Similarity solutions for laminar free convection flow of a thermomicropolar fluid past a non-isothermal vertical flat plate

Laminar free convection boundary layer flow of a thermomicropolar fluid past a non-isothermal vertical flat plate has been studied in detail. It has been established that the flow problem has similarity solutions when the variation in the temperature of the plate is a linear function of the distance from the leading edge measured along the plate. The resulting system of the nonlinear ordinary differential equations has been solved numerically by “Shooting Method” for various values of the material parameters. The effects of these parameters has been studied on the velocity and microrotation fields graphically. Also “Tables” have been given for the values of temperature, skin-friction parameter, microrotation gradient on the wall and Nusselt number. Two types of boundary conditions are prescribed for the microrotation on the wall.     

Laminar free convection on a vertical nonisothermal plate with vigorous blowing

Asymptotic solutions are obtained for the equations of a dynamically similar boundary layer in the case of natural convection on a vertical nonisothermal plate with vigorous blowing.Translated from Inzhenerno Fizicheskii Zhurnal, Vol. 43, No. 6, pp. 909–913, December, 1982.     

A numerical study of similarity solutions for combined forced and free convection

Summary The similarity equations for combined forced and free convection flow over a horizontal plate when the wall temperature is inversely proportional to the square root of the distance from the leading edge are solved by introducing a scaling similar to that for the Blasius equation. The technique is also applied to the local similarity equations for the case of a constant wall temperature.With 7 Figures     

Effect of plate temperature oscillations on free convection boundary layers along a vertical plate

The laminar free convection flow and heat transfer along a semi-infinite vertical plate is analysed when the plate temperature $\text{T}{}_{\mathrm{p}}$ takes the form $\text{T}{}_{\mathrm{p}}\text{=T}{}_{\mathrm{w}}\text{+α(T}{}_{\mathrm{w}}\text{?T}{}_{\mathrm{\infty }}\text{sin}\text{ω}\text{t}\text{?}$ where$\text{0 ? α < 1}$. Thus the plate temperature consists of a basic steady distribution $\text{T}{}_{\mathrm{w}}$ with a super-imposed oscillatory distribution $\text{α(T}{}_{\mathrm{w}}\text{?T}{}_{\mathrm{\infty }}\text{)}\text{sin}\text{ω}\text{t}\text{?}$. It is to be noted that the main difference between this and the earlier works is that here the magnitude of oscillations α is not required to be very small. The skin friction and the rate of heat transfer from the wall are caleulated by means of two asymptotic expansions. A regular expansion is obtained for small values of the frequency parameter $\text{? (x) = [x/gβ(T}{}_{\mathrm{w}}\text{?T}{}_{\mathrm{\infty }}\text{)]}\text{ω}$ while for large values of ?, a singular perturbation technique is developed. Numerical computations are made for various values of α and it is demonstrated that there are values of ? at which the small and large ? exppnsions for the skin friction and the rate of heat transfer overlap satisfactorily.     

Hall effects on magnetohydrodynamic free convection about a semi-infinite vertical flat plate

A theoretical analysis is presented for the problem of free convection flow of a conducting fluid along a semi-infinite vertical flat plate when the fluid is permeated by a transverse magnetic field and the Hall effects are taken into account. The derived fundamental equations on the assumption of small magnetic Reynolds number are solved numerically by employing the difference-differential method in combination with the Simpson's rule. The velocity and temperature profiles as well as the local Nusselt number are computed for various values of the Hall and magnetic parameters. The results are compared with those known from the literature.     

Transient free convection about a vertical flat plate embedded in a porous medium

The problems of transient free convection in a porous medium adjacent to a vertical semi-infinite flat plate with a step increase in wall temperature and surface heat flux are considered in this paper. By assuming a temperature profile in each case, the governing equation for the boundary layer thickness is obtained by an integral method. These governing equations are first-order partial differential equations of the hyperbolic type that can be solved exactly by the method of characteristics and approximately by the method of integral relations. The results based on the method of characteristics clearly indicate that during the initial stage when the leading edge effect is not being felt, heat is transferred as if by transient 1-dimensional heat conduction. At a later time, depending on the vertical location, the heat transfer characteristics change from transient 1-dimensional heat conduction to steady 2-dimensional convection. The thickness of the boundary layer is shown to be increasing with time until it reaches steady state where its value remains constant thereafter. The growth rate of the boundary layer thickness exhibits a discontinuity at the end of the transient period and the beginning of the steady state period. On the other hand, the results based on the method of integral relations show that the boundary layer thickness grows continuously with time and approaches the steady state value asymptotically; the growth rate of the boundary layer thickness decreases from a finite value to zero continuously as the steady state is approached. Except between the end of the transient period and the beginning of the steady state period, the results based on the method of integral relations are in good agreement with those based on the method of characteristics.     

