Couette flow of viscoelastic fluid with constitutive relation involving general Caputo-type fractional derivative

In this paper, the general Caputo-type fractional differential operator introduced by Pr. Anatoly N. Kochubei is applied to the linear theory of viscoelasticity. Firstly, using the general Caputo-type derivative, a generalized linear viscoelastic constitutive equation is proposed for the first time. Secondly, the momentum equation for the plane Couette flow of viscoelastic fluid with the constitutive relation is given as an integrodifferential equation and the analytical solution of the equation is established by employing the separation of variables method. Lastly, for special cases of the general constitutive relation, the analytical solutions are obtained in terms of the Mittag-Leffler functions.     

Couette flow of dipolar fluids

An exact solution for the problem of steady flow of an incompressible, isothermal and homogeneous dipolar fluid in the annular space between two concentric circular cylinders rotating with uniform angular velocities is obtained. The effect of the material constants on the torque acting on the cylinders is studied particular cases.     

Time-dependent MHD Couette flow in a porous annulus

This study presents the solution for the MHD transient Couette flow in an annulus formed by two concentric porous cylinders of infinite length. The fluid flow is induced by either the impulsive or the accelerated movements of the outer cylinder. A uniform magnetic field is assumed to be applied perpendicular to the direction of flow. General solution of the governing equations is obtained using a combination of Laplace transform and the Riemann-sum approximation method of Laplace inversion. The expressions for the skin friction at the two walls are obtained in both cases. The variations of the velocity and the skin friction with respect to the Hartmann number and suction/injection parameter have been discussed. It is found out that suction accelerates the flow whereas injection retards the flow.     

Numerical simulation of flow of viscoelastic fluids

The paper deals with the numerical calculation of highly viscous, non-newtonian fluid flows in apparatus of mechanical engineering. We use a differential constitutive equation to approximate the real behaviour of technical fluids like polymer melts. By means of calculated flow examples in two and three dimensional geometries we demonstrate the influence of the non-newtonian behaviour. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unsteady flow of viscoelastic fluid with the fractional K-BKZ model between two parallel plates

In the present paper, we investigate the unsteady flow of a viscoelastic fluid between two parallel plates which is generated by the impulsively accelerated motion of the bottom plate. Based on the result of (Jaishankar and McKinley, 2014), the fractional K-BKZ constitutive equation is obtained from the fractional Maxwell model. Using respectively the fractional Maxwell model and fractional K-BKZ model, the unidirectional flows between two plates are simulated and compared. The velocity field and shear stress of the flows are calculated by developing efficient finite difference schemes. The results show that the fluid with the fractional Maxwell model gradually loses the viscoelasticity, but the fluid with the fractional K-BKZ model continues to preserve the viscoelasticity. The dependence of the flow velocity on various parameters of the fractional K-BKZ model is analyzed graphically.     

Analysis of viscoelastic fluid flow with temperature dependent properties in plane Couette flow and thin annuli

The steady state flow in very thin annuli has been studied analytically for the case where the annular gap is much smaller than the radius of the inner cylinder and for the outer cylinder rotating at constant angular speed and the inner cylinder at rest. The cylinders were subjected to two different thermal boundary conditions. The exponential effect of temperature on the relaxation time and the viscosity coefficient was accounted into the governing differential equations using Nahme’s law. Effects of viscous dissipation as well as εDe² (viscoelastic index for SPTT constitutive equation) on the dimensionless velocity and temperature profiles have been investigated. Results show that while the properties of the fluid depend on temperature, the velocity and temperature profiles are different compared to those obtained with constant physical properties. The Nahme–Griffith number increases whereas εDe² as a viscoelastic index decreases when temperature dependent physical properties are considered. In addition, the results indicate that the viscous dissipation has a sensible effect on heat transfer and the Nusselt number decreases with an increase in the Nahme–Griffith number.     

Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels

Transient electro-osmotic flow of viscoelastic fluids in rectangular micro-channels is investigated. The general twofold series solution for the velocity distribution of electro-osmotic flow of viscoelastic fluids with generalized fractional Oldroyd-B constitutive model is obtained by using finite Fourier and Laplace transforms. Under three limiting cases, the generalized Oldroyd-B model simplifies to Newtonian model, fractional Maxwell model and generalized second grade model, where all the explicit exact solutions for velocity distribution are found through the discrete Laplace transform of the sequential fractional derivatives. These exact solutions may be able to predict the flow behavior of viscoelastic biological fluids in BioMEMS and Lab-on-a-chip devices and thus could benefit the design of these devices.     

