Effect of uniform suction or injection on a magnetohydrodynamic boundary layer flow along a flat plate with pressure gradient

Summary The effect of uniform suction or injection on the flow of an incompressible electrically conducting fluid past a flat plate with pressure gradient in the presence of a transverse magnetic field was theoretically investigated. The boundary layer equations were transformed into non-similar ones, and the numerical calculations of the resulting equations were performed by the difference differential method. The velocity profiles, the coefficient of skin friction, and the displacement thickness were given for various values of the pressure gradient, the magnetic and suction/injection parameters. The neutral stability curves for wavelike disturbances of Tollmien-Schlichting type and the critical Reynolds numbers were then presented for the velocity profiles obtained above.With 13 Figures     

Influence of pressure gradient upon boundary layer stability and transition

Summary The present paper deals with the effect of adverse pressure gradient on boundary layer stability and transition in an incompressible fluid qualitatively as well as quantitatively. In order to produce various kinds of pressure gradients along a flat plate located horizontally, the cross sectional area of the working chamber is varied by the vertical side walls. Changes in velocity profiles and in intensity of streamwise velocity fluctuation in a boundary layer are measured in the transition region. The distribution of dynamic pressure at 0.43 mm from the surface is correlated with that of intensity of velocity fluctuation. Moreover, transition length is correlated with a shape parameter. Further, the process of transition is shown to be different from the case of zero pressure gradient when the half included angle is 3.6 degree. Finally, mention is made of the relation between transition Reynolds number and a characteristic frequency from the standpoint of linear stability theory.With 11 Figures     

Theoretical analysis on mixed convection boundary layer flow over a wedge with uniform suction or injection

Summary Forced and free mixed convection boundary layer flow over a wedge with uniform suction or injection is theoretically investigated. Nonsimilar partial differential equations are transformed into ordinary differential equations by means of difference-differential method. The solutions of the resulting equations are obtained in integral forms and are calculated by iterative numerical procedures. The results were given for velocity profiles, temperature profiles, friction and heat transfer parameters for various values of suction/injection parameter, pressure gradient parameter and buoyancy parameter.     

Effect of pressure gradient on MHD boundary layer over a flat plate

Summary The magnetohydrodynamic (MHD) boundary layer flow over a flat plate is examined here for two cases, viz. a uniform free-stream velocity and a uniform hydrostatic pressure. The nonlinear boundary layer equations are solved using a reliable finite-difference method. The boundary layer physical parameters such as skin-friction coefficient, displacement, momentum and energy thicknesses of the boundary layer are determined. It is found that the normal surface velocity gradient decreases with the local magnetic interaction parameter for the cases of a uniform hydrostatic pressure, whereas in the case of a uniform free-stream volocity it increases with the interaction parameter.     

The effect of a longitudinal positive pressure gradient on the critical injection parameter

On the basis of conclusions from the asymptotic theory of the boundary layer we have derived a theoretical relationship between the magnitude of the critical injection parameter and the positive longitudinal pressure gradient, as well as velocity profiles at the point of displacement for these conditions. Results of experimental work are in satisfactory agreement with theoretical data.     

Magnetohydrodynamic flow past a semi-infinite moving plate

Summary The nonlinear boundary layer equations of the title problem are solved numerically for obtaining the coefficient of skin-friction. It is found that the magnitude of the normal surface velocity gradient increases with the magnetic interaction parameter.     

Similarity analysis in magnetohydrodynamics: Hall effects on forced convective heat and mass transfer of non–Newtonian power law fluids past a semi-infinite vertical flat plate

Summary The forced convective heat and mass transfer along a semi-infinite vertical flat plate is investigated for non–Newtonian power law fluids in the presence of a strong nonuniform magnetic field, and the Hall currents are taken into account. The similarity solutions are obtained using transformations group theory. These are the only symmetry transformations admitted by the field equations. The application of one-parameter groups reduces the number of independent variables by one, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with the appropriate boundary conditions. Furthermore the similarity equations are solved numerically by using a fourth-order Runge-Kutta scheme with the shooting method. Numerical results for the velocity profiles, the temperature profiles and the concentration profiles are presented graphically for various values of the power-law viscosity index n, generalized Schmidt number Sc, generalized Prandtl number Pr, the magnetic parameter M and the Hall parameter m.     

