Integral transform analysis of MHD flow and heat transfer in parallel-plates channels

A hybrid solution through the so-called Generalized Integral Transform Technique (GITT) is obtained for the MHD flow and heat transfer of a Newtonian fluid in parallel-plates channels. A simple mathematical formulation for the problem is adopted, which evidences both the transient regime flow sustainable only by a constant pressure gradient; and the steady state situation that considers both a constant pressure gradient and a movement of the upper plate, as well as the action of an inflow and outflow perpendicular to the porous plates. Results for the velocity and temperature fields are computed within the governing parameters, namely, pressure gradient, suction velocity, upper plate velocity and Hartmann numbers, for typical situations. A convergence analysis is also performed showing the consistency of the results. In addition, the present results are confronted with those previously reported ones in the literature showing excellent agreements.     

Free and forced convection boundary layer flow through a porous medium with large suction

An analytical study is made of the free and forced convection boundary layer flow past a porous medium bounded by a semi-infinite vertical porous plate. Locally similar solutions are then obtained by a perturbation method for large suction. Solutions for the velocity and temperature distributions are shown graphically for various suction velocities and values of the driving parameter G_r/R, where G_r is the Grashof number and R_e is the Reynolds number. The corresponding values of the skin friction coefficient and the Nusselt number are finally shown in tabular form.     

Natural convection in three-dimensional porous cavities: integral transform method

A transient three-dimensional Darcy model of natural convection in porous medium filled cavities is studied, using a vorticity-vector potential formulation and the generalized integral transform technique (GITT). A general formulation and solution methodology for vertical cavities (insulated vertical walls with differential horizontal wall temperatures) is developed. Results for cubic cavities are presented while evaluating the Rayleigh number effects for stable situations, observing the transient evolution of the heat transfer process. The convergence behavior of the proposed eigenfunction expansion solution is investigated and comparisons with previously reported steady-state solutions are critically performed.     

The boundary element method applied to forced convection heat problems

An integral equation formulation for steady flow of a viscous fluid is presented based on the boundary element method. The continuity, Navier–Stokes and energy equations are used for calculation of the flow and temperature fields. The governing differential equations, in terms of primitive variables, are derived using velocity–pressure–temperature parameters. The calculation of fundamental solutions and solutions tensor is shown. Applications to simple flow cases, such as driven cavity, forward facing step, deep cavity and channel are presented. Convergence difficulties are indicated, which have limited the applications to flows of low Reynolds numbers.     

Mixed convection in laminar film flow of a micropolar fluid

In this paper, the steady mixed convection boundary layer in laminar film flow of a micropolar fluid is considered. The resulting nonlinear coupled ordinary differential equations are solved numerically using an efficient implicit finite-difference scheme. The numerical results obtained for the skin friction coefficient and the local Nusselt number, as well as the velocity, angular velocity or microrotation and temperature profiles are presented in tables and figures for different values of the material parameter K and the Richardson number Ri when the Prandtl number Pr = 0.7 and Pr = 1.     

Mixed convection boundary layer flow along vertical thin needles: Assisting and opposing flows

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical thin needles for both assisting and opposing flow cases is considered in this paper. The transformed boundary layer equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. Numerical computations are carried out for various values of the dimensionless parameters of the problem, namely the mixed convection parameter λ and the parameter a representing the needle size, with Prandtl number, Pr = 0.7. It has been found that the flow and heat transfer characteristics are significantly influenced by these parameters.     

Integral treatment of coupled heat and mass transfer by natural convection from a cylinder in porous media

This paper deals with the study of the buoyancy induced heat and mass transfer from a slender body of revolution embedded in a saturated porous medium. The study has reported the important case of a cylinder with linear temperature and concentration distributions. The governing parameters for the problem under study are buoyancy ratio (N) and Lewis number (Le). The numerical values of local Nusselt and local Sherwood numbers have also been computed for a wide range of N and Le. The results pertaining to the variations of local Nusselt number, local Sherwood number, N and Le with one another have been studied graphically, and it has been concluded that the local Nusselt number decreases while the local Sherwood number increases along with N > 0 for increasing Lewis number. The local Nusselt number decreases while the local Sherwood number increases along with Le for positive values of N. Also the boundary layer thickness ratio decreases along with Le for N > = 0. In this study, an integral method of Von-Karman type has been used in order to obtain mathematical expressions for local Nusselt and local Sherwood numbers.     

