Magnetohydrodynamic mixed convection in a vertical channel

The problem of combined free and forced convective magnetohydrodynamic flow in a vertical channel is analysed by taking into account the effect of viscous and ohmic dissipations. The channel walls are maintained at equal or at different constant temperatures. The velocity field and the temperature field are obtained analytically by perturbation series method and numerically by finite difference technique. The results are presented for various values of the Brinkman number and the ratio of Grashof number to the Reynolds number for both equal and different wall temperatures. Nusselt number at the walls is determined. It is found that the viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. It is also found that the analytical and numerical solutions agree very well for small values of ε.     

Mass transfer effects on flow past an impulsively started infinite vertical plate with constant mass flux — an exact solution

An exact solution to the problem of flow past an impulsively started infinite vertical plate in the presence of a foreign mass and constant mass flux at the plate is presented by the Laplace-transform technique. The velocity, the temperature and the concentration profiles are shown on graphs. The skin-friction and the Sherwood number are also shown on graphs. The effects of different parameters likeG (the Grashof number),Gc (the modified Grashof number),Pr (the Prandtl number) andSc (the Schmidt number) are discussed.     

Natural convection in an infinite square under low-gravity conditions

In order to study natural convection effects on fluid flows under low-gravity in space, we have expanded variables into a power series of Grashof number by using perturbation theory to reduce the Navier-Stokes equations to the Poisson equation for temperature T and biharmonic equation for stream function φ. Suppose that a square infinite closed cylinder horizontally imposes a specified temperature of linear distribution on the boundaries, we investigate the two dimensional steady flows in detail. The results for stream function φ, velocity u and temperature T are gained. The analysis of the influences of some parameters such as Grashof number G_r and Prandtl number P_r on the fluid motion lead to several interesting conclusions. At last, we make a comparison between two results, one from approximate equations, the other from the original version. It shows that the approximate theory correctly simplifies the physical problem, so that we can expect the theory will be applied to unsteady or three-dimensi     

Homogenization of a contact problem for a system of densely situated punches

A linear contact problem of an elastic half space with rigid punches ε-periodically situated on a bounded part of the boundary of the elastic solid is investigated. Using the method of homogenization theory and the method of matched asymptotic expansions, the leading terms of the asymptotic solution are constructed as ε→0. The general capacity of the contact spot is introduced and some its properties are described.     

The effect of buoyancy forces on laminar convection with viscous dissipation in a vertical channel

The interaction of the viscous dissipation effect with the presence of buoyancy forces is investigated for laminar-flow heat transfer in a parallel-plate vertical channel. One of the channel walls is considered as isothermal with a prescribed temperature, while the other wall is considered as insulated. The velocity field is assumed to be parallel. The velocity field, the temperature field and the Nusselt number are obtained by a perturbation series method which employs the ratio between the Grashof number and the Reynolds number as the perturbation parameter. The radius of convergence of the perturbation series is estimated. Received on 10 December 1997     

Peristaltic flow of a nanofluid in a non-uniform tube

The present analysis discusses the peristaltic flow of a nanofluid in a diverging tube. This is the first article on the peristaltic flow in nanofluids. The governing equations for nanofluid are modelled in cylindrical coordinates system. The flow is investigated in a wave frame of reference moving with velocity of the wave c. Temperature and nanoparticle equations are coupled so Homotopy perturbation method is used to calculate the solutions of temperature and nanoparticle equations, while exact solutions have been calculated for velocity profile and pressure gradient. The solution depends on Brownian motion number N _b , thermophoresis number N _t , local temperature Grashof number B _r and local nanoparticle Grashof number G _r . The effects of various emerging parameters are investigated for five different peristaltic waves. It is observed that the pressure rise decreases with the increase in thermophoresis number N _t . Increase in the Brownian motion parameter N _b and the thermophoresis parameter N _t temperature profile increases. Streamlines have been plotted at the end of the article.     

