Oberbeck convection flow of a couple stress fluid through a vertical porous stratum

The flow and heat transfer characteristics of Oberbeck convection of a couple stress fluid in a vertical porous stratum is investigated. The perturbation method of solution is obtained in terms of buoyancy parameter N valid for small values of N. This limitation is relaxed through numerical solutions using the finite difference technique with an error of 0.1×10^-7. The effect of increase in the values of temperature difference between the plates, permeability parameter and couple stress parameter on velocity, temperature, mass flow rate, skin friction and rate of heat transfer are reported. A new achievement is explored to analyse the flow for strong, weak and comparable porosity with the couple stress parameter. It is noted that both the porous parameter and the couple stress parameter suppress the flow. Higher-temperature difference is required to achieve the mass flow rate equivalent to that of viscous flow.     

非稳态分数阶Oldroyd-B流体绕楔形体流动研究

研究了非稳态分数阶Oldroyd-B流体在多孔介质中通过楔形拉伸板的驻点流动问题。基于分数阶Oldroyd-B流体的本构模型建立了动量方程,并在其中引入了浮升力和驻点流动特征。此外,考虑了具有热松弛延迟时间的修正的分数阶Fourier定律,并将其应用于能量方程和对流换热边界条件。接着,采用与L1算法相结合的有限差分法求解控制偏微分方程。最后,分析了相关物理参数对流动的影响。结果表明,随着楔角参数的增加,流体受到的浮升力增大,导致速度加快;达西数越大,介质的孔隙度变大,流体的流动越快;此外,温度分布先略有上升后明显下降,这表明Oldroyd-B流体具有热延迟特性。     

Magnetoconvection in a vertical channel with heat source or sink

The problem of steady, laminar mixed convective flow and heat transfer of an electrically conducting fluid through a vertical channel with heat source or sink is analyzed. The effects of viscous and Ohmic dissipations are included in the energy equation. Both walls are kept either at the same or different temperatures such as isothermal-isothermal, isoflux-isothermal and isothermal-isoflux conditions. Analytical solutions are found using regular perturbation technique and numerical solutions are found using finite difference method. A selected set of graphical results illustrating the effects of various parameters involved in the problem on the flow as well as flow reversal situation and Nusselt numbers are presented and discussed. It is also found that both the analytical and numerical solutions agree very well for small values of the perturbation parameter.     

Numerical solution of oscillatory flow of Maxwell fluid in a rectangular straight duct

A numerical analysis is presented for the oscillatory flow of Maxwell fluid in a rectangular straight duct subjected to a simple harmonic periodic pressure gradient.The numerical solutions are obtained by a finite difference scheme method. The stability of this finite difference scheme method is discussed. The distributions of the velocity and phase difference are given numerically and graphically. The effects of the Reynolds number, relaxation time, and aspect ratio of the cross section on the oscillatory flow are investigated. The results show that when the relaxation time of the Maxwell model and the Reynolds number increase, the resonance phenomena for the distributions of the velocity and phase difference enhance.     

Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux

 The effect of uniform suction on the steady two-dimensional laminar forced flow of a viscous incompressible fluid of temperature dependent viscosity past a wedge with uniform surface heat flux is considered. The governing equations for the flow are obtained by using suitable transformations and are solved by using an implicit finite difference method. Perturbation solutions are also obtained near the leading edge and in the downstream regime. The results are obtained in terms of the local skin friction coefficient and the rate of heat transfer for various values of the pertinent parameters, such as the Prandtl number, Pr, the velocity gradient parameter, m, the local suction parameter, ξ, and the viscosity variation parameter, ɛ. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The effect of ξ, m and ɛ on the dimensionless velocity profiles and viscosity distribution are also presented graphically for Pr = 0.7 and 7.0, which are the appropriate values for gases and water respectively. Received on 22 July 1999     

A numerical model for thermal systems with helically-coiled tubes and its experimental verification

 A conjugate numerical model proposed by Nakayama et al. for the steady problem of cooling a fluid flowing through a coiled tube, has been successfully extended to investigate two distinctive thermal problems, namely, the transient cooling processes associated with a beer dispenser, and the transient processes of heat storage and recovery associated with a packed bed saturated with a molten salt. An axisymmetric numerical procedure is adopted for determining the velocity and temperature fields within the chilled water bath of the beer dispenser. A simplified one-dimensional heat transfer model is introduced for coupling the tube flow with the recirculating flow in the bath. A similar axisymmetric finite difference procedure is applied for the heat transfer analysis of the packed bed saturated with a molten salt. For the heat recovery process, a one-dimensional heat balance equation for the two-phase flow with a helically-coiled tube is introduced to update the wall surface temperatures, which are needed to calculate the temperature field in the saturated packed bed. The numerical results for both thermal systems associated with coiled tubes agree very well with the corresponding velocity and temperature data obtained from the experiments. Received on 28 August 2000 / Published online: 29 November 2001     

Non-linear Oberbeck-electroconvection in a poorly conducting fluid through a vertical channel in the presence of an electric field

