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.     

Unsteady MHD-boundary-layer of a source and vortex flow adjacent to a stationary surface

Summary The steady laminar incompressible flow of an electrically conducting fluid over an infinite permeable disk in the presence of an axial magnetic field has been investigated, and a self-similar solution of the boundary-layer equations is obtained numerically. For large values of the suction parameter, a closed form solution is obtained. Also, an asymptotic solution is found for large values of the independent variable. The surface-shear stresses in the radial and tangential directions and the surface heat transfer strongly depend on the suction parameter, the ratio of the source and vortex flow and the magnetic field except the surface heat transfer which weakly depends on the magnetic field. The similarity solution of the boundary-layer equations exists only when a certain minimum suction or magnetic field is applied. The results of the analytical solution are in good agreement with those of the numerical solution for the suction parameterf _w3.     

Unsteady MHD forced flow due to a point sink

Summary An analysis is performed to study the unsteady laminar incompressible boundary-layer flow of an electrically conducting fluid in a cone due to a point sink with an applied magnetic field. The unsteadiness in the flow is considered for two types of motion, viz. the motion arising due to the free stream velocity varying continuously with time and the transient motion occurring due to an impulsive change either in the strength of the point sink or in the wall temperature. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The magnetic field increases the skin friction but reduces heat transfer. The heat transfer and temperature field are strongly influenced by the viscous dissipation and Prandtl number. The velocity field is more affected at the early stage of the transient motion, caused by an impulsive change in the strength of the point sink, as compared to the temperature field. When the transient motion is caused by a sudden change in the wall temperature, both skin friction and heat transfer take more time to reach a new steady state. The transient nature of the flow and heat transfer is active for a short time in the case of suction and for a long time in the case of injection. The viscous dissipation prolongs the transient behavior of the flow.     

Resonance instabilities in the boundary-layer flow over a rotating-disk under the influence of a uniform magnetic field

Instabilities due to resonating waves in the three-dimensional incompressible viscous/inviscid rotating-disk boundary-layer flow are investigated numerically. The influence of a normal magnetic field on the resonances leading, respectively, to the direct spatial instability, the direct temporal instability and the absolute instability are worked out, in an attempt to determine the physically most significant one. It is found that the magnetic field has a stabilizing influence on all the resonance mechanisms outlined. However, the direct spatial-resonance-instability mechanism persists for low Reynolds numbers, when the flow is still in the laminar regime. Thus, attention should be directed to this specific instability mechanism, which offers a strong route to transition to turbulence by means of triggering the nonlinearity.     

MHD boundary-layer flow of a micropolar fluid past a wedge with variable wall temperature

Summary The steady laminar MHD boundary-layer flow past a wedge immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solved numerically using a finite-difference scheme known as the Keller-box method. Numerical results show that the micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics.     

Hydromagnetic flow and heat transfer on a continuously moving vertical surface

Summary A theoretical solution for hydromagnetic convection over a continuously moving vertical surface with uniform suction is obtained. A flow of this types represents a new class of boundary-layer flow at a surface of finite length. The solutions for the velocity and temperature profiles are obtained. It is observed that the velocity decreases considerably in the presence of a magnetic field, as compared to its absence.     

Hydromagnetic couette flow of an Oldroyd-B fluid in a rotating system

The motion of electrically conducting, Oldroyd-B and incompressible fluid between two infinitely extended non-conducting parallel plates under a uniform transverse magnetic field, fixed relative to the fluid has been considered. The lower plate is at rest and the upper plate is oscillating in its own plane. The governing partial differential equation of this problem, subject to boundary conditions are solved analytically. The expressions for the steady and unsteady velocity fields for the conducting Oldroyd-B fluid are obtained. The graphs are plotted for different values of dimensionless parameters of the problem and the analysis of the results showed that the flow field is appreciably influenced by the applied magnetic field, the rotation and the material parameters of the fluid.     

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.     

An effective numerical method for solving viscous-inviscid interaction problems

This paper presents a new numerical method to solve the equations of the asymptotic theory of separated flows. A number of measures was taken to ensure fast convergence of the iteration procedure, which is employed to treat the nonlinear terms in the governing equations. Firstly, we selected carefully the set of variables for which the nonlinear finite difference equations were formulated. Secondly, a Newton-Raphson strategy was applied to these equations. Thirdly, the calculations were facilitated by utilizing linear approximation of the boundary-layer equations when calculating the corresponding Jacobi matrix.The performance of the method is illustrated, using as an example, the problem of laminar two-dimensional boundary-layer separation in the flow of an incompressible fluid near a corner point of a rigid body contour. The solution of this problem is non-unique in a certain parameter range where two solution branches are possible.     

