infinite flat plate with heat transfer

The group theoretic method is applied for solving problem of the flow of an elastico-viscous liquid past an infinite flat plate in the presence of a magnetic field normal to the plate. The application of one-parameter transformation group reduces the number of independent variables, by one, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate corresponding conditions. Numerical solution of the velocity field and heat transfer have been obtained. The effect of the magnetic parameter M on velocity field, shear stress, temperature fields and heat transfer has been discussed.     

Heat transfer for flow past permeable bed

A theoretical analysis of the temperature distribution is presented for a viscous liquid through a two dimensional parallel channel formed by an impermeable upper plate and permeable lower plate under the action of a constant pressure gradient past a porous bed. It is shown that the flow properties are dependent on the parametersα, β andσ characteristic of the permeable material. The effects of these parameters on temperature distribution and Nusselt number at both the plates are studied in detail.     

Numerical study of transient free convective mass transfer in a Walters-B viscoelastic flow with wall suction

A transient model for the free convective, nonlinear, steady, laminar flow and mass transfer in a viscoelastic fluid from a vertical porous plate is presented. The Walters-B liquid model is employed which introduces supplementary terms into the momentum conservation equation. The transformed conservation equations are solved using the finite difference method (FDM). The influence of viscoelasticity parameter (Γ), species Grashof number (Gc), Schmidt number (Sc), distance (Y) and time (t) on the velocity (U) and also concentration distribution (C) is studied graphically. Velocity is found to increase with a rise in viscoelasticity parameter (Γ) with both time and distances close to the plate surface. An increase in Schmidt number is observed to significantly decrease both velocity and concentration in time and also with separation from the plate. Increasing species Grashof number boosts the flow velocity through all time and causes a significant rise primarily near the plate surface. The study has applications in polymer materials processing.     

Effect of variable density on hydromagnetic mixed convection flow of a non-Newtonian fluid past a moving vertical plate

The effect of temperature-dependent density on MHD mixed convection flow of power-law fluid past a moving semi-infinite vertical plate for high temperature differences between the plate and the ambient fluid is studied. The fluid density is assumed to decrease exponentially with temperature. The usual Boussinesq approximations are not considered due to the large temperature differences. The surface temperature of the moving plate was assumed to vary according to a power-law form, that is, T_w(x) = T_∞ + Ax^γ. The fluid is permeated by a uniform magnetic field imposed perpendicularly to the plate on the assumption of small magnetic Reynolds number. A numerical shooting algorithm for two unknown initial conditions with fourth-order Runge–Kutta integration scheme has been used to solve the coupled non-linear boundary value problem. The effects of various parameters on the velocity and temperature profiles as well as the local skin-friction coefficient and the local Nusselt number are presented graphically and in the tabular form. The results show that application of Boussinesq approximations in a non-Newtonian fluid subjected to high temperature differences gives a significant error in the values of the skin-friction coefficient and the application of an external magnetic field reduces this error markedly in the case of shear-thickening fluid.     

Effect of sudden suction on the hydromagnetic natural convection from a vertical plate

The effects of step function change in suction velocity are investigated for a natural convection flow from a vertical porous flat plate of infinite length in the presence of transverse magnetic field for two cases namely (i) when the plate is suddenly raised to a uniform higher temperature, (ii) when the plate suddenly begins to generate a uniform heat flux at its surface. In (i) the coefficient of heat transfer becomes independent of the Hartmann numberM. In either case for a fixed time t except for correct steady state t→ ∞, the effect of the Hartmann number is to decrease both the velocity of the fluid and the skin-friction at the plate. In the correct steady state the velocity field and the skin-fiiction become independent of the Hartmann number.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Effects of mass transfer on free convective flow of a dissipative,incompressible fluid past an infinite vertical porous plate with suction

An analysis of the mass transfer effects on the free convective flow of an incompressible, dissipative, viscous fluid past an infinite, vertical porous plate with constant suction, has been carried out. Approximate solutions to coupled non-linear equations governing the flow are derived. The velocity and the temperature profiles are shown graphically for air (P=0·71). The effects of Gr (Grashof number), Gc (the modified Grashof number), Sc (Schmidt number), E (Eckert number) are discussed qualitatively during the course of discussion. It is observed that due to the addition of the foreign mass, there is a rise in the velocity and a fall in the temperature. But the skin-friction increases when Sc?1 and it decreases when Sc ～1. The rate of heat transfer increases for Sc<1 and decreases for Sc>1.     

