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.     

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.     

Exact solution for the rotational flow of a generalized second grade fluid in a circular cylinder

This paper deals with the rotational flow of a generalized second grade fluid, within a circular cylinder, due to a torsional shear stress. The fractional calculus approach in the constitutive relationship model of a second grade fluid is introduced. The velocity field and the resulting shear stress are determined by means of the Laplace and finite Hankel transforms to satisfy all imposed initial and boundary conditions. The solutions corresponding to second grade fluids as well as those for Newtonian fluids are obtained as limiting cases of our general solutions. The influence of the fractional coefficient on the velocity of the fluid is also analyzed by graphical illustrations.     

Convection, stability and uniqueness for a fluid of third grade

The equations for a fluid of third grade derived by Fosdick and Rajagopal are first studied on an exterior region in three-dimensional space. A uniqueness theorem and a pointwise continuous dependence theorem (on the initial data) are proved. The conditions at infinity are weak, certainly L² integrability is not required. Then, the equations for the third grade fluid are adapted to the problem of thermal convection due to heating from below. It is shown that the linear instability problem reduces to that of a second grade fluid. Interestingly, a study of non-linear stability for the same problem reveals that the constitutive inequalities obtained by Fosdick and Rajagopal play a very important role; there may be stronger asymptotic stability than for a second grade fluid, although in certain cases the stability may be much weaker.     

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.     

Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates

The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced. Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus. The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates. In addition, the solutions of the shear stresses at the plates are also determined. The project supported by the National Natural Science Foundation of China (10372007, 10002003) and CNPC Innovation Fund     

Start-up flows of second grade fluids in domains with one finite dimension

A number of unidirectional transient flows of a second grade fluid in a domain with one finite dimension are studied. The method of integral transforms (Fourier, Hankel or Laplace) is applied to obtain exact solutions. A general theorem on start-up flows for second grade fluids is presented that allows us to determine unidirectional flows of second grade fluids once the corresponding solution is known within the context of the Navier-Stokes theory. In the process of obtaining solutions for the fluid of second grade, we find several new exact solutions within the context of the classical Navier-Stokes theory.     

MHD squeezing flow of second‐grade fluid between two parallel disks

This paper examines the unsteady two‐dimensional flow of a second‐grade fluid between parallel disks in the presence of an applied magnetic field. The continuity and momentum equations governing the unsteady two‐dimensional flow of a second‐grade fluid are reduced to a single differential equation through similarity transformations. The resulting differential system is computed by a homotopy analysis method. Graphical results are discussed for both suction and blowing cases. In addition, the derived results are compared with the homotopy perturbation solution in a viscous fluid (Math. Probl. Eng., DOI: 10.1155/2009/603916 ). Copyright © 2011 John Wiley & Sons, Ltd.     

Decay of a potential vortex and propagation of a heat wave in a second grade fluid

The propagation of a heat wave in an incompressible second grade fluid within the context of a potential vortex is studied. The solutions for the Newtonian fluid can be obtained from those for fluids of second grade as a limiting case.     

Uniqueness and drag for fluids of second grade in steady motion

For incompressible fluids of second grade that are compatible with the Clausius-Duhem inequality, non-uniqueness of steady flows with small Reynolds number (i.e. creeping flows) is possible provided the ‘absorption number’ is also small. We discuss this uniqueness question, generalize a well-known theorem of Tanner concerning how solutions of the Stokes equations may be used to generate solutions of the creeping flow equations for fluids of second grade, and give a new uniqueness theorem appropriate to a class of problems for the steady creeping flow of fluids of second grade. Under the conditions for uniqueness, we obtain a simple formula for the drag force on a fixed body which is immersed in a fluid of second grade which is undergoing uniform creeping flow. For bodies with certain geometric symmetries, the non-Newtonian nature of the fluid has no effect upon the drag.     

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.     

The Rayleigh-Stokes problem for heated second grade fluids

The temperature distribution in a second grade fluid subject to a linear flow on a heated flat plate and within a heated edge is determined using the simple and double Fourier sine transforms. At rest, it is the same both for a second grade fluid and for a Newtonian one.     

The Rayleigh-Stokes-Problem for a fluid of Maxwellian type

The velocity fields corresponding to an incompressible fluid of Maxwellian type subjected to a linear flow on an infinite flat plate and within an infinite edge are determined by means of the Fourier sine transforms. They are in close proximity of those of a second grade fluid. The well known solutions for a Navier-Stokes fluid appear as a limiting case of our solutions.     

Hodograph transformation methods in incompressible second grade fluid

Solutions for the equations of motion of a steady plane flow for a second grade fluid are obtained by using hodograph transformation techniques and the results are compared with the corresponding solutions for viscous fluids.     

SIMILARITY SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A SPECIAL NON-NEWTONIAN FLUID IN A SPECIAL COORDINATE SYSTME

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the phicoordinates and velocity potential lines are psi-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As a first approximation, assumption that second grade terms are negligible compared to viscous and third grade terms. Second grade terms spoil scaling transformation which is only transformation leading to similarity solutions for third grade fluid. By ~sing Lie group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions of outcoming nonlinear differential equations are found by using combination of a Runge-Kutta algorithm and shooting technique.     

The effects of the side walls on the flow of a second grade fluid in ducts with suction and injection

The effects of the side walls on the flow in ducts with suction and injection are examined. Three illustrative examples are given. The first example considers the effect of the side walls on the flow over a porous plate. The second example considers the flow between two parallel porous plates and the third example is devoted to the investigation of the flow in a rectangular duct with two porous walls. Exact solution of the governing equation using the no-slip boundary condition and an additional condition are obtained. The expression of the velocity, the volume flux and the vorticity are given. It is found that for large values of the cross-Reynolds number near the suction region the flow for a Newtonian fluid does not satisfy the boundary condition, but it does not behave in the same way for a second grade fluid. Three examples considered show that there are pronounced effects of the side walls on the flows of a second grade fluid in ducts with suction and injection.     

Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

The velocity field and the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. Initially the fluid is at rest, and at time t = 0⁺, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary second grade and Newtonian fluids are obtained as limiting cases of the general solutions. Finally, some characteristics of the motion, as well as the influences of the material and fractional parameters on the fluid motion and a comparison between models, are underlined by graphical illustrations.     

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.     

Certain inverse solutions of a non-Newtonian fluid

Inverse solutions of the equations of motion of an incompressible second grade or order fluid are obtained by assuming certain forms for the stream functions a priori. The equations considered are in plane polar coordinates, axisymmetric polar coordinates and in axisymmetric spherical polar coordinates. Expressions for stream lines, veiocity components and pressure distributions are given explicitly, in each case, and are compared with the corresponding results of a viscous fluid.     

Flow of a generalized second grade non-Newtonian fluid with variable viscosity

A modified constitutive equation for a second grade fluid is proposed so that the model would be suitable for studies where shear-thinning (or shear-thickening) may occur. In addition, the dependence of viscosity on the temperature follows the Reynolds equation. In this paper, we propose a constitutive relation, (18), which has the basic structure of a second grade fluid, where the viscosity is now a function of temperature, shear rate, and concentration. As a special case, we solve the fully developed flow of a non-Newtonian fluid given by (11), where the effects of concentration are neglected.Received: 28 August 2003, Accepted: 3 March 2004, Published online: 25 June 2004 Correspondence to: M. Massoudi Dedicated to Professor Brian Straughan