Three-dimensional channel flow of second grade fluid in rotating frame

An analysis is performed for the hydromagnetic second grade fluid flow between two horizontal plates in a rotating system in the presence of a magnetic field.The lower sheet is considered to be a stret...     

Hydromagnetic flow in a rotating channel

The steady flow in a parallel plate channel rotating with an angular velocity Ω and subjected to a constant transverse magnetic field is analysed. An exact solution of the governing equations is obtained. The solution in the dimensionless form contains two parameters: the Hartmann number, M ², and K ² which is the reciprocal of the Ekman number. The effects of these parameters on the velocity and magnetic field distributions are studied. For large values of the parameters, there arise thin boundary layers on the walls of the channel.     

MHD flow of a third-grade fluid due to eccentric rotations of a porous disk and a fluid at infinity

The problem of magnetohydrodynamics (MHD) flow of a conducting, incompressible third-grade fluid due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field is considered. An exact analysis is carried out to model the governing non-linear partial differential equation. A numerical solution of the third-order non-linear partial differential equation has been obtained. Several graphs and tables have been drawn to show the influence of porosity ε, magnetic parameter N, material parameters α and β on the velocity distribution.     

MHD mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface: a comprehensive report of dual solutions

The steady laminar magnetohydrodynamic mixed convection boundary layer flow of a nanofluid near the stagnation-point on a vertical permeable plate with prescribed external flow and surface temperature is investigated in this study. Here, both assisting and opposing flows are considered and studied. Using appropriate similarity variables, the governing equations are transformed into nonlinear ordinary differential equations in the dimensionless stream function, which is solved numerically using the Runge–Kutta scheme coupled with a conventional shooting procedure. Three different types of nanoparticles, namely copper Cu, alumina Al₂O₃ and titania TiO₂ with water as the base fluid are considered. Numerical results are obtained for the skin-friction coefficient and Nusselt number as well as for the velocity and temperature profiles for some values of the governing parameters, namely, the volume fraction of nanoparticles ?, permeability parameter f _o , magnetic parameter M and mixed convection parameter λ. It is found that dual solutions exist for both assisting and opposing flows, and the range of the mixed convection parameter for which the solution exists, increases with suction, magnetic field and volume fraction of nanoparticles.     

Similarity analysis in magnetohydrodynamics: Hall effects on free convection flow and mass transfer past a semi-infinite vertical flat plate

Heat and mass transfer along a semi-infinite vertical flat plate under the combined buoyancy force effects of thermal and species diffusion is investigated in the presence of a strong non-uniform magnetic field and the Hall currents are taken into account. The induced magnetic field due to the motion of the electrically conducting fluid is negligible. This assumption is valid for a small magnetic Reynolds number. The similarity solutions are obtained using the scale group of transformations. These are the only symmetry transformations admitted by the field equations. The non-linear boundary layer equations with the boundary conditions are transferred to a system of non-linear 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 magnetic parameter M in the range of 0-1 with the Hall parameter m taking the values 0.5, 1, 2, and 3.     

Motion of a sphere in an electrically conducting rotating fluid

The combined effect of rotation and magnetic field is investigated for the axisymmetric flow due to the motion of a sphere in an inviscid, incompressible electrically conducting fluid having uniform rotation far upstream. The steady-state linearized equations contain a single parameter α=1/2βR _m, β being the magnetic pressure number and R _m the magnetic Reynolds number. The complete solution for the flow field and magnetic field is obtained and the distribution of vorticity and current density is found. The induced vorticity is O(α⁴) and the current density is O(R _m) on the sphere.     