Pressure and velocity of air during drying and storage of cereal grains

A common method of drying cereal grains is to ventilate a large static mass of grain with an even flow of air at near ambient temperature. After the grain has been dried it is often stored in the same container and kept cool by aeration with a lower velocity of air than is used in drying. To analyse the airflow through this mass of grain a nonlinear momentum equation for flow through porous media is used where the resistance to flow is a + b ¦ν¦. This equation, together with the assumption that the air is incompressible, defines the problem which is solved numerically, using the finite element method, and the results compared with experimental values. The small parameter ε = bν _r /a, where ν _r is the velocity scale, is used in a perturbation analysis to examine the nonlinear effects of the resistance on the airflow. When ε = 0 the equations reduce to those for potential flow, while for small values of ε there are first-order corrections to the pressure p ₁ and the stream function χ ₁. The nonlinear problem is simplified by changing to curvilinear coordinates (s, t) where s is constant on the potential flow isobars while t is constant on the streamlines. General conclusions are derived for p ₁ and χ ₁, for example that they depend on the curvature of the potential flow solution with a large curvature of the isobars leading to larger values of p ₁ and similarly for the streamlines. The potential flow solution p ₀ and the first order solution p ₀ + εp ₁ are close to the solution of the full nonlinear problem when ε is small. To illustrate this for a typical grain storage problem, the solution p ₀ is shown to be very close to the finite element solution (with a difference of less than 1%) when ε < 0.03 while for the first order solution p ₀ + εp ₁ the difference is less than 1% when ε < 0.1.     

Thermo-diffusion impact on immiscible flow characteristics of convectively heated vertical two-layered Baffle saturated porous channels in a suspension of nanoparticles: an analytical study

The heat and mass transfer of two immiscible fluids in an inclined channel with thermal diffusion, vicious, and Darcy dissipation is studied. The first region consists of a clear fluid, and the second one is filled with a nanofluid saturated with a porous medium. The behaviors of Cu-H₂O, In-H₂O, and Au-H₂O nanofluids are analyzed. The transport properties are assumed to be constant. The coupled non-linear equations of the flow model are transformed into the dimensionless form, and the solutions for the velocity, temperature, and concentration are obtained by the regular perturbation technique. Investigations are carried out on the flow characteristics for various values of the material parameters. The results show that the velocity and temperature of the fluids enhance with the thermal Grashof number, solutal Grashof number, and Brinkman number while decrease with the porosity parameter and solid volume fraction.     

Finite difference analysis of mass transfer effects on flow past an impulsively started infinite vertical plate in dissipative fluid and constant heat flux

A finite-difference solution to the flow past an impulsively started infinite vertical plate is derived by assuming 1) presence of species concentration like water vapour, CO₂ etc. and 2) constant heat flux at the plate. The velocity and the temperature profiles, the skin-friction and the rate of heat transfer are shown graphically. The effects of the modified Grashof number,Gm, the Eckert numberE, the Schmidt numberSc on the flow of air are discussed.     

Investigation of nanofluid flow in a vertical channel considering polynomial boundary conditions by Akbari-Ganji's method

In this research, a vertical channel containing a laminar and fully developed nanofluid flow is investigated. The channel surface's boundary conditions for temperature and volume fraction functions are considered qth-order polynomials. The equations related to this problem have been extracted and then solved by the AGM and validated through the Runge-Kutta numerical method and another similar study. In the study, the effect of parameters, including Grashof number, Brownian motion parameter, etc., on the motion, velocity, temperature, and volume fraction of nanofluids have been analyzed. The results demonstrate that increasing the Gr number by 100% will increase the velocity profile function by 78% and decrease the temperature and fraction profiles by 20.87% and 120.75%. Moreover, rising the Brownian motion parameter in five different sizes (0.1, 0.2, 0.3, 0.4, and 0.5) causes lesser velocity, about 24.3% at first and 4.35% at the last level, and a maximum 52.86% increase for temperature and a 24.32% rise for Ψ occurs when N_b rises from 0.1 to 0.2. For all N_t values, at least 55.44%, 18.69%, for F(η), and Ω(η), and 20.23% rise for Ψ(η) function is observed. Furthermore, enlarging the N_r parameter from 0.25 to 0.1 leads F(η) to rise by 199.7%, fluid dimensionless temperature, and dimensional volume fraction to decrease by 18% and 92.3%. In the end, a greater value of q means a more powerful energy source, amplifying all velocity, temperature, and volume fraction functions. The main novelty of this research is the combined convection qth-order polynomials boundary condition applied to the channel walls. Moreover, The AMG semi-analytical method is used as a novel method to solve the governing equations.     

