BOUNDARY LAYER FLOW AND DOUBLE DIFFUSION OVER AN UNSTEADY STRETCHING SURFACE WITH HALL EFFECT

The present investigation is concerned with the effect of Hall currents on boundary layer flow, and heat and mass transfer of an electrically conducting fluid over an unsteady stretching sheet in the presence of a strong magnetic field. The electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist, while the induced magnetic field is neglected. The governing time-dependent boundary layer equations for momentum, thermal energy, and concentration are reduced using a similarity transformation to a set of coupled ordinary differential equations. The similarity ordinary differential equations are then solved numerically by the successive linearization method together with the Chebyshev pseudo-spectral collocation method. Effects of the Prandtl number, Pr, Schmidt number, Sc, magnetic field, M, Hall parameter, m, and the unsteadiness parameter, A, on the velocity, temperature, and concentration profiles as well as the local skin friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form. Favorable comparisons with previously published work on various special cases of the problem are also obtained.     

Mixed convection flow of non‐Newtonian fluid from a slotted vertical surface with uniform surface heat flux

In the present paper, the combined convection flow of an Ostwald–de Waele type power‐law non‐Newtonian fluid past a vertical slotted surface has been investigated numerically. The boundary condition of uniform surface heat flux is considered. The equations governing the flow and the heat transfer are reduced to local non‐similarity form. The transformed boundary layer equations are solved numerically using implicit finite difference method. Solutions for the heat transfer rate obtained for the rigid surface compare well with those documented in the published literature. From the present analysis, it is observed that, an increase in χ leads to increase in skin friction as well as reduction in heat transfer at the surface. As the power‐law index n increases, the friction factor as well as heat transfer increase.     

Thermal Diffusion and MHD Effects on Combined Free‐Forced Convection and Mass Transfer of a Viscous Fluid Flow Through a Porous Medium with Heat Generation

The problem of thermal diffusion and magnetic field effects on combined free‐forced convection and mass transfer flow past a vertical porous flat plate, in the presence of heat generation is studied numerically. The governing momentum, energy and concentration equations are converted into a system of nonlinear ordinary differential equations by means of similarity transformations. The resulting system of coupled nonlinear ordinary differential equations is solved numerically by using the Shooting method. Numerical results are presented for velocity, temperature and concentration profiles within the boundary layer for different parameters of the problem including suction parameter, heat generation parameter, Soret number, Dufour number, magnetic parameter, etc. In addition, the effects of the pertinent parameters on the skin friction and the rates of heat and mass transfer are discussed numerically and illustrated graphically.     

Free Convection Flow from a Vertical Isothermal Cone with Temperature-Dependent Viscosity

This paper reports a study of the effect of variable viscosity, expressed as an inverse linear function of temperature, on the free convective heat and fluid flow past a vertical isothermal cone. By similarity analysis together with the boundary layer assumptions, this complex problem is reduced to a system of coupled nonlinear ordinary differential equations. The resultant boundary layer equations are integrated numerically to obtain the flow field and the temperature distribution for selected influence parameters such as the viscosity parameter θ _r and the Prandtl number Pr. The viscosity parameter θ _r plays a prominent role in both flow field and heat transfer. In this paper, θ _r relates to the reciprocal of the temperature difference between the heated surface and the ambient. The calculated results indicate that some variations will be introduced in the heat transfer rate as well as the skin friction coefficient.     

Mixed convection heat transfer in porous media in the non-darcy regime

In this study mixed convection heat transfer in a homogeneous porous duct of square cross section in a horizontal orientation is examined. Results from a generalized Forchheimer model are compared with that from the Darcy model. The heat transfer rate and the flow behavior depend on the following parameters: Grashof number, Gr = Q'gβKa/kv², an axial flow pressure drop parameter, ζ = (aK/vμ)dp'/dz', an inertial parameter ξ = mK/a, appearing in the Forccheimer model and the Prandtl number, Pr = C_pμ/k. In the Darcy limit, ξ → 0, the role of the axial flow parameter, λ is reduced to a mere scale factor and the flow behavior is determined by a single parameter, λ = Gr · Pr. Both the Darcy and the Forchheimer models exhibit dual solutions and a hysteresis behavior over a certain range of Gr. Such parametric dependence can be used as an additional tool along with carefully designed experiments to determine the importance of inertial and Prandtl number effects on convective heat transfer in porous media.     