Hall effects on MHD flow in a rotating system with heat transfer characteristics

Closed-form solutions are derived for the steady magnetohydrodynamic (MHD) viscous flow in a parallel plate channel system with perfectly conducting walls in a rotating frame of reference, in the presence of Hall currents, heat transfer and a transverse uniform magnetic field. A mathematical analysis is described to evaluate the velocity, induced magnetic field and mass flow rate distributions, for a wide range of the governing parameters. Asymptotic behavior of the solution is analyzed for large M ² (Hartmann number squared) and K ² (rotation parameter). The heat transfer aspect is considered also with Joule and viscous heating effects present. Boundary layers arise close to the channel walls for large K ², i.e. strong rotation of the channel. For slowly rotating systems (small K ²), Hall current parameter (m) reduces primary mass flow rate (Q _x /R ρ v). Heat transfer rate at the upper plate (d θ/d η) _η=1 decreases, while at the lower plate (d θ/d η) _η=−1 increases, with increase in either K ² or m. For constant values of the rotation parameter, K ², heat transfer rate at both plates exhibits an oscillatory pattern with an increase in Hall current parameter, m. The response of the primary and secondary velocity components and also the primary and secondary induced magnetic field components to the control parameters is also studied graphically. Applications of the study arise in rotating MHD induction machine energy generators, planetary and solar plasma fluid dynamics systems, magnetic field control of materials processing systems, hybrid magnetic propulsion systems for space travel etc.     

A note on magnetoconvection in a vertical enclosure

The combined effect of viscous and ohmic dissipations on magnetoconvection in a vertical enclosure heated at the vertical side walls in the presence of applied electric field parallel to gravity and magnetic field normal to gravity is investigated. The coupled non-linear equations governing the motion are solved both analytically valid for small buoyancy parameter N and numerically valid for large N. Solutions for large N reveal a marked change in velocity profile, mass flow rate, skin friction and rate of heat transfer. These results are presented for various Hartmann number M, electric field loading parameter E and buoyancy parameter N. It is shown in the case of open circuit (i.e. E ≠ 0) that the effect of magnetic field is to increase both the velocity and temperature in contrast with the short circuit case (i.e. E = 0). The results for the case when the walls are maintained at the same temperatures (i.e. T₁ = T₂) are obtained as a particular case.     

Stagnation point flow of a micropolar fluid towards a stretching sheet

The steady two-dimensional stagnation point flow of an incompressible micropolar fluid over a stretching sheet when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation point, has been studied in this paper. The resulting equations of non-linear ordinary coupled differential equations are solved numerically using the Keller-box method. The results obtained for velocity, microrotation and skin friction are shown in tables and graphs. Comparison with the recent results of Mahapatra and Gupta {Heat Mass Transfer 38 (2002) 517} for the corresponding problem of a viscous fluid (K=0) has been done and it has been shown that the results are in excellent agreement.     

Effect of temperature dependent viscosity on revolving axi-symmetric ferrofluid flow with heat transfer

The prime objective of the present study is to examine the effect of temperature dependent viscosity μ(T) on the revolving axi-symmetric laminar boundary layer flow of an incompressible, electrically non-conducting ferrofluid in the presence of a stationary plate subjected to a magnetic field and maintained at a uniform temperature. To serve this purpose, the non-linear coupled partial differential equations are firstly converted into the ordinary differential equations using well-known similarity transformations. The popular finite difference method is employed to discretize the non-linear coupled differential equations. These discretized equations are then solved using the Newton method in MATLAB, for which an initial guess is made with the help of the Flex PDE Solver. Along with the velocity profiles, the effects of temperature dependent viscosity are also examined on the skin friction, the heat transfer, and the boundary layer displacement thickness. The obtained results are presented numerically as well as graphically.     

Combined effect of viscous dissipation and thermal radiation on fluid flows over a non-linearly stretched permeable wall

An analysis is presented for the steady non-linear viscous flow of an incompressible viscous fluid over a horizontal surface of variable temperature with a power-law velocity under the influences of suction/blowing, viscous dissipation and thermal radiation. Numerical results are illustrated by means of tables and graphs. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation. The effects of the stretching parameter n, suction/blowing parameter b, Prandtl number σ, Eckert number E_c(E_c ^* )E_{c}(E_{c}^{ *} ) and radiation parameter N _R are discussed. Two cases are studied, namely, (i) Prescribed surface temperature (PST case) and, (ii) Prescribed heat flux at the sheet (PHF case).     

