The effect of a nonuniform basic temperature gradient on the convective instability of a fluid-saturated porous layer with general velocity and thermal conditions

Summary The effects of a nonuniform basic temperature gradient and bounding permeable walls on Rayleigh-Bénard convection in a sparsely packed porous medium are investigated using the single term Galerkin method. The nature of the boundaries dictates general boundary conditions on velocity and temperature. The classical results of free-free, rigid-free and rigid-rigid boundaries with isothermal or adiabatic boundaries are recovered as limiting cases of the present study. The multi-layer porous media problem has practical implications in situations where fluids are used as a working medium and porous media are used as dampers.     

Influence of boundary conditions on the cooling effect of crushed-rock embankment in permafrost regions of Qinghai–Tibetan Plateau

The crushed-rock layer is a highly porous medium that has been used to ensure the stability of embankment in permafrost regions. At present, depending on different boundary conditions (impermeable and permeable) of crushed-rock layer in embankment, the crushed-rock embankments are divided into two kinds of structures in the construction of Qinghai–Tibetan railway in China. One is a closed-boundary crushed-rock embankment; the other is an open-boundary crushed-rock embankment. In order to investigate the influence of boundary conditions (impermeable and permeable) on the cooling effect of a crushed-rock embankment, two numerical models of the unsteady two-dimensional hydrokinetic equations for incompressible fluid are presented to analyze the velocity and temperature characteristics of crushed-rock embankment with different embankment heights under impermeable and permeable boundary conditions for a period of 50 years. The results indicate: (1) the boundary conditions (impermeable and permeable) of crushed-rock embankment can have a very large impact on the heat transfer pattern within it in windy permafrost regions of Qinghai–Tibetan Plateau. The cooling effect of the closed crushed-rock embankment mainly relies on natural convection within crushed-rock layer, which is caused by the thermal boundary condition, but the cooling effect of the open crushed-rock embankment is due to the heat transfer enhancement because of internal forced convection induced by the external low temperature air flow (wind); (2) from the temperature distributions of crushed-rock embankments, it can be found that, under the assumption that the air temperature will be warmed up by 2.6 °C in a period of 50 years and in the areas where the mean annual air temperature is − 4.0 °C, when embankment is low, the cooling effects of crushed-rock embankment have no obvious difference under the two boundary conditions, and the cooling effect of closed crushed-rock embankment is only a little better than that of open one; however, when embankment is high, the boundary conditions cause a distinct influence on the temperature distribution of crushed-rock embankment, and the cooling effect under the permeable boundary condition is far better than that under the impermeable boundary condition. However, the asymmetric temperature distribution problem of the high crushed-rock embankment, caused by permeable boundary and external wind, must be considered when it is designed and constructed.     

Heat transfer in boundary layer flow of a micropolar fluid past a curved surface with suction and injection

Van Dyke's singular perturbation technique has been used to study the heat transfer in the flow of a micropolar fluid past a curved surface with suction and injection. The conditions for similar solutions of the thermal boundary layer equations have been obtained. In addition to the usual “no slip” condition for velocity, the two types of boundary conditions used for microrotation are: (i) no relative spin on the boundary; (ii) the anti-symmetric part of the stress tensor vanishes at the boundary. The effect of suction or injection on velocity, microrotation, temperature, skin friction coefficient, wall couple stress coefficient, displacement and momentum thicknesses, rate of heat transfer and adiabatic wall temperature have been studied. It is observed that with the increase of injection velocity, the thickness of the boundary layer is increased and the local drag is reduced. A comparison with the results obtained for a Newtonian fluid reveals that the microelements present in the fluid reduce the velocity and frictional drag, and cool the boundary.     

Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption

The problem of unsteady, two-dimensional, laminar, boundary-layer flow of a viscous, incompressible, electrically conducting and heat-absorbing fluid along a semi-infinite vertical permeable moving plate in the presence of a uniform transverse magnetic field and thermal and concentration buoyancy effects is considered. The plate is assumed to move with a constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. Time-dependent wall suction is assumed to occur at the permeable surface. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The obtained analytical results reduce to previously published results on a special case of the problem. Numerical evaluation of the analytical results is performed and some graphical results for the velocity, temperature and concentration profiles within the boundary layer and tabulated results for the skin-friction coefficient, Nusselt number and the Sherwood number are presented and discussed.     

Asymptotic analysis of natural convection through horizontal porous layer

Two-dimensional cellular convection in a fluid saturated horizontal porous layer is studied using a simple asymptotic analysis. Based on the physical grounds, supported by experiments, the entire region of study is divided into no-slip regime governed by the Brinkinan equations and slip regime governed by the Darcy equations. Self-consistent solutions for velocity and temperature are obtained in a closed form using the technique of matched asymptotic expansion. The transition from the no-slip regime to the slip regime is followed exactly and the maximum value of velocity is obtained. It is shown that the point of maximum velocity gets close to the bounding walls as the permeability parameter decreases. This exhibits the boundary layer behaviour. It is: also shown that permeability has negligible effect on the temperature distribution. Close agreement between analytical and numerical results is observed.     

