Oscillatory flow of second grade fluid in cylindrical tube

The unsteady oscillatory flow of an incompressible second grade fluid in a cylindrical tube with large wall suction is studied analytically. Flow in the tube is due to uniform suction at the permeable walls, and the oscillations in the velocity field are due to small amplitude time harmonic pressure waves. The physical quantities of interest are the velocity field, the amplitude of oscillation, and the penetration depth of the oscillatory wave. The analytical solution of the governing boundary value problem is obtained, and the effects of second grade fluid parameters are analyzed and discussed.     

On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

Laminar flow through a channel with uniformly porous walls of different permeability

Summary The steady laminar flow of a viscous incompressible fluid through a two-dimensional channel, having fluid sucked or injected with different velocities through its uniformly porous parallel walls is considered. A solution for small suction Reynolds number has been given by the authors in a previous paper. The purpose of this paper is to present a solution valid for large Reynolds numbers for the cases of (i) suction at both walls, and (ii) suction at one wall and injection at the other. A technique of matching outer and inner expansions is used to obtain an asymptotic solution for both of these cases. Further a perturbation solution for the case of suction at one wall and injection at the other is obtained by choosing the difference between two wall velocities as the perturbation parameter. Both asymptotic and perturbation solutions are confirmed by exact numerical solutions. As expected, the resulting solutions show the presence of the usual suction boundary layers in both types of flow considered in this paper.     

Effects of the side walls on starting flows in ducts

This article considers the effects of the side walls on the unsteady flow of an incompressible viscous fluid in a duct of uniform cross-section. In order to show the effects of the side walls, three illustrative examples are given. They are: the starting flow in a duct of semicircular cross-section, the starting flow in a duct of rectangular cross-section and the starting flow in a duct of circular cross-section. The velocity distributions and the volume fluxes obtained for these flows are compared and it is shown that the flow in a duct of semicircular cross-section reaches steady state earlier than those for the flow in a duct of circular cross-section and for the flow in a duct of square cross-section. It is found that there are remarkable effects of the side walls of a duct on the required time to attain the asymptotic values of flow properties.     

MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium

Exact analytical solutions for magnetohydrodynamic (MHD) flows of an incompressible second grade fluid in a porous medium are developed. The modified Darcy's law for second grade fluid has been used in the flow modelling. The Hall effect is taken into account. The exact solutions for the unsteady flow induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane has been constructed by means of Fourier sine transforms. The similar solutions for a Newtonian fluid, performing the same motion, appear as limiting cases of the solutions obtained here. The influence of various parameters of interest on the velocity and shear stress at the bottom wall has been shown and discussed through several graphs. A comparison between a Newtonian and a second grade fluids is also made.     

MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel

Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel. The walls of the channel are taken as porous. Using the similarity transformations and boundary layer approximations, the nonlinear partial differential equations are reduced to an ordinary differential equation. The developed nonlinear equation is solved analytically using the homotopy analysis method. An expression for the analytic solution is derived in the form of a series. The convergence of the obtained series is shown. The effects of the Reynolds number Re, Deborah number De and Hartman number M are shown through graphs and discussed for both the suction and injection cases.     

Unsteady flow of a fourth-grade fluid due to an oscillating plate

The governing non-linear high-order, sixth-order in space and third-order in time, differential equation is constructed for the unsteady flow of an incompressible conducting fourth-grade fluid in a semi-infinite domain. The unsteady flow is induced by a periodically oscillating two-dimensional infinite porous plate with suction/blowing, located in a uniform magnetic field. It is shown that by augmenting additional boundary conditions at infinity based on asymptotic structures and transforming the semi-infinite physical space to a bounded computational domain by means of a coordinate transformation, it is possible to obtain numerical solutions of the non-linear magnetohydrodynamic equation. In particular, due to the unsymmetry of the boundary conditions, in numerical simulations non-central difference schemes are constructed and employed to approximate the emerging higher-order spatial derivatives. Effects of material parameters, uniform suction or blowing past the porous plate, exerted magnetic field and oscillation frequency of the plate on the time-dependent flow, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviour of the fourth-grade non-Newtonian fluid is also compared with those of the Newtonian fluid.     

