Squeeze film flow of ideal elastic liquids

An exact solution is presented for the squeeze film flow of an Oldroyd B. fluid. The solution demonstrates that the flow kinematics is similar to the Newtonian (or Maxwellian) one. Theoretical predictions for constant velocity squeezing are compared to experimental observation for well characterized non-shear thinning elastic fluids. It is shown both theoretically and experimentally that the effect of elasticity in a constant velocity squeeze film flow is to always reduce the load relative to the inelastic (Newtonian) prediction and that this load reduction falls between the upper and lower asymptote prediction by the exact solution for the Oldroyd B fluid. The upper load asymptote is given by the Stefan solution for the viscosity of the polymer solution and the lower asymptote is given by the Stefan solution for the viscosity of the solvent. Experimental observations agree with the theoretical prediction for the Oldroyd B fluid at low shear rates where it is shown that the steady and dynamic flow properties of the test fluids used in the experimental program are well represented by the Oldroyd B constitutive equation. With the exception of the work of Lee et al. [6] for constant load squeezing of a Maxwell fluid, this work represents one of the few cases where experimental observation of large effects due to elasticity are indeed predicted with a constitutive equation which actually describes the steady and dynamic shear properties of the fluids used in the experimental program.     

Response of viscoelastic fluids in extensional flow generated by a pulsating sphere

A simple analysis of the periodic extensional flow generated by a pulsating sphere in an infinite sea of viscoelastic fluid has been carried out. The general procedure is illustrated by two specific constitutive equations: the corotational Jeffreys fluid and the Oldroyd fluid model B. The response of these fluids is reflected in the temporal variation of the pressure on the surface of the sphere, with Reynolds and Deborah numbers and parameters of the constitutive equations as independent variables. For the case of pulsation with infinitesimal amplitude the fluid response is summarised in the form of pressure amplitude and phase lag versus Deborah number plots. The role of the pulsating flow in the characterisation of viscoelastic fluids and the extension of the procedure to other constitutive equations are briefly discussed.     

Ion slip effect on a steady flow through a circular pipe of a dusty conducting Oldroyd 8-constant fluid

In this paper, a steady magnetohydrodynamic (MHD) flow of a dusty incompressible electrically conducting Oldroyd 8-constant fluid through a circular pipe is examined with considering the ion slip effect. A constant pressure gradient in the axial direction and an external uniform magnetic field in the perpendicular direction are applied. A numerical solution is obtained for the governing nonlinear momentum equations by using finite differences. The effect of the ion slip, the non-Newtonian fluid characteristics, and the particle-phase viscosity on the velocity, volumetric flow rates, and skin friction coefficients of both the fluid and particle phases is reported.     

Front‐tracking by the level‐set and the volume penalization methods in a two‐phase microfluidic network

Two‐phase immiscible fluids in a two‐dimensional micro‐channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd‐B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro‐channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level‐set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two‐phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.     

Steady spinning of the oldroyd fluid B: I: Theory

Equations are derived for the isothermal, low-speed fiber spinning of the Oldroyd fluid B. These equations involve three dimensionless groups whose numerical values are restricted to narrow ranges by the boundary conditions that have to be imposed. Asymptotic results are obtained for limiting values of these groups by means of a perturbation analysis. Also, numerical solutions are presented both in terms of velocity profiles and the admissible values of the three dimensionless groups for different fiber draw ratios. These results are discussed in the context of their application to Boger fluids and are also contrasted with the predictions of the Maxwell fluid model. It is found that under appropriate conditions the predictions of the Oldroyd model B can be quite different from those of the upper convected Maxwell model. In particular, the fluid behaves in a Newtonian manner for both very low and very high Deborah number situations.     

Peristaltic flow of a Johnson-Segalman fluid through a deformable tube

To understand theoretically the flow properties of physiological fluids we have considered as a model the peristaltic motion of a Johnson–Segalman fluid in a tube with a sinusoidal wave traveling down its wall. The perturbation solution for the stream function is obtained for large wavelength and small Weissenberg number. The expressions for the axial velocity, pressure gradient, and pressure rise per wavelength are also constructed. The general solution of the governing nonlinear partial differential equation is given using a transformation method. The numerical solution is also obtained and is compared with the perturbation solution. Numerical results are demonstrated for various values of the physical parameters of interest.      

MHD free convective flow past semi-infinite vertical permeable wall

In this paper,the basic equations governing the flow and heat transfer of an incompressible viscous and electrically conducting fluid past a semi-infinite vertical permeable plate in the form of partial differential equations are reduced to a set of non-linear ordinary differential equations by applying a suitable similarity transformation.Approximate solutions of the transformed equations are obtained by employing the perturbation method for two cases,i.e.,small and large values of the suction parameter.From the numerical evaluations of the solution,it can be seen that the velocity field at any point decreases as the values of the magnetic and suction parameters increase.The effect of the magnetic parameter is to increase the thermal boundary layer.It is also found that the velocity and temperature fields decrease with the increase in the sink parameter.     