Mixed convection of a viscous dissipating fluid about a vertical flat plate

In this study, the effect of the viscous dissipation in steady, laminar mixed convection heat transfer from a heated/cooled vertical flat plate is investigated in both aiding and opposing buoyancy situations. The external flow field is assumed to be uniform. The governing systems of partial differential equations are solved numerically using the finite difference method. A parametric study is performed in order to illustrate the interactive influences of the governing parameters, mainly, the Richardson number, Ri (also known as the mixed convection parameter) and the Eckert number, Ec on the velocity and temperature profiles as well as the friction and heat transfer coefficients. Based on the facts the free stream is either in parallel or reverse to the gravity direction and the plate is heated or cooled, different flow situations are identified. The influence of the viscous dissipation on the heat transfer varied according to the situation. For some limiting cases, the obtained results are validated by comparing with those available from the existing literature. An expression correlating Nu in terms of Pr, Ri and Ec is developed.     

Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field

The two-dimensional, steady, laminar, forced and free convective boundary layer flow of a magnetic fluid over a semi-infinite vertical plate, under the action of a localized magnetic field, is numerically studied. The magnetic fluid is considered to be water-based with temperature dependent viscosity and thermal conductivity. The study of the boundary layer is separated into two cases. In case I the boundary layer is studied near the leading edge, where it is dominated by the large viscous forces, whereas in case II the boundary layer is studied far from the leading edge of the plate where the effects of buoyancy forces increase. The numerical solution, for these two different cases, is obtained by an efficient numerical technique based on the common finite difference method. Numerical calculations are carried out for the value of Prandl number Pr =  49.832 (water-based magnetic fluid) and for different values of the dimensionless parameters entering into the problem and especially for the magnetic parameter Mn, the viscosity/temperature parameter Θ _r and the thermal/conductivity parameter S*. The analysis of the obtained results show that the flow field is influenced by the application of the magnetic field as well as by the variation of the viscosity and the thermal conductivity of the fluid with temperature. It is hoped that they could be interesting for engineering applications.     

Flow engendered by the time-varying motion of a porous plate in the presence of a magnetic field

The problem of flow of an electrically conducting viscous liquid due to the time-varying motion of an infinite porous plate has been studied. There is a uniform magnetic field imposed transversely to the plate and the magnetic lines of force are taken to be fixed relative to the fluid. Exact solutions for the velocity field and the skin-friction have been obtained and some particular cases have been discussed. The effect of suction parameter and magnetic field strength on the flow characteristics have been depicted through several graphs and tables.     

Viscous dissipation and Joule heating effects on MHD-free convection from a vertical plate with power-law variation in surface temperature in the presence of Hall and ion-slip currents

An analysis is presented to study the effects of viscous dissipation and Joule heating on MHD-free convection flow past a semi-infinite vertical flat plate in the presence of the combined effect of Hall and ion-slip currents for the case of power-law variation of the wall temperature. The fluid is permeated by a strong transverse magnetic field imposed perpendicularly to the plate on the assumption of a small magnetic Reynolds number. The governing differential equations are transformed by introducing proper non-similarity variables and solved numerically. The effects of various parameters on the velocity and temperature profiles as well as the local wall shear stresses and the local Nusselt number are presented graphically and in tabular form. It is found that the magnetic field acts as a retarding force on the tangential flow but has a propelling effect on the induced lateral flow. The skin-friction factor for the tangential flow and the local Nusselt number decrease but the skin-friction factor for the lateral flow increases as the magnetic field increases. The skin-friction factor for the tangential and lateral flows are increased while the local Nusselt number is decreased if the effect of viscous dissipation, Joule heating and heat generation are considered. The opposite trend was observed as the temperature power coefficient n is increased. Also, the skin-friction factor for the tangential flow and the local Nusselt number are increased due to the Hall and ion-slip currents, whereas the skin-friction factor for the tangential flow increases when Hall values increase to one and decreases for values of Hall greater than one, but reduces by rising ion-slip values.     

Free convection flow of an elastico-viscous liquid through a vertical circular cylinder

The problem of free convection flow of an elastico-viscous liquid through a vertical circular cylinder at constant temperature including the effects of frictional heating and distributed heat sources or sinks has been considered. The equations of motion and energy have been solved by an iteration technique. The effects of the elasticity of the liquid, buoyancy forces and heat sources or sinks on the velocity and temperature fields, skin-friction and Nusselt number have been discussed.     

Similarity solutions for mixed convection boundary layer flow over a permeable horizontal flat plate

The effects of suction and injection on steady laminar mixed convection boundary layer flow over a permeable horizontal flat plate in a viscous and incompressible fluid is investigated in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction and injection parameter f₀, the constant exponent n of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using a finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the reduced local Nusselt number, and the velocity and temperature profiles are obtained for various values of the parameters considered. Dual solutions are found to exist for the opposing flow.     

