MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium

Exact analytical solutions for magnetohydrodynamic (MHD) flows of an incompressible second grade fluid in a porous medium are developed. The modified Darcy's law for second grade fluid has been used in the flow modelling. The Hall effect is taken into account. The exact solutions for the unsteady flow induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane has been constructed by means of Fourier sine transforms. The similar solutions for a Newtonian fluid, performing the same motion, appear as limiting cases of the solutions obtained here. The influence of various parameters of interest on the velocity and shear stress at the bottom wall has been shown and discussed through several graphs. A comparison between a Newtonian and a second grade fluids is also made.     

Effects of the side walls on unsteady flow of a second grade fluid over a plane wall

The effects of the side walls on unsteady flow of a second grade fluid over a plan wall are considered. The solution of the governing equation for velocity is obtained by the sine transform method. This gives a correct result for the shear stress at the bottom wall. The shear stress at the bottom wall is minimum at the middle of the plate and it increases near the side walls. It is shown that the mean thickness of the layer of the liquid over the plate increases with time and the ratio of the mean thickness to the distance between the side walls becomes ultimately 0.2714.     

Axial Couette flow of an Oldroyd-B fluid in an annulus

This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for 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 motion are graphically illustrated.     

Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

The velocity field and the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. Initially the fluid is at rest, and at time t = 0⁺, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary second grade and Newtonian fluids are obtained as limiting cases of the general solutions. Finally, some characteristics of the motion, as well as the influences of the material and fractional parameters on the fluid motion and a comparison between models, are underlined by graphical illustrations.     

Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders

The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and at t = 0＋ both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies Ω1 and Ω2. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.     

On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.     

Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates

The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced. Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus. The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates. In addition, the solutions of the shear stresses at the plates are also determined. The project supported by the National Natural Science Foundation of China (10372007, 10002003) and CNPC Innovation Fund     

Unsteady flow of an Oldroyd-B fluid generated by a constantly accelerating plate between two side walls perpendicular to the plate

The velocity field and the adequate tangential stresses corresponding to the unsteady flow of an Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate are established by means of Fourier sine transforms. The solutions corresponding to Maxwell, second grade and Newtonian fluids, performing the same motion, appear as limiting cases of the solutions obtained here. In the absence of the side walls, namely when the distance between walls tends to infinity, all solutions that have been determined reduce to those corresponding to the flow over an infinite plate. Finally, for comparison, the velocity field at the middle of the channel as well as the shear stress on the bottom wall is plotted as a function of y for several values of t and of the material constants. The influence of the side walls on the motion of the fluid is also emphasized by graphical illustrations.     

Non-Newtonian flow over a wedge with suction

A pseudo-similarity solution is obtained for the flow of an incompressible fluid of second grade past a wedge with suction at the surface. The non-linear differential equation is solved using quasi-linearization and orthonormalization. The numerical method developed for this purpose enables computation of the flow characteristics for any values of the parameters K, a and b, where K is the dimensionless normal stress modulus of the fluid, a is related to the wedge angle and b is the suction parameter. A significant effect of suction on the wall shear stress is observed. The present results match exactly those from an earlier perturbation analysis for Kx^2a ? 0·01 but differ significantly as Kx^2a increases.     

Unsteady motion of a non-linear viscoelastic fluid

In this paper, we study the unsteady flow of a generalized second grade fluid. Specifically, we solve numerically the linear momentum equations for the flow of this viscoelastic shear-thinning (shear-thickening) fluid surrounding a solid cylindrical rod that is suddenly set into longitudinal and torsional motion. The equations are made dimensionless. The results are presented for the shear stresses at the wall, related to the drag force; these are physical quantities of interest, especially in oil-drilling applications.     

Model-free inversion of squeeze-flow rheometer data

A method for extracting rheological data from squeeze-flow tests is proposed. The analysis, based upon the lubrication approximation for a generalized Newtonian fluid, differentiates experimental data in order to obtain an estimate of the wall shear rate (as in the Weissenberg–Rabinowitsch correction for the capillary rheometer) and of the wall shear stress. Two examples are discussed. The first is based on an approximate expression for the force required to squeeze a Herschel–Bulkley fluid. The second example concerns a power-law fluid with partial slip at the plates (but non-zero wall shear stress). Second derivatives of the experimental data are required: the interpretation of noisy results is therefore likely to be difficult.     

Exact solutions for the unsteady rotational flow of an Oldroyd-B fluid with fractional derivatives induced by a circular cylinder

In this research article, the unsteady rotational flow of an Oldroyd-B fluid with fractional derivative model through an infinite circular cylinder is studied by means of the finite Hankel and Laplace transforms. The motion is produced by the cylinder, that after time t=0⁺, begins to rotate about its axis with an angular velocity Ωt ^p . The solutions that have been obtained, presented under series form in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The corresponding solutions that have been obtained can be easily particularized to give the similar solutions for Maxwell and Second grade fluids with fractional derivatives and for ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian fluids) performing the same motion, are obtained as limiting cases of general solutions. The most important things regarding this paper to mention are that (1) we extracted the expressions for the velocity field and the shear stress corresponding to the motion of Second grade fluid with fractional derivatives as a limiting case of our general solutions corresponding to the Oldroyd-B fluid with fractional derivatives, this is not previously done in the literature to the best of our knowledge, and (2) the expressions for the velocity field and the shear stress are in the most simplified form, and the point worth mentioning is that these expressions are free from convolution product and the integral of the product of the generalized G-functions. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.     