Unsteady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface

The unsteady mixed convection boundary-layer flow of a micropolar fluid near the region of the stagnation point on a double-infinite vertical flat plate is studied. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase or sudden decrease in the surface temperature from the uniform ambient temperature. The problem is reduced to a system of non-dimensional partial differential equations, which is solved numerically using the Keller-box method. This method may present well-behaved solutions for the transient (small time) solution and those of the steady-state flow (large time) solution. It was found that there is a smooth transition from the small-time solution (initial unsteady-state flow) to the large-time solution (final steady-state flow). Further, it is shown that for both assisting and opposing cases and a fixed value of the Prandtl number, the reduced steady-state skin friction and the steady-state heat transfer from the wall (or Nusselt number) decrease with the increase of the material parameter. On the other hand, it is shown that with the increase of the Prandtl number and a fixed value of the material parameter, the reduced steady-state skin friction decreases when the flow is assisting and it increases when the flow is opposing.     

Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate

The combined free convection boundary layer flow with thermal radiation and mass transfer past a permeable vertical plate is studied when the plate moves in its own plane. The plate is maintained at a uniform temperature with uniform species concentration and the fluid is considered to be gray, absorbing–emitting. The coupled unsteady non-linear momentum, energy and concentration equations governing the problem is obtained and made similar by introducing a time-dependent length scale. The similarity equations are solved numerically using superposition method. The resulting velocity, temperature and concentration distributions are shown graphically for different values of parameters entering into the problem. The numerical values of the local wall shear stress, local surface heat and mass flux are shown in tabular form.     

Explicit analytical solutions of 2-D laminar natural convection

There are many natural convection processes in various fields, and it is still a hot topic to investigate the fluid dynamics and heat transfer of natural convection. The analytical solutions are meaningful in both theoretical investigation and practical applications. Specially, they are very useful to computational fluid dynamics and heat transfer as the benchmark solutions to check the numerical solutions and to develop numerical differencing schemes, grid generation methods and so forth. Two explicit analytical solutions of 2-D steady laminar natural convection along a vertical porous plate and between two vertical plates were derived for better understanding the flow and heat transfer as well as promoting the computational fluid dynamics and computational heat transfer.     

Bypass flow computations on the LOFA transient in a VHTR

Bypass flow in the prismatic gas-cooled very high temperature reactor (VHTR) is not intentionally designed to occur, but is present in the gaps between graphite blocks. Previous studies of the bypass flow in the core indicated that the cooling provided by flow in the bypass gaps had a significant effect on temperature and flow distributions for normal operating conditions. However, the flow and heat transports in the core are changed significantly after a Loss of Flow Accident (LOFA). This study aims to study the effect and role of the bypass flow after a LOFA in terms of the temperature and flow distributions and for the heat transport out of the core by natural convection of the coolant for a 1/12 symmetric section of the active core which is composed of images and mirror images of two sub-region models. The two sub-region models, 9 × 1/12 and 15 × 1/12 symmetric sectors of the active core, are employed as the CFD flow models using computational grid systems of 70.2 million and 117 million nodes, respectively. It is concluded that the effect of bypass flow is significant for the initial conditions and the beginning of LOFA, but the bypass flow has little effect after a long period of time in the transient computation of natural circulation.     

Mixed convection boundary layer flow past an isothermal horizontal circular cylinder with temperature-dependent viscosity

The problem of steady laminar mixed convection boundary layer flow past an isothermal horizontal circular cylinder placed in a viscous and incompressible fluid of temperature-dependent viscosity is theoretically considered in this paper. The partial differential equations governing the flow and heat transfer are shown to be non-similar. Full numerical solutions of these governing equations are obtained using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for various values of the Prandtl number Pr, the mixed convection parameter λ and the viscosity/temperature parameter θ_r. The obtained results show that the flow and heat transfer characteristics are significantly influenced by these parameters.     

Effect of convection on boundary layer structures in finite thermoelasticity

Abstract

Till now, all of the research on boundary layer structures in thermoelasticity has focused on conduction as the primary mode of heat transfer. In this article, we investigate the additional effect of convection on the deformation field boundary layer structures formed within a thin infinite slab made of a neo-Hookean material. We find that additionally introducing convection in finite thermos-elasticity shifts the boundary layer vertically, while retaining its shape.     

Natural convection boundary layer flow over a truncated cone in a porous medium saturated by a nanofluid

This work studies the natural convection boundary layer flow over a truncated cone embedded in a porous medium saturated by a nanofluid with constant wall temperature and constant wall nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. A suitable coordinate transformation is performed, and the obtained nonsimilar equations are solved by the cubic spline collocation method. The effect of the Brownian motion parameter and thermophoresis parameter on the temperature, nanoparticle volume fraction and velocity profiles are discussed. The effects of the thermophoresis parameter, Brownian parameter, Lewis number, and buoyancy ratio on the local Nusselt number have been studied. Results show that an increase in the thermophoresis parameter or the Brownian parameter tends to decrease the local Nusselt number. Moreover, the local Nusselt number increases as the buoyancy ratio or the Lewis number is decreased.     