Effects of mass transfer on a resonance instability in the laminar boundary layer flow over a rotating-disk

In this paper the linear stability properties of the magnetohydrodynamic flow of an incompressible, viscous and electrically conducting fluid are investigated for the boundary-layer due to an infinite permeable rotating-disk. The fluid is subjected to an external magnetic field perpendicular to the disk. The interest lies also in finding out the effects of uniform suction or injection. In place of the traditional linear stability method, a theoretical approach is adopted here based on the high-Reynolds-number triple-deck theory. It is demonstrated that the nonstationary perturbations evolve in accordance with an eigenrelation analytically obtained.     

Asymptotic stability of linearly evolving non-stationary modes in a uniform hydromagnetic field over a rotating-disk flow with suction and injection

In this paper the linear stability properties of the magnetohydrodynamic flow of an incompressible, viscous and electrically conducting fluid are investigated for the boundary-layer due to an infinite permeable rotating-disk. The fluid is subjected to an external magnetic field perpendicular to the disk. The interest lies also in finding out the effects of uniform suction or injection. In place of the traditional linear stability method, a theoretical approach is adopted here based on the high-Reynolds-number triple-deck theory. It is demonstrated that the nonstationary perturbations evolve in accordance with an eigenrelation analytically obtained.     

Analytical solutions of hydromagnetic boundary-layer flow of a non-Newtonian power-law fluid past a continuously moving surface

Summary The boundary-layer flow of a power-law non–Newtonian fluid over a continuously moving surface in the presence of a magnetic field B(x) applied perpendicular to the surface has been investigated. An analytical solution is obtained and compared with the numerical solution of the resulting non linear ordinary differential equation. The effects of the Stewart number (N) and the power law-index (n) on the velocity profiles and the skin-friction are studied.     

磁流变液在圆筒间的粘塑性流动

磁流变液具有粘性和粘塑性流体特性.基于Navier-Stokes方程,分析了外加磁场对在两圆筒间隙间的磁流变液粘塑性流动的影响,得到了速度和流量表达式,为圆筒式磁流变器件的设计提供了理论依据.研究结果表明:速度沿磁场方向呈抛物线和直线分布;流量可由外加磁场连续调控.     

Capillary magnetic field flow fractionation and analysis of magnetic nanoparticles

This paper reports the purification and analysis of magnetic nanoparticles using capillary magnetic field flow fractionation, which utilizes an applied magnetic field oriented orthogonal to the capillary flow. To validate this approach as a separation method for nanometer-scale particles, samples of magnetic nanoparticles composed of either gamma-Fe2O3 (maghemite) or CoFe2O4 with average diameters ranging from 4 to 13 nm were prepared and characterized by transmission electron microscopy and SQUID magnetometry. Retention of the samples on the capillary was investigated as a function of solvent flow rate and the nanoparticle size and composition; the elution times of the nanoparticles are strongly dependent on their magnetic moments. We demonstrate the use of this method to separate a mixture of nanoparticles into size-monodisperse fractions. The magnetic moments of the particles are calculated based on analysis of the retention parameters and correlate with values obtained in separate SQUID magnetometry measurements.     

Generalized Crane flow induced by continuous surfaces stretching with arbitrary velocities

The boundary-layer flow induced by a permeable sheet stretching with general polynomial velocity distribution is considered. This generalizes the work of Kumaran and Ramanaiah (Acta Mech. 116: 229–233, 1996) who were the first to observe that a Crane-type solution exists for wall motion composed of arbitrary linear and quadratic stretching terms, as long as an appropriate lateral transpiration is applied. We solve explicitly the problem to an arbitrarily high degree of the polynomial stretching. This motivates the second part of our study which provides explicit boundary-layer solutions for arbitrary wall stretching, with suitable transpiration. These solutions describe generalized Crane flows whose reciprocal (dimensionless) thicknesses always coincide with the negative of their (dimensionless) entrainment velocities. The associated heat transfer problem is solved explicitly for arbitrary stretching when an appropriate surface temperature distribution is prescribed.     

