An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching

In this paper, an exact analytical solution of the famous Falkner-Skan equation is obtained. The solution involves the boundary layer flow over a moving wall with mass transfer in presence of a free stream with a power-law velocity distribution. Multiple solution branches are observed. The effects of mass transfer and wall stretching are analyzed. Interesting velocity profiles including velocity overshoot and reversal flows are observed in the presence of both mass transfer and wall stretching. These solutions greatly enrich the analytical solution for the celebrated Falkner-Skan equation and the understanding of this important and interesting equation.     

Unsteady magnetohydrodynamic stagnation point flow — closed-form analytical solutions

This paper investigates the unsteady stagnation point flow and heat transfer of magnetohydrodynamic(MHD) fluids over a moving permeable flat surface. The unsteady Navier-Stokes(NS) equations are transformed into a similarity nonlinear ordinary differential equation, and a closed form solution is obtained for the unsteadiness parameter of 2. The boundary layer energy equation is transformed into a similarity equation,and is solved for a constant wall temperature and a time-dependent uniform wall heat flux case. The solution domain, velocity, and temperature profiles are calculated for different combinations of parameters including the Prandtl number, mass transfer parameter, wall moving parameter, and magnetic parameter. Two solution branches are obtained for certain combinations of the controlling parameters, and a stability analysis demonstrates that the lower solution branch is not stable. The present solutions provide an exact solution to the entire unsteady MHD NS equations, which can be used for validating the numerical code of computational fluid dynamics.     

Gravity-driven laminar film flow of power-law fluids along vertical walls

A theoretical analysis is presented which brings steady laminar film flow of power-law fluids within the framework of classical boundary layer theory. The upper part of the film, which consists of a developing viscous boundary layer and an external inviscid freestream, is treated separately from the viscous dominated part of the flow, thereby taking advantage of the distinguishing features of each flow region. It is demonstrated that the film boundary layer developing along a vertical wall can be described by a generalized Falkner-Skan type equation originally developed for wedge flow. An exact similarity solution for the velocity field in the film boundary layer is thus made available.Downstream of the boundary layer flow regime the fluid flow is completely dominated by the action of viscous shear, and fairly accurate solutions are obtained by the Von Karman integral method approach. A new form of the velocity profile is assumed, which reduces to the exact analytic solution for the fully-developed film. By matching the downstream integral method solution to the upstream generalized Falkner-Skan similarity solution, accurate estimates for the hydrodynamic entrance length are obtained. It is also shown that the flow development in the upstream region predicted by the approximate integral method closely corresponds to the exact similarity solution for that flow regime. An analytical solution of the resulting integral equation for the Newtonian case is compared with previously published results.     

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.     

Flow and heat transfer due to impulsive motion of a cone in a viscous fluid

Numerical solution has been obtained for the development of the flow over a cone which is impulsively set into motion. Initially the flow is described by the solution of Rayleigh and then it tends to the ultimate steady state solution of Falkner-Skan equation. But due to the leading edge effect the semi-similar equation describing the transient flow changes its character after certain time and the solution depends also on the ultimate steady state solution of the Falkner-Skan equation. A second-order upwind difference scheme has been used for discretisation. The temperature distribution and heat transfer has also been obtained for constant wall temperature as well as for constant heat flux at the wall. With the increase ofm, Falkner-Skan parameter, the magnitude of skin friction and wall heat transfer increases. It has been found that form≥?0.275 flow separation does not occur.     

Approximate solutions to MHD Falkner-Skan flow over permeable wall

The magnetohydrodynamic (MHD) Falkner-Skan boundary layer flow over a permeable wall in the presence of a transverse magnetic field is examined. The approximate solutions and skin friction coefficients of the MHD boundary layer flow are obtained by using a method that couples the differential transform method (DTM) with the Padé approximation called DTM-Padé. The approximate solutions are expressed in the form of a power series that can be easily computed with an iterative procedure. The approximate solutions are tabulated, plotted for the values of different parameters and compared with the numerical ones obtained by employing the shooting technique. It is found that the approximate solution agrees very well with the numerical solution, showing the reliability and validity of the present work. Moreover, the effects of various physical parameters on the boundary layer flow are presented graphically and discussed.     

DTM-BF method and dual solutions for unsteady MHD flow over permeable shrinking sheet with velocity slip

An unsteady magnetohydrodynamic (MHD) boundary layer flow over a shrinking permeable sheet embedded in a moving viscous electrically conducting fluid is investigated both analytically and numerically. The velocity slip at the solid surface is taken into account in the boundary conditions. A novel analytical method named DTMBF is proposed and used to get the approximate analytical solutions to the nonlinear governing equation along with the boundary conditions at infinity. All analytical results are compared with those obtained by a numerical method. The comparison shows good agreement, which validates the accuracy of the DTM-BF method. Moreover, the existence ranges of the dual solutions and the unique solution for various parameters are obtained. The effects of the velocity slip parameter, the unsteadiness parameter, the magnetic parameter, the suction/injection parameter, and the velocity ratio parameter on the skin friction, the unique velocity, and the dual velocity profiles are explored, respectively.     