On the free convection boundary layer on a vertical plate with prescribed surface heat flux

The free convection boundary layer on a vertical plate with a prescribed surface heat flux proportional to (1 +x ²)^µ (µ a constant) is discussed. For µ > –12 the boundary-layer solution develops from a similarity solution valid forx small to the one valid forx large. However, with µ –12 the similarity equations forx large are not solvable and the behaviour for largex in this case is discussed. It is found that there are two cases to consider, namely µ < –12 and µ = –12. In both cases the leading-order problem is homogeneous involving an arbitrary constant which is determined from an integral property of the full boundary-layer problem. However, in the former case the asymptotic behaviour is algebraic, with the perturbation to the leading-order solution, arising from the heat flux boundary condition, being ofO[x ^1+2µ]. The latter case also involves logarithmic terms, with the perturbation to be leading-order solution now being ofO[(logx)^–1].     

A note on free convection in turbine cavities

A study is made of free convection in axisymmetric cavities with variable acceleration forces. It is shown that convective heat transfer should be taken into account when dealing with turbine rotor cavities.     

Finite difference analysis of free convection effects on stokes problem for a vertical plate in dissipative fluid

The flow past an infinite vertical isothermal plate started impulsively in its own plane in a viscous incompressible fluid has been considered on taking into account the viscous dissipative heat. The coupled non-linear equations governing the flow are solved by finite-difference method. The velocity and temperature field have been shown graphically for G 0 (G, the Grashof number, G 0, cooling of the plate by the free convection currents, G < 0, heating of the plate by the free convection currents) and the numerical values of the skin-friction and the rate of heat transfer are entered in tables. The effects of G and E (the Eckert number) are discussed on the flow field.     

Finite element analysis of transient free convection flow over vertical plate

A finite element method is used to study the heat-transfer response of an incompressible, laminar, transient free convection flow over a semi-infinite vertical plate. The non-dimensional governing equations are solved by the finite element method (FEM). The resulting non-linear integral equations are linearized and solved using the Newton–Raphson iteration. The resulting first-order ordinary differential equations with respect to time are solved using the implicit Euler scheme. Numerical results for the details of the velocity and temperature contours and profiles as well as heat transfer rate in terms of Nusselt number which are shown on graphs have been presented.     

Hydromagnetic convection at a heated semi-infinite vertical plate

Numerical results are presented for the transient and steady-state velocity field, temperature field, and heat transfer characteristics. These results are obtained by solving the highly nonlinear partial differential equations describing the conservation of mass, momentum and energy (with variable fluid properties) by an explicit finite-difference method in time-dependent form. Two cases are studied, namely the hydromagnetic (HM) case and the hydrodynamic (HD) case. Values of the velocity components u and v (absolute) in the HD case are larger than those in the HM case. Quite opposite is the phenomenon in the case of the temperature field θ. Further, the velocity and temperature fields are more prominent in the presence of heat sources than in their absence. Numerical results indicate that the presence of magnetic field delays attainment of the steady-state condition. Furthermore, in the hydromagnetic case, it is observed that the variable fluid-property has almost negligible effect on the flow and heat transfer characteristics, which is contrary to that in the hydrodynamic case.     

Transient free convection flow of water at 4°C past a semi-infinite vertical plate

Transient free convection flow of water at 4°C past an isothermal semi-infinite vertical plate, when its density is maximum, is analysed by employing an explicit finite-difference technique. The transient velocity and temperature, the average skin-friction and the average Nusselt number are shown graphically for Prandtl number Pr=11.4 (4°C) and 7.0 (20°C). It has ben observed that when the density of water is maximum, it takes more time to reach its steady state. Also the average skin-friction decreases considerably when the water is at 4°C, as compared to that at 20°C. The average Nusselt number increases at small values of time and decreases at large values of time when the water is at 4°C.     

Finite-difference analysis of transient free convection with mass transfer on an isothermal vertical flat plate

Implicit finite-difference analysis of transient free convective flow past a semi-infinite vertical flat plate, on taking into account mass transfer, is carried out. Presence of gases like H, He, H₂O, NH₃, CO₂ in air are considered. Transient velocity, temperature and species concentration profiles are shown on graphs. Also values of local and average skin-friction, Nusselt and Sherwood number are calculated and shown graphically. During the course of discussion, the effects of Sc (Schmidt number), N (buoyancy ratio parameter) on the flow are discussed.     