Thermal radiation effects on magnetohydrodynamic mixed convection flow of a micropolar fluid past a continuously moving semi-infinite plate for high temperature differences

Summary An analysis is presented to study the effect of radiation on magnetohydrodynamic mixed convective steady laminar boundary layer flow of an optically thick electrically conducting viscous micropolar fluid past a moving semi-infinite vertical plate for high temperature differences. A uniform magnetic field is applied perpendicular to the moving plate. The density of the micropolar fluid is assumed to reduce exponentially with temperature. The usual Boussinesq approximation is neglected because of the high temperature differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting governing equations are transformed using a similarity transformation and then solved numerically by applying an efficient technique. The effects of radiation parameter R, magnetic parameter M, couple parameter Δ and density/temperature parameter n on the velocity, angular velocity and temperature profiles as well as the local skin friction coefficient, wall couple stress and the local Nusselt number are presented graphically and in tabular form.     

连续排水边界下考虑起始坡降的软黏土固结解

Terzaghi一维固结理论中边界条件和初始条件相矛盾的逻辑问题以及软黏土中水的渗流在低水力坡降下可忽略均已逐渐被认识。但在连续排水边界条件下能考虑起始水力坡降的黏性土一维固结理论,尤其是固结模型的解析解还鲜见报道。基于此,该文引入了连续排水边界,并同时考虑黏性土中渗流存在的起始水力坡降,建立了均质地基的单面排水一维固结模型。采用有限傅里叶变换方法求解了所建立固结模型的孔压、固结度以及沉降解析解答,分析了土层在恒载下同时考虑连续排水边界和起始水力坡降的一维固结性状。结果表明:考虑起始水力坡降后,连续排水界面参数b对固结性状的影响与达西定律下相同,即b值越大,排水面透水情况越好,孔压消散速率越快,固结完成时间越短;相反b值越小,排水面透水情况越差,孔压消散越慢,固结完成时间越长。连续排水条件下无量纲R(起始水力坡降、水的重度与土层厚度的乘积除以外荷载)对固结性状影响与完全透水边界下相比无明显改变。R值越大,渗流前锋到达土层底部的时间越长,固结完成时土中超静孔压残留值越大,土层按孔压定义的平均固结度越小,最终沉降量也越小。     

Nonaxisymmetric unsteady motion over a rotating disk in the presence of free convection and magnetic field

The nonaxisymmetric unsteady motion produced by a buoyancy-induced cross-flow of an electrically conducting fluid over an infinite rotating disk in a vertical plane and in the presence of an applied magnetic field normal to the disk has been studied. Both constant wall and constant heat flux conditions have been considered. It has been found that if the angular velocity of the disk and the applied magnetic field squared vary inversely as a linear function of time (i.e. as (1−λt*)⁻¹, the governing Navier-Stokes equation and the energy equation admit a locally self-similar solution. The resulting set of ordinary differential equations has been solved using a shooting method with a generalized Newton's correction procedure for guessed boundary conditions. It is observed that in a certain region near the disk the buoyancy induced cross-flow dominates the primary von Karman flow. The shear stresses induced by the cross-flow are found to be more than these of the primary flow and they increase with magnetic parameter or the parameter λ characterizing the unsteadiness. The velocity profiles in the x- and y-directions for the primary flow at any two values of the unsteady parameter λ cross each other towards the edge of the boundary layer. The heat transfer increases with the Prandtl number but reduces with the magnetic parameter.     