Non-linear Oberbeck-electroconvection (OBEC) in a poorly electrically conducting fluid through a vertical channel, when the walls are held at different temperatures with temperature difference perpendicular to gravity, is studied using the modified Navier stokes equation in the presence of both induced and an applied electric field. Both analytical and numerical solutions for the non-linear coupled equations governing the motion are obtained and found that analytical solutions agree well with numerical solutions for values of the buoyancy parameter N<1. It is shown that OBEC can be controlled by maintaining the temperature difference either in the same direction or opposing the potential difference with a suitable value of electric number W. The effect of W on velocity, temperature, rate of heat transfer, skin friction and mass flow rate are computed and the results are depicted graphically. We found that analytical results agree well with numerical results for small values of N. We also found that an increase in W accelerates the flow and hence increases linearly the skin friction and mass flow rate.     

Effect of ion slip on the time-varying Hartmann flow of a non-Newtonian viscoelastic fluid with heat transfer

Ion slip in a time-varying Hartmann flow of a conducting incompressible non-Newtonian viscoelastic fluid between two parallel horizontal insulating porous plates is studied with allowance for heat transfer. A uniform and constant pressure gradient is applied in the axial direction. An external uniform magnetic field and uniform suction and injection through the surface of the plates are applied in the normal direction. The two plates are maintained at different but constant temperatures; the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained with the use of finite differences, and the effect of various physical parameters on both the velocity and temperature fields is discussed.     

A coupled solution of the vorticity–velocity formulation of the incompressible Navier–Stokes equations

A relatively novel formulation of the Navier-Stokes equations is used for obtaining solutions of two dimensional incompressible fluid flow and convective heat transfer problems. A vorticity transport equation along with two Poisson equations for the velocity components and the energy equation are solved by a finite difference scheme. A coupled solution procedure is used for solving simultaneously the dependent variables along a line, using a block tridiagonal matrix algorithm. The formulation is found to be stable and has features that may be desirable for solving a wide variety of flow and heat transfer problems.     

Stability analysis of pipes conveying fluid with fractional viscoelastic model

Divergence and flutter instabilities of pipes conveying fluid with fractional viscoelastic model has been investigated in the present work. Attention is concentrated on the boundaries of the stability. Based on the Euler–Bernoulli beam theory for structural dynamics, viscoelastic fractional model for damping and, plug flow model for fluid flow, equation of motion has been derived. The effects of gravity, and distributed follower forces are also considered. By transferring the equation of motion to the Laplace domain and using the Galerkin method, the characteristic equations are obtained. By solving the eigenvalue problem, frequencies and dampings of the system have been obtained versus flow velocity. Some numerical test cases have been studied with viscoelastic fractional model and the effect of the fractional derivative order and the retardation time is investigated for various boundary conditions.

     

Laminar natural convection in a fully partitioned enclosure containing fluid with nonlinear thermophysical properties

The effects of a heat conducting partition on the laminar natural convection heat transfer and fluid flow were obtained by comparing the numerical and experimental results for a cubic enclosure without and with a partition. The two opposite vertical walls of the enclosure were isothermal at different temperatures. The working fluid was glycerol. The complete vertical partition, made of Plexiglass, was positioned in the middle of the enclosure. The visualizations of the velocity and temperature fields were obtained by using respectively, Plexiglass and liquid crystal particles as tracers. A middle plane perpendicular to the partition was numerically modeled. The steady two-dimensional model accounted for the variable thermophysical properties of the fluid. The finite volume method based on the finite difference approach was applied. The convective terms were approximated using a deferred correction central difference scheme. The velocity and temperature fields and the distribution of the local and average Nusselt numbers were found as a function of the Rayleigh (38 000 <Ra <369 000) and Prandtl (2700 < Pr < 7000) numbers.     

Numerical and experimental study of driven flow in a polar cavity

A numerical and an experimental study of the flow of an incompressible fluid in a polar cavity is presented. The experiments included flow visualization, in two perpendicular planes, and quantitative measurements of the velocity field by a laser Doppler anemometer. Measurements were done for two ranges of Reynolds numbers; about 60 and about 350. The stream function-vorticity form of the governing equations was approximated by upwind or central finite-differences. Both types of finite-difference approximations were solved by a multi-grid method. Numerical solutions were computed on a sequence of grids and the relative accuracy of the solutions was studied. Our most accurate numerical solutions had an estimated error of 0.1 per cent and 1 per cent for Re = 60 and Re = 350, respectively. It was also noted that the solution to the second order finite difference equations was more accurate, compared to the solution to the first order equations, only if fine enough meshes were used. The possibility of using extrapolations to improve accuracy was also considered. Extrapolated solutions were found to be valid only if solutions computed on fine enough meshes were used. The numerical and the experimental results were found to be in very good agreement.     

