Three-dimensional flow with heat transfer of a viscoelastic fluid over a stretching surface with a magnetic field

The present research study deals with the steady flow and heat transfer of a viscoelastic fluid over a stretching surface in two lateral directions with a magnetic field applied normal to the surface. The fluid far away from the surface is ambient and the motion in the flow field is caused by stretching surface in two directions. This result is a three-dimensional flow instead of two-dimensional as considered by many authors. Self-similar solutions are obtained numerically. For some particular cases, closed form analytical solutions are also obtained. The numerical calculations show that the skin friction coefficients in x- and y-directions and the heat transfer coefficient decrease with the increasing elastic parameter, but they increase with the stretching parameter. The heat transfer coefficient for the constant heat flux case is higher than that of the constant wall temperature case.     

Stagnation flow and heat transfer in a zonal magnetic field

Summary The problem stated in the title is studied for small values of the diffusivity ratio and the magnetic force coefficient , the magnetic field being of internal origin. Uniformly valid expansions are derived for the velocity and magnetic fields in the fluid. It is found that as 1, the viscous layer is brought to rest and the current in the layer is uniform and normal to the wall.The heat transfer is next calculated at a uniformly heated wall on the assumption of small temperature variations. It is deduced that when log(^–1) approaches a certain value depending on the wall temperature etc., the thermal boundary layer separates at the stagnation point and, if dissipation is neglected, along the whole wall.     

Unsteady laminar incompressible second-order boundary-layer flow at a three-dimensional stagnation point

Summary The unsteadely laminar incompressible second-order boundary-layer flow at the stagnation point of a three-dimensional body has been studied for both nodal and saddle point regions. The effects of mass transfer and Prandtl number have been taken into account. The equations governing the flow have been solved numerically using an implicit finite-difference scheme. It has been found that the parameter characterizing the unsteadiness in the velocity of the free stream, the nature of the stagnation point, the mass transfer and Prandtl number strongly affect the second-order skin friction and heat transfer. The overall skin friction becomes less due to second-order effects but the heat transfer has the opposite behaviour. For large injection, the second-order skin-friction and heat-transfer results prevail over the first-order boundary layer results whereas for the case of large suction the behaviour is just the opposite.With 14 Figures     

Unsteady viscous flow in the vicinity of an axisymmetric stagnation point on a circular cylinder

An analysis is presented to investigate the unsteady fluid dynamic characteristics of an axisymmetric stagnation flow on a circular cylinder performing a harmonic motion in its own plane. Different solutions are presented for the small and high values of the reduced frequency of oscillation. The range of Reynolds numbers considered was from 0.01 to 100. Numerical solutions for the velocity functions are presented and the wall values of the velocity gradients are tabulated.     

Compressible axially symmetric laminar boundary layer flow in the stagnation region of a blunt body in the presence of magnetic field

Summary In this paper, the steady laminar viscous hypersonic flow of an electrically conducting fluid in the region of the stagnation point of an insulating blunt body in the presence of a radial magnetic field is studied by similarity solution approach, taking into account the variation of the product of density and viscosity across the boundary layer. The two coupled non-linear ordinary differential equations are solved simultaneously using Runge-Kutta-Gill method. It has been found that the effect of the variation of the product of density and viscosity on skin friction coefficient and Nusselt number is appreciable. The skin friction coefficient increases but Nusselt number decreases as the magnetic field or the total enthalpy at the wall increases.

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Kompressible axialsymmetrische laminare Grenzschichtströmung in der Nähe des Staupunktes eines stumpfen Körpers unter Einwirkung eines Magnetfeldes

Zusammenfassung In dieser Arbeit wird die stationäre laminare viskose Überschallströmung einer elektrisch leitenden Flüssigkeit in der Nähe des Staupunktes eines isolierten stumpfen Körpers unter Einwirkung eines Magnetfeldes studiert unter Berücksichtigung der Veränderlichkeit des Produktes Dichte—Zähigkeit über die Grenzschicht. Die zwei gekoppelten nichtlinearen Gleichungen wurden mit Hilfe des Verfahrens von Runge-Kutta-Gill gelöst. Es zeigte sich, daß der Einfluß der Ortsabhängigkeit des Produktes Dichte—Zähigkeit auf den örtlichen Reibungsbeiwert und die Nusselt-Zahl keineswegs gering ist. Mit Zunahme des Magnetfeldes, oder der totalen Enthalpie an der Wand, nimmt der örtliche Reibungsbeiwert zu, die Nusselt-Zahl jedoch ab.

