Heat transfer in laminar flow in a uniformly porous channel with an applied transverse magnetic field

Summary The steady laminar flow of an incompressible, viscous, and electrically conducting fluid between two parallel porous plates with equal permeability has been discussed by Terrill and Shrestha [6]. In this paper, using the solution of [6] for the velocity field, the heat transfer problems of (i) uniform wall temperature and (ii) uniform heat flux at wall are solved.For small suction Reynolds numbers we find that the Nusselt number, with increasing Reynolds number, increases for case (i) and decreases for (ii).Nomenclature stream function - 2h channel width - x, y distances measured parallel, perpendicular to the channel walls - U velocity of fluid in the x direction at x=0 - V constant velocity of suction at the wall - nondimensional distance, y/h - nondimensional distance, x/h - f() function defined in (1) - density - coefficient of kinematic viscosity - R suction Reynolds number, V h/ - Re channel Reynolds number, 4U h/ - B ₀ magnetic induction - electrical conductivity - M Hartmann number, B ₀ h(/)^1/2 - K constant defined in (3) - A constant defined in (5) - 4R/Re - q local heat flux per unit area at the wall - k thermal conductivity - T temperature of the fluid - X –1/ ln(1–) - C _p specific heat at constant pressure - j current density - Pr Prandtl number, C _p/k - P mass transfer Péclet number, R Pr - Pe mass transfer Péclet number, P/ - T ₀ temperature at x=0 - T _H() temperature in the fully developed region - T _h(X, ) temperature in the entrance region - Y _n () eigenfunctions, uniform wall temperature - _n eigenvalues - e() function defined by (24) - B _n 2/3 _n ² - A _n constants defined by (28) - a _2m constants defined by (30) - F _n () eigenfunctions, uniform wall heat flux - a _n , b _n , c _n , d _n , e _n constants defined by (45) and (48) - S a parameter, U ²/q - h ₁ heat transfer coefficient - T _m mean temperature - Nu Nusselt number - Nu _T Nusselt number, uniform wall temperature - Nu _q Nusselt number, uniform wall heat flux     

Self-similar channel flow of a viscous gas with heat transfer

We examine self-similar flows of a viscous gas in long, smooth channels with a special heat transfer law at the wall, corresponding to the same Mach number profile at all cross sections.     

Heat transfer in flow of nonlinear fluids with viscous dissipation

The rotational flow of viscoplastic fluids between concentric cylinders is examined while dissipation due to viscous effects through the energy balance. The viscosity of fluid is simultaneously dependent on shear rate and temperature. Exponential dependence of viscosity on temperature is modeled through Nahme law, and the shear dependency is modeled according to the Carreau equation. Hydrodynamically, stick boundary conditions are applied, and thermally, both constant temperature and constant heat flux on the exterior of cylinders are considered. The governing motion and energy balance equations are coupled adding complexity to the already highly correlated set of differential equations. Introduction of Nahme number has resulted in a nonlinear base flow between the cylinders. As well, the condition of constant heat flux has moved the point of maximum temperature toward the inner cylinder. Taking viscous heating into account, the effects of parameters such as Nahme and Brinkman numbers, material time and pseudoplasticity constant on the stability of the flow are investigated. Moreover, the study shows that the total entropy generation number decreases as the fluid elasticity increases. It, however, increases with increasing Nahme and Brinkman numbers.     

Heat transfer in flow past a continuously moving semi-infinite flat plate in transverse magnetic field with heat flux

Thermal boundary layer on a continuously moving semi-infinite flat plate in the presence of transverse magnetic field with heat flux has been examined. Similarity solutions have been derived and the resulting equations are integrated numerically. This investigation has indicated a fall in the temperature of the thermal boundary layer with increase in magnetic field parameter.

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Wärmeübertragung in Strömungen an einer gleichmäßig bewegten, halbunendlichen ebenen Platte in einem quergerichteten Magnetfeld mit Wärmefluß

Zusammenfassung Es ist die thermische Grenzschicht auf einer gleichmäßig bewegten, halbunendlichen ebenen Platte in einem quergerichteten Magnetfeld mit Wärmefluß untersucht worden. Ähnlichkeitslösungen sind abgeleitet und die erhaltenen Gleichungen numerisch integriert worden. Diese Untersuchung hat einen Rückgang der Temperatur in der thermischen Grenzschicht mit steigendem Magnetfeldparameter nachgewiesen.

     

Fluid flow and heat transfer in a two-dimensional wavy channel

A numerical study is presented for the laminar fully developed flow and heat transfer in a two-dimensional wavy channel. The effects of the geometry, Reynolds and Prandtl number on the flow field and heat transfer are investigated. The channel is characterized by a wavy wall, heated at uniform heat flux, and an opposite wall, being plane and adiabatic. The extent of the wall waviness and the distance between the channel walls are found to significantly affect the streamlines contours as well as the heat transfer coefficients. Comparisons with the straight channel, in the same flow rate and heat transfer conditions, have been performed. Pressure drop of the wavy channel is found to be always larger than the value characteristic of a straight channel, while heat transfer performance decreases or increases depending on the values of the parameters (geometry, Reynolds and Prandtl numbers).     