A three-dimensional non-linear constitutive law for magnetorheological fluids, with applications

In this paper we first summarize the magnetic and mechanical balance equations for magnetorheological fluids undergoing steady motion in the presence of a magnetic field. A general three-dimensional non-linear constitutive law for such a fluid is given for the case in which the magnetic induction vector is used as the independent magnetic variable. The equations are needed for the analysis of boundary-value problems involving fluids with dispersed micron-sized ferrous particles subjected to a time-independent magnetic field. For illustration, the equations are applied, in the case of an incompressible fluid, to the solution of some basic problems. We consider unidirectional flow in a region confined by two infinite parallel plates with a magnetic field applied perpendicular to the plates. Next, we examine two problems involving a circular cylindrical geometry with the fluid occupying the region between two concentric cylinders: axial flow subjected to an axial magnetic field and circumferential flow with a circumferential field. After making some simplifying assumptions on the constitutive law and choosing material parameters, numerical solutions for the velocity profiles are illustrated.     

On axisymmetric flow of Sisko fluid over a radially stretching sheet

This study presents an analysis of the axisymmetric flow of a non-Newtonian fluid over a radially stretching sheet. The momentum equations for two-dimensional flow are first modeled for Sisko fluid constitutive model, which is a combination of power-law and Newtonian fluids. The general momentum equations are then simplified by invoking the boundary layer analysis. Then a non-linear ordinary differential equation governing the axisymmetric boundary layer flow of Sisko fluid over a radially stretching sheet is obtained by introducing new suitable similarity transformations. The resulting non-linear ordinary differential equation is solved analytically via the homotopy analysis method (HAM). Closed form exact solution is then also obtained for the cases n=0 and 1. Analytical results are presented for the velocity profiles for some values of governing parameters such as power-law index, material parameter and stretching parameter. In addition, the local skin friction coefficient for several sets of the values of physical parameter is tabulated and analyzed. It is shown that the results presented in this study for the axisymmetric flow over a radially non-linear stretching sheet of Sisko fluid are quite general so that the corresponding results for the Newtonian fluid and the power-law fluid can be obtained as two limiting cases.     

On non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid

In this work we develop a mathematical model to predict the velocity profile for an unidirectional, incompressible and steady flow of an Oldroyd 6-constant fluid. The fluid is electrically conducting by a transverse magnetic field. The developed governing equation is non-linear. This equation is solved analytically to obtain the general solution. The governing non-linear equation is also solved numerically subject to appropriate boundary conditions (three cases of typical plane shearing flows) by an iterative technique with the finite-difference discretizations. A parametric study of the physical parameters involved in the problems such as the applied magnetic field and the material constants is conducted. The obtained results are illustrated graphically to show salient features of the solutions. Numerical results show that the applied magnetic field tends to reduce the flow velocity. Depending on the choice of the material parameters, the fluid exhibits shear-thickening or shear-thinning behaviours.     

Numerical and analytical solutions of an Oldroyd 8‐constant MHD fluid with nonlinear slip conditions

The flow of an Oldroyd 8‐constant non‐Newtonian MHD fluid is investigated analytically and numerically. The governing equations for the flow field are derived for a steady one‐dimensional flow. The effect of constant applied magnetic field is included and its influence on the flow field is studied. The nonlinear governing equation along with nonlinear boundary conditions is solved analytically and the solution is obtained in an elegant way. Numerical solutions are also obtained using higher order Chebyshev spectral methods. The influence of various non‐Newtonian parameters and applied magnetic field is investigated. Results showing the effect of various physical parameters of the flow are presented and investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

Viscoplastic boundary layer

Boundary layer (BL) solutions around a flat plate for viscoplastic fluids are re-examined, after the precursory study by Oldroyd of low inertia Bingham fluid mechanics. Due consideration is paid to the admissible stress fields far from the obstacle. Normalized Cauchy equations are introduced in the flowing regions. They initially contain inertia, pressure, and viscoplastic terms obtained by adding the yield value and a viscous stress excess in the flowing regions. New similarity solutions for the BL along a flat plate are derived, and the VPBL properties are given, in the limiting case of creeping flows with a dominant yield value. Improvements with respect to Oldroyd solution and relevant aspects are presented and discussed.     

Transient Couette flows of Oldroyd's fluids under imposed torques

We study the transient Couette flow of an Oldroyd fluid that fills the gap between two circular cylinders when a constant torque is suddenly applied to the inner cylinder, the outer one being kept motionless. Contrarily to most former studies, the inertia of the moving boundary is not neglected. We give the exact solutions of this problem for a wide class of initial conditions and we present a rigorous asymptotic analysis for small gap devices when the initial state is stationary. The case of Grade 2 fluids is also considered and treated. We also show in some experimental tests, that the knowledge of the relaxation curve of the angular velocity of the rotor can be used to identify the parameters of the model.     

