Exact solutions for some simple flows of an Oldroyd-B fluid
TL;DR: In this paper, two simple but elegant solutions for the flow of an Oldroyd-B fluid are presented for flow past an infinite porous plate and find that the problem admits an asymptotically decaying solution in the case of suction at the plate, and that in case of blowing it admits no such solution.
Abstract: We present two simple but elegant solutions for the flow of an Oldroyd-B fluid. First, we consider the flow past an infinite porous plate and find that the problem admits an asymptotically decaying solution in the case of suction at the plate, and that in the case of blowing it admits no such solution. Second, we study the longitudinal and torsional oscillations of an infinitely long rod of finite radius. The solutions are found in terms of Bessel functions.
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TL;DR: In this paper, the effect of viscoelasticity on the unsteady flow in porous media is investigated and a y-dependent steady state solution for an Oldroyd-B fluid in the porous half space was obtained by using Fourier sine transform.
Abstract: Based on a modified Darcy’s law for a viscoelastic fluid, Stokes’ first problem was extended to that for an Oldroyd-B fluid in a porous half space. By using Fourier sine transform, an exact solution was obtained. In contrast to the classical Stokes’ first problem for a clear fluid, there is a y-dependent steady state solution for an Oldroyd-B fluid in the porous half space, which is a damping exponential function with respect to the distance from the flat plate. The thickness of the boundary layer, which tends to be a limited value, is also different from that of a clear fluid. The effect of viscoelasticity on the unsteady flow in porous media is investigated. It was found if α>1∕4[(αt∕Re)+Re]2, oscillations in velocity occur obviously and the system exhibits viscoelastic behaviors, where α and αt are nondimensional relaxation and retardation times, respectively, Re is Reynold number in porous media. Some previous solutions of Stokes’ first problem corresponding to Maxwell fluid and Newtonian fluid in por...
304 citations
TL;DR: In this paper, the fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced, and the exact solutions of some unsteady flows of a viscous fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional algebra.
Abstract: The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.
278 citations
TL;DR: In this article, the Fourier sine transforms are used to obtain exact solutions corresponding to the motions of a second grade fluid, due to the cosine and sine oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient.
224 citations
TL;DR: In this paper, exact solutions for the following five problems of an Oldroyd-B fluid are obtained: (i) Stokes problem (ii) modified Stokes Problem (iii) the time-periodic Poiseuille flow due to an oscillating pressure gradient (iv) the non-periodical flows between two boundaries, and (v) symmetric flow with an arbitrary initial velocity.
193 citations
TL;DR: In this article, the homotopy analysis method (HAM) was used to construct the solutions for Couette, Poiseuille and generalized Couette flows for all values of non-Newtonian parameters.
191 citations
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01 Jan 1955
TL;DR: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part, denoted as turbulence as discussed by the authors, and the actual flow is very different from that of the Poiseuille flow.
Abstract: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part. These actual flows show a special characteristic, denoted as turbulence. The character of a turbulent flow is most easily understood the case of the pipe flow. Consider the flow through a straight pipe of circular cross section and with a smooth wall. For laminar flow each fluid particle moves with uniform velocity along a rectilinear path. Because of viscosity, the velocity of the particles near the wall is smaller than that of the particles at the center. i% order to maintain the motion, a pressure decrease is required which, for laminar flow, is proportional to the first power of the mean flow velocity. Actually, however, one ob~erves that, for larger Reynolds numbers, the pressure drop increases almost with the square of the velocity and is very much larger then that given by the Hagen Poiseuille law. One may conclude that the actual flow is very different from that of the Poiseuille flow.
17,321 citations
TL;DR: The invariant forms of rheological equations of state for a homogeneous continuum, suitable for application to all conditions of motion and stress, are discussed in this article, where the right invariance properties can most readily be recognized if the frame of reference is a co-ordinate system convected with the material.
Abstract: The invariant forms of rheological equations of state for a homogeneous continuum, suitable for application to all conditions of motion and stress, are discussed. The right invariance properties can most readily be recognized if the frame of reference is a co-ordinate system convected with the material, but it is necessary to transform to a fixed frame of reference in order to solve the equations of state simultaneously with the equations of continuity and of motion. An illustration is given of the process of formulating equations of state suitable for universal application, based on non-invariant equations obtained from a simple experiment or structural theory. Anisotropic materials, and materials whose properties depend on previous rheological history, are included within the scope of the paper.
1,714 citations
TL;DR: Etude du probleme d'un barreau infiniment long subissant des oscillations longitudinales and de torsion dans un fluide incompressible and homogene du second ordre as discussed by the authors.
Abstract: Etude du probleme d'un barreau infiniment long subissant des oscillations longitudinales et de torsion dans un fluide incompressible et homogene du second ordre
79 citations
"Exact solutions for some simple flo..." refers background in this paper
...The longitudinal and torsional oscillations of a rod in a fluid of the differential type of second grade were studied by Rajagopal [ 4 ]....
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TL;DR: In this article, an analytical expression for the viscous drag on a smooth circular cylindrical rod-like cable oscillating with longitudinal and torsional motion is obtained based on a linear damping law, and an approximate value for the drag coefficient is also presented.
Abstract: An analytical expression for the viscous drag on a smooth circular cylindrical rodlike cable oscillating with longitudinal and torsional motion is obtained. Based on a linear damping law, an approximate value for the drag coefficient is also presented. The analysis and con- clusions are useful when dealing with many ocean engineering problems of practical importance. the analysis of many practical problems related to the sea-state excitation of long rods and cable-body systems for salvage, buoy, oil drilling, and towing operations, an expression for the viscous drag force acting on the longi- tudinal rod or cable is required. It has been shown that the inclusion of this term can have a pronounced effect on the tension in the rod or cable and subsequent response of the system. For example, the propagation and attenuation of internal stress waves and snap loads through the cable in a cable lifting system are affected by the external viscous damping.1 The problem is also of considerable interest in offshore oil drilling problems since a similar analysis is usually performed.2 It is a common practice to postulate a linear relationship between the external damping force and the surface velocity of the rod when investigating the longitudinal vibrations based on the wave equation. The solution of this damped wave equation with an arbitrary damping coefficient can show the qualitative effects of external damping. However, the magnitude and frequency dependence of the damping coefficient can significantly alter the nature of the solution. The purpose of this investigation is to formulate an analyti- cal expression for the viscous drag on a smooth circular cylindrical rodlike cable oscillating with longitudinal and torsional motion. An examination of the hypothesis of the linear damping law will be made and the damping coefficient estimated.
53 citations
"Exact solutions for some simple flo..." refers background in this paper
...The problem of torsional oscillations of a rod in a Navier-Stokes fluid was studied by Stokes [2], and that of torsional and longitudinal oscillations by Casarella and Laura [ 3 ]....
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