Heat transfer in unsteady laminar flow through a channel

The temperature distribution in unsteady laminar flow of a viscous incompressible fluid in a flat channel is investigated, when the pressure gradient is an arbitrary function of time. Two techniques are presented (i) explicit finite difference scheme, and (ii) Chebyshev polynomial solution. Using both the techniques, several cases of pressure gradient are considered, with special attention paid to the linearly varying case.     

Heat transfer analysis of a particle-containing channel flow

This paper describes an analytical model of heat transfer in a two-dimensional, steady, nonreacting particle-containing channel flow. An idealized gas flow of specified uniform velocity between insulated parallel plates is assumed and the nonvaporizing particles are conceptualized as contained within an thin sheet injected at the symmetry plane. Two dimensionless parameters that affect the solution are described. These are the effective gas diffusivityK and the dimensionless particle number densityP. The linear, coupled differential equations governing the energy exchange between the gas and liquid phases are solved by means of the Green's function technique. This procedure yields a Volterra integral-series equation as the solution of the gas-phase energy equation. A series solution of this integral equation is obtained by the method of successive substitutions and terms up to second order are calculated.     

Laminar flow in a uniformly porous channel with an applied transverse magnetic field

Summary The flow of a viscous incompressible and electrically conducting fluid in a two-dimensional uniformly porous channel, having fluid sucked or injected with a constant velocity through its walls, is considered in the presence of a transverse magnetic field. A solution for small Reynolds number has been given by the authors in a previous paper. A solution valid for large suction Reynolds number and all values of Hartmann number is presented here and the resulting boundary layer is discussed. Also Yuan's solution for large negativeR is extened to include small values ofM ₂/R.Nomenclature x, y distances parallel and perpendicular to the channel walls - u, v velocity components inx, y directions - p pressure - density - U(0) entrance velocity atx=0 - V suction velocity at the wall - V velocity field - J current density - E electric field - H magnetic field - H ₀ applied magnetic field - electrical conductivity - _m magnetic permeability - 2h distance between the porous walls - kinematic viscosity - y/h - B _m H - B _{0 m}H₀ - R Vh/, Reynolds number - M _mH₀ h(/)^1/2, Hartmann number - M/R - a - b - z 1–     

Heat transfer in a radiating fluid with slug flow in a parallel-plate channel

Summary As a step towards a better understanding of combined conduction, convection, and radiation, fully developed heat transfer in slug flow in a flat duct between two parallel plates is considered. The flat duct consists of two diffuse, nonblack, isothermal surfaces. The gray radiating fluid between them is capable of emitting and absorbing thermal radiation. The problem is formulated in terms of a nonlinear integrodifferential equation, and the solution is obtained by an approximate method. The differences between heating and cooling the fluid are examined. The effects of the optical thickness of the fluid, the ratio which determines the relative role of energy transport by conduction to that by radiation, and the emissivity of the duct surfaces on the temperature distribution and the heat transfer characteristics are investigated. An approximate method for calculating the radiant heat flux at the wall is presented, and the accuracy of the approximation is tested.Work done under the auspices of the U.S. Atomic Energy Commission.     

Heat transfer in magnetohydrodynamic laminar radial channel flow

Heat transfer in the steady axisymetrical laminar source flow of an incompressible electrically conducting fluid between two parallel disks in the presence of a transverse applied magnetic field is analyzed. The energy equation is solved numerically for the temperature distribution, where both Joulean and viscous heating are included. Both local and average Nusselt numbers for the case of constant wall temperature are evaluated. For fluids of moderate and high Prandtl numbers, Nusselt number is seen to be a strong function of both Hartmann number and a heat generation parameter together with a modified Peclet number. However, for fluids of small Prandtl number, Joulean heating and viscous dissipation can be neglected without appreciable error.     

Time-dependent viscous flow in a straight channel

An infinite straight channel, filled with an incompressible viscous fluid, is closed at one end by a piston. This is set in motion with finite acceleration and then maintained at constant velocity until the flow pattern in the fluid reaches a steady state. The development of velocity profiles, stream lines, and streak lines is investigated by direct numerical solution of the complete Navier Stokes equations. It is found that the nonconvex velocity profiles found in previous work on steady-state problems appear from the beginning, and their development is studied. In the downstream region alternative methods can be used which serve as a check on the accuracy of the numerical procedures. The asymptotic behaviour downstream is studied in some detail.Nomenclature a acceleration of piston - f(t) nondimensional distance travelled by piston up to time t - 2l width of channel - p pressure (in units of u ₀ ² ) - R Reynolds number, lu ₀/ - t ₀ time during which piston is accelerated - u ₀ final velocity of piston - (u, v) (x, y) components of fluid velocity (relative to piston, in units of u ₀) - x distance measured downstream from piston (in units of l) - y distance from central axis of channel (in units of l) - vorticity - density of fluid - kinematic viscosity - stream function     

Heat transfer at the surface of a triangular body in a hypersonic viscous gas flow

The boundary layer on a semi-infinite triangular body of power-law shape is calculated for viscous interaction with an external hypersonic flow. The results of calculating the characteristics of the three-dimensional boundary layer are presented. The formation of secondary return flows and zones of intensified heat transfer on the surface of the body in the neighborhood of lines of flow divergence is noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 77–82, January–February, 1988.     