Unsteady flow of an elastico-viscous fluid between two coaxial circular cylinders

Unsteady flow of an Oldroyd fluid between two coaxial circular cylinders is investigated, the fluid being set in motion as the inner cylinder moves from rest for a certain period with linearly growing speed and then stops suddenly. The Laplace transform technique is used to derive the solution. For the case when the gap between the cylinders is small, a simplified solution is obtained. The expression for the shear stress on the wall of the outer cylinder is obtained and particular cases are discussed.     

Peristaltic motion of a non-Newtonian fluid

Summary To understand theoretically the flow properties of physiological fluids, we have considered as a model the peristaltic motion of a power law fluid in a tube, with a sinusoidal wave of small amplitude travelling down its wall. The solution for the stream function is obtained as a power series in terms of the amplitude of the wave. The stream function and the velocity components are evaluated by solving numerically two point boundary value problems with a singular point at the origin. The influence of the applied pressure gradient along with non-Newtonian parameters on the streamlines and velocity profiles are discussed in detail.     

The Rayleigh and Stokes problems with an incompressible non-Newtonian fluid

Summary The Rayleigh problem or impulsive motion of a flat plate has been solved using a perturbation scheme when the surrounding fluid is representable by the constitutive equations of Oldroyd or Coleman and Noll. The shear stress and normal stress at the wall were expressed analytically for this unsteady motion. Further, an exact solution of the equations was found for a special case of the constitutive equations.The motion of the fluid above a harmonically oscillating plate or the Stokes problem has been determined for a special non-Newtonian fluid. The penetration of the shear wave into the fluid, the energy dissipation, the velocity profiles and the shear and normal stresses at the wall were expressed and compared to an equivalent Newtonian fluid.Some of the features of these non-Newtonian fluids were examined in simple shearing flows, and techniques to calculate some of the material constants discussed.     

Viscous flow between two rotating nonconcentric cylinders for small eccentricity

The flow of a viscous and incompressible fluid between two rotating nonconcentric cylinders is investigated. An approximate solution of the Navier-Stokes equations is obtained by a perturbation method for the case of small eccentricity. A second solution of the basic flow is obtained by imposing the additional geometric restriction of small gap between the two cylinders and employing the asymptotic expansion of Bessel functions by Meissel's series. This second solution is also obtained by formulating a small gap boundary value problem. The transverse velocity profiles are presented for the case when the eccentricity and gap are small and the outer cylinder is stationary.     

Axisymmetric flow and heat transfer of the Carreau fluid due to a radially stretching sheet: Numerical study

The prime objective of this article is to study the axisymmetric flow and heat transfer of the Carreau fluid over a radially stretching sheet. The Carreau constitutive model is used to discuss the characteristics of both shear-thinning and shear-thickening fluids. The momentum equations for the two-dimensional flow field are first modeled for the Carreau fluid with the aid of the boundary layer approximations. The essential equations of the problem are reduced to a set of nonlinear ordinary differential equations by using local similarity transformations. Numerical solutions of the governing differential equations are obtained for the velocity and temperature fields by using the fifth-order Runge–Kutta method along with the shooting technique. These solutions are obtained for various values of physical parameters. The results indicate substantial reduction of the flow velocity as well as the thermal boundary layer thickness for the shear-thinning fluid with an increase in the Weissenberg number, and the opposite behavior is noted for the shear-thickening fluid. Numerical results are validated by comparisons with already published results.     

Similar solution for three‐dimensional flow in an Oldroyd‐B fluid over a stretching surface

The present investigation deals with the three‐dimensional flow of an Oldroyd‐B fluid over a stretching surface. The governing equations for the three‐dimensional flow are developed. Similarity transformations are invoked for the conversion of nonlinear partial differential equations into the coupled system of ordinary differential equations. Computations for the series solution are presented through implementation of homotopy analysis method. The salient features of the involved parameters have been pointed out. Copyright © 2011 John Wiley & Sons, Ltd.     

Surface tension driven oscillations of a bubble in a viscoelastic liquid

Small amplitude surface tension driven oscillations of a spherical bubble in a dilute polymer solution are considered. The rheological properties of the liquid are modelled by using a 3-constant constitutive equation of the Oldroyd type. The Laplace transform of the solution of the initial value problem is inverted numerically. As in the Newtonian fluid case, both a discrete and a continuous spectrum occurs. In addition to the non-dimensional parameters in the corresponding problem for a Newtonian fluid, the results depend on two other parameters: the ratio of the relaxation time of the polymer solution and the time scale of the flow (the Deborah number) and the product of the polymer concentration and the intrinsic viscosity. For small bubbles in an aqueous solution having a small relaxation time, significant additional damping is found even for dilute solutions.