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.     

Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative

This paper deals with the unsteady helical flows of a generalized Oldroyd-B fluid between two infinite coaxial cylinders and within an infinite cylinder. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions are obtained with the help of integral transforms (Laplace transform, Weber transform and finite Hankel transform). The corresponding solutions for generalized second grade and Maxwell fluids as well as those for the Newtonian and ordinary Oldroyd-B fluids are also given in limiting cases. Finally, the influence of model parameters on the velocity field is also analyzed by graphical illustrations.     

Helical flows of second grade fluid due to constantly accelerated shear stresses

The helical flows of second grade fluid between two infinite coaxial circular cylinders is considered. The motion is produced by the inner cylinder that at the initial moment applies torsional and longitudinal constantly accelerated shear stresses to the fluid. The exact analytic solutions, obtained by employing the Laplace and finite Hankel transforms and presented in series form in term of usual Bessel functions of first and second kind, satisfy both the governing equations and all imposed initial and boundary conditions. In the limiting case when α → 0, the solutions for Newtonian fluid are obtained for the same motion. The large-time solutions and transient solutions for second grade fluid are also obtained, and effect of material parameter α and kinematic viscosity ν is discussed. In the last, the effects of various parameters of interest on fluid motion as well as the comparison between second grade and Newtonian fluids are analyzed by graphical illustrations.     

Exact solutions for some oscillating flows of a second grade fluid with a fractional derivative model

In the present work, the exact analytic solutions for some oscillating flows of a generalized second grade fluid are investigated using Fourier sine and Laplace transforms. A more appropriate model is presented for fluid material between viscous and elastic to introduce the fractional calculus approach into the constitutive relationship. This paper employs the fractional calculus approach to study second grade fluid flows. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method has been used. Similar solutions for second grade fluid appear as the limiting cases of our solutions. The influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn.     

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.     

Exact solutions of starting flows for a fractional Burgers’ fluid between coaxial cylinders

This paper deals with the unsteady flows of a viscoelastic fluid between two infinitely long concentric circular cylinders. The fractional calculus approach in the constitutive relationship model of a Burgers’ fluid is introduced. With the help of integral transforms (the Laplace transform and the Weber transform), exact solutions are constructed for the following two problems: (i) when the outer cylinder makes a simple harmonic oscillation; and (ii) when the outer cylinder suddenly begins rotating while the inner cylinder remains stationary. Some previous and classical results can be recovered from the presented results, such as starting solutions for second grade, Maxwell, Oldroyd-B, and Burgers’ fluids.     

On some generalizations of the second grade fluid model

In this article, we provide a brief review of some generalizations of the second grade fluid model. We discuss certain similarities between these fluids and fluids of higher grades, while also describing certain limitations of these models. The new models that we put forth are based upon some interesting experimental results which suggest that not only can normal stress coefficients depend upon the shear rate, but that this dependency is in fact not the same rate as that of shear viscosity variation with shear rate. We then discuss some steady flows of these generalized second grade fluid models.     

Exact solutions for the unsteady axial flow of Non-Newtonian fluids through a circular cylinder

The velocity field and the shear stress corresponding to the motion of a generalized Oldroyd-B fluid due to an infinite circular cylinder subject to a longitudinal time-dependent shear stress are established by means of the Laplace and finite Hankel transforms. The exact solutions, written under series form, can be easily specialized to give the similar solutions for generalized Maxwell and generalized second grade fluids as well as for ordinary Oldroyd-B, Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid are shown by graphical illustrations.     

Some analytical solutions for second grade fluid flows for cylindrical geometries

This paper deals with some unsteady flow problems of a second grade fluid. The flows are generated by the sudden application of a constant pressure gradient or by the impulsive motion of a boundary. The velocities of the flows are described by the partial differential equations. Exact analytic solutions of these differential equations are obtained. The well known solutions for a Navier–Stokes fluid in the hydrodynamic case appear as the limiting cases of our solutions.     

On some unsteady flows of a non-Newtonian fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. Flows considered are: unsteady flow over a plane wall, unsteady Couette flow, flow between two parallel plates suddenly set in motion with the same speed, flow due to one rigid boundary moved suddenly and one being free, unsteady Poiseuille flow and unsteady generalized Couette flow. The results obtained are compared with those of the exact solutions of the Navier–Stokes equations. It is found that the stress at time zero on the stationary boundary for the flows generated by impulsive motion of a boundary or two boundaries is finite for a fluid of second grade and infinite for a Newtonian fluid. Furthermore, it is shown that for unsteady Poiseuille flow the stress at time zero on the boundary is zero for a Newtonian fluid, but it is not zero for a fluid of second grade.     

On some axial Couette flows of non-Newtonian fluids

The velocity fields corresponding to some flows of second grade and Maxwell fluids, induced by a circular cylinder subject to a constantly accelerating translation along its symmetry axis, are presented as Fourier-Bessel series in terms of the eigenfunctions of some suitable boundary value problems. These solutions satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For α or λ → 0, they are going to those for a Newtonian fluid. Finally, for comparison, some diagrams corresponding to the solutions for the flow through a circular cylinder are presented for different values of t and of the material constants.     