Some recent developments in the theory of boundary layer instability

The stability of boundary-layer flow over a flat-plate is investigated after taking into account the effects of boundary-layer growth. A critical review and analysis of earlier work is presented, mainly for those works that use an inhomogeneous Orr-Sommerfeld equation and a solvability condition to obtain corrections for the growth-rate based on the quasi-parallel (qp) approximation. During the course of this review and analysis an important result is proven; viz., for a ray, the basic spatial periodicity of the disturbance wave is indimensional space. Thereafter the energy integral equation is invoked, and an optimal monitorable property is found that has the same growth-rate as given by the eigensolution of the associated homogeneous problem. This also leads to the optimal normalisation of the eigenfunction at different downstream stations along the plate. A surprising result found is that the past non-parallel results can be virtually totally reproduced based on theqp-approximation and using the present methodology. And, by using the present methodology for the full non-parallel problem, the results obtained are in very much better agreement with past experimental results, and with the results of Smith based on the triple-deck theory. The students and co-workers who have worked on this problem are Dr T K Vashist and Ms R Verma.     

Numerical investigation of two-dimensional turbulent boundary-layer compressible flow with adverse pressure gradient and heat and mass transfer

Summary The effect of heat and mass transfer on the steady turbulent compressible boundary-layer flow with adverse pressure gradient are numerically studied. The Reynolds-averaged boundary-layer equations and their boundary conditions are transformed, in a suitable form for numerical solution, by using the compressible version of the Falkner-Skan transformation and the resulting coupled and nonlinear system of partial differential equations is solved using the Keller's-box method and a modified version of it. For the eddy kinematic viscosity the model developed by Cebeci and Smith is employed whereas for the turbulent Prandlt number model a modification of the extended Kays and Crawford's model is used. Numerical calculations are carried out for the case of air, at about free stream temperature of 300°K, and for a linearly retarded flow, known as Howarth's flow when the porous limiting surface is adiabatic, heated or cooled. The porous surface is subjected to a continuous or localized suction/injection velocity and the influence of this velocity as well as of the free-stream Mach number and of the heat-transfer parameter on the turbulent boundary-layer and the separation point is examined. It is hoped that in the absence of detailed investigations into this problem, the obtained results, presented in the figures, are very interesting and give a clearer insight into the mechanism of controlling a turbulent boundary-layer compressible flow.     

On the spatial stability of free-convection flows in a saturated porous medium

We consider the free-convection boundary-layer flow in a saturated porous medium adjacent to an impermeable vertical surface. It is assumed that the surface is supplying heat to the porous medium in a prescribed way, which varies along the surface. The problem, which relates to the spatial stability of the known similarity solutions of the boundary-layer equations, is formulated and certain analytical results presented for special cases. For this special class of flows we are able to determine analytically the first eigenvalue for all relevant parameter values and thereby show that such flows are spatially stable.     

An eigenvalue study of a viscous flow separated by mass addition

Summary Eigenvalue problem methods, developed for boundary-layer flow are used to consider the spatial stability of the viscous flow past a flat plate which has been separated by mass addition at the surface. A study is made of the rate of approach of a slightly disturbed initial profile to the interaction similarity solution found by Kassoy [1] and Klemp and Acrivos [2]. It is shown that eigenfunctions generated in the separated viscous layer (free shear layer) propagate into the inviscid rotational layer adjacent to the wall. Thus by the inherent interaction process involved, these disturbances affect the external flow as well. The results indicate a relatively slow rate of decay when compared to an attached boundary-layer flow on an impermeable surface.     

A finite difference-Galerkin finite element solution of compressible laminar boundary-layer flows

A finite difference-Galerkinfinite element method is presented for the solution of the two-dimensional compressible laminar boundary-layer flow problem. The streamwise derivatives in the momentum and energy equations are approximated by finite differences. An iterative scheme, due to the non-linearity of the problem, in conjunction with the Galerkin finite element method is then proposed for the solution of the problem through the boundary-layer thickness. Numerical results are presented and these are compared with other numerical and analytical solutions in order to show the applicability and the effectiveness of the proposed formulation. In all the cases here examined, the results obtained attained the same accuracy of other numerical methods for a much smaller number of points in the boundary-layer.