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With 3 Figures     

Unsteady compressible flow in the stagnation region of two-dimensional and axi-symmetric bodies

Summary This paper deals with a new similarity solution of unsteady laminar compressible two-dimensional and axi-symmetric boundary layers. It has been shown that a self-similar solution is possible when the free stream velocity varies inversely with time. The two-point boundary value problems governed by self-similar equations have been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique. It is observed that the effect of the acceleration parameter (A) in the free stream velocity on the skin friction is more pronounced compared to the heat transfer. For certain values of the acceleration parameter and the total enthalpy at the wall, the surface shear stress (skin friction) vanishes. The skin friction and heat transfer increase due to suction, and the effect of injection is found to be just opposite. Velocity profiles are presented with reverse flow and without reverse flow depending on the values of toal enthalpy at the wall and the acceleration parameter.     

Unsteady axisymmetric stagnation-point flow of a viscous fluid on a cylinder

The unsteady viscous flow in the vicinity of an axisymmetric stagnation point of an infinite circular cylinder is investigated when both the free stream velocity and the velocity of the cylinder vary arbitrarily with time. The cylinder moves either in the same direction as that of the free stream or in the opposite direction. The flow is initially (t=0) steady and then at t>0 it becomes unsteady. The semi-similar solution of the unsteady Navier–Stokes equations has been obtained numerically using an implicit finite-difference scheme. Also the self-similar solution of the Navier–Stokes equations is obtained when the velocity of the cylinder and the free stream velocity vary inversely as a linear function of time. For small Reynolds number, a closed form solution is obtained. When the Reynolds number tends to infinity, the Navier–Stokes equations reduce to those of the two-dimensional stagnation-point flow. The shear stresses corresponding to stationary and the moving cylinder increase with the Reynolds number. The shear stresses increase with time for the accelerating flow but decrease with increasing time for the decelerating flow. For the decelerating case flow reversal occurs in the velocity profiles after a certain instant of time.     

Heat transfer in the stagnation point flow of a micropolar fluid

Summary Similar solutions of the equations describing the thermal boundary layer of a micropolar fluid on a plane wall are found to exist for the stagnation point flow when the wall temperature variation is parabolic. The two types of boundary conditions used for microrotation are: (a) the relative spin of the particles on the boundary is related to the skew symmetric part of the stress on the boundary by a parameter which is a measure of the concentration of microelements, and (b) the couple stress on the boundary is related to the relative spin of the particles on the boundary by a friction factor which accounts for the rotational slip of the fluid along the boundary. The velocity, microrotation and temperature fields have been presented graphically for various values of the boundary condition parameters. The skin friction coefficient, wall couple stress coefficient, displacement and momentum thicknesses and rate of heat transfer have been tabulated. A comparison with the corresponding results for a Newtonian fluid has been made.

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Wärmeübergang in der Staupunktsströmung eines mikropolaren Fluids

Zusammenfassung Es werden ähnliche Lösungen der Gleichungen, die die thermische Randschicht eines mikropolaren Fluids längs einer ebenen Wand beschreiben, für die Staupunktsströmung bei parabolischer Änderung der Wandtemperatur gefunden. Zwei Typen von Randbedingungen werden für die Mikrorotation verwendet: (a) Der relative Spin der Teilchen am Rand ist verknüpft mit dem schiefsymmetrischen Anteil der Spannungen am Rand über einen Parameter, der ein Maß für die Konzentration der Mikroelemente darstellt. (b) Die Momentenspannung an der Berandung ist mit dem relativen Spin der Teilchen am Rand mit einem Reibungsfaktor verknüpft, der den Drehslip des Fluids längs der Berandung beschreibt. Die Geschwindigkeits-, Mikrorotations- und Temperaturfelder werden graphisch für verschiedene Werte des Parameters für die Randbedingungen dargestellt. Der Wandreibungskoeffizient, der Koeffizient der Wandmomentenspannung, Verschiebung und Impulsflußdicke und die Wärmeübergangsrate werden tabelliert. Ein Vergleich mit den entsprechenden Ergebnissen der Newtonschen Flüssigkeit wird angestellt.

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With 4 Figures     

Unsteady incompressible boundary layer flow of a micropolar fluid at a stagnation point

The flow, heat and mass transfer on the unsteady laminar incompressible boundary layer in micropolar fluid at the stagnation point of a 2-dimensional and an axisymmetric body have been studied when the free stream velocity and the wall temperature vary arbitrarily with time. The partial defferential equations governing the flow have been solved numerically using a quasilinear finite-difference scheme. The skin friction, microrotation gradient and heat transfer parameters are found to be strongly dependent on the coupling parameter, mass transfer and time, whereas the effect of the microrotation parameter on the skin friction and heat transfer is rather weak, but microrotation gradient is strongly affected by it. The Prandtl number and the variation of the wall temperature with time affect the heat-transfer very significantly but the skin friction and micrortation gradient are unaffected by them.     