Direct numerical simulation of turbulent flow and heat transfer in a square duct with natural convection

In this paper, a direct numerical simulation of a fully developed turbulent flow and heat transfer are studied in a square duct with an imposed temperature difference between the vertical walls and the perfectly insulated horizontal walls. The natural convection is considered on the cross section in the duct. The numerical scheme employs a time-splitting method to integrate the three dimensional incompressible Navier-Stokes equation. The unsteady flow field was simulated at a Reynolds number of 400 based on the Mean friction velocity and the hydraulic diameter (Re _m = 6200), while the Prandtl number (Pr) is assumed 0.71. Four different Grashof numbers (Gr = 10⁴, 10⁵, 10⁶ and 10⁷) are considered. The results show that the secondary flow and turbulent characteristics are not affected obviously at lower Grashof number (Gr ≤ 10⁵) cases, while for the higher Grashof number cases, natural convection has an important effect, but the mean flow and mean temperature at the cross section are also affected strongly by Reynolds stresses. Compared with the laminar heat transfer at the same Grashof number, the intensity of the combined heat transfer is somewhat decreased.     

Mixed convection flow in vertical channel with boundary conditions of third kind in presence of heat source/sink

The effects of viscous dissipation and heat source/sink on fully developed mixed convection for the laminar flow in a parallel-plate vertical channel are investigated.The plate exchanges heat with an external fluid.Both conditions of equal and different reference temperatures of the external fluid are considered.First,the simple cases of the negligible Brinkman number or the negligible Grashof number are solved analytically.Then,the combined effects of buoyancy forces and viscous dissipation in the presence of heat source/sink are analyzed by a perturbation series method valid for small values of the perturbation parameter.To relax the conditions on the perturbation parameter,the velocity and temperature fields are solved by using the Runge-Kutta fourth-order method with the shooting technique.The velocity,temperature,skin friction,and Nusselt numbers at the plates are discussed numerically and presented through graphs.     

Heat transfer by laminar flow of an elastico-viscous fluid in posttreatment analysis of wire coating with linearly varying temperature along the coated wire

The present study focuses on the heat transfer by the laminar flow of an elastico-viscous fluid in posttreatment of wire coating analysis with linearly varying temperature on the surface of coated wire. The surface of wire (uncoated) and the surface of coated wire were subjected to two thermal boundary conditions. The constitutive equation of motion and equation of energy have been solved by using perturbation theory for velocity, pressure distribution along the radial direction and temperature distribution. The theoretical analysis of flow rate, average velocity, shear stress, thickness of coated wire, and force on the total wire were also derived. Moreover, the flow phenomenon has been studied under the influence of elastic number R _e velocity ratio U and the dimensionless number S in the ranges 0?≤?R _e ?≤?20, 0.2?≤?U?≤?1.4 and 0?≤?S?≤?20. We noticed that with the increase in elastic number R _e velocity decreases whereas thickness of the coated wire and force on the total wire increases. Also temperature profile decreases with the increase of non-dimensional parameter S.     

Three dimensional unsteady convection and mass transfer flow through porous medium

The problem of three dimensional unsteady convection flow through a porous medium, with effect of mass transfer bounded by an infinite vertical porous plate is discussed, when the suction at the plate is transverse sinusoidal and the plate temperature oscillates in time about a constant mean. Assuming the free stream velocity to be uniform, approximate solutions are obtained for the flow field, the temperature field, the skin-friction and the rate of heat transfer. The dependence of solution on Pr (Prandtl number), Gr (Grashof number based on temperature), Gc (modified Grashof number based on concentration difference), Sc (Schimdt number), the frequency and the permeability parameter is also investigated.     