Stability of flow of viscoelastic fluids between arbitrarily spaced cylinders

Résumé On a étudié le problème de la stabilité d'un fluide viscoélastique du troisième ordre contenu entre deux cylindres concentriques. L'approximation habituelle que l'écart entre les cylindres est petit n'est pas faite. On a démontré que pour un fluide viscoélastique le nombre de Taylor critique dépend considerablement de l'écart entre les cylindres.     

Rheological and flow modelling of viscoelastic fluids between eccentric cylinders

Viscoelastic fluid flows within eccentric rotating cylinders are simulated using a finite element based commercial software POLYFLOW, and an approach based on the stream-tube method (STM) and domain decomposition.     

Steady-state flow of an incompressible viscoelastic polymer fluid between two coaxial cylinders

A boundary value problem for a quasi-linear equation determining the velocity profile of a flow of a polymer fluid in a pipe formed by two coaxial cylinders is considered. On the basis of methods of approximation without saturation, a computational algorithm of increased accuracy is developed, making it possible to solve the problem in a wide range of parameters, including record-low values of r ₀, the radius of the inner cylinder.     

Analytic solutions for MHD flow in an annulus

This work aims to determine analytic solutions for a nonlinear problem governing the magnetohydrodynamic (MHD) flow of a third grade fluid in the annulus of rotating concentric cylinders. The influence of the embedded flow parameters on the velocity profile is analyzed.     

Axial Couette flow of a generalized Oldroyd-B fluid due to a longitudinal time-dependent shear stress

Abstract

The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid in an infinite circular cylinder are determined by means of Hankel and Laplace transforms. The solutions that have been obtained, written in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The similar solutions for generalized Maxwell fluids as well as those for ordinary fluids are obtained as limiting cases of our general solutions.     

Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel

The paper presents the transportation of viscoelastic fluid with fractional Maxwell model by peristalsis through a channel under long wavelength and low Reynolds number approximations. The propagation of wall of channel is taken as sinusoidal wave propagation (contraction and relaxation). Homotopy perturbation method (HPM) and Adomian decomposition method (ADM) are used to obtain the analytical approximate solutions of the problem. The expressions of axial velocity, volume flow rate and pressure gradient are obtained. The effects of fractional parameters (α), relaxation time (λ₁) and amplitude (?) on the pressure difference and friction force across one wavelength are calculated numerically for different particular cases and depicted through graphs.     

Laminar swirling flow in an annulus with porous walls

Flow of an incompressible viscous fluid under constant pressure gradient in a porous annulus is considered, with surface mass transfer, both in the radial and circumferential directions. Expressions for the fully developed velocities are then obtained. With these fully developed velocities on a section of the annulus, on which the origin of co-ordinates lies, the downstream decay of the swirl velocity is analysed, subject to the condition that the velocities on the boundaries are now purely radial. The perturbations in the axial and radial velocities, caused by the decaying swirl, are also determined. It is observed that the decay of the swirl velocity is faster for suction at the inner boundary and injection at the outer andvice versa.     

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     

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.     

On hydrodynamics of viscoelastic fluids

In this paper, we study a hydrodynamic system describing fluids with viscoelastic properties. After a brief examination of the relations between several models, we shall concentrate on a few analytical issues concerning them. In particular, we establish local existence and global existence (with small initial data) of classical solutions for an Oldroyd system without an artificially postulated damping mechanism. © 2005 Wiley Periodicals, Inc.     

Effect of Hall current on transient hydromagnetic Couette–Poiseuille flow of a viscoelastic fluid with heat transfer

The unsteady Couette–Poiseuille flow of an electrically conducting incompressible non-Newtonian viscoelastic fluid between two parallel horizontal non-conducting porous plates is studied with heat transfer considering the Hall effect. A sudden uniform and constant pressure gradient, an external uniform magnetic field that is perpendicular to the plates and uniform suction and injection through the surface of the plates are applied. The two plates are kept at different but constant temperatures while the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained using finite difference approximations. The effect of the Hall term, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions is examined.     

Stability of oscillatory two-phase Couette flow

The authors investigate the stability of two-phase Couette flowof different liquids bounded between plane parallel plates.One of the plates has a time-dependent velocity in its own plane,which is composed of a constant steady part and a time-harmoniccomponent. In the absence of time-harmonic modulations, theflow can be unstable to an interfacial instability if the viscositiesare different, and the more viscous fluid occupies the thinnerof the two layers. Using Floquet theory, it is shown analyticallyin the limit of long waves that time-periodic modulations inthe basic flow can have a significant influence on flow stability.In particular, flows which are otherwise unstable for extensiveranges of viscosity ratios can be stabilized completely by theinclusion of background modulations, a finding that can haveuseful consequences in many practical applications.