3D flow of a generalized Oldroyd-B fluid induced by a constant pressure gradient between two side walls perpendicular to a plate

This paper deals with the 3D flow of a generalized Oldroyd-B fluid due to a constant pressure gradient between two side walls perpendicular to a plate. The fractional calculus approach is used to establish the constitutive relationship of the non-Newtonian fluid model. Exact analytic solutions for the velocity and stress fields, in terms of the Fox H-function, are established by means of the finite Fourier sine transform and the Laplace transform. Solutions similar to those for ordinary Oldroyd-B fluid as well as those for Maxwell and second-grade fluids are also obtained as limiting cases of the results presented. Furthermore, 3D figures for velocity and shear stress fields are presented for the first time for certain values of the parameters, and the associated transport characteristics are analyzed and discussed.     

Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate is investigated. Exact solutions for the velocity field are established by means of the Fourier and Laplace transforms. The similar solutions for Maxwell and Newtonian fluids can be obtained as limiting cases of our results. In the absence of side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.     

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.     

Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model, induced by a constantly accelerating plate between two side walls perpendicular to the plate, is investigated by means of the integral transforms. Exact solutions for the velocity field are presented under integral and series forms in terms of the derivatives of generalized Mittag–Leffler functions. The corresponding solutions for Maxwell fluids are obtained as limiting cases for β → 1. In the absence of the side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.      

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.     

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.     

Flow of a Maxwell fluid between two side walls induced by a constantly accelerating plate

The unsteady flow of a Maxwell fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate is studied. Exact solutions for the velocity field are established by means of the Fourier sine transforms. The adequate tangential stresses are also determined. The similar solutions for a Newtonian fluid are obtained as limiting cases of our solutions. In the absence of the side walls, the similar solutions for the unsteady flow over an infinite flat plate are recovered. Finally, for comparison, the velocity field in the middle of the channel and the shear stresses at the bottom wall and on the side walls are plotted for different values of the material constants.      

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 a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium

This paper presents an analysis for helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium, where the motion is due to the longitudinal time-dependent shear stress and the oscillating velocity in boundary. The exact solutions are established by using the sequential fractional derivatives Laplace transform coupled with finite Hankel transforms in terms of generalized G function. Moreover, the effects of various parameters (relaxation time, fractional parameter, permeability and porosity) on the flow and heat transfer are analyzed in detail by graphical illustrations.     

Natural convection flow of a generalized second grade fluid between two vertical walls

We study the flow due to natural convection of a non-Newtonian fluid, modeled as a generalized second grade fluid, between two vertical parallel walls. The flow results from the two walls being held at different temperatures. The viscosity of the fluid is taken to be a function of temperature according to Reynolds’ exponential law. We solve for the dimensionless velocity and temperature profiles and study their dependence upon certain material parameters.     

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.     

Unsteady longitudinal flow of a generalized Oldroyd-B fluid in cylindrical domains

Considering a fractional derivative model the unsteady flow of an Oldroyd-B fluid between two infinite coaxial circular cylinders is studied by using finite Hankel and Laplace transforms. The motion is produced by the inner cylinder which is subject to a time dependent longitudinal shear stress at time t = 0⁺. The solution obtained under series form in terms of generalized G and R functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, generalized and ordinary Maxwell, and Newtonian fluids are obtained as limiting cases of our general solutions. The influence of pertinent parameters on the fluid motion as well as a comparison between models is illustrated graphically.     

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.