Experimental investigation of heat transfer in the region of a stagnation point

An investigation has been made of heat transfer to a body from a gas stream in the region of the stagnation point, as a function of enthalpy and stagnation pressure of the gas, and the velocity gradient at the stagnation point. The enthalpy of the gas (nitrogen) was varied from 6.5 · 10³ to 115 · 10³ kJ/kg, and the pressure from 10.3 to 14.7 N/cm². The upper enthalpy limit corresponded to an ionization level of x_N+=0.43. The investigation was carried out in a gas jet heated by means of an electric arc.     

Unsteady convective heat transfer in potential flow

Analytical relations are obtained for the unsteady temperature field in potential flow over a flat plate and a cylinder.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 1, pp. 109–115, January, 1977.     

Time dependent rotational flow of a viscous fluid over an infinite porous disk with a magnetic field

Both the semi-similar and self-similar flows due to a viscous fluid rotating with time dependent angular velocity over a porous disk of large radius at rest with or without a magnetic field are investigated. For the self-similar case the resulting equations for the suction and no mass transfer cases are solved numerically by quasilinearization method whereas for the semi-similar case and injection in the self-similar case an implicit finite difference method with Newton's linearization is employed. For rapid deceleration of fluid and for moderate suction in the case of self-similar flow there exists a layer of fluid, close to the disk surface where the sense of rotation is opposite to that of the fluid rotating far away. The velocity profiles in the absence of magnetic field are found to be oscillatory except for suction. For the accelerating freestream, (semi-similar flow) the effect of time is to reduce the amplitude of the oscillations of the velocity components. On the other hand the effect of time for the oscillating case is just the opposite.     

Pressure correction DRBEM solution for heat transfer and fluid flow in incompressible viscous fluids

The Dual Reciprocity Boundary Element Method (DRBEM) is used to solve incompressible laminar viscous fluid flows and heat transfer. The DRBEM is extended to develop a pressure correction scheme to solve the incompressible Navier-Stokes equations. The velocity field is then used as input to the DRBEM solution of the energy transport equation, thereby retaining the boundary only discretization feature of the BEM for the solution of this problem. Numerical results for the proposed DRBEM solution for laminar flow and heat transfer in a channel are obtained for several Reynolds numbers and compare well with previously published data.     

Flow and heat transfer in a jet near the stagnation point of a concave body

Results are presented of calculations of flow and heat transfer near the stagnation point of a concave body in a two-dimensional subsonic jet, using a flow establishment method.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 30, No. 2, pp. 301–309, February, 1976.     

Unsteady heat transfer in a micropolar fluid flowing in a plane channel

Heat transfer in a micropolar fluid flowing in a plane channel following an abrupt change in the wall temperature is investigated. The obtained results indicate that in several cases the fluid microstructure has a considerable effect on the main heat-transfer characteristics.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 6, pp. 994–999, December, 1979.     

Unsteady boundary layer flow of a micropolar fluid near the forward stagnation point of a plane surface

The growth of the boundary-layer flow of a micropolar fluid started impulsively from rest near the forward stagnation point of a two-dimensional plane surface is studied theoretically. The transformed non-similar boundary-layer equations are solved numerically using a very efficient finite-difference method known as Keller-box method. This method may present well-behaved solutions for the transient (small time) solution and those of the steady-state flow (large time) solution. Numerical results are given for the reduced velocity and microrotation profiles, as well as for the skin friction coefficient when the material parameter K takes the value K=0 (Newtonian fluid), 0.5, 1, 1.5, 2, 2.5 and 3. Important features of these flow characteristics are shown on graphs and in tables.     

Unsteady free-convection heat transfer during vortex flow in a horizontal tube

We conducted a theoretical and experimental investigation of the hydrodynamic structure and unsteady free-convection heat transfer in a round horizontal tube under different conditions of the second kind. During the experiment the Grashof number varied from Gr=1.54·10⁵ to Gr=7.9·10⁵. Using the interferometric method we investigated the distribution of the temperature field for different structures of free-convection flow in the tube.Academic Scientific Complex A. V. Luikov Heat and Mass Transfer Institute of the Academy of Sciences of Belarus, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 1, pp. 3–8, January–February, 1995.