The effects of side walls on axial flow in rectangular ducts with suction and injection

Summary.  The effects of the side walls on the flows in ducts with suction and injection are examined. Three illustrative examples are considered. The first example considers the effect of the side walls on the flow over a porous plate. It is shown that the presence of the side walls provides a solution for both injection and suction, although, in the absence of the side walls, a solution exists only in the case of suction. The second example considers the flow between two porous plates and the third example the flow in a rectangular duct with two porous walls. Analytical solutions are obtained for the velocity, the volume flux across a plane normal to the flow and the vorticity. In order to show the effects of the side walls for the flow on a rectangular duct, a comparison of these quantities with those in the flow between two parallel porous plates is established. These three examples show that there are pronounced effects of the side walls on the flows in ducts with suction and injection. Received January 10, 2002; revised September 27, 2002 Published online: May 8, 2003 The author is grateful to Prof. D. Poulikakos for his suggestions and to a referee for careful corrections of an earlier version of this paper.     

Laminar flow with injection in a straight duct

The velocity distribution and pressure drop associated with injection flow in a straight duct are analyzed on the basis of exact solutions of the Navier-Stokes equations.     

Laminar flow of a non-Newtonian fluid in channels with wall suction or injection

The one-dimensional approximate equation in the rectangular Cartesian coordinates governing flow of a non-Newtonian fluid confined in two large plates separated by a small distance of h, with the upper plate stationary while the lower plate is uniformly porous and moving in the x-direction with constant velocity, is derived by accounting for the order of magnitude of terms as well as the accompanying approximations to the full-blown three-dimensional equations by using scaling arguments, asymptotic techniques and assuming the cross-flow velocity is much less than the axial velocity. The one-dimensional governing equation for a power-law fluid flow confined between parallel plates, with the upper plate is stationary and the bottom plate subjected to sudden acceleration with a constant velocity in the x-direction and uniformly porous, is solved analytically for a Newtonian fluid case (n = 1) and numerically for various values of power-law index to determine the transient velocity and thus the overall transient velocity distribution. The effects of mass suction/injection at the porous bottom plate on the flow of non-Newtonian fluids are examined for various values of time and power-law index. The results obtained from the present analysis are compared with the data available in the literature.     

Magnetohydrodynamic flow for a non-Newtonian power-law fluid having a pressure gradient and fluid injection

Summary The nonlinear partial differential equation of motion for an incompressible, non-Newtonian power-law fluid flowing over flat plate under the influence of a magnetic field and a pressure gradient, and with or without fluid injection or ejection, is transformed to a nonlinear third-order ordinary differential equation by using a stream function and a similarity transformation.The necessary boundary conditions are developed for flow with and without fluid injection (or ejection), and a solution for four different power-law fluids, including a Newtonian fluid, is presented.The controlling equation includes, as special cases, the Falkner-Skan equation and the Blasius equation.     

Finite element method for solving mhd flow in a rectangular duct

A finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady MHD (magnetohydrodynamic) flow through a rectangular pipe having arbitrarily conducting walls. Linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and Hartmann number M (M ≤ 100). A special solution procedure has been used for the resulting block tridiagonal system of equations. Computations have been carried out for several values of Hartmann number (5 ≤ M ≤ 100) and wall conductivity. It is also found that, if the wall conductivity increases, the flux decreases. The same is the effect of increasing the Hartmann number. Selected graphs are given showing the behaviour of the velocity field and induced magnetic field.     

Heat transfer in MHD flow with pressure gradient,suction and injection

Summary Numerical solutions to the MHD Falkner-Skan equation and the corresponding heat transfer equation have been obtained on taking into consideration the effects of suction and injection and the pressure gradient parameter . Velocity and temperature profiles are shown on graphs and the numerical values of the skin friction and the rate of heat transfer are given in the form of tables. It has been observed that an increase inN _m (magnetic field parameter) leads to an increase in velocity, skin friction, rate of heat transfer and a fall in temperature. Also an increase in suction leads to a fall in the value of the skin friction and the rate of heat transfer, opposite to the case of injection.     

Variational approach to free convection boundary-layer flow with suction and injection

The Gyarmati variational principle — a significant development in the field of the thermodynamics of irreversible processes — is employed to study suction and injection effects in flow and heat transfer in a free convection boundary layer over a cone. The velocity and temperature distributions inside respective boundary layers are considered as simple polynomial functions, and with the use of the perturbation procedure the variational principle is formulated. The Euler-Lagrange equations are reduced to coupled polynomial equations in terms of boundary-layer thicknesses. The skin-friction (shear-stress) and heat-transfer (Nusselt number) values with constant wall temperature are computed for various values of the suction and injection parameters and the cone-angle parameter. The comparison of the present solution with an available numerical solution shows good agreement. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 6, pp. 109–115, November–December, 2007.     