Flow,heat, and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer

In this paper, we investigate the flow, heat and mass transfer of a viscous fluid flow over a stretching sheet by including the blowing effects of mass transfer under high flux conditions. Mass transfer in this work means species transfer and is different from mass transpiration for permeable walls. The new contribution from this work is, for the first time, to consider the coupled blowing effects from massive species transfer on flow, heat, and species transfer for a stretching plate. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy and mass transfer equations are solved exactly and the effects of the blowing parameter, the Schmidt number, and the Prandtl number on the flow, heat and mass transfer are presented and discussed. The solution is given in terms of an incomplete Gamma function. It is found the coupled blowing effects due to mass transfer can have significant influences on velocity profiles, drag, heat flux, as well as temperature and concentration profiles. These solutions provide rare results with closed form analytical expressions and can be used as benchmark problem for numerical code validation.     

The nonsimilar laminar wall jet along a moving wall,in a free stream and in a free stream/moving wall

The flow of a laminar wall jet along either a moving plate, or in a free stream, or in combined moving plate and free stream is considered. The plate is isothermal and its temperature is different from that of the ambient fluid. The governing boundary-layer equations are converted into non-dimensional form and are solved numerically. Velocity and temperature profiles as well as the variation of wall shear stress and wall heat transfer are presented for all cases considered. For the case of a moving plate new results have been found although this problem has been investigated in the past. For the case of the free stream and the combined moving plate/free stream all the results are new and are presented for the first time in the literature. The three cases have been tackled with a unified way.     

Inverse modeling of a solar collector involving Fourier and non-Fourier heat conduction

This article applies the golden section search method (GSSM), simplex search method (SSM) and differential evolution (DE) for predicting the unknown Fourier number (Fo), Vernotte number (Ve) and non-dimensional solar heat flux (S¹) in a flat-plate solar collector when subjected to a given temperature requirement. The required temperature field is calculated using an analytical forward method by considering Fourier and non-Fourier heat conduction, and using this, the inverse problem is solved to predict the Fo, Ve and S¹ which are assumed to be the unknown parameters. The study reveals that the temperature field is highly sensitive to the Fo, thus even a small error in the temperature measurement can result in an unrealistic estimation of heating time of the collector. The present study is proposed to be useful in determining the time, the time lag and solar heat flux for controlled heating of an absorber plate within a stipulated time, which will be required to attain a prescribed/desired temperature distribution. Additionally, the study also shows that subjected to different time levels, the same temperature distribution is possible through different absorber plate materials. It has been observed from the present study that apart from SSM and DE, GSSM fails to estimate the unknown parameters at large value of Ve and small value of Fo, due to the associated fluctuation in the measured temperature field. The present study further discusses the computational performance of direct search method (e.g. GSSM and SSM) with that of the evolutionary method (DE) in terms of the maximum number of iteration and CPU time required to achieve the desired objective.     

Combined free and forced convection effects on the magnetohydrodynamic flow through a porous channel

The combined effect of free and forced convection on the flow of an electrically conducting liquid between two horizontal parallel porous walls has been studied. There is a transverse magnetic field at the walls. The equations of motion and energy have been solved by a small perturbation method. The flow phenomenon has been characterized by the non-dimensional numbers like R (cross-flow Reynolds number), K (Brinkman number), G (Grashof number), M (magnetic number) and the effects of these numbers on the velocity and temperature fields, induced magnetic field, electric field and shearing stress at the walls have been studied.     

Investigation of viscous flow phenomena in a rotating system

In this work we have investigated viscous flow in a rotating system. The system consists of a horizontally oriented cylindrical drum rotating around the center and a fixed plate as a scraping internal tool. By this plate the inner volume is divided into two separate chambers. No mass transport between the chambers can take place. The drum is completely or partially filled with a highly viscous, optically transparent silicon oil. In the second case the remaining space is filled with air as a passive phase. Depending on the settings in the system, which are the rotational speed of the drum, material properties of the liquid and the volume ratio of the two phases, different flow phenomena can be observed [1, 2]. For the characterization of the flow dimensionless ratios given by the Reynolds number Re, the Froude number Fr, the Capillary number Ca and the volumetric filling degree F are used. The system is investigated applying the experimental method of Particle Image Velocimetry (PIV) as well as suitable models of Computational Fluid Dynamics (CFD). The goal of this investigation is to identify the velocity fields in the liquid phase for a better understanding of the observed phenomena. The results obtained by the different methods are presented and discussed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study of unsteady flow of a second order liquid past a porous infinite flat plate with uniform suction

The two-dimensional unsteady flow of a second order visco-elastic liquid of finite depth over an infinite porous flat plate has been studied. The suction velocity normal to the plate is taken to be uniform and directed towards the plate and the external flow velocity is considered to be exponentially decaying with time. The governing equation of motion has been solved numerically over an IBM 1130 electronic computer. Shearing stress at the plate has also been sought.     

Free convection flow of a second order fluid past a hot vertical plate

Laminar free convection flow of a second order fluid past a hot vertical plate with varying wall temperature has been studied in this paper. Exact solutions for the velocity and temperature fields have been obtained. The effects of elastic coefficient on the velocity and temperature fields, rate of heat transfer and the skin-friction have been studied.