Forced and free mixed convection boundary layer flow with uniform suction or injection on a vertical flat plate

Summary A laminar forced and free mixed convection flow on a flat plate with uniform suction or injection was theoretically investigated. Nonsimilar partial differential equations are transformed into nonsimilar ordinary ones by means of difference-differential method. The solutions of the resulting equations are obtained with integral forms, and are calculated by the method of successive iteration. The velocity profiles, temperature profiles, friction coefficient and heat transfer coefficient are obtained for various values of suction/injection parameter and buoyancy parameter.     

Free convection on a heated vertical plate: The solution for small Prandtl number

The solution of the equations for the free-convection boundary-layer flow on a vertical plate with a prescribed power-law heating is considered for small values of the Prandtl number σ. It is shown that the boundary layer divides up into two regions. There is a thin inner region, of thickness O{ie273-1}, in which, to leading order, the temperature is a constant, but which is not determined from the inner solution. This gives rise to a large temperature on the plate of O{ie273-2}. This inner region drives a flow in a much thicker inviscid outer region, of thickness of O{ie273-3}. At the outer edge of this outer region the ambient conditions are attained, and it is the matching between the two regions which determines the plate temperature.     

Free convection and radiation characteristics for a vertical plate embedded in a porous medium

Steady two‐dimensional free convection flow due to combined effect of radiation and convection through a porous medium bounded by a vertical infinite plate is considered. The behaviour of Darcy and non‐Darcy flow is investigated. The flow of water through different porous media under different environmental conditions is discussed. Effect of four non‐dimensional parameters, i.e. Prandtl number (Pr), modified Grashof number (G), permeability parameter (K) and radiation parameter (N) has been studied. Effect of these parameters on Nusselt number is analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

Heat transfer response of free convection flow from a vertical heated plate to an oscillating surface heat flux

Summary An investigation is undertaken of the unsteady response of two-dimensional laminar free convection boundary layer flow of a viscous incompressible fluid along a semi-infinite vertical heated plate where the mean surface heat flux oscillates with a small amplitude about a steady profile. The buoyancy forces are favourable, resulting from a positive flux of heat from the surface of the plate into the fluid. The interaction of the time-periodic heat flux with the usual boundary-layer flow is examined by using a linearized theory. Solutions are obtained using three distinct methods, namely an extended series expansion method for low frequencies, an asymptotic series expansion method for high frequencies and a fully numerical finite difference method for general frequencies. Calculations have been carried out for a wide range of parameters to examine the solutions in terms of the amplitude and phase angle of the fluctuating parts of the surface shear stress and the surface temperature. It has been found that the amplitude and phase angle of both the shear stress and the surface temperature predicted by these three methods are in very good agreement in their respective ranges of validity.     

Unsteady free convection flow over a continuous moving vertical surface

Summary The problem of heat transfer in the unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid has been investigated. Both constant surface temperature and constant surface heat flux conditions have been considered. The nonlinear coupled partial differential equations governing the flow have been solved numerically using the Keller box method and the Nakamura method which both give closely similar solutions. The results indicate that the cooling rate of the sheet can be enhanced by increasing the buoancy parameter or the velocity of the sheet. It is found that a better cooling performance could be achieved by using a liquid as a cooling medium rather than a gas. The overshoot in the velocity occurs near the surface when the buoyancy parameter exceeds a certain critical value.     

Oscillatory free convection from a disk rotating in a vertical plane

Unsteady free convective flow of a viscous incompressible fluid from a heated disk rotating in a vertical plane induced by periodic variation in the mean temperature of the disk is analysed for low and high frequencies of oscillation. The oscillatory solutions of the energy equation are also obtained. The solutions are matched at a suitable frequency parameter which increases with the Prandtl number of the fluid.     

A note on the boundary value problem arising in free convection in porous media

By similarity transformation and governing equations of free convection on a heated vertical plate embedded in porous medium are reduced to coupled nonlinear equations. The equations are numerically integrated using the boundary conditions at the plate and at ‘infinity’. Assuming that the plate is subjected to a prescribed temperature [1–3] or to a prescribed heat flux [4, 5], the boundary value problems have been solved independently. These researchers seem to have not noted that the solutions for the two cases are dependent on each other. In the present note we consider yet another thermal boundary condition, namely, radiation boundary condition [6] at the plate and show that the solutions for the three cases are dependent and one can pass from one solution to the other easily.