MIXED CONVECTION BOUNDARY LAYER FLOW OF A MICROPOLAR FLUID ON A HORIZONTAL PLATE

An analysis is performed to study the heat transfer characteristics of laminar mixed convection boundary layer flow of a micropolar fluid past a semi-infinite horizontal fiat plate with variable surface heat flux. A nonsimilar mixed convection parameter £ and a pseudosimilarity variable v are introduced to cast the governing boundary layer equations into a system of dimensionless equations, which are solved numerically using the finite difference method. A single mixed convection parameter is used to cover the entire regime of mixed convection from the pure forced convection limit to the pure free convection limit. The effect of material parameters, the power-law variation of surface heat flux, nonsimilar mixed convection parameter and Prandtl number are considered. The micropolar fluids are observed to display drag reduction and reduced surface heat transfer rate when compared to Newtonian fluids. The effect of the buoyancy force results in the enhancements of friction factor, heat transfer rate and wall couple stress.     

NUMERICAL MODELING OF MHD CONVECTIVE HEAT AND MASS TRANSFER IN PRESENCE OF FIRST-ORDER CHEMICAL REACTION AND THERMAL RADIATION

An analysis was carried out numerically to study unsteady heat and mass transfer by free convection flow of a viscous, incompressible, electrically conducting Newtonian fluid along a vertical permeable plate under the action of transverse magnetic field taking into account thermal radiation as well as homogeneous chemical reaction of first order. The fluid considered here is an optically thin gray gas, absorbing-emitting radiation, but a non-scattering medium. The porous plate was subjected to a constant suction velocity with variable surface temperature and concentration. The dimensionless governing coupled, nonlinear boundary layer partial differential equations were solved by an efficient, accurate, extensively validated, and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields were studied for the effects of Hartmann number (M), radiation parameter (R), chemical reaction (K), and Schmidt number (Sc). The local skin friction, Nusselt number, and Sherwood number are also presented and analyzed graphically. It is found that velocity is reduced considerably with a rise in the magnetic body parameter (M), whereas the temperature and concentration are found to be markedly boosted with an increase in the magnetic body parameter (M). An increase in the conduction-radiation parameter (R) is found to escalate the local skin friction (τ), Nusselt number, and concentration, whereas an increase in the conduction-radiation parameter (R) is shown to exert the opposite effect on either velocity or temperature field. Similarly, the local skin friction and the Sherwood number are both considerably increased with an increase in the chemical reaction parameter. Possible applications of the present study include laminar magneto-aerodynamics, materials processing, and MHD propulsion thermo-fluid dynamics.     

MHD‐Buoyancy Aiding and Opposing Flows with Viscous Dissipation Effects from Radiate Vertical Surfaces

The combined effects of forced and natural convection heat transfer in the presence of transverse magnetic field form a vertical surfaces with radiation heat transfer is studied. The buoyancy aided flow and the buoyancy opposing flows are investigated with viscous dissipation effects included in the governing equations. It is found that four parameters can describe the problem under consideration, the mixed parameter, χ, the radiation‐conduction parameter, R_d, the magnetic field parameter, Ha²_x/Re_x, and the Eckert number, Ec. The local wall shear stresses and the local Nusselt number variations are drawn for different dimensionless groups.     

A bifurcation study of convective heat transfer in a hele-shaw cell

Steady-state multiplicity characteristics of convective heat transfer within a Hele-Shaw cell are investigated. The Navier-Stokes equations and the energy equation are averaged across the narrow gap, d, of the cell. The resulting two-dimensional, stationary equations depend on the following parameters: (i) the length to height aspect ratio γ, (ii) the tilt anle ? (iii) the Prandtl number Pr, (iv) an inertia parameter ξ = d²/ 12a², and (v) the Grashof number. Gr = Q_gβga⁵/kv². Here a is the height of the cell and Q, is the heat generation rate per unit volume. The complete structure of symmetric and asymmetric stationary solutions are traced using recent algorithms from bifurcation theory. In the double limit of ξ → 0 and Gr → ∞ such that Ra = 4GrPrξ remains finite (where Ra is the Rayleigh number for the Darcy model) the Hele-Shaw model reduces to that of the Darcy model.     

RADIATION EFFECT ON FREE CONVECTION LAMINAR FLOW FROM AN ISOTHERMAL SPHERE

Natural convection laminar flow from an isothermal sphere immersed in a viscous incompressible optically dense fluid in the presence of radiation effects has been investigated. The governing boundary layer equations are transformed into a non dimensional form and the resulting nonlinear systems of partial differential equations are reduced to local non-similarity boundary layer equations, which are solved numerically by two distinct, efficient methods, namely, (i) implicit finite difference method together with the Keller box scheme and (ii) local non-similarity method. Numerical results of the velocity and temperature profiles of the fluid are presented. The results of the shearing stress and the heat transfer rate in terms of skin-friction coefficient and Nusselt number respectively are also presented for a wide range of the radiation-conduction parameter or Planck number R _d (=0.0, 1.0, 2.0, 3.0), the surface heating parameter θ _w (=1.1, 1.2, 1.4), and Prandtl number Pr (=7.0, 10.0, 15.0, 20.0).     

