Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier–Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (, ). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.     

Unsteady mixed convection boundary layer flow over a vertical cone with non-uniform slot suction (injection)

The non-similar solution of an unsteady mixed convection laminar boundary layer flow over a vertical cone in the presence of non-uniform surface mass transfer through slot has been obtained while the axis of cone is inline with the flow. The unsteadiness is caused by the time dependent free stream velocity. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar transformations. The resulting coupled non-linear partial differential equations have been solved numerically by the combination of quasi-linearization technique and an implicit finite difference scheme. Numerical computations are performed for different values of the parameters to display the velocity and temperature profiles graphically. Both accelerating and decelerating free stream velocities are considered. Numerical results are reported to display the effects of non-uniform single and double slot suction (injection) on skin friction and heat transfer coefficients at the wall. Further, the effects of Prandtl number, buoyancy and mass transfer (suction or injection) parameters at different stream-wise locations for various times on velocity and temperature profiles, and on skin friction and heat transfer coefficients are also presented in this paper.     

Unsteady laminar mixed convection about a spinning sphere with a magnetic field

The unsteady laminar boundary-layer flow over an impulsively started translating and spinning isothermal body of revolution in the presence of buoyancy force and magnetic field applied normal to the surface are investigated. Velocity components and temperature are obtained as series of functions in powers of time. Leading and first order functions are obtained analytically and second order functions are determined numerically. The general results are applied to a sphere to investigate the effects of magnetic field and buoyancy force on the velocity and temperature fields and the onset of separation. The magnetic field and buoyancy force are more effective for small rotational speeds and the presence of magnetic field retards the onset of separation. The effect of magnetic field on the temperature field and surface heat flux is weak, indirect and through the velocity field. The magnetic field is observed to initially increase the surface heat flux on the upstream face of the sphere and decrease it on the downstream face.     

Unsteady mixed convection flow in stagnation region adjacent to a vertical surface

The unsteady two-dimensional laminar mixed convection flow in the stagnation region of a vertical surface has been studied where the buoyancy forces are due to both the temperature and concentration gradients. The unsteadiness in the flow and temperature fields is caused by the time-dependent free stream velocity. Both arbitrary wall temperature and concentration, and arbitrary surface heat and mass flux variations have been considered. The Navier-Stokes equations, the energy equation and the concentration equation, which are coupled nonlinear partial differential equations with three independent variables, have been reduced to a set of nonlinear ordinary differential equations. The analysis has also been done using boundary layer approximations and the difference between the solutions has been discussed. The governing ordinary differential equations for buoyancy assisting and buoyancy opposing regions have been solved numerically using a shooting method. The skin friction, heat transfer and mass transfer coefficients increase with the buoyancy parameter. However, the skin friction coefficient increases with the parameter , which represents the unsteadiness in the free stream velocity, but the heat and mass transfer coefficients decrease. In the case of buoyancy opposed flow, the solution does not exist beyond a certain critical value of the buoyancy parameter. Also, for a certain range of the buoyancy parameter dual solutions exist.Die zweidimensionale laminare Mischkonvektionsströmung im Staubereich einer vertikalen Oberfläche, in der Temperatur- und Konzentrationsgradienten die Auftriebskräfte erzeugen, wurde untersucht. Die Unstetigkeiten im Strömungs- und im Temperaturfeld liegen in der zeitabhängigen freien Strömungsge-schwindigkeit begründet. Die willkürliche Wandtempereratur und Konzentration sowie die willkürliche Oberflächenwärme- und die Massenstromschwankungen wurden in Betracht gezogen. Die Navier-Stokes-Gleichung, die Energiegleichung und die Konzentrationsgleichung, die drei nicht lineare, partielle Differentialgleichungen mit drei unabhängigen Variablen darstellen, sind auf eine Gruppe von nicht linearen, gewöhnlichen Differentialgleichungen reduziert worden. Die Berechnung ist auch mit der Grenzflächenap-proximation gemacht worden und die Unterschiede der Ergebnisse wurden diskutiert.Die bestehenden gewöhnlichen Differentialgleichungen für auftriebsuntertützte und auftriebshemmende Bereiche sind numerisch mit dem Shooting-Verfahren gelöst worden. Die Oberflächenreibung, die Wärme- und Stoffübertragungskoeffizienten steigen mit dem Auftriebsparameter. Der Oberflächenreibungskoeffizient steigt mit dem Parameter, der auch für die Unstetigkeit der freien Strömungsgeschwindigkeit verantwortlich ist. Die Wärme- und Stoffübertragungskoeffizienten sinken dann. Im Fall der auftriebsgehemmten Strömung, existiert nach einem bestimmten kritischen Wert des Auftriebsparameters keine Lösung mehr. Für eine bestimmte Reihe von Auftriebsparametern gibt es zwei Lösungen.     