Transient forced convection heat transfer in a channel

A numerical analysis is made of incompressible transient turbulent flow heat transfer between two parallel plates when there is a step jump in space along the channel in wall heat flux or wall temperature. The variation of the fluid velocity and effective diffusivity over the channel cross section are accounted for. The fluid is assumed to have a fully-developed turbulent velocity profile throughout the length of the channel. The thermal responses of the system are obtained by solving energy equation for air by a digital computer. The results are presented in graphical forms. The stability of the finite difference solution is studied and condition for the stability of the difference solution is derived. A method is given to obtain velocity distributions from the distribution of turbulent eddy diffusivity of momentum. Variations of Nusselt numbers are obtained as a function of time and space. Steady-state values are also given and compared with the published results.     

Coupling model for unsteady MHD flow of generalized Maxwell fluid with radiation thermal transform

This paper introduces a new model for the Fourier law of heat conduction with the time-fractional order to the generalized Maxwell fluid. The flow is influenced by magnetic field, radiation heat, and heat source. A fractional calculus approach is used to establish the constitutive relationship coupling model of a viscoelastic fluid. We use the Laplace transform and solve ordinary differential equations with a matrix form to obtain the velocity and temperature in the Laplace domain. To obtain solutions from the Laplace space back to the original space, the numerical inversion of the Laplace transform is used. According to the results and graphs, a new theory can be constructed. Comparisons of the associated parameters and the corresponding flow and heat transfer characteristics are presented and analyzed in detail.     

Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

The velocity field and the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. Initially the fluid is at rest, and at time t = 0⁺, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary second grade and Newtonian fluids are obtained as limiting cases of the general solutions. Finally, some characteristics of the motion, as well as the influences of the material and fractional parameters on the fluid motion and a comparison between models, are underlined by graphical illustrations.     

Ion slip effect on a steady flow through a circular pipe of a dusty conducting Oldroyd 8-constant fluid

In this paper, a steady magnetohydrodynamic (MHD) flow of a dusty incompressible electrically conducting Oldroyd 8-constant fluid through a circular pipe is examined with considering the ion slip effect. A constant pressure gradient in the axial direction and an external uniform magnetic field in the perpendicular direction are applied. A numerical solution is obtained for the governing nonlinear momentum equations by using finite differences. The effect of the ion slip, the non-Newtonian fluid characteristics, and the particle-phase viscosity on the velocity, volumetric flow rates, and skin friction coefficients of both the fluid and particle phases is reported.     

Exact solutions for the unsteady rotational flow of an Oldroyd-B fluid with fractional derivatives induced by a circular cylinder

In this research article, the unsteady rotational flow of an Oldroyd-B fluid with fractional derivative model through an infinite circular cylinder is studied by means of the finite Hankel and Laplace transforms. The motion is produced by the cylinder, that after time t=0⁺, begins to rotate about its axis with an angular velocity Ωt ^p . The solutions that have been obtained, presented under series form in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The corresponding solutions that have been obtained can be easily particularized to give the similar solutions for Maxwell and Second grade fluids with fractional derivatives and for ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian fluids) performing the same motion, are obtained as limiting cases of general solutions. The most important things regarding this paper to mention are that (1) we extracted the expressions for the velocity field and the shear stress corresponding to the motion of Second grade fluid with fractional derivatives as a limiting case of our general solutions corresponding to the Oldroyd-B fluid with fractional derivatives, this is not previously done in the literature to the best of our knowledge, and (2) the expressions for the velocity field and the shear stress are in the most simplified form, and the point worth mentioning is that these expressions are free from convolution product and the integral of the product of the generalized G-functions. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.     

Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium

The melting phenomenon in two-dimensional (2D) flow of fourth-grade material over a stretching surface is explored. The flow is created via a stretching surface. A Darcy-Forchheimer (D-F) porous medium is considered in the flow field. The heat transport is examined with the existence of the Cattaneo-Christov (C-C) heat flux. The fourth-grade material is electrically conducting subject to an applied magnetic field. The governing partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by appropriate transformations. The solutions are constructed analytically through the optimal homotopy analysis method (OHAM). The fluid velocity, temperature, and skin friction are examined under the effects of various involved parameters. The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter, porosity parameter, and Forchheimer number. The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number, magnetic parameter, and thermal relaxation parameter. Furthermore, the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters, thermal relaxation parameter, and Forchheimer number.

     

Effect of duct velocity profile and buoyancy‐induced flow on efficiency of transient hydrodynamic removal of a contaminant from a cavity

This paper describes a preliminary numerical analysis of the effect of duct velocity profile and buoyancy‐induced flow generated by the heat source on hydrodynamic removal of contaminants contained in cavities. The process of fluid renewal in a cavity is modelled via a numerical solution of the Navier–Stokes equations coupled with the energy equation for transient flows. The foulant has the same density as the fluid in the duct and the duct velocity profile is considered to be Poiseuille flow and Couette flow, respectively. The results show that the change in Grashof number and duct flow velocity profile causes a dramatic difference in the observed flow patterns and cleaning efficiency. From a cleaning perspective, the results suggest that Couette flow at higher value of Grashof number becomes more effective in further purging of contaminated fluid from a cavity. Copyright © 2004 John Wiley & Sons, Ltd.