On accelerated flows of an Oldroyd-B fluid in a porous medium

In this work, the problems dealing with unsteady unidirectional flows of an Oldroyd-B fluid in a porous medium are investigated. By using modified Darcy's law of an Oldroyd-B fluid, the equations governing the flow are modelled. Employing Fourier sine transform, the analytic solutions of the modelled equations are developed for the following two problems: (i) constant accelerated flow, (ii) variable accelerated flow. Explicit expressions for the velocity field and adequate tangential stress are obtained in each case. The solutions for Newtonian, second grade and Maxwell fluids in a porous medium appear as the limiting cases of the present analysis.     

On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid

This paper deals with exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. The fractional calculus approach is used in the constitutive relationship of fluid model. Analytical expressions for the velocity field and the corresponding shear stress for flows due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined using Fourier sine and Laplace transforms. The obtained solutions are presented under integral and series forms in terms of the Mittag–Leffler functions. For α = β = 1, our solutions tend to the similar solutions for ordinary Oldroyd-B fluid. A comparison between generalized and ordinary Oldroyd-B fluids is shown by means of graphical illustrations.     

Exact solution for rotating flows of a generalized Burgers’ fluid in a porous space

This article considers the oscillatory flows of a generalized Burgers’ fluid on an infinite insulating plate when the fluid is permeated by a transverse magnetic field. The effects of Hall current are taken into account. Modified Darcy’s law for a generalized Burgers’ fluid has been used to discuss the flows in a porous medium. The governing time dependent equations in a rotating frame are first developed and then solved for the two problems. The influence of various emerging parameters is discussed through various graphs. The solutions for the Newtonian, second grade, Maxwell, Oldroyd-B and Burgers’ fluids can be obtained from our solutions as the limiting cases.     

Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses

Exact solutions corresponding to the unsteady helical flow of an Oldroyd-B fluid due to an infinite circular cylinder subject to torsional and longitudinal time-dependent shear stresses are established using Hankel transforms. These solutions, presented under series form in terms of Bessel functions J ₀(·), J ₁(·) and J ₂(·), can be easily specialized to give the similar solutions for Maxwell, Second grade and Newtonian fluids performing the same motion. Some characteristics of the motion, as well as the influence of pertinent parameters on the velocity profiles, are underlined by graphical illustrations.     

On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid

This paper deals with exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. The fractional calculus approach is used in the constitutive relationship of fluid model. Analytical expressions for the velocity field and the corresponding shear stress for flows due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined using Fourier sine and Laplace transforms. The obtained solutions are presented under integral and series forms in terms of the Mittag–Leffler functions. For α = β = 1, our solutions tend to the similar solutions for ordinary Oldroyd-B fluid. A comparison between generalized and ordinary Oldroyd-B fluids is shown by means of graphical illustrations.     

Exact solutions of accelerated flows for a Burgers’ fluid II. The cases γ = λ2/4 and γ > λ2/4

The purpose of this study is to provide the exact analytic solutions of accelerated flows for a Burgers’ fluid when the relaxation times satisfy the conditions γ = λ²/4 and γ > λ²/4. The velocity field and the adequate tangential stress that is induced by the flow due to constantly accelerating plate and flow due to variable accelerating plate are determined by means of Laplace transform. All the solutions that have been obtained are presented in the form of simple or multiple integrals in terms of Bessel functions. A comparison between Burgers’ and Newtonian fluids for the velocity and the shear stress is also made through several graphs.     

Exact solutions for the flow of an Oldroyd-B fluid due to an infinite flat plate

The unsteady flow of an Oldroyd-B fluid due to an infinite flat plate, subject to a translation motion of linear time-dependent velocity in its plane, is studied by means of the Laplace transform. The velocity field and the associated tangential stress corresponding to the flow induced by the constantly accelerating plate as well as those produced by the impulsive motion of the plate are obtained as special cases. The solutions that have been determined, in all accordance with the solutions established using the Fourier transform, reduce to those for a Newtonian fluid as a limiting case. The similar solutions for a Maxwell fluid are also obtained.     

Starting solutions for some unsteady unidirectional flows of Oldroyd-B fluids

Analytical expressions for the velocity fields corresponding to the motions of an Oldroyd-B fluid due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient and pressure jumps are determined by means of the Fourier sine transform. The corresponding solutions for a Maxwell and a Newtonian fluid appear as limiting cases of the solutions established here. Relevant physical properties of the flows and their dependence on material and geometry parameters are discussed.     