Unsteady free convection flow under the influence of a magnetic field

Summary The unsteady free convection boundary layer at the stagnation point of a two-dimensional body and an axisymmetric body with prescribed surface heat flux or temperature has been studied. The magnetic field is applied parallel to the surface and the effect of induced magnetic field has been considered. It is found that for certain powerlaw distribution of surface heat flux or temperature and magnetic field with time, the governing boundary layer equations admit a self-similar solution locally. The resulting nonlinear ordinary differential equations have been solved using a finite element method and a shooting method with Newton's corrections for missing initial conditions. The results show that the skin friction and heat transfer coefficients, andx-component of the induced magnetic field on the surface increase with the applied magnetic field. In general, the skin friction, heat transfer andx-component of the induced magnetic field for axisymmetric case are more than those of the two-dimensional case. Also they change more when the surface heat flux or temperature decreases with time than when it increases with time. The skin friction, heat transfer andx-component of the induced magnetic field are significantly affected by the magnetic Prandtl number and they increase as the magnetic Prandtl number decreases. The skin friction andx-component of the magnetic field increase with the dissipation parameter, but heat transfer decreases.

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Instationäre freie Konvektionsströmung unter dem Einfluß eines magnetischen Feldes

Übersicht Untersucht wurde die instationäre freie Konvektionsgrenzschicht am Ruhepunkt eines zweidimensionalen und achsensymmetrischen umströmten Körpers bei vorgegebenem Wärmefluß bzw. bei vorgegebener Temperatur an der Oberfläche. Das magnetische Feld wird parallel zur Oberfläche angelegt, und der Einfluß des induzierten magnetischen Feldes wurde berücksichtigt. Es stellt sich heraus, daß bei bestimmter, zeitlicher Potenzgesetzverteilung des Wärmeflusses bzw. der Temperatur und des magnetischen Feldes an der Oberfläche die geltenden Grenzschichtgleichungen örtlich eine selbstähnliche Lösung erlauben. Die sich ergebenden nichtlinearen gewöhnlichen Differentialgleichungen wurden mittels einer Finite-Element-Methode und einer Shooting-Methode mit Newtonschen Korrekturen für fehlende Anfangsbedingungen gelöst. Die Ergebnisse zeigen, daß die Oberflächenreibung und die Wärmeübergangskoeffizienten sowie diex-Komponente des induzierten magnetischen Feldes an der Oberfläche mit dem angelegten magnetischen Feld zunehmen. Im allgemeinen sind die Oberflächenreibung, der Wärmeübergang und diex-Komponente des induzierten magnetischen Feldes im achsensymmetrischen Fall größer als die entsprechenden Werte im zweidimensionalen Fall. Außerdem verändern sich diese Werte beim zeitlichen Abfallen des Wärmeflusses an der Oberfläche bzw. der Temperatur in höherem Maße als bei der zeitlichen Zunahme dieser Werte. Die Oberflächenreibung, der Wärmeübergang und diex-Komponente des induzierten magnetischen Feldes werden durch die magnetische Prandtl-Zahl erheblich beeinflußt; sie nehmen mit abfallender magnetischer Prandtl-Zahl zu. Die Oberflächenreibung und diex-Komponente des magnetischen Feldes nehmen mit dem Wärmeableitungsparameter zu, der Wärmeübergang jedoch fällt ab.

     

Effect of magnetic field on thermal instability in mixed convection flow over a rotating boundary layer

Effect of magnetic field on the formation of longitudinal vortices in mixed convection flow over a rotating heated flat plate is presented. The onset position is characterized by the Grashof number, the rotational number, the Prandtl number, the Eckert number, the magnetic field parameter, and the wave number. Negative rotation (clockwise) and external magnetic field stabilize the boundary layer flow. On the contrary, positive rotation (anti-clockwise), the Eckert number, and the Prandtl number destabilize the flow. The numerical data show agreement with the experimental data with the case of zero Hartmann number in the literature.     