Nonuniform slot injection (suction) into a compressible flow

Summary The steady nonsimilar compressible laminar boundary-layer flow over a yawed infinite circular cylinder with nonuniform slot injection (suction) has been studied up to the point of separation. The finite discontinuities arising at the leading and trailing edges of the slot for the uniform slot injection (suction) are removed by choosing appropriate nonuniform mass transfer distributions in the slot. The difficulties arising at the starting point of the streamwise co-ordinate, at the edges of the slot and at the point of separation are overcome by applying the method of quasilinear implicit finite difference scheme with an appropriate selection of finer step size along the streamwise direction. It is found that the nonuniform slot injection moves the point of separation downstream but the nonuniform slot suction has the reverse effect. The increase of Mach number shifts the point of separation upstream due to the adverse pressure gradient. The increase of total enthalpy at the wall causes the separation to occur earlier while cooling delays it. The yaw angle has very little effect on the location of the point of separation.     

Thermal boundary layers over a wedge with uniform suction or injection in forced flow

Summary The behavior of incompressible laminar boundary layers in forced flow over a wedge with uniform suction or injection was theoretically investigated. The boundary layer equations along a wedge are transformed into non-similar partial differential ones, and the ordinary differential equations were obtained by means of the difference-differential method. The solutions of the resulting equations are expressed in a form of integral equations which are in turn solved by iterative numerical quadratures. The numerical results are given for the velocity distribution, temperature distribution and the coefficient of skin friction and heat transfer for various values of suction/injection parameter.     

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.     

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.     

Magnetohydrodynamic stratified flow past a sphere

This paper considers the uniform, steady, horizontal flow of a vertically stratified, electrically conducting, non-diffusive fluid over a non-conducting sphere in the presence of a uniform magnetic field. The force exerted on the sphere is investigated on the basis of the method of matched asymptotic expansions, for small values of a stratification parameter α, $\text{R}{}_{\mathrm{e}}\text{? ¦α¦}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$, $\text{F}{}_{\mathrm{r}}{}^2\text{? ¦α¦}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>3</mtext>}}$ and for $\text{M}{}^2\text{= 0(α}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Up to the first order of calculations the drag is computed for a few typical values of magnetic interaction parameter when, (a) The applied magnetic field lies in the vertical plane and inclined with the ambient flow direction. (b) The applied magnetic field lies in the horizontal plane and perpendicular to the flow direction. Further it is shown that the sphere has no tendency to rotate nor it experiences a lift force upto the order of calculations which we have made. The drag experienced by the sphere is found to be increased due to the combined effects of stratification and magnetic field.     

Optimization of the geometry of a rectangular duct for low-gradient flow generation

A numerical experiment is carried out to determine the feasibility of generating low-gradient flow around samples of materials in a rectangular duct.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 54, No. 6, pp. 930–934, June, 1988.     

Stability of fluid flow in a plane channel with uniform injection or suction through porous walls

The stability of laminar flow in a channel with porous walls is analyzed within the scope of the linear theory.Notation x distance from entrance cross section - y transverse coordinate measured from axis - u_x, u_y longitudinal and transverse velocity components of main flow - h half-width of channel - kinematic viscosity coefficient - U₀ average velocity in entrance cross section - V suction or injection rate (positive for suction) - U=U₀–V_x/h local average velocity - amplitude of flow disturbances - wave number - c complex phase velocity of disturbances - c_r real propagation velocity of disturbances, =y/h - Re=Uh/ Reynolds number of main flow - R=Vh/ injection or suction Reynolds number - m=U/¦V¦ injection rate parameter Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 436–440, September, 1981.     

On the steady flow produced by a rotating disc with either surface suction or injection

Summary A series solution is presented for the steady, laminar flow produced by a rotating disc. The series consists solely of exponential terms with negative exponents. It is shown that this approach yields uniformly valid solutions of high accuracy for all cases of suction and for low values of injection at the disc surface. The radius of convergence of the series is determined. For those injection cases for which the direct series method fails, an integral method is presented which is based on the properties of the series. The latter method consists of obtaining differential equations which represent the behaviour of the sums of the series. This method allows the solution of the governing differential equations as an initial value problem.     

Magnetohydrodynamic stagnation-point flow towards a stretching sheet

Steady two-dimensional stagnation-point flow of an incompressible viscous electrically conducting fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown 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. Temperature distribution in the flow is obtained when the surface is held at a constant temperature.     

Magnetohydrodynamic poiseuille flow through a porous medium

Abstract

This paper is concerned with formulating equations for the flow of an electrically conducting fluid through a non‐conducting porous medium with non‐porous and non‐conducting boundaries. Equations are developed for the general case of the flow of a solid‐fluid suspensions; flow through a porous medium is treated as a special case by letting the velocity of the particle phase goes to zero. Two cases are considered. Exact solution is obtained for the case of flow between parallel plates, but for flow in pipes of square and circular cross sections, the equations have to be solved numerically. The numerical technique developed can treat elliptical cross sections as well. The flow in all cases is assumed to be steady, laminar, incompressible, viscous, and fully developed. The results are presented in terms of a parameter which measures the resistance of the porous medium.     

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.     

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.