On the stability of mixed convective motions in a vertical layer with wave-like boundaries

The stability of the stationary and oscillatory convective motions which develop in a vertical layer with periodically curved boundaries is studied for the case of longitudinal fluid injection. The amplitude of the boundary undulations and the flow of fluid along the layer are both assumed to be small, and methods of perturbation theory are used. The characteristic properties of the incremental spectrum of the spatially periodic motions are studied and the most dangerous types of perturbations as well as the forms of the stability regions are determined.

Theoretical investigations of the effect of spatial inhomogeneity of the boundary conditions on the stability of convection were sparse, and they deal mainly with horizontal layers of fluid /1–3/. Stationary, spatially periodic motions in a vertical layer with curved boundaries were investigated in /4/ for the case of free convection (when the flow was closed), and their stability was investigated in /5/. It was established that the presence of a small but finite flow of fluid along the layer leads to an increase in the number of different modes of flow, and to the appearance of non-stationary convective motions in the region near the threshold.     

Nonsimilar solutions for double-diffusion boundary layers on a sphere in micropolar fluids with constant wall heat and mass fluxes

This work presents nonsimilar boundary layer solutions for double-diffusion natural convection near a sphere with constant wall heat and mass fluxes in a micropolar fluid. A coordinate transformation is employed to transform the governing equations into nondimensional nonsimilar boundary layer equations and the obtained boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt number and the local Sherwood number are presented as functions of the vortex viscosity parameter, Schmidt number, buoyancy ratio, and Prandtl number. Higher vortex viscosity tends to retard the flow, and thus decreases the local convection heat and mass transfer coefficients, raising the wall temperature and concentration. Moreover, the local convection heat and mass transfer coefficients near a sphere in Newtonian fluids are higher than those in micropolar fluids.     

A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition

This paper considers the classical problem of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid. It is well known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, no such solution has been attempted for the convective surface boundary condition. The paper demonstrates that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x^?1/2. Numerical solutions of the resulting similarity energy equation are provided for representative Prandtl numbers of 0.1, 0.72, and 10 and a range of values of the parameter characterizing the hot fluid convection process. For the case of constant heat transfer coefficient, the same data provide local similarity solutions.     

Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Study of free convective boundary layer of isothermal lateral surface of axisymmetrical horizontal body

Approximate analytical solution of simplified Navier–Stokes and Fourier–Kirchhoff equations describing free convective heat transfer from isothermal surface has been presented. It is supposed that the surface has the horizontal axis of symmetry and its axial cross-section lateral boundary is a concave function. The equation for the boundary layer thickness is derived for typical for natural convection assumptions. The most important are that the convective fluid flow is stationary and the normal to the surface component of velocity is negligibly small in comparison with the tangential one. The theoretical results are verified by two characteristic cases of the revolution surfaces namely for horizontal conic and vertical round plate. Both limits of presented solution coincide with known formulas.     

Magnetohydrodynamic stationary and oscillatory convective stability in a mushy layer during binary alloy solidification

For the case of solidification of a bottom cooled binary alloy, the magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer is investigated analytically using normal mode linear stability analysis. In the limit of large Stefan number (S_t), a near–eutectic approximation with large far field temperature is considered in the present research. To ascertain the instability in the mushy layer, the strength of the superimposed magnetic field is so chosen that it corresponds to a given mush Hartmann number (Ha_m) of the problem. The results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha_m ≤ 50. The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha_m. The magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of the stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of β (β = S_t/℘² ϒ², ℘ is the compositional ratio, ϒ = 1 + S_t/℘), and vice versa for lower β values. Analogous to the behavior for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer.     

Numerical study of mixed convection flow of a micropolar fluid towards permeable vertical plate with convective boundary condition

In this article, the mixed convective flow of a micropolar fluid along a permeable vertical plate under the convective boundary condition is analyzed. The scaling group of transformations is applied to get the similarity representation of the system of partial differential equations of the problem and then the resulting equations are solved by using Spectral Quasi-Linearisation Method. This study reveals that the dual solutions exists for certain values of mixed convection parameter. The outcomes are analyzed with dual solutions in detail. Effects of micropolar parameter, Biot number and suction/injection parameters on different flow profiles are discussed and depicted graphically.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

滑移垂直壁面驻点附近的混合对流边界层流动

就粘性不可压缩流体,研究垂直壁面的滑移,对壁面驻点附近稳定混合对流边界层流动的影响.假定表面温度和外部流动速度与到驻点的距离呈线性变化.首先,将偏微分的控制方程,转变为常微分方程组,然后应用打靶法进行数值求解.对不同数值的控制参数,按分顺流和逆流两种情况,分析和讨论了流动特性和热传导特征.结果表明,逆流时,在浮力参数的某一范围内出现双解;顺流时,解是唯一的.一般而言,速度滑移导致壁面热传导率增大,而热滑移使之减小.     