Turbulent flow past a transverse cavity with inclined side walls. 2. Heat transfer

Convective heat transfer in a transverse cavity with a small aspect ratio, angle of wall inclination ϕ = 30–90°, and heated bottom, frontal, and rear walls of the cavity is studied experimentally. Temperature distributions are measured in longitudinal and transverse sections on three walls; temperature fields are measured over the entire heated surface. Local and mean heat-transfer coefficients are calculated. The highest intensification of heat transfer is found to occur on the rear wall for low values of ϕ Reconstruction of the one-cell structure to the two-cell structure of the primary vortex in the cavity leads to a drastic decrease in heat transfer over the cavity span from the end faces toward the center in the case with ϕ = 60 and 70°. A certain increase in the mean heat-transfer coefficient averaged over the entire heated surface is noted for ϕ = 60°. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 23–29, July–August, 2007.     

Computer simulation of a viscous channel flow

Summary A numerical solution of the two — dimensional Navier — Stokes equations for the flow in a duct between parallel plates in the presence of a diaphragm with an orifice at Reynolds number Re=1800 is presented. The plots of the resulting streamlines are illustrated and the computed discharge coefficient is compared with the experimental results.

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Sommario Si presenta una soluzione numerica delle equazioni di Navier-Stokes bidimensionali in un condotto tra lastre parallelel in presenza di un diaframma forato al numero di Reynolds Re=1800. Si illustrano i grafici delle linee di corrente ottenute e si fa un confronto tra il coefficiente di portata calcolato e i risultati sperimentali.

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This work was done under the auspices of the CNR Research Groups.     

Viscoelastic flow in a two-dimensional collapsible channel

We compute the flow of three viscoelastic fluids (Oldroyd-B, FENE-P, and Owens blood model) in a two-dimensional channel partly bounded by a tensioned membrane, a benchmark geometry for fluid–structure interactions. The predicted flow patterns are compared to those of a Newtonian liquid. We find that computations fail beyond a limiting Weissenberg number. Flow fields and membrane shape differ significantly because of the different degree of shear thinning and molecular extensibility underlying the three different microstructural models.     

Finite element computations of two-dimensional arterial flow in the presence of a transverse magnetic field

A finite element solution of the Navier-Stokes equations for steady flow under the magnetic effect through a double-branched two-dimensional section of a three-dimensional model of the canine aorta is discussed. The numerical scheme involves transforming the physical co-ordinates to a curvilinear boundary-fitted co-ordinate system. The shear stress at the wall is calculated for a Reynolds number of 1000 with the branch-to-main aortic flow rate ratio as a parameter. The results are compared with earlier works involving experimental data and found to be in reasonable qualitative agreement. The steady flow, shear stress and branch flow under the effect of a magnetic field have been discussed in detail.     

Laminar slip flow of an electrically conducting incompressible rarefied gas in a channel with a transverse magnetic field

Summary The effects of a constant external magnetic field on the laminar, fully developed flow of an electrically conducting incompressible rarefied gas in a nonconducting parallel-plate channel are studied. Consideration is given to the slip-flow regime, wherein a gas velocity discontinuity occurs at the channel walls. It is found that the magnitude of the slip velocity is unaffected by the magnetic-field strength for a given pressure drop, but that the mean gas velocity and wall friction coefficient are functions of both the velocity slip coefficient and the magnetic-field strength. The effect of a second-order slip-flow boundary condition is briefly discussed.     

Asymptotic solutions for the asymmetric flow in a channel with porous retractable walls under a transverse magnetic field

The self-similarity solutions of the Navier-Stokes equations are constructed for an incompressible laminar flow through a uniformly porous channel with retractable walls under a transverse magnetic field. The flow is driven by the expanding or contracting walls with different permeability. The velocities of the asymmetric flow at the upper and lower walls are different in not only the magnitude but also the direction. The asymptotic solutions are well constructed with the method of boundary layer correction in two cases with large Reynolds numbers, i.e., both walls of the channel are with suction, and one of the walls is with injection while the other one is with suction. For small Reynolds number cases, the double perturbation method is used to construct the asymptotic solution. All the asymptotic results are finally verified by numerical results.