Flow of a generalized second grade non-Newtonian fluid with variable viscosity

A modified constitutive equation for a second grade fluid is proposed so that the model would be suitable for studies where shear-thinning (or shear-thickening) may occur. In addition, the dependence of viscosity on the temperature follows the Reynolds equation. In this paper, we propose a constitutive relation, (18), which has the basic structure of a second grade fluid, where the viscosity is now a function of temperature, shear rate, and concentration. As a special case, we solve the fully developed flow of a non-Newtonian fluid given by (11), where the effects of concentration are neglected.Received: 28 August 2003, Accepted: 3 March 2004, Published online: 25 June 2004 Correspondence to: M. Massoudi Dedicated to Professor Brian Straughan     

Decay of a potential vortex and propagation of a heat wave in a second grade fluid

The propagation of a heat wave in an incompressible second grade fluid within the context of a potential vortex is studied. The solutions for the Newtonian fluid can be obtained from those for fluids of second grade as a limiting case.     

The normal stress behaviour of suspensions with viscoelastic matrix fluids

 We investigate the variations in the shear stress and the first and second normal stress differences of suspensions formulated with viscoelastic fluids as the suspending medium. The test materials comprise two different silicone oils for the matrix fluids and glass spheres of two different mean diameters spanning a range of volume fractions between 5 and 25%. In agreement with previous investigations, the shear stress–shear rate functions of the viscoelastic suspensions were found to be of the same form as the viscometric functions of their matrix fluids, but progressively shifted along the shear rate axis to lower shear rates with increasing solid fraction. The normal stress differences in all of the suspensions examined can be conveniently represented as functions of the shear stress in the fluid. When plotted in this form, the first normal stress difference, as measured with a cone and plate rheometer, is positive in magnitude but strongly decreases with increasing solid fraction. The contributions of the first and the second normal stress differences are separated by using normal force measurements with parallel plate fixtures in conjunction with the cone-and-plate observations. In this way it is possible for the first time to quantify successfully the variations in the second normal stress difference of viscoelastic suspensions for solid fractions of up to 25 vol.%. In contrast to measurements of the first normal stress difference, the second normal stress difference is negative with a magnitude that increases with increasing solid content. The changes in the first and second normal stress differences are also strongly correlated to each other: The relative increase in the second normal stress difference is equal to the relative decrease of the first normal stress difference at the same solid fraction. The variations of the first as well as of the second normal stress difference are represented by power law functions of the shear stress with an unique power law exponent that is independent of the solid fraction. The well known edge effects that arise in cone-and-plate as well as parallel-plate rheometry and limit the accessible measuring range in highly viscoelastic materials to low shear rates could be partially suppressed by utilizing a custom- designed guard-ring arrangement. A procedure to correct the guard-ring influence on torque and normal force measurements is also presented. Received: 20 December 2000 Accepted: 7 May 2001     

填隙二阶流体下圆球平行于壁平移的粘性阻力

离散元法是分析散体力学行为的数值方法。存在填隙流体时，颗粒之间或颗粒与壁之间产生的法向挤压力和切向阻力、阻力矩，是湿颗粒离散元法的理论基础。二阶流体是以微小偏离牛顿流体本构而考虑时间影响的一种流体。它具有常粘度，并且第一和第二法向应力差正比于剪切率的平方。根据Reynolds润滑理论，采用小参数法，导出了存在填隙二阶流体时，圆球沿平行于平壁缓慢移动时流体的速度场和压力方程，进而求出切向阻力和阻力矩的解析解。有趣的是在推导时所得的速度场和压力方程形式比牛顿流体要复杂得多，但最终结果表明圆球沿平行于平壁移动时因填隙二阶流体引起的切向阻力和阻力矩与牛顿流体时的结果相同。     

The yield surface of viscoelastic and plastic fluids in a vane viscometer

Vane viscometers are often used to investigate the low shear rate properties of plastic fluids. The shear stress is determined by assuming that the material is held in the space between the vane blades so that it behaves like a rigid cylinder. Experimental evidence supports this assumption and the aim of the present study is to model numerically the yield process in a vane rheometer using viscoelastic and plastic fluids. The finite element method has been used to model the behavior of Herschel-Bulkley (Bingham), Casson and viscoelastic (Maxwell type) fluids. The penalty function approach for the pressure approximation and a rotating reference frame are used together with fine meshes containing more than 1300 elements. The results show that for Herschel-Bulkley (Bingham), and Casson fluids a rotating rigid cylinder of fluid is trapped inside the periphery of the vane, the shear stress is uniformly distributed over the surface of the cylinder. Finally a modified second order fluid is used to simulate the viscoelastic behaviour, anticipated to be an intermediate between the elastic deformation and the plastic flow, to provide a more realistic simulation of the yield process about a vane. In this case, contrast with the concentration of the elastic strain rate at the blade tips, a nearly uniform distribution of the plastic shear rate is still found. This implies that the plastic shear always distributes uniformly during the entire yielding process. Evidently the assumption of uniform shear on a rotating cylinder of material occluded in the blades of a vane is a valid and useful model for many types of fluid possessing a yield stress.     

On the unsteady rotational flow of a fractional second grade fluid through a circular cylinder

Here the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized second grade fluid within an infinite circular cylinder are determined by means of the Laplace and finite Hankel transforms. At time t=0 the fluid is at rest and the motion is produced by the rotation of the cylinder around its axis. The solutions that have been obtained are presented under series form in terms of the generalized G-functions. The similar solutions for ordinary second grade and Newtonian fluids are obtained from general solution for β→1, respectively, β→1 and α ₁→0. Finally, the influences of the pertinent parameters on the fluid motion, as well as a comparison between models, is underlined by graphical illustrations.     

磁流变液体的流变力学特性试验和建模

从唯象角度讨论磁流变液体的流变力学特性。基于对磁流变液体的试验 ,讨论了磁感强度、温度和剪切应变速率等对磁流变液剪切应力的影响 ;建立了描述磁流变液剪切应力的 Bingham模型、广义 Bingham模型和非线性模型。