Perturbation analysis of a modified second grade fluid over a porous plate

A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The flow of this fluid is considered over a porous plate. Equations of motion in dimensionless form are derived. When the power-law effects are small compared to second grade effects, a regular perturbation problem arises which is solved. The validity criterion for the solution is derived. When second grade effects are small compared to power-law effects, or when both effects are small, the problem becomes a boundary layer problem for which the solutions are also obtained. Perturbation solutions are contrasted with the numerical solutions. For the regular perturbation problem of small power-law effects, an excellent match is observed between the solutions if the validity criterion is met. For the boundary layer solution of vanishing second grade effects however, the agreement with the numerical data is not good. When both effects are considered small, the boundary layer solution leads to the same solution given in the case of a regular perturbation problem.     

Two-layer flows of generalized immiscible second grade fluids in a rectangular channel

Unsteady one-dimensional flows of two incompressible and immiscible generalized second grade fluids in a rectangular channel are studied. A constant pressure gradient acts in the flow direction, while the channel walls have oscillating translational motions in their planes. The generalization considered in this paper consists into a mathematical model based on constitutive equations of second grade fluid with Caputo time-fractional derivative in which the history of the shear stress influences the velocity gradient. The velocity and shear stress fields in the Laplace transform domain are obtained. Numerical solutions for the real velocity and shear stress have been found by employing the Stehfest numerical algorithm for the inverse Laplace transform. The influence of the fractional parameters on the velocity and shear stress has been studied by numerical simulations and graphical illustrations. It is found that the memory effects are significant only for small values of the time t.     

On some axial Couette flows of non-Newtonian fluids

The velocity fields corresponding to some flows of second grade and Maxwell fluids, induced by a circular cylinder subject to a constantly accelerating translation along its symmetry axis, are presented as Fourier-Bessel series in terms of the eigenfunctions of some suitable boundary value problems. These solutions satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For α or λ → 0, they are going to those for a Newtonian fluid. Finally, for comparison, some diagrams corresponding to the solutions for the flow through a circular cylinder are presented for different values of t and of the material constants. Received: March 18, 2004; revised: October 28, 2004     

Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative

The aim of this paper is to present the analytical solutions corresponding to two types of unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative between two parallel plates. The fractional calculus approach is used in solving the problems. The velocity distributions are determined by means of discrete Laplace transform and finite Fourier sine transform. The obtained results indicate that some well known solutions for the generalized second grade fluid, the generalized Maxwell fluid as well as the ordinary Oldroyd-B fluid appear as the limiting cases of the presented results.     

Some exact solutions for fractional generalized Burgers’ fluid in a porous space

This work is concerned with deriving the equation for describing the magnetohydrodynamic (MHD) flow of a fractional generalized Burgers’ fluid in a porous space. Modified Darcy's law has been taken into account. Closed form solutions for velocity are obtained in three problems. The solutions for Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the obtained solutions. A parametric study of some physical parameters involved in the problems is performed to illustrate the influence of these parameters on the velocity profiles.     

Heat transfer in unsteady axisymmetric second grade fluid

This work looks at the heat transfer effects on the flow of a second grade fluid over a radially stretching sheet. The axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet. Mathematical analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST case) and (ii) prescribed surface heat flux (PHF case). The modelled non-linear partial differential equations in two dependent variables are reduced into a partial differential equation with one dependent variable. The resulting non-linear partial differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence is properly discussed. The series solutions and graphs of velocity and temperature are constructed. Particular attention is given to the variations of emerging parameters such as second grade parameter, Prandtl and Eckert numbers.     

特殊坐标系中特殊非牛顿流边界层方程的相似解

给出了在一个特殊坐标系中三阶流体的二维定常运动方程组．该坐标系中由无粘流体的势流确定，即以环绕任意物体的非粘性流动的流线为Ф-坐标，速度势线为ψ-坐标，构成正交曲线坐标系．结果表明，边界层方程与浸没在流体中的物体的形状无关．第一次近似假定第二梯度项与粘性项和第三梯度项相比，可以忽略不计．第二梯度项的存在，将防碍第三梯度流相似解的比例变换的导出．利用李群方法计算了边界层方程的无穷小生成元．将边界层方程组变换为常微分方程组．利用Runge-Kutta法结合打靶技术求解了该非线性微分方程组的数值解．     

Oscillatory flows of second grade fluid in a porous space

The exact analytical solutions are developed for the magnetohydrodynamic (MHD) flows of the second grade fluid in a porous medium. The analysis is performed using modified Darcy’s law and takes into account the effect of the Hall current. Closed form solutions are given for three problems using the Fourier sine transform. Comparison has been made with the existing results and are found to be in excellent agreement. The graphs are plotted for various emerging parameters and discussed.     

New exact solutions for magnetohydrodynamic flows of an Oldroyd-B fluid

This paper presents the new exact analytical solutions for magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid. The explicit expressions for the velocity field and the associated tangential stress are established by using the Laplace transform method. Three characteristic examples: (i) flow due to impulsive motion of plate, (ii) flow due to uniformly accelerated plate, and (iii) flow due to non-uniformly accelerated plate are considered. The solutions for the hydrodynamic flows are special cases of the presented solutions. Moreover, the similar solutions corresponding to Maxwell and Newtonian fluids in the presence as well as absence of a magnetic field appear as the limiting cases of our solutions. The influences of the exerted magnetic field on the flow are also graphically presented and discussed. In particular, graphical results for the Oldroyd-B fluid are compared with those of a Newtonian fluid.