An effect of sudden circumferential strain on the laminar boundary layer on a rotating cylinder in an axial flow

Summary The laminar boundary layer which develops on a rotating thin cylinder fitted with an aft-section rotating with an angular velocity different from that of a fore-section is examined numerically. This problem concerns a relaxation process of the boundary layer subjected to a sudden circumferential rate of strain. Two methods are adopted: one is to regard the flow on the aft-cylinder as perturbations of that on the fore-section, and the other is to approximate the discontinuous change in the angular velocity by the cumulative normal distribution function. It is shown that the flow fields are largely influenced by the degree of favorable pressure gradient produced in the boundary layer. Especially, in case of flow passing onto the aft-section with a smaller angular velocity, adverse pressure gradient is induced immediately after the junction between two cylinders, and when the degree of the discontinuity is increased, flow separation can be provoken.With 11 Figures     

A singular perturbation problem of non-newtonian flow between porous disks

The non-Newtonian flow between parallel porous stationary disks due to uniform suction at the disks is considered for both small and large suction Reynolds numbers. In the case of small suction Reynolds number the Navier-Stokes equations have been solved by a regular perturbation technique. The solution obtained is valid for both suction and injection Reynolds numbers. The velocity, pressure and shear distributions have been obtained and are compared with those of the Newtonian flow. We find, in the case of injection, that the combined effect of cross viscosity and visco-elastic co-efficients is to increase the maximum velocity at the centre of the channel and to decrease the magnitude of the velocity gradient at the disks. Whereas in the case of suction, velocity profile is flatter with higher magnitude for the velocity gradients at the disks.In the case of large suction, the Navier-Stokes equations have been solved by the method of matched asymptotic expansions. We find that the effect of large suction at the disks is to flatten the velocity profiles considerably and thereby to push the boundary layer towards the disks. The combined effect of visco-elastic and cross-viscosity terms is to decrease the radial velocity and to increase the axial velocity distributions.     

MHD Boundary Layer Flow of a Power-Law Nanofluid Containing Gyrotactic Microorganisms Over an Exponentially Stretching Surface

This study focusses on the numerical investigations of boundary layer flow for magnetohydrodynamic (MHD) and a power-law nanofluid containing gyrotactic microorganisms on an exponentially stretching surface with zero nanoparticle mass flux and convective heating. The nonlinear system of the governing equations is transformed and solved by Runge-Kutta-Fehlberg method. The impacts of the transverse magnetic field, bioconvection parameters, Lewis number, nanofluid parameters, Prandtl number and power-law index on the velocity, temperature, nanoparticle volume fraction, density of motile microorganism profiles is explored. In addition, the impacts of these parameters on local skin-friction coefficient, local Nusselt, local Sherwood numbers and local density number of the motile microorganisms are discussed. The results reveal that the power law index is considered an important factor in this study. Due to neglecting the buoyancy force term, the bioconvection and nanofluid parameters have slight effects on the velocity profiles. The resultant Lorentz force, from increasing the magnetic field parameter, try to decrease the velocity profiles and increase the rescaled density of motile microorganisms, temperature and nanoparticle volume fraction profiles. Physically, an augmentation of power-law index drops the reduced local skin-friction and reduced Sherwood number, while it increases reduced Nusselt number and reduced local density number of motile microorganisms.     

A perturbation solution for the Laminar boundary layer on a continuous moving surface

Summary A perturbation solution is presented for the nonsimilar boundary layer flow past a moving surface with constant wall velocity. A zero pressure gradiant is assumed and the tangential velocity and stagnation enthalpy profiles are prescribed at the initial station. The eigen values obtained for the first order perturbation solution are integers and the eigen functions are the derivatives of the error function. The same Green's function is obtained for all the higher order perturbation equations, and the higher order perturbation solutions are given in integral form in terms of the Green's function. The analyses include both the momentum equation and the energy equation which is uncoupled with the momentum equation, and which is subjected to a nonsimilar velocity flow field.     

An experimental investigation of liquid metal flow past a cylinder in longitudinal magnetic field

The distribution of pressure on the surface of a cylinder is obtained as a result of experimental investigation; this distribution significantly varies with increasing magnetic induction compared to the flow past a cylinder in the absence of magnetic field. In so doing, the pressure drag coefficient of the cylinder significantly increases. The measurements of velocity profiles reveal that the extent of the region of stagnant flow before the cylinder (the so-called “leading” wake) increases with magnetic induction. The dependence of axial defect of velocity on the MHD interaction parameter is obtained; this dependence under conditions of flow in a strong magnetic field is unaffected by the shape of the body subjected to flow, as is confirmed by the results of experiment involving the flow past a plate.     