Cross flow on transient double‐diffusive natural convection in inclined porous trapezoidal enclosures

This paper investigates the cross‐diffusion effects subject to exponential variable boundary conditions on transient double‐diffusive natural convection flow in an enclosure. The flow domain is a two‐dimensional inclined trapezoidal cavity filled with a porous medium. The top wall is assumed to be insulated and permeable, while the enclosure's bottom wall is subject to exponential varying temperature and concentration. The prescribed temperature and concentration are different at the vertical walls. Conservation equations are used as the governing equations. The finite element Galerkin weighted residual method, in association with the Newton‐Raphson scheme is employed to solve the system of coupled nondimensional equations. The numerical tests are confirmed with existing literature and are found to be in excellent agreement. The simulations results for stream functions, isotherms, and isoconcentrations are discussed for the various flow parameters. A sensitivity analysis using the response surface method suggests that the average Nusselt and Sherwood numbers are more sensitive to the cross‐diffusion effects. It is further observed that the cross‐diffusion terms stabilize the sensitivity to the angle of inclination.     

Non-axisymmetric forced and free flow in a vertical circular duct

The fully developed laminar mixed convection in a vertical circular duct is studied analytically, with reference to non-axisymmetric boundary conditions such that the fluid temperature does not change along the axial direction. The Boussinesq approximation is applied by taking the average temperature in a duct section as the reference fluid temperature. The dimensionless momentum and energy balance equations are solved by employing Fourier series expansions of the temperature and the velocity fields. The solution shows that the temperature field is not influenced by the velocity distribution and that the Fanning friction factor is not affected by buoyancy. On the other hand, the velocity field is strongly influenced by the buoyancy forces and may display flow reversal phenomena. Two special cases are studied in detail: a duct with a sinusoidal wall temperature distribution; a duct subjected to an external convection heat transfer with two environments having different reference temperatures.     

A general analytical solution to the equation of transient forced convection with fully developed flow

A general solution to the energy equation under zero wall temperature or zero heat flux boundary condition for the decay of an inlet and initial temperature distribution of an incompressible transient turbulent flow heat transfer between two parallel plates is given. It is shown that these solutions may then be used to obtain solutions due to unit steps in wall temperature or wall heat flux which is sufficient to sort out prescribed wall temperature and prescribed wall heat flux boundary condition. The results are confirmed experimentally by the frequency method. An experimental apparatus has been designed, built and used for this purpose.     

Effect of variable viscosity on mixed convection boundary layer flow over a vertical surface embedded in a porous medium

The steady mixed convection boundary layer flow over a vertical impermeable surface embedded in a porous medium when the viscosity of the fluid varies inversely as a linear function of the temperature is studied. Both cases of assisting and opposing flows are considered. The transformed boundary layer equations are solved numerically by a finite difference method. Numerical results for the flow and heat transfer characteristics are obtained for various values of the mixed convection parameter ε and the variable viscosity parameter θ_e. It has been found that in the opposing flow case, dual solutions exist and boundary separation occurs.     

Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids

The steady mixed convection boundary layer flow past a vertical flat plate embedded in a porous medium filled with nanofluids is studied using different types of nanoparticles as Cu (cuprom), Al₂O₃ (aluminium) and TiO₂ (titanium). The model used for the nanofluid is the one which incorporates only the nanoparticle volume fraction parameter. The basic partial equations are reduced to an ordinary differential equation which is solved numerically for some values of the volume fraction and mixed convection parameters. It is shown that the solution has two branches in a certain range of the parameters. The effects of these parameters on the velocity distribution are presented graphically.     

Exact analytical solution of forced convection through a porous-saturated duct

In the current paper, the effect of a magnetic field on the fully developed forced convective flow and heat transfer is studied. An exact solution is extracted when the flow in the porous medium is governed by the Brinkman–Forchheimer Extended Darcy model. First, the problem formulation is explained to obtain a new system of mathematical formulation. Then, by utilizing the properties which are imposed into the problem, the exact closed-form analytical solution of the problem is explored. Finally, the main results are illustrated to show the impact of the porous media-shaped parameter, magnetic parameter, Forchheimer number, and viscosity ratio. It should be mentioned that the asymptotic results achieved in this study were compared with the exact results and it is found that they are in good agreement.