UNSTEADY FLUID AND HEAT FLOW INDUCED BY A STRETCHING SHEET WITH MASS TRANSFER AND CHEMICAL REACTION

The effect of chemical reaction on the flow, heat, and mass transfer within a viscous fluid on an unsteady stretching sheet is examined. The stretching rate, temperature and concentration of the sheet, and the chemical reaction rate are assumed to vary with time. The time-dependent boundary layer equations governing the flow are reduced through a convenient similarity transformation to a set of ordinary differential equations, which are numerically solved by applying the fourth-order Runge-Kutta-Fehlberg scheme with the shooting technique. Results for the velocity, temperature, and concentration distributions as well as the wall temperature and concentration gradients are presented graphically for various values of the unsteadiness parameter A, Prandtl number Pr, Schmidt number Sc, and chemical reaction parameter γ.     

Numerical studies of a tube-in-tube helically coiled heat exchanger

In the present study a tube-in-tube helically coiled (TTHC) heat exchanger has been numerically modeled for fluid flow and heat transfer characteristics for different fluid flow rates in the inner as well as outer tube. The three-dimensional governing equations for mass, momentum and heat transfer have been solved using a control volume finite difference method (CVFDM). The renormalization group (RNG) k– model is used to model the turbulent flow and heat transfer in the TTHC heat exchanger. The fluid considered in the inner tube is compressed air at higher pressure and cooling water in the outer tube at ambient conditions. The inner tube pressure is varied from 10 to 30 bars. The Reynolds numbers for the inner tube ranged from 20,000 to 70,000. The mass flow rate in the outer tube is varied from 200 to 600 kg/h. The outer tube is fitted with semicircular plates to support the inner tube and also to provide high turbulence in the annulus region. The overall heat transfer coefficients are calculated for both parallel and counter flow configurations. The Nusselt number and friction factor values in the inner and outer tubes are compared with the experimental data reported in the literature. New empirical correlations are developed for hydrodynamic and heat-transfer predictions in the outer tube of the TTHC.     

Unsteady flow and heat transfer on an accelerating surface with blowing or suction in the absence and presence of a heat source or sink

The problem of unsteady flow and heat transfer in the laminar boundary layer on a linearly accelerating surface with suction or blowing in the absence and presence of a heat source or sink is considered. The governing partial differential equations for this investigation are transformed into the non-dimensional equations by using pseudo-similarity time and pseudo-similarity coordinate. The resulting two points boundary-value problem is solved numerically by the central finite difference method associated with Newton's iteration from the initial stage (ξ=0) to a steady state (ξ=1) completely. A parametric study is performed to illustrate the effects of Prandtl number, power-law surface temperature (PLST) or power-law heat flux (PLHF), heat sink or heat source, and suction or blowing parameter on the dynamic velocity and temperature fields as well as the transient development of the skin-friction coefficients and the Nusselt number. These results are depicted graphically to display special aspects of unsteady flow and heat transfer characteristics in all time.     

A NEW CLASS OF SIMILARITY SOLUTIONS OF AN UNSTEADY ELECTRICALLY CONDUCTING FREE-FORCED CONVECTIVE FLOW IN A VERTICAL POROUS SURFACE WITH DUFOUR AND SORET EFFECTS

The 2-D unsteady magnetohydrodynamic free-forced convective boundary layer flow of a viscous incompressible fluid is studied numerically taking into account heat and mass transfer. The fluid is subjected to uniform heat and mass fluxes embedded in a porous medium by the presence of coupled Dufour and Soret effects. A new class of similarity equations has been obtained by introducing a time-dependent length scale and a corresponding similarity variable. The resulting equations are then integrated numerically using the Nachtsheim-Swigert shooting iteration technique along with the sixth-order Runge-Kutta integration scheme. By developing locally similar solutions of the fluid flow, the behavior of the velocity, temperature, and concentration fields as well as the rate of heat transfer, wall temperature gradient, rate of mass transfer, and skin friction coefficient have been investigated. The effects of Grashof number (Gr), modified Grashof number (Gm), combined effects of the porous and magnetic parameter (S), suction/injection parameter Fw, Brinkman number (Br), Soret number (Sr), and Dufour number (D_f) have been observed on the flow field and discussed.     