Effects of the side walls on unsteady flow of a second grade fluid over a plane wall

The effects of the side walls on unsteady flow of a second grade fluid over a plan wall are considered. The solution of the governing equation for velocity is obtained by the sine transform method. This gives a correct result for the shear stress at the bottom wall. The shear stress at the bottom wall is minimum at the middle of the plate and it increases near the side walls. It is shown that the mean thickness of the layer of the liquid over the plate increases with time and the ratio of the mean thickness to the distance between the side walls becomes ultimately 0.2714.     

Unsteady Motion of a Viscous Liquid between Rotating Parallel Walls in the Presence of a Crossflow

The paper considers the unsteady flow of a viscous incompressible fluid inside an infinitely long slot with uniform injection or suction of the fluid through the porous walls of the slot. The plates with the fluid are rotated rigidly with constant angular velocity. The unsteady flow is induced by nontorsional vibrations of the upper plate. The flowvelocity field and the tangential stress vectors exerted by the fluid on the upper and lower walls of the slot are determined. In this case, one can find an exact solution of the threedimensional nonstationary Navier–Stokes equations. No restrictions are imposed on the motion pattern of the plate.     

Radiation effect on forced convective flow and heat transfer over a porous plate in a porous medium

Heat transfer analysis has been presented for the boundary layer forced convective flow of an incompressible fluid past a plate embedded in a porous medium. The similarity solutions for the problem are obtained and the reduced nonlinear ordinary differential equations are solved numerically. In case of porous plate, fluid velocity increases for increasing values of suction parameter whereas due to injection, fluid velocity is noticed to decrease. The non-dimensional temperature increases with the increasing values of injection parameter. A novel result of this investigation is that the flow separation occurred due to suction/injection may be controlled by increasing the permeability parameter of the medium. The effect of thermal radiation on temperature field is also analyzed.     

Asymptotic solutions for the asymmetric flow in a channel with porous retractable walls under a transverse magnetic field

The self-similarity solutions of the Navier-Stokes equations are constructed for an incompressible laminar flow through a uniformly porous channel with retractable walls under a transverse magnetic field. The flow is driven by the expanding or contracting walls with different permeability. The velocities of the asymmetric flow at the upper and lower walls are different in not only the magnitude but also the direction. The asymptotic solutions are well constructed with the method of boundary layer correction in two cases with large Reynolds numbers, i.e., both walls of the channel are with suction, and one of the walls is with injection while the other one is with suction. For small Reynolds number cases, the double perturbation method is used to construct the asymptotic solution. All the asymptotic results are finally verified by numerical results.     

Unsteady flow of an Oldroyd-B fluid generated by a constantly accelerating plate between two side walls perpendicular to the plate

The velocity field and the adequate tangential stresses corresponding to the unsteady flow of an Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate are established by means of Fourier sine transforms. The solutions corresponding to Maxwell, second grade and Newtonian fluids, performing the same motion, appear as limiting cases of the solutions obtained here. In the absence of the side walls, namely when the distance between walls tends to infinity, all solutions that have been determined reduce to those corresponding to the flow over an infinite plate. Finally, for comparison, the velocity field at the middle of the channel as well as the shear stress on the bottom wall is plotted as a function of y for several values of t and of the material constants. The influence of the side walls on the motion of the fluid is also emphasized by graphical illustrations.     

Series solution for steady flow of a third grade fluid through porous space

In this paper steady flow of a third grade fluid through porous space is considered. Modified Darcy’s law for third grade fluid in a porous space has been introduced. The governing non-linear equation is first modelled and then solved using homotopy analysis method (HAM). The convergence of the obtained series solution is discussed. The effects of the emerging parameters on the velocity field are seen. It is noted that meaningful solution exists only in the case of suction.     

Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary layer flow of an electrically conducting fluid of second grade in a porous medium subject to a transverse uniform magnetic field past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux. The effects of viscous dissipation, internal heat generation of absorption and work done due to deformation are considered in the energy equation. The variations of surface temperature gradient for the prescribed surface temperature case (PST) and surface temperature for the prescribed heat flux case (PHF) with various parameters are tabulated. The asymptotic expansions of the solutions for large Prandtl number are also given for the two heating conditions. It is shown that, when the Eckert number is large enough, the heat flow may transfer from the fluid to the wall rather than from the wall to the fluid when Eckert number is small. A physical explanation is given for this phenomenon.     

On the comparison of two different solutions in the form of series of the governing equation of an unsteady flow of a second grade fluid

In this paper, two different solutions in the form of series of the governing equation of unsteady flow of a second grade fluid are considered. These are series expansions with respect to inverse power of time and a perturbation expansion. Two illustrative examples are given. One of them is the unsteady flow of a second grade fluid over a plane wall suddenly set in motion and the other is the diffusion of a line vortex in a fluid of second grade. It is a remarkable fact that the expression of the series expansion with respect to inverse power of time is exactly in the same form as that of the perturbation expansion. Thus, it is possible to replace a series expansion with respect to inverse power of time with a perturbation expansion.     

Effects of thermal stratification on flow and heat transfer past a porous vertical stretching surface

The aim of this paper is to present the boundary layer flow of viscous incompressible fluid due to a porous vertical stretching surface with a power-law stretching velocity in a thermally stratified medium. Using a special form of Lie group transformations viz. scaling group of transformations, similarity solutions for this problem are obtained. The equations are then solved numerically. With increasing values of the stratification parameter, the velocity as well as temperature decreases. At a particular point of the porous stretching sheet, the velocity decreases with the increasing suction parameter. The dimensionless temperature at a point of the sheet decreases due to suction but increases due to injection. The findings of this study reveal that stratification and suction can be used as means of cooling the boundary layer flow region.     

Analytical investigation of the fluid flow in the interface region between a porous medium and a clear fluid in channels partially filled with a porous medium

In this paper analytical solutions for the steady fully developed laminar fluid flow in the parallel-plate and cylindrical channels partially filled with a porous medium and partially with a clear fluid are presented. The Brinkman-extended Darcy equation is utilized to model the flow in a porous region. The solutions account for the boundary effects and for the stress jump boundary condition at the interface recently suggested by Ochoa-Tapia and Whitaker. The dependence of the velocity on the Darcy number and on the adjustable coefficient in the stress jump boundary condition is investigated. It is shown that accounting for a jump in the shear stress at the interface essentially influences velocity profiles.     

Slip effects on the flow of a third order fluid with variable suction

The effect of slip condition on the flow of third order fluid past a porous plate with variable suction is investigated. Perturbation solution of the resulting problem is derived. Several limiting solutions have been deduced. Graphs are plotted and discussed.     

Laminar flow through channels with porous walls and with an applied transverse magnetic field

Summary The problem of two-dimensional steady laminar flow of a viscous incompressible and electrically conducting fluid through a channel with two equally porous walls in the presence of a transverse magnetic field has been extended to include all values of Hartmann number and small suction velocity at the walls. Expressions for the velocity components, the pressure and the wall friction in terms of the Hartmann number and the suction Reynolds number are given. It is found that the pressure drop in the major flow direction and the wall friction decrease with the increase in suction and increase with the increase in the strength of the magnetic field.     

Effects of suction or blowing on the velocity and temperature distribution in the flow past a porous flat plate of a power-law fluid

An analysis is made of the steady flow of a non-Newtonian fluid past an infinite porous flat plate subject to suction or blowing. The incompressible fluid obeys Ostwald-de Waele power-law model. It is shown that steady solutions for velocity distribution exist only for a pseudoplastic (shear-thinning) fluid for which the power-law index n satisfies 0<n<1 provided that there is suction at the plate. Velocity at a point is found to increase with increase in n. No steady solution for velocity distribution exists when there is blowing at the plate. The solution of the energy equation governing temperature distribution in the flow of a pseudoplastic fluid past an infinite porous plate subject to uniform suction reveals that temperature at a given point near the plate increases with n but further away, temperature decreases with increase in n. A novel result of the analysis is that both the skin-friction and the heat flux at the plate are independent of n.