Natural convection over a vertical flat plate due to absorption of thermal radiation

Heat transfer by simultaneous free convection and radiation in a participating fluid has received some attention during the past few years. However most of the previous work has been focussed on gases. The present work investigates the problem of combined radiation and natural convection in liquids. Analysis are given for an optically thick cold fluid layer adjacent to a non-emitting and non-reflecting radiation-transmitting plate. The external surface of the plate is subjected to heat loss to surroundings. The governing differential equations are transformed to a dimensionless form where the solution becomes dependent on the following parameters: the plate absorpitivity,α _p; the dimensionless distance along the plate,ζ; the fluid Prandtl number,Pr; and dimensionless heat loss coefficient to surrounding,N _c. A local non-similar technique is adopted to obtain solutions atPr=6.5 and at a wide range ofα _p,ζ, andN _c. The results showed that both velocity and temperature are non-similar and they are greatly affected by the value ofα _p whenζ is small. At large values of f the effect ofα _p diminishes and for a plate without heat loss the velocity becomes similar, i.e. independent of C The heat loss from the external surface of the plate causes the maximum temperature of the fluid to depart far from the plate. The results also showed that for plates without heat loss the local heat transfer coefficient from the plate depends on the local Grashof number to the power 0.185.     

Numerical investigation of convective motion in a closed cavity

The investigation of thermal convection in a closed cavity is of considerable interest in connection with the problem of heat transfer. The problem may be solved comparatively simply in the case of small characteristic temperature difference with heating from the side, when equilibrium is not possible and when slow movement is initiated for an arbitrarily small horizontal temperature gradient. In this case the motion may be studied using the small parameter method, based on expanding the velocity, temperature, and pressure in series in powers of the Grashof number—the dimensionless parameter which characterizes the intensity of the convection [1–4]. In the problems considered it has been possible to find only two or three terms of these series. The solutions obtained in this approximation describe only weak nonlinear effects and the region of their applicability is limited, naturally, to small values of the Grashof number (no larger than 10³).With increase of the temperature difference the nature of the motion gradually changes—at the boundaries of the cavity a convective boundary layer is formed, in which the primary temperature and velocity gradients are concentrated; the remaining portion of the liquid forms the flow core. On the basis of an analysis of the equations of motion for the plane case, Batchelor [4] suggested that the core is isothermal and rotates with constant and uniform vorticity. The value of the vorticity in the core must be determined as the eigenvalue of the problem of a closed boundary layer. A closed convective boundary layer in a horizontal cylinder and in a plane vertical stratum was considered in [5, 6] using the Batchelor scheme. The boundary layer parameters and the vorticity in the core were determined with the aid of an integral method. An attempt to solve the boundary layer equations analytically for a horizontal cylinder using the Oseen linearization method was made in [7].However, the results of experiments in which a study was made of the structure of the convective motion of various liquids and gases in closed cavities of different shapes [8–13] definitely contradict the Batchelor hypothesis. The measurements show that the core is not isothermal; on the contrary, there is a constant vertical temperature gradient directed upward in the core. Further, the core is practically motionless. In the core there are found retrograde motions with velocities much smaller than the velocities in the boundary layer.The use of numerical methods may be of assistance in clarifying the laws governing the convective motion in a closed cavity with large temperature differences. In [14] the two-dimensional problem of steady air convection in a square cavity was solved by expansion in orthogonal polynomials. The author was able to progress in the calculation only to a value of the Grashof numberG=10⁴. At these values of the Grashof numberG the formation of the boundary layer and the core has really only started, therefore the author's conclusion on the agreement of the numerical results with the Batchelor hypothesis is not justified. In addition, the bifurcation of the central isotherm (Fig. 3 of [14]), on the basis of which the conclusion was drawn concerning the formation of the isothermal core, is apparently the result of a misunderstanding, since an isotherm of this form obviously contradicts the symmetry of the solution.In [5] the method of finite differences is used to obtain the solution of the problem of strong convection of a gas in a horizontal cylinder whose lateral sides have different temperatures. According to the results of the calculation and in accordance with the experimental data [9], in the cavity there is a practically stationary core. However, since the authors started from the convection equations in the boundary layer approximation they did not obtain any detailed information on the core structure, in particular on the distribution of the temperature in the core.In the following we present the results of a finite difference solution of the complete nonlinear problem of plane convective motion in a square cavity. The vertical boundaries of the cavity are held at constant temperatures; the temperature varies linearly on the horizontal boundaries. The velocity and temperature distributions are obtained for values of the Grashof number in the range 0<G4·10⁵ and for a value of the Prandtl number P=1. The results of the calculation permit following the formation of the closed boundary layer and the very slowly moving core with a constant vertical temperature gradient. The heat flux through the cavity is found as a function of the Grashof number.     