Impingement heat transfer under a confined slot jet. Part II: Effects of surface motion and throughflow

Local and average heat transfer coefficients were measured for a confined turbulent slot jet impinging on a moving surface at which there may be throughflow. Profiles of the local convective coefficient at the impingement surface were obtained using a fast responding, highly sensitive porous heat flux sensor. The decrease in average heat transfer with surface motion is not negligible, ～20%, at values of the surface motion parameter, M_vs, comparable to those used in industry. The enhancement of heat transfer by throughflow at a moving impingement surface is linearly additive and, when expressed as δSt, is proportional to only the throughflow parameter, M_us, with a proportionality constant of 0.17 which is independent of Re, M_vs or extent of the heat transfer surface.     

Combined forced and free convection flow past a horizontal flat plate

The problem of simultaneous forced and free convection flow of a Newtonian fluid past a hot or cold horizontal flat plate is investigated by means of numerical solutions of the full equations of motion and thermal energy subject only to the Boussinesq approximation. These solutions span the parameter ranges 10 ≤ Re ≤ 100, 0.1 ≤ Pr ≤ 10, and –2.215 ≤ Gr/Re^5/2 ≤ 2.215 where Re, Pr, and Gr are based on the ambient free stream fluid properties and the overall plate length l. When Gr > 0, the boundary flow near the plate surface is accelerated relative to the corresponding forced convection flow, with a resulting increase in both the local skin friction and heat transfer coefficients. When Gr < 0, the boundary flow is decelerated, the local skin friction and heat transfer are decreased, and the flow actually separates for Gr/Re^5/2 < –0.8 when Pr = 0.7. In the latter circumstance, an increasing degree of upstream influence is observed as Gr/Re^5/2 is further decreased.     

Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet

In this work, we study the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation, included in the energy equation, and variable wall temperature. A similarity transformation was used to transform the governing partial differential equations to a system of nonlinear ordinary differential equations. An efficient numerical shooting technique with a fourth-order Runge-Kutta scheme was used to obtain the solution of the boundary value problem. The variations of dimensionless surface temperature, as well as flow and heat-transfer characteristics with the governing dimensionless parameters of the problem, which include the nanoparticle volume fraction ϕ, the nonlinearly stretching sheet parameter n, the thermal radiation parameter N_R, and the viscous dissipation parameter Ec, were graphed and tabulated. Excellent validation of the present numerical results has been achieved with the earlier nonlinearly stretching sheet problem of Cortell for local Nusselt number without taking the effect of nanoparticles.     

Heat Transfer from a Continuous Surface to a Non-Newtonian Fluid

Boundary layer solutions are presented to investigate the heat transfer characteristics of a continuous flat surface moving in a parallel stream of power-law-type non-Newtonian fluid. Analytical results have been given for the friction factor and heat transfer rates. The velocity difference between the surface and the free stream is treated as an important parameter.     

Unsteady conjugate mass transfer of a 2D deformable droplet in a modest extensional flow in across-slot

This work aims to investigate the unsteady conjugate interphase mass transfer between a stationary deformed drop and the modest extensional flow in a cross-intersected 2D channel. It is very difficult to accurately quantify the transient mass transfer rate of solute in such a geometry. Therefore, we established a mathematical model on the basic of the Stokes equation and solved it by the boundary element method, which could deal precisely with a two-phase flow system with a deformable interface; meanwhile, the convection-diffusion equation was solved by the finite difference method to calculate the unsteady conjugate interphase mass transfer. The simulation results showed that the mass transfer rate, analyzed and characterized in terms of mean concentration variation and Sherwood number Sh, was affected by capillary number Ca, Peclet number Pe, viscosity ratio λ , interior-to-exterior diffusivity ratio K, distribution coefficient m, and wall effect factor W.     

Combined buoyancy effects of thermal and mass diffusion on laminar forced convection in the thermal entrance region of horizontal square channels

Heat and mass transfer in laminar mixed convection in the thermal entrance region of horizontal square channels is investigated by using the vorticity-velocity formulation of the Naiver-Stokes equations. The numerical results, including the developments of temperature and concentration contours, Nu_z and friction coefficient ratios, fRe / (fRe)₀, are presented for an air-water system. The effects of bottom wall temperature and the relative humidity on the momentum, heat and mass transfer in the flow are examined in detail. Results show that the influences of the evaporation of the water vapor along the wetted wall on the heat and mass transfer and the ratio fRe / (fRe)₀ are rather substantial.