Free convection boundary layer flow with uniform suction or injection over a cone

Summary The effect of uniform suction or injection on free convection boundary layer over a cone was theoretically investigated. The non-linear ordinary differential equations were obtained by the difference-differential method after transforming it to an equivalent two-dimensional problem by Mangler's transformation. The solutions of the resulting equations can be expressed in the form of integral equations. Numerical calculations were performed solving the integral equations by the iterative numerical quadrature. The velocity profiles, temperature profiles, skin friction parameters and heat transfer parameters with constant wall temperature were computed for various values of suction/injection parameter and cone angle parameter.     

Buildup of particles on fibers in a high field-high gradient separator

In previous publications the performance of a high field-high gradient magnetic filter was calculated using a particle trajectory model to obtain the capture cross section of the fibers. The configuration of the particle buildup was assumed to be constant without considering the forces on the particle after impinging on the fiber. We now calculate the equilibrium configuration of particle buildup on the edge of a flat ribbon considering only the magnetic and viscous forces and including the effect of the fluid boundary layer. The configurations obtained as a function of field and flow are consistent with the few approximate direct observations reported in the literature. The performance of the filter, calculated using the particle trajectory model, now modified by this equilibrium buildup configuration, is compared to previously reported experimental results on separation of CuO from Al2O3slurries. A somewhat better fit is now obtained over the entire range of fluid velocity and fields, but an adjustable parameter affecting the viscous drag on the collected particles is required.     

Effects of suction-injection-combination (SIC) on the onset of Rayleigh-Benard magnetoconvection in a micropolar fluid

The effects of suction-injection-combination (SIC) and magnetic field on the linear stability analysis of Rayleigh-Benard convection in a horizontal layer of an Boussinesq micropolar fluid is studied using a Rayleigh-Ritz techinque. The eigenvalues are obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal and adiabatic temperature conditions on the spin-vanishing boundaries. The eigenvalues are also obtained for lower rigid isothermal and upper free adiabatic boundaries with vanishing spin. The influence of various micropolar fluid parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pe-sensitivity is found in respect of the critical wave number. It is observed that the micropolar fluid layer heated from below is more stable compared to the classical fluid layer.     

MHD compressible turbulent boundary-layer flow with adverse pressure gradient

Summary The effects of the magnetic field and localized suction on the steady turbulent compressible boundary-layer flow with adverse pressure gradient are numerically studied. The magnetic field is constant and applied transversely to the direction of the flow (global or local). The fluid flow is subjected to a constant velocity of localized suction, and there is no heat transfer between the fluid and the plate (adiabatic plate). The Reynolds-Averaged Boundary-Layer (RABL) equations and their boundary conditions are transformed using the compressible Falkner-Skan transformation. The resulting coupled and nonlinear system of PDEs is solved using the Keller’s box method. For the eddy-kinematic viscosity the turbulent models of Cebeci-Smith and Baldwin-Lomax are employed. For the turbulent Prandtl number the extended Kays-Crawford’s model is used. The flow is subjected to an adverse pressure gradient. The obtained results show that the flow field can be controlled by the applied magnetic field as well as by localized suction.     

Generalised vortex motion of compressible fluid with and without magnetic field and suction

The laminar boundary layer over a stationary infinite disk induced by a rotating compressible fluid is considered. The free stream velocity has been taken as tangential and varies as a power of radius, i.e. v_∞ ˜ r⁻ⁿ. The effect of the axial magnetic field and suction is also included in the analysis. An implicit finite difference scheme is employed to the governing similarity equations for numerical computations. Solutions are studied for various values of disk to fluid temperature ratio and for values of n between 1 and −1. In the absence of the magnetic field and suction, velocity profiles exhibit oscillations. It has been observed that for a hot disk in the presence of a magnetic field the boundary layer solutions decay algebraically instead of decaying exponentially. In the absence of the magnetic field and suction, the solution of the similarity equations exists only for a certain range of n.