Effects of surface mass transfer on unsteady mixed convection flow over a vertical cone with chemical reaction

The unsteady mixed convection boundary layer flow over a vertical cone is considered to investigate the combined effects of the buoyancy force, thermal and mass diffusion in the presence of the first order chemical reaction and surface mass transfer. The unsteadiness is caused by the time dependent free stream velocity varying arbitrarily with time. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar transformations. The resulting system of coupled non-linear partial differential equations is solved numerically by the combination of quasi-linearization technique and an implicit finite difference scheme. Numerical computations are performed for different values of the parameters to display the velocity, temperature and concentration profiles graphically. Both accelerating and decelerating free stream velocities are considered. Numerical results are presented for the velocity, temperature and concentration profiles as well as for the skin-friction coefficient, local Nusselt number and local Sherwood number. The obtained results are compared with previously reported ones and are found to be in excellent agreement.     

Unsteady natural convection from a vertical flat plate with suction

Summary Exact solutions of the Navier-Stokes equations and the energy equation neglecting dissipation terms for natural convection from a porous vertical flat plate have been obtained in the following two cases: (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. Expressions for the skin friction and coefficient of heat transfer at the plate have been obtained. Skin friction decreases with suction in both cases.     

Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field

The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.     

Unsteady MHD mixed convection flow over an impulsively stretched permeable vertical surface in a quiescent fluid

The unsteady mixed convection flow of an incompressible laminar electrically conducting fluid over an impulsively stretched permeable vertical surface in an unbounded quiescent fluid in the presence of a transverse magnetic field has been investigated. At the same time, the surface temperature is suddenly increased from the surrounding fluid temperature or a constant heat flux is suddenly imposed on the surface. The problem is formulated in such a way that for small time it is governed by Rayleigh type of equation and for large time by Crane type of equation. The non-linear coupled parabolic partial differential equations governing the unsteady mixed convection flow under boundary layer approximations have been solved analytically by using the homotopy analysis method as well as numerically by an implicit finite difference scheme. The local skin friction coefficient and the local Nusselt number are found to decrease rapidly with time in a small time interval and they tend to steady-state values for t*≥5. They also increase with the buoyancy force and suction, but decrease with injection rate. The local skin friction coefficient increases with the magnetic field, but the local Nusselt number decreases. There is a smooth transition from the unsteady state to the steady state.     

Non-uniform slot suction/injection into mixed convection boundary layer flow over vertical cone

The aim of this work is to study the effect of non-uniform single and double slot suction/injection into a steady mixed convection boundary layer flow over a vertical cone, while the axis of the cone is inline with the flow. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar trans- formations. The resulting coupled non-linear partial differential equations are solved nu- merically by employing the quasi-linearization technique and an implicit finite-difference scheme. Numerical computations are performed for different values of the dimensionless parameters to display the velocity and temperature profiles graphically. Also, numerical results are presented for the skin friction and heat transfer coefficients. Results indicate that the skin friction and heat transfer coefficients increase with non-uniform slot suction, but the effect of non-uniform slot injection is just opposite.     

Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.     

Hydromagnetic convection in a rotating annulus with an azimuthal magnetic field

The problem of convection induced by radial buoyancy in an electrically conducting fluid contained by a rotating cylindrical annulus (angular frequency, ) in the presence of a homogeneous magnetic field (B) in the azimuthal direction is considered. The small gap approximation is used together with rigid cylindrical boundaries. The onset of convection occurs in the form of axial, axisymmetric or oblique rolls. The angle between the roll axis and the axis of rotation depends of the ratio between the Chandrasekhar number, QB², and the Coriolis number, . Fully three-dimensional numerical simulations as well as Galerkin representations for roll patterns including the subsequent stability analysis are used in the theoretical investigation. At finite amplitudes, secondary transitions to 3D-hexarolls and to spatio-temporal chaos are found. Overlapping regions of pattern stability exist such that the asymptotically realized state may depend on the initial conditions. PACS 47.27.-i, 47.65.+a     

Finite-difference solution of unsteady natural convection flow past a nonisothermal vertical cone under the influence of a magnetic field and thermal radiation

An analysis is performed to study the heat transfer characteristics of natural convection over a vertical cone under the combined effects of a magnetic field and thermal radiation. The cone surface is subjected to a variable surface temperature. The fluid considered is a gray absorbing/emitting, but non-scattering medium. The boundary layer equations governing the flow are reduced to non-dimensional equations using non-dimensional quantities valid in the free-convection regime. The resulting non-dimensional governing equations are solved by an implicit finite-difference method of the Crank-Nicolson type, which is rapidly convergent and unconditionally stable. Numerical results are obtained for velocity, temperature, local and average skin friction, and local and average Nusselt numbers for various values of parameters occurring in the problem and are presented in the graphical form. Excellent agreement of the results obtained with available data is demonstrated.     