On the origin of convection

The method of investigating bifurcations developed in [1 and 2] is applicable to many hydrodynamic problems. In the present paper it is applied to investigate the origin of convection in a horizontal fluid layer heated from below.

Secondary stationary flows are of particular interest in the convection problem since the loss of stability is associated with these flows: “the principle of the change in stability” is not only valid here but has been proved rigorously [3]. It has also been proved that secondary stationary flows are generated by branching off from the state of rest [4 and 5].

The problem under consideration is invariant relative to the group of motions of a horizontal plane.

The single solution invariant relative to this whole group is the rest solution. When this solution is unstable, it is natural to expect the occurrence of solutions invariant relative to some subgroup of the group of motions. If the mentioned subgroup is generated by a pair of translations (in perpendicular directions), we arrive at doubly-periodic solutions (Section 1), and if invariance relative to rotation through a certain angle is required in addition, we arrive at solutions of hexagonal type (Section 2). As is known, precisely these latter are realized in convection experiments [6]. Deductions on the existence of doubly-pertodic convection flows are elucidated in Theorem 1.1, and the existence of solutions of hexagonal convection type is asserted in Theorem 1.2. The applied method has slight connection with the boundary conditions. Only for definiteness is it assumed that the boundaries of the layer are solid walls on each of which the temperature is specified.     

Nonsimilar boundary layer analysis of double-diffusive convection from a vertical truncated cone in a porous medium with variable viscosity

This work presents a boundary layer analysis about variable viscosity effects on the double-diffusive convection near a vertical truncated cone in a fluid-saturated porous medium with constant wall temperature and concentration. The viscosity of the fluid is assumed to be an inverse linear function of the temperature. A boundary layer analysis is employed to derive the nondimensional nonsimilar governing equations, and the transformed boundary layer governing equations are solved by the cubic spline collocation method to yield computationally efficient numerical solutions. The obtained results are found to be in good agreement with previous papers on special cases of the problem. Results for local Nusselt and Sherwood numbers are presented as functions of viscosity-variation parameter, buoyancy ratio, and Lewis number. For a porous medium saturated with a Newtonian fluid with viscosity proportional to an inverse linear function of temperature, higher value of viscosity-variation parameter leads to the decrease of the viscosity in fluid flow, thus increasing the fluid velocity as well as the local Nusselt number and the local Sherwood number.     

Incompressible reacting flows

We establish steady-state convergence results for a system of
reaction-convection-diffusion equations that model in particular combustion phenomena in the presence of nontrivial incompressible fluid motion. Despite the presence of the convection terms, we find that the asymptotic behavior of the system is identical to the case we have previously considered in which the velocity field was set equal to zero. In particular we are again able to establish the convergence of solutions to steady-states and to explicitly calculate the steady-states from the initial and boundary data. Key to our analysis is the establishment of high-order uniform bounds on the temperature and mass fraction components, a process significantly complicated by the presence of the convection terms.

     

Free convection effects on the oscillatory flow past a vertical porous plate in the presence of radiation for an optically thin fluid

The purpose of this work is to study the effect of transverse sinusoidal suction velocity on the flow and mass transfer on free convective oscillatory viscous and optically thin grey fluid over a porous vertical plate in the presence of radiation. The flow becomes three-dimensional due to the variation of suction velocity in the transverse direction. Analytical expressions for velocity and temperature fields are obtained using the perturbation technique. The governing equations has been transformed to ordinary differential equations. Numerical solutions are obtained for different values of radiation parameter, Grashof number and Schmidt number. It is found that non-dimensional velocity decreases with increase of radiation parameter, increases with increase of Grashof number, decreases with increase of Schmidt number and non-dimensional temperature decreases with the increase of radiation parameter.     

Natural convection boundary layer flow of fluid with temperature-dependent viscosity from a horizontal elliptical cylinder with constant surface heat flux

This study deals with the temperature-dependent viscosity effects on the natural convection boundary layer on a horizontal elliptical cylinder with constant surface heat flux. The mathematical problem is reduced to a pair of coupled partial differential equations for the temperature and the stream function, and the resulting nonlinear equations are solved numerically by cubic spline collocation method. Results for the heat transfer characteristics are presented as functions of eccentric angle for various values of viscosity variation parameters, Prandtl numbers and aspect ratios. Results show that an increase in the viscosity variation parameter tends to accelerate the fluid flow near the surface and increase the maximum velocity, thus decreasing the velocity boundary layer thickness. As the viscosity variation parameter is increased, the surface temperature tends to decrease, thus increasing the local Nusselt number. Moreover, the local Nusselt number of the elliptical cylinder increases as the Prandtl number of the fluid is increased.     