MHD flow and heat transfer of a UCM fluid over a stretching surface with variable thermophysical properties

In this paper we investigate the effects of temperature-dependent viscosity, thermal conductivity and internal heat generation/absorption on the MHD flow and heat transfer of a non-Newtonian UCM fluid over a stretching sheet. The governing partial differential equations are first transformed into coupled non-linear ordinary differential equation using a similarity transformation. The resulting intricate coupled non-linear boundary value problem is solved numerically by a second order finite difference scheme known as Keller-Box method for various values of the pertinent parameters. Numerical computations are performed for two different cases namely, zero and non-zero values of the fluid viscosity parameter. That is, 1/?? _r ??0 and 1/?? _r ??0 to get the effects of the magnetic field and the Maxwell parameter on the velocity and temperature fields, for several physical situations. Comparisons with previously published works are presented as special cases. Numerical results for the skin-friction co-efficient and the Nusselt number with changes in the Maxwell parameter and the fluid viscosity parameter are tabulated for different values of the pertinent parameters. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study on the non-Newtonian fluid phenomena, especially the UCM fluid phenomena. Maxwell fluid reduces the wall-shear stress.     

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier–Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (, ). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.     

Thermal radiation effects on MHD flow past a semi-infinite inclined plate in the presence of mass diffusion

MHD mixed free-forced heat and mass convective steady incompressible laminar boundary layer flow of a gray optically thick electrically conducting viscous fluid past a semi-infinite inclined plate for high temperature and concentration differences is studied. A uniform magnetic field is applied perpendicular to the plate. The density of the fluid is assumed to reduce exponentially with temperature and concentration. The usual Boussinesq approximation is neglected due to the high temperature and concentration differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The boundary layer equations governing the flow are reduced to ordinary differential equations, which are numerically solved by applying an efficient technique. The effects of the density/temperature parameter n, the density/concentration parameter m, the local magnetic parameter M_x and the radiation parameter R are examined on the velocity, temperature and concentration distributions as well as the coefficients of skin-friction, heat flux and mass flux.     

Magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface

Steady two-dimensional stagnation-point flow of an electrically conducting power-law fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. We have discussed the uniqueness of the solution except when the ratio of free stream velocity and stretching velocity is equal to 1. The effect of magnetic field on the flow characteristic is explored numerically and it is concluded that the velocity at a point decreases/increases with increase in the magnetic field when the free stream velocity is less/greater than the stretching velocity. It is further observed that for a given value of magnetic parameter M, the dimensionless shear stress coefficient |F^″(0)| increases with increase in power-law index n when the value of the ratio of free stream velocity and stretching velocity is close to 1 but not equal to 1. But when the value of this ratio further differs from 1, the variation of |F^″(0)| with n is non-monotonic.     

Mixed convection boundary layer flow past a wedge with permeable walls

The effects of suction/injection on steady laminar mixed convection boundary layer flow over a permeable horizontal surface of a wedge in a viscous and incompressible fluid is considered in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction/injection parameter f ₀, the constant exponent m of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using an implicit finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the local Nusselt number, and the velocity and temperature profiles are obtained for various values of parameters considered. Dual solutions are found to exist for the case of opposing flow.     

Series solutions for steady three-dimensional stagnation point flow of a nanofluid past a circular cylinder with sinusoidal radius variation

This article deals with the study of the steady three-dimensional stagnation point flow of a nanofluid past a circular cylinder that has a sinusoidal radius variation. By means of similarity transformation, the governing partial differential equations are reduced into highly non-linear ordinary differential equations. The resulting non-linear system has been solved analytically using an efficient technique namely homotopy analysis method (HAM). Expressions for velocity and temperature fields are developed in series form. In this study, three different types of nanoparticles are considered, namely alumina (Al₂O₃), titania (TiO₂), and copper (Cu) with water as the base fluid. For alumina–water nanofluid, graphical results are presented to describe the influence of the nanoparticle volume fraction φ and the ratio of the gradient of velocities c on the velocity and temperature fields. Moreover, the features of the flow and heat transfer characteristics are analyzed and discussed for foregoing nanofluids. It is found that the skin friction coefficient and the heat transfer rate at the surface are highest for copper–water nanofluid compared to the alumina–water and titania–water nanofluids.     