A class of exact solutions for the incompressible viscous magnetohydrodynamic flow over a porous rotating disk

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow due to a porous disk rotating with a constant angular speed.The three-dimensional hydromagnetic equations of motion are treated analytically to obtained exact solutions with the inclusion of suction and injection.The well-known thinning/thickening flow field effect of the suction/injection is better understood from the constructed closed form velocity equations.Making use of this solution,analytical formulas for the angular velocity components as well as for the permeable wall shear stresses are derived.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation.The temperature field is shown to accord with the dissipation and the Joule heating.As a result,exact formulas are obtained for the temperature field which take different forms corresponding to the condition of suction or injection imposed on the wall.     

Flow and heat transfer over a generalized stretching/shrinking wall problem—Exact solutions of the Navier-Stokes equations

In this paper, we investigate the steady momentum and heat transfer of a viscous fluid flow over a stretching/shrinking sheet. Exact solutions are presented for the Navier-Stokes equations. The new solutions provide a more general formulation including the linearly stretching and shrinking wall problems as well as the asymptotic suction velocity profiles over a moving plate. Interesting non-linear phenomena are observed in the current results including both exponentially decaying solution and algebraically decaying solution, multiple solutions with infinite number of solutions for the flow field, and velocity overshoot. The energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking strength on the temperature profiles and wall heat flux are also presented and discussed. The exact solution of this general flow configuration is a rare case for the Navier-Stokes equation.     

Heat transfer analysis of MHD and electroosmotic flow of non-Newtonian fluid in a rotating microfluidic channel: an exact solution

The heat transfer of the combined magnetohydrodynamic(MHD) and electroosmotic flow(EOF) of non-Newtonian fluid in a rotating microchannel is analyzed. A couple stress fluid model is scrutinized to simulate the rheological characteristics of the fluid. The exact solution for the energy transport equation is achieved. Subsequently,this solution is utilized to obtain the flow velocity and volume flow rates within the flow domain under appropriate boundary conditions. The obtained analytical solution results are compared with the previous data in the literature, and good agreement is obtained.A detailed parametric study of the effects of several factors, e.g., the rotational Reynolds number, the Joule heating parameter, the couple stress parameter, the Hartmann number, and the buoyancy parameter, on the flow velocities and temperature is explored. It is unveiled that the elevation in a couple stress parameter enhances the EOF velocity in the axial direction.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

A solution for the vertical variation of stress,rather than velocity,in a three-dimensional circulation model

A simple technique is presented that allows a numerical solution to be sought for the vertical variation of shear stress as a substitute for the vertical variation of velocity in a three-dimensional hydrodynamic model. In its most general form the direct stress solution (DSS) method depends only upon the validity of an eddy viscosity relation between the shear stress and the vertical gradient of velocity. The rationale for preferring a numerical solution for shear stress to one for velocity is that shear stress tends to vary more slowly over the vertical than velocity, particularly near boundaries. Consequently, a numerical solution can be obtained much more efficiently for shear stress than for velocity. When needed, the velocity profile can be recovered from the stress profile by solving a one-dimensional integral equation over the vertical. For most practical problems this equation can be solved in closed form. Comparisons are presented between the DSS technique, the standard velocity solution technique and analytical solutions for wind-driven circulation in an unstratified, closed, rectangular channel governed by the linear equations of motion. In no case was the computational effort required by the velocity solution competitive with the DSS when a physically realistic boundary layer was included. The DSS technique should be particularly beneficial in numerical models of relatively shallow water bodies in which the bottom and surface boundary layers occupy a significant portion of the water column.     

Magnetohydrodynamic Rotating Flow of a Generalized Burgers’ Fluid in a Porous Medium with Hall Current

This study concentrates on the unsteady magnetohydrodynamics (MHD) rotating flow of an incompressible generalized Burgers’s fluid past a suddenly moved plate through a porous medium. Modified Darcy’s law for generalized Burgers’s fluid in a rotating frame has been used to model the governing flow problem. The closed form solution of the governing flow problem has been obtained by employing Laplace transform technique. The integral appearing in the inverse Laplace transform has been evaluated numerically. The influence of various parameters on the velocity profile has been delineated through several graphs and discussed in detail. It was found that the fluid is decelerated with increasing Hartmann number M and porosity parameter K. However, for large Hall parameter m, the real part of velocity decreases and the imaginary part of velocity increases.     