A new wall-damping function for large eddy simulation employing Kolmogorov velocity scale

A new wall-damping function, based on the Kolmogorov velocity scale, for large eddy simulation (LES) is proposed, which accounts for the near-wall effect. To calculate the Kolmogorov velocity scale, u_ε, the dissipation rate of turbulent energy, ε, is needed. In LES, however, the dissipation rate is generally not solved, unlike in the Reynolds averaged Navier-Stokes (RANS) simulations, e.g., k-ε models. Although, in some previous studies, the dissipation rate of the subgrid-scale (SGS) turbulent energy, ε_SGS, is used instead of ε in calculating the Kolmogorov velocity scale, the scale obtained using such a method overly depends on the grid resolution employed and is generally inappropriate. Accordingly, the wall-damping function using the incorrect velocity scale also depends on the grid resolution and gives an inadequate wall effect. This is because ε_SGS contains only the components in the scale smaller than the grid-filter width, which obviously varies with the grid resolution employed. In this study, to overcome this problem, we propose a method for estimating the Kolmogorov velocity scale with a technique of conversion in LES, and the estimated one is utilized in the wall-damping function. The revised wall-damping function for LES is tested in channel flows and a backward-facing step flow. The results show that it yields a proper near-wall effect in all test cases which cover a wide range of grid resolution and Reynolds numbers. It is also shown that all three kinds of SGS models incorporating the present wall-damping function provide good predictions, and it is effective both in one-equation and 0-equation SGS models. These results suggest that the use of the proposed wall-damping function is a refined and versatile near-wall treatment in LES with various kinds of SGS models.     

Adaptive and anisotropic mesh strategy for thin shell problems. Case of inhibited parabolic shells

The aim of this paper is to show the reliability of an adaptive and anisotropic mesh procedure for thin shell problems. We consider singular perturbation problems only for parabolic shells whose behavior is described by the Koiter model. The corresponding system of equations, which depends on the relative thickness ε of the shell, is elliptic except at the limit for ε = 0 where it is parabolic. In a first part of this paper, we study theoretically the phenomena of internal layers appearing during the singular perturbation process, when the loading is somewhat singular. These layers have very different structures either they are along or across the asymptotic lines of the middle surface of the shell. In a second part, numerical computations are performed using a finite element software coupled with an adaptive anisotropic mesh generator. This technique enables to approach accurately the singularities and the layers predicted by the theory especially for very small values of the thickness. The efficiency of such a procedure in comparison with uniform meshes is put in a prominent position.     

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

A numerical study of micropolar flow inside a lid-driven triangular enclosure

A study is carried out to analyze the mixed convection flow and heat transfer inside a lid-driven triangular conduit under the effects of micro-gyration boundary conditions. The micropolar constitutive equation characterizes the fluid inside the cavity. The lower boundary is at a uniform temperature and sliding in its plane with constant velocity u₀, while the inclined walls are cold. Dual cases are considered here, namely the intense concentration (d) and the weak concentration of microelements (\(m = 0.5\)). The governing nonlinear equations are simulated employing the Galerkin finite element method, where the pressure term is handled via the Penalty approach. Using the numerical data, graphical results are produced to illustrate the effects of physical parameters. Specifically, this refers to the effects of the Grashof number (Gr), Prandtl number (Pr), Reynolds number (Re) and vortex viscosity parameter (K) on the streamlines, mid-section velocity profiles, temperature contours, and local and average Nusselt numbers on the cold and heated boundaries of the conduit. Particular emphasis is given on the identification of the set of parameters for which simultaneous symmetry in streamlines and isotherms prevails. The grid independence test is also performed by comparing the average Nusselt numbers (on the hot and cold boundaries of the conduit) for various mesh sizes, and the optimal solution is found. Moreover, the results are also benchmarked with the previously published data.