Numerical study of transient 2-D compressible flow with heat and mass transfer using the finite volume method

In this paper, we used a pressure-based finite volume method to investigate the problem of transient 2-D compressible flow with heat and mass transfer in a rectangular domain. We have used this method to solve the governing equations with given initial and wall slip boundary conditions. We implemented the SIMPLE-TS algorithm in order to compute the numerical solutions for the flow variables, viz., velocity, pressure, temperature, concentration, density. The variation of density of the fluid along the horizontal and vertical line through geometric center of the domain has been studied. The transient solutions of temperature and concentration indicate that, the transient flow though dominates initially, it finally settles down to steady states solutions after elapse of some time. Nusselt number and Sherwood numbers were used to predict the behavior of heat transfer and mass transfer, respectively, at the center line of the rectangular domain.     

Flow with convective acceleration through a porous medium

Summary The flow streaming into a porous and permeable medium with arbitrary but smooth wall surface is considered on the basis of the Euler equation (in the pure fluid region) and a generalized Darcy's law in which the convective acceleration is taken into account. The asymptotic behavior of the flow for small permeability of the medium is investigated. It is shown that the flow in the porous medium is irrotational except in the boundary layer next to the surface. The velocity distribution in the boundary layer is given in a universal form. Proper boundary conditions connecting the potential flow in the pure fluid region and the potential flow in the porous medium are obtained when the boundary layer is neglected.On leave from Department of Aeronautical Engineering, Kyoto University, Kyoto, Japan.     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

Nonsimilar second order viscoelastic boundary layer flow at a stagnation point over a moving wall

An analysis is presented to investigate the fluid dynamic characteristics of a steady, laminar second order viscoelastic boundary layer flow at a two-dimensional stagnation point over a moving wall. The governing boundary layer equations have been solved by means of a series solution approach. Numerical solutions for the series functions have been given in tabular form. The development of the velocity distribution has been illustrated for several positive and negative values of the wall velocity. The values of the Weissenberg numbers ranged from 0 to 0.3.     

BEM simulations of potential flow with viscous effects as applied to a rising bubble

A model for the unsteady rise and deformation of non-oscillating bubbles under buoyancy force at high Reynolds numbers has been implemented using a boundary element method. Results such as the evolution of the bubble shape, variations of the transient velocity with rise height and the terminal velocity for different size bubbles have been compared to recent experimental data in clean water and to numerical solutions of the unsteady Navier–Stokes equation. The aim is to capture the essential physical ingredients that couple bubble deformation and the transient approach towards terminal velocity. This model requires very modest computational resources and yet has the flexibility to be extended to more general applications.     

Nonsimilar axisymmetric stagnation flow on a moving cylinder

An analysis is presented to investigate the fluid dynamic characteristics of an axisymmetric stagnation flow on a moving circular cylinder. The governing boundary layer equations have been solved numerically. The range of Reynolds numbers considered was from 0.01 to 100. The development of the velocity distribution has been illustrated for various positive and negative values of the wall velocity.     

Three-dimensional stagnation point flow of a second grade fluid towards a moving plate

The problem dealing with the steady three-dimensional flow of a second grade fluid near the stagnation point of an infinite plate moving parallel to itself with constant velocity has been investigated. By using the appropriate transformations for the velocity components and temperature, the basic equations governing flow and heat transfer have been reduced to a set of ordinary differential equations. These equations have been solved approximately subject to the relevant boundary conditions by employing a numerical technique. The effect of a nondimensional elastic parameter on the velocity components, wall shear stress, temperature and heat transfer has been examined carefully.     

饱和多孔介质一维瞬态波动问题的解析分析

采用基于混合物理论的多孔介质模型,提出了饱和多孔介质一维动力响应的初边值问题。利用拉氏变换和卷积定理,分别得到了边界自由排水时在任意应力边界条件和任意位移边界条件下瞬态波动过程的解析表达。几种典型的数值算例同时给出了两类边界条件下瞬态波动过程中多孔固体的位移场、应力场和孔隙流体的速度场、压力场。结果表明,饱和多孔介质的波动过程是多孔固体和孔隙流体中以同一速度传播的两种波动的耦合过程,时效特性分析也揭示了饱和多孔介质固有的表观粘弹性性质。     

Free convection in the laminar boundary layer flow of a thermomicropolar fluid past a vertical flat plate with suction/injection