Transient MHD stagnation flow of a non-Newtonian fluid due to impulsive motion from rest

The transient boundary layer flow and heat transfer of a viscous incompressible electrically conducting non-Newtonian power-law fluid in a stagnation region of a two-dimensional body in the presence of an applied magnetic field have been studied when the motion is induced impulsively from rest. The non-linear partial differential equations governing the flow and heat transfer have been solved by the homotopy analysis method and by an implicit finite-difference scheme. For some cases, analytical or approximate solutions have also been obtained. The special interest are the effects of the power-law index, magnetic parameter and the generalized Prandtl number on the surface shear stress and heat transfer rate. In all cases, there is a smooth transition from the transient state to steady state. The shear stress and heat transfer rate at the surface are found to be significantly influenced by the power-law index N except for large time and they show opposite behaviour for steady and unsteady flows. The magnetic field strongly affects the surface shear stress, but its effect on the surface heat transfer rate is comparatively weak except for large time. On the other hand, the generalized Prandtl number exerts strong influence on the surface heat transfer. The skin friction coefficient and the Nusselt number decrease rapidly in a small interval 0<t^*<1 and reach the steady-state values for t^*≥4.     

Computational study of combined effects of conduction-radiation and hydromagnetics on natural convection flow past magnetized permeable plate

The computational study of the combined effects of radiation and hydromagnetics on the natural convection flow of a viscous,incompressible,and electrically conducting fluid past a magnetized permeable vertical plate is presented.The governing non-similar equations are numerically solved by using a finite difference method for all values of the suction parameter ξ and the asymptotic solution for small and large values of ξ.The effects of varying the Prandtl number P r,the magnetic Prandtl number P r m,the magnetic force parameter S,the radiation parameter R d,and the surface temperature θ w on the coefficients of the skin friction,the rate of heat transfer,and the current density are shown graphically and in tables.An attempt is made to examine the effects of the above mentioned physical parameters on the velocity profile,the temperature distribution,and the transverse component of the magnetic field.     

Computational modeling of biomagnetic micropolar blood flow and heat transfer in a two-dimensional non-Darcian porous medium

We study theoretically and computationally the incompressible, non-conducting, micropolar, biomagnetic (blood) flow and heat transfer through a two-dimensional square porous medium in an (x,y) coordinate system, bound by impermeable walls. The magnetic field acting on the fluid is generated by an electrical current flowing normal to the x–y plane, at a distance l beneath the base side of the square. The flow regime is affected by the magnetization B ₀ and a linear relation is used to define the relationship between magnetization and magnetic field intensity. The steady governing equations for x-direction translational (linear) momentum, y-direction translational (linear) momentum, angular momentum (micro-rotation) and energy (heat) conservation are presented. The energy equation incorporates a special term designating the thermal power per unit volume due to the magnetocaloric effect. The governing equations are non-dimensionalized into a dimensionless (ξ,η) coordinate system using a set of similarity transformations. The resulting two point boundary value problem is shown to be represented by five dependent non-dimensional variables, f _ξ  (velocity), f _η (velocity), g (micro-rotation), E (magnetic field intensity) and θ (temperature) with appropriate boundary conditions at the walls. The thermophysical parameters controlling the flow are the micropolar parameter (R), biomagnetic parameter (N _H ), Darcy number (Da), Forchheimer (Fs), magnetic field strength parameter (Mn), Eckert number (Ec) and Prandtl number (Pr). Numerical solutions are obtained using the finite element method and also the finite difference method for Ec=2.476×10⁻⁶ and Prandtl number Pr=20, which represent realistic biomagnetic hemodynamic and heat transfer scenarios. Temperatures are shown to be considerably increased with Mn values but depressed by a rise in biomagnetic parameter (N _H ) and also a rise in micropolarity (R). Translational velocity components are found to decrease substantially with micropolarity (R), a trend consistent with Newtonian blood flows. Micro-rotation values are shown to increase considerably with a rise in R values but are reduced with a rise in biomagnetic parameter (N _H ). Both translational velocities are boosted with a rise in Darcy number as is micro-rotation. Forchheimer number is also shown to decrease translational velocities but increase micro-rotation. Excellent agreement is demonstrated between both numerical solutions. The mathematical model finds applications in blood flow control devices, hemodynamics in porous biomaterials and also biomagnetic flows in highly perfused skeletal tissue. Dedicated to Professor Y.C. Fung (1919-), Emeritus Professor of Biomechanics, Bioengineering Department, University of California at San Diego, USA for his seminal contributions to biomechanics and physiological fluid mechanics over four decades and his excellent encouragement to the authors, in particular OAB, with computational biofluid dynamics research.