An analytical theory of heated duct flows in supersonic combustors

One-dimensional analytical theory is developed for supersonic duct flow with variation of cross section, wall friction, heat addition, and relations between the inlet and outlet flow parameters are obtained. By introducing a selfsimilar parameter, effects of heat releasing, wall friction, and change in cross section area on the flow can be normalized and a self-similar solution of the flow equations can be found. Based on the result of self-similar solution, the sufficient and necessary condition for the occurrence of thermal choking is derived. A relation of the maximum heat addition leading to thermal choking of the duct flow is derived as functions of area ratio, wall friction, and mass addition, which is an extension of the classic Rayleigh flow theory, where the effects of wall friction and mass addition are not considered. The present work is expected to provide fundamentals for developing an integral analytical theory for ramjets and scramjets.     

Magnetohydrodynamic flow on a half-plane

We investigate the magnetohydrodynamic flow (MHD) on the upper, half of a non-conducting plane for the case when the flow is driven by the current produced by an electrode placed in the middle of the plane. The applied magnetic field is perpendicular to the plane, the flow is laminar, uniform, steady and incompressible. An analytical solution has been developed for the velocity field and the induced magnetic field by reducing the problem to the solution of a Fredholm's integral equation of the second kind, which has been solved numerically. Infinite integrals occurring in the kernel of the integral equation and in the velocity and magnetic field were approximated for large Hartmann numbers by using Bessel functions. As the Hartmann number M increases, boundary layers are formed near the non-conducting boundaries and a parabolic boundary layer is developed in the interface region. Some graphs are given to show examples of this behaviour.     

The Laminar Boundary Layer over a Permeable Wall

An analysis is given of the laminar boundary layer over a permeable/porous wall. The porous wall is passive in the sense that no suction or blowing velocity is imposed. To describe the flow inside and above the porous wall a continuum approach is employed based on the Volume-Averaging Method (S. Whitaker The method of volume averaging). With help of an order-of-magnitude analysis the boundary-layer equations are derived. The analysis is constrained by: (a) a low wall permeability; (b) a low Reynolds number for the flow inside the porous wall; (c) a sufficiently high Reynolds number for the freestream flow above the porous wall. Two boundary layers lying on top of each other can be distinguished: the Prandtl boundary layer above the porous wall, and the Brinkman boundary layer inside the porous wall. Based on the analytical solution for the Brinkman boundary layer in combination with the momentum transfer model of Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). for the interface region, a closed set of equations is derived for the Prandtl boundary layer. For the stream function a power series expansion in the perturbation parameter is adopted, where is proportional to ratio of the Brinkman to the Prandtl boundary-layer thickness. A generalization of the Falkner–Skan equation for boundary-layer flow past a wedge is derived, in which wall permeability is incorporated. Numerical solutions of the Falkner–Skan equation for various wedge angles are presented. Up to the first order in wall permeability causes a positive streamwise velocity at the interface and inside the porous wall, but a wall-normal interface velocity is a second-order effect. Furthermore, wall permeability causes a decrease in the wall shear stress when the freestream flow accelerates, but an increase in the wall shear stress when the freestream flow decelerates. From the latter it follows that separation, as indicated by zero wall shear stress, is delayed to a larger positive pressure gradient.     

Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet

The present article investigates the dual nature of the solution of the magneto- hydrodynamic （MHD） stagnation-point flow of a Prandtl fluid model towards a shrinking surface. The self-similar nonlinear ordinary differential equations are solved numerically by the shooting： method. It is found that the dual solutions of the flow exist for cer- tain values of tile velocity ratio parameter. The special case of the first branch solutions （the classical Newtonian fluid model） is compared with the present numerical results of stretching flow. The results are found to be in good agreement. It is also shown that the boundary layer thickness for the second solution is thicker than that for the first solution.     

Magnetohydrodynamic flow in an infinite channel

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.     

Hydrodynamics of viscous fluid through porous slit with linear absorption

The exact solutions for the viscous fluid through a porous slit with linear ab-sorption are obtained. The Stokes equation with non-homogeneous boundary conditions is solved to get the expressions for the velocity components, pressure distribution, wall shear stress, fractional absorption, and leakage flux. The volume flow rate and mean flow rate are found to be useful in obtaining a convenient form of the longitudinal velocity component and pressure difference. The points of the maximum velocity components for a fixed axial distance are identified. The value of the linear absorption parameter is ran-domly chosen, and the rest available data of the rat kidney to the tabulate pressure drop and fractional absorption are incorporated. The effects of the linear absorption, uniform absorption, and flow rate parameters on the flow properties are discussed by graphs. It is found that forward flow occurs only if the volume flux per unit width is greater than the absorption velocity throughout the length of the slit, otherwise back flow may occur. The leakage flux increases with the increase in the linear absorption parameter. Streamlines are drawn to help the analysis of the flow behaviors during the absorption of the fluid flow through the renal tubule and purification of blood through an artificial kidney.     

MHD Falkner-Skan flow of Maxwell fluid by rational Chebyshev collocation method

The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.