A study on preconditioning iterative methods and relativity of pressure in the numerical calculation of fluid flow

This article presents the effect of preconditioning iterative methods on boundary conditions of the pressure‐correction in the numerical computation of fluid flow with known velocity components on all boundaries using the SIMPLE algorithm. In such computation, a set of solutions of the pressure‐correction is indefinite, because only the Neumann condition is imposed on all boundaries. However, solutions become unique if the value of pressure‐correction is fixed at least on one boundary point, and the Dirichlet condition is additionally imposed. Though both conditions must give exactly the same velocity and temperature fields, this problem arises from the relativity of the pressure. The mathematical illustration for this problem is provided using the numerical computation of the natural convection in an enclosure. It is concluded that the preconditioner adopted and the condition that only the Neumann condition on all boundaries is given are effective to reduce the number of iterations in solving the linear system of equations of the pressure‐correction at the computation of the natural convection. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Starting solutions for oscillating motions of an Oldroyd-B fluid over a plane wall

In this paper, we establish the starting solutions for oscillating motions of an Oldroyd-B fluid between two side walls perpendicular to a plane wall. The expressions for the velocity field and the associated tangential stress at the bottom wall are obtained, presented under integral and series form. These satisfy all imposed initial and boundary conditions. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. In the absence of side walls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate. Moreover, the obtained solutions can be specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motions.     

Free convection from a point heat source in a stratified fluid

A non-stationary problem of free convection from a point heat source in a stratified fluid is considered. The system of equations is reduced to a single equation for a special scalar function which determinos the velocity field, and the temperature and salinity distribution. Relations are found connecting the spatial and temporal scales of the phenomenon with the parameters of the medium and the intensity of the heat source. The magnitude of the critical source intensity at which the fluid begins to move in a jet-flow mode is established.The structure of convective flows above the heat sources depends, in the stratified media, essentially on the nature of the stratification /1/ which may be caused by a change in the temperature of the medium /2, 3/ or its salinity /4–7/, and by the form of the heat source. When a temperature gradient exists within the medium, an ascending jet forms above the point source, mushrooming outwards near the horizon of the hydrostatic equilibrium. In the case of a fluid with salinity gradient, the jet is surrounded by a sheet of descending salty fluid, and a regular system of annular convective cells is formed around it /1/.The height of the stationary jet computed in /2, 3/ on the basis of conservative laws agrees with experiment. However, this approach does not enable the temperature and velocity distribution over the whole space to be found and does not enable the problem of determining the flow to be investigated. A stationary solution of the linearized convection equations /8/ does not correspond to detail to the observed flow pattern /1, 5–7/. In this connection the study of the non-linear, non-stationary convection equations is of interest.The purpose of this paper is to construct a non-linear, non-stationary free convection equation above a point heat source, and to analyse the scales of the resulting structure and the critical conditions under which the flow pattern changes.     

Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions

In this paper, we investigate the heat transfer of a viscous fluid flow over a stretching/shrinking sheet with a convective boundary condition. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameter on the temperature profiles and wall heat flux are presented and discussed. The solution is given as an incomplete Gamma function. It is found the convective boundary conditions results in temperature slip at the wall and this temperature slip is greatly affected by the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameters. The temperature profiles in the fluid are also quite different from the prescribed wall temperature cases.     

Mixed convection in the stagnation-point flow of a Maxwell fluid towards a vertical stretching surface

In the present analysis, we study the steady mixed convection boundary layer flow of an incompressible Maxwell fluid near the two-dimensional stagnation-point flow over a vertical stretching surface. It is assumed that the stretching velocity and the surface temperature vary linearly with the distance from the stagnation-point. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. Analytical and numerical solutions of the derived system of equations are developed. The homotopy analysis method (HAM) and finite difference scheme are employed in constructing the analytical and numerical solutions, respectively. Comparison between the analytical and numerical solutions is given and found to be in excellent agreement. Both cases of assisting and opposing flows are considered. The influence of the various interesting parameters on the flow and heat transfer is analyzed and discussed through graphs in detail. The values of the local Nusselt number for different physical parameters are also tabulated. Comparison of the present results with known numerical results of viscous fluid is shown and a good agreement is observed.     

On thermal stability of a reactive third-grade fluid in a channel with convective cooling the walls

In this paper, the thermal stability of a reactive third-grade liquid flowing steadily between two parallel plates with symmetrical convective cooling at the walls is investigated. The system is assumed to exchange heat with the ambient following Newton’s cooling law and the reaction is exothermic under Arrhenius kinetics, neglecting the consumption of the material. Approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of Hermite-Padé approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed. It is observed that a combined increase in non-Newtonian parameter and convective cooling enhances the thermal stability of the material.