Starting solutions for some unsteady unidirectional flows of Oldroyd-B fluids

Analytical expressions for the velocity fields corresponding to the motions of an Oldroyd-B fluid due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient and pressure jumps are determined by means of the Fourier sine transform. The corresponding solutions for a Maxwell and a Newtonian fluid appear as limiting cases of the solutions established here. Relevant physical properties of the flows and their dependence on material and geometry parameters are discussed. (Received: August 16, 2005; revised: December 19, 2005)     

On exact solutions of Stokes second problem for a Burgers’ fluid, I. The case γ < λ2/4

In the present work, the exact solutions of Stokes second problem for a Burgers’ fluid are investigated. The expressions for the velocity field and the corresponding tangential stress are obtained when the relaxation times satisfy the condition γ < λ²/4. The solutions have been determined by means of Laplace transform. Only one initial condition is necessary for velocity and these solutions presented in the forms of simple or multiple integrals in terms of Bessel functions. The corresponding solutions for a Newtonian fluid as well as Oldroyd-B fluid appear as the limiting cases of the presented results. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. Moreover, a comparison for velocity is made with Oldroyd-B and Newtonian fluids.     

Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses

Exact solutions corresponding to the unsteady helical flow of an Oldroyd-B fluid due to an infinite circular cylinder subject to torsional and longitudinal time-dependent shear stresses are established using Hankel transforms. These solutions, presented under series form in terms of Bessel functions J ₀(·), J ₁(·) and J ₂(·), can be easily specialized to give the similar solutions for Maxwell, Second grade and Newtonian fluids performing the same motion. Some characteristics of the motion, as well as the influence of pertinent parameters on the velocity profiles, are underlined by graphical illustrations.     

A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid

The velocity field and the adequate tangential stress that is induced by the flow due to a constantly accelerating plate in an Oldroyd-B fluid, are determined by means of Fourier sine transforms. The solutions corresponding to a Maxwell, Second grade and Navier–Stokes fluid appear as limiting cases of the solutions obtained here. However, in marked contrast to the solution for a Navier–Stokes fluid, in the case of an Oldroyd-B fluid oscillations are set up which decay exponentially with time.     

On hydromagnetic rotating flow of an Oldroyd-B fluid near an oscillating plate

An initial value investigation is made of the motion of an incompressible, viscoelastic, conducting Oldroyd-B fluid bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with a constant angular velocity about an axis normal to the plate. The flow is generated from rest in the rotating viscoelastic system due to harmonic oscillations of a given frequency superimposed on the plate in presence of a transverse magnetic field. The exact solutions for the velocity field and the wall shear stress are obtained. The results are examined quantitatively for a particular case of an impulsively moved plate and the effects of various flow parameters on them are discussed. Many known results are found to emerge as limiting cases of the present analysis.     

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.     

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.     

Starting solutions for oscillating motions of an Oldroyd-B fluid over a plane wall

In this paper, we establish the starting solutions for oscillating motions of an Oldroyd-B fluid between two side walls perpendicular to a plane wall. The expressions for the velocity field and the associated tangential stress at the bottom wall are obtained, presented under integral and series form. These satisfy all imposed initial and boundary conditions. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. In the absence of side walls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate. Moreover, the obtained solutions can be specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motions.     

Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field

In this paper, we develop a set of differential equations describing the steady flow of an Oldroyd 6-constant magnetohydrodynamic fluid. The fluid is electrically conducting in the presence of a uniform transverse magnetic field. The developed non-linear differential equation takes into account the effect of the material constants and the applied magnetic field. We presented the solution for three types of steady flows, namely,

(i)

Couette flow

(ii)

Poiseuille flow and

(iii)

generalized Couette flow.

Homotopy analysis method (HAM) is used to solve the non-linear differential equation analytically. It is found from the present analysis that for steady flow the obtained solutions are strongly dependent on the material constants (non-Newtonian parameters) which is different from the model of Oldroyd 3-constant fluid. Numerical solutions are also given and compared with the solutions by HAM.     

Some new exact analytical solutions for helical flows of second grade fluids

The helical flow of a second grade fluid, between two infinite coaxial circular cylinders, is studied using Laplace and finite Hankel transforms. The motion of the fluid is due to the inner cylinder that, at time t = 0⁺ begins to rotate around its axis, and to slide along the same axis due to hyperbolic sine or cosine shear stresses. The components of the velocity field and the resulting shear stresses are presented in series form in terms of Bessel functions J₀(•), Y₀(•), J₁(•), Y₁(•), J₂(•) and Y₂(•). The solutions that have been obtained satisfy all imposed initial and boundary conditions and are presented as a sum of large-time and transient solutions. Furthermore, the solutions for Newtonian fluids performing the same motion are also obtained as special cases of general solutions. Finally, the solutions that have been obtained are compared and the influence of pertinent parameters on the fluid motion is discussed. A comparison between second grade and Newtonian fluids is analyzed by graphical illustrations.     

Unsteady helical flows of Oldroyd-B fluids

The unsteady helical flow of an Oldroyd-B fluid, in an infinite circular cylinder, is studied by using finite Hankel transforms. The motion is produced by the cylinder that, at time t = 0⁺, is subject to torsional and longitudinal time-dependent shear stresses. The solutions that have been obtained, presented under series form, satisfy all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting cases of general solutions. Finally, the influence of the pertinent parameters on the fluid motion is underlined by graphical illustrations.