Summary The effect of suction/injection in the laminar free convection flow of a thermomicropolar fluid past a nonuniformly heated vertical flat plate has been considered. The conditions under which similarity exists have been examined. The resulting system of non-linear ordinary differential equations has been solved numerically after transforming the infinite domain of boundary layer coordinate into a finite domain. The effects of variation of the boundary condition parameter and suction/injection parameter on the velocity, microrotation and temperature fields and the heat transfer coefficient have been studied graphically. The skin-friction parameter and the gradient of microrotation on the wall have been tabulated. It is found that there is significant increase in velocity, skin-friction and the heat transfer coefficient with the decreasing concentration of microelements.With 4 Figures     

Flow and heat transfer in a micropolar fluid past a flat plate with suction and heat sources

The laminar boundary layer flow of a micropolar fluid past a flat plate subject to uniform suction has been examined. The heat transfer study has been made in the presence of both constant as well as temperature dependent heat sources. The governing equations of momentum, first stress momentum and energy have been solved using numerical integration and Gauss-Seidel iterative procedure. For a particular value of suction parameter λ, as compared to the Newtonian fluid, the velocity decreases on increasing the value of micropolar parameter R. The skin friction and the Nusselt number have been calculated and presented in tables.     

A time-dependent incompressible viscous BEM for moderate Reynolds numbers

The boundary element method is applied to transient viscous incompressible flow. The time-domain formulation allows a boundary-only solution for linear Stokes flow. For higher speed flows in which the non-linear convective effects cannot be ignored, a volume integral must be retained. However, the introduction of reference velocities often limits the non-linear region to the vicinity of obstacles or bounding surfaces. Additionally, the volume terms are rewritten to eliminate the need for the calculation of velocity gradients. A general purpose numerical implementation of this new formulation then produces a very attractive tool for engineering analysis. This implementation includes a Newton-Raphson algorithm, permitting accurate solutions up to the moderate Reynolds number range. Several numerical examples are provided to validate the present approach.     

Momentum and heat transfer from a continuous moving surface to a power-law fluid

Summary Momentum and heat transfer from a continuous moving surface with an arbitrary surface velocity distribution and uniform surface temperature in a power-law fluid have been considered. Using a coordinate transformation, the boundary layer equations are reduced to a simple form. Modified Merk's series method has been used for momentum equation and universal function approach for energy equation. The resulting equations have been integrated numerically by using fourth-order Runge-Kutta method and method of continuation. Two types of plate velocity distributions are considered: (i) surface velocity proportional to positive power of distance from the slot, (ii) linearly stretched velocity distribution with nonzero slot velocity. It is found that the displacement thickness is much thicker for pseudoplastic fluids than for Newtonian and dilatant fluids for both cases. The local Nusselt number, obtained by the universal function method, has been compared with non-similar results. The results are in good agreement.     

A spin-vorticity relation for unidirectional plane flows of micropolar fluids

The field equations of micropolar fluid theory are applied to consider transient unidirectional plane flows. The relative angular velocity is introduced and maximum and minimum principles of parabolic partial differential equations are utilized to establish a spin-vorticity relation which is valid for very general boundary and initial conditions on spin.     

Free convection boundary layer flow of a micropolar fluid past slender bodies

The transverse curvature effects on axisymmetric free convection boundary layer flow of a micropolar fluid past vertical cylinders are investigated using the theory of micropolar fluids formulated by Eringen. The governing equations for momentum, angular momentum and energy have been solved numerically. Missing values of the velocity, angular velocity and thermal functions are tabulated for a wide range of the material parameters, transverse curvature parameter and Prandtl number of the fluid. A comparison has been made with the corresponding results for Newtonian fluids. Micropolar fluids display drag reduction and reduced surface heat transfer rate as compared with Newtonian fluids.     

A Fractal-Fractional Model for the MHD Flow of Casson Fluid in a Channel

An emerging definition of the fractal-fractional operator has been used in this study for the modeling of Casson fluid flow. The magnetohydrodynamics flow of Casson fluid has cogent in a channel where the motion of the upper plate generates the flow while the lower plate is at a static position. The proposed model is non-dimensionalized using the Pi-Buckingham theorem to reduce the complexity in solving the model and computation time. The non-dimensional fractal-fractional model with the power-law kernel has been solved through the Laplace transform technique. The Mathcad software has been used for illustration of the influence of various parameters, i.e., Hartman number, fractal, fractional, and Casson fluid parameters on the velocity of fluid flow. Through graphs and tables, the results have been implemented and it is shown that the boundary conditions are fully satisfied. The results reveal that the flow velocity is decreasing with the increasing values of the Hartman number and is increasing with the increasing values of the Casson fluid parameter. The findings of the fractal-fractional model have elucidated that the memory effect of the flow model has higher quality than the simple fractional and classical models. Furthermore, to show the validity of the obtained closed-form solutions, special cases have been obtained which are in agreement with the already published solutions.