Transient mixed convection flow instabilities in a vertical pipe

The present paper investigates a numerical study of flow instabilities in transient mixed convection in a vertical pipe. At the entrance of the pipe, the flow is suddenly submitted to a temperature step. The convection heat transfer on the outer surface of the pipe is taken into account. The governing equations are solved using a finite difference explicit scheme. The numerical results show that the time development of streamlines and isotherms is strongly dependent on the inlet temperature steps. For positive temperature steps, the unsteady vortex is significant in the vicinity of the wall and the reversal flow appears below the wave instability. In the case of negative temperature steps and especially for the low Reynolds number, the reversal flow appears on top of the wave instability. During the transient, the apparition of the vortical structures along the wall leads to the wall boundary layer instability. This phenomenon is due to the transient mixed convection flows. The temperature step effects on the heat transfer of the flow are presented in our paper.     

Steady laminar mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface in the presence of a magnetic field

A similarity solution for a steady laminar mixed convection boundary layer flow of a nanofluid near the stagnation point on a vertical permeable plate with a magnetic field and a buoyancy force is obtained by solving a system of nonlinear ordinary differential equations. These equations are solved analytically by using a new kind of a powerful analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). Three different types of nanoparticles, namely, copper (Cu), alumina (Al₂O₃), and titanium oxide (TiO₂), with water as the base fluid are considered. The influence of the volume fraction of nanoparticles, permeability parameter, magnetic parameter, and mixed convection parameter on the surface shear stress and surface heat transfer, as well as on the velocity and temperature profiles, is considered. It is observed that the skin friction coefficient and the local Nusselt number increase with the nanoparticle volume fraction for all types of nanoparticles considered in this study. The greatest values of the skin friction coefficient and the local Nusselt number are obtained for Cu nanoparticles.     

Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field

The steady MHD mixed convection flow of a viscoelastic fluid in the vicinity of two-dimensional stagnation point with magnetic field has been investigated under the assumption that the fluid obeys the upper-convected Maxwell (UCM) model. Boundary layer theory is used to simplify the equations of motion, induced magnetic field and energy which results in three coupled non-linear ordinary differential equations which are well-posed. These equations have been solved by using finite difference method. The results indicate the reduction in the surface velocity gradient, surface heat transfer and displacement thickness with the increase in the elasticity number. These trends are opposite to those reported in the literature for a second-grade fluid. The surface velocity gradient and heat transfer are enhanced by the magnetic and buoyancy parameters. The surface heat transfer increases with the Prandtl number, but the surface velocity gradient decreases.     

The laminar free‐convection boundary‐layer flow about a heated and rotating down‐pointing vertical cone in the presence of a transverse magnetic field

This paper deals with the study of the laminar free‐convection boundary‐layer flow about a heated and rotating down‐pointing vertical cone in the presence of a transverse magnetic field. Two cases of heat transfer analysis are discussed. These are: (i) the rotating cone with prescribed surface temperature and (ii) the rotating cone with prescribed surface heat flux. By means of similarity transformation, the governing partial differential equations are reduced into highly non‐linear ordinary differential equations. The resulting non‐linear system has been solved analytically using a very efficient technique, namely homotopy analysis method. Expressions for velocity and temperature fields are developed in a series form. The influence of various pertinent parameters is also seen on the velocity and temperature fields. Copyright © 2010 John Wiley & Sons, Ltd.     

Radiation and magnetic field effects on the unsteady mixed convection flow of a second grade fluid over a vertical stretching sheet

This present analysis discusses the mixed convection boundary layer flow of a magnetohydrodynamic second grade fluid over an unsteady permeable stretching sheet. The time‐dependent stretching velocity and the surface temperature are chosen. Series solutions of the governing boundary value problems are obtained by employing homotopy analysis method. Convergence of the obtained solution is explicitly discussed. The dependence of velocity and temperature profiles on the various quantities is shown and discussed by plotting graphs. Skin friction coefficient and the local Nusselt number tabulated and analyzed. Copyright © 2010 John Wiley & Sons, Ltd.     

Unsteady MHD free convection flow past a vertical permeable flat plate in a rotating frame of reference with constant heat source in a nanofluid

The unsteady magnetohydrodynamic flow of a nanofluid past an oscillatory moving vertical permeable semi-infinite flat plate with constant heat source in a rotating frame of reference is theoretically investigated. The velocity along the plate (slip velocity) is assumed to oscillate on time with a constant frequency. The analytical solutions of the boundary layer equations are assumed of oscillatory type and they are obtained by using the small perturbation approximations. The influence of various relevant physical characteristics are presented and discussed.