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     

Transient flows of a second grade fluid

Exact analytical solutions for a class of unsteady unidirectional flows of an incompressible second-order fluid are constructed. The flows are generated impulsively from rest by motion of a plate or two plates or by sudden application of a pressure gradient. Expressions for velocity, flux and skin friction are obtained for both large and small times. It is found that large and small times solutions are dependent on the coefficient of viscoelasticity. The solutions corresponding to Newtonian fluids can be easily obtained from those for fluids of second order by letting the viscoelastic parameter to be zero.     

Particle motions in non-Newtonian media

Summary An experimental study of the behaviour of rigid and deformable particles suspended in pseudoplastic and elasticoviscous liquids undergoing slowCouette flow was undertaken. The velocity profiles deviated slightly from those obtained forNewtonian fluids, but the measured angular velocities of rigid spheres showed that the rotation of the field was equal to half the velocity gradient. While the measured angular velocities of rods and discs were in accord with theory applicable toNewtonian liquids, in both non-Newtonian media there was a steady drift in the orbit towards an asymptotic value corresponding to minimum energy dissipation in the flow. Furthermore, discs in elasticoviscous solutions of polyacrylamide at higher shear stresses aligned themselves in the direction of the flow and ceased to rotate.Migration of rigid particles across the planes of shear in the annul us of theCouette was also observed. In pseudoplastic liquids, the migration was towards the region of higher shear, whereas the opposite was true in elasticoviscous liquids.The deformation, orientation and burst of pseudoplastic drops inNewtonian liquids and that ofNewtonian drops in pseudoplastic fluids were similar to those previously in completelyNewtonian systems. With elasticoviscous drops, however, the deformation was smaller than given by theory.As in elasticoviscous fluids, two-body collisions of rigid uniform spheres in the pseudoplastic liquids were unsymmetrical and irreversible, thus differing from collisions inNewtonian systems where complete reversibility is observed.While some of the observed phenomena in elasticoviscous suspensions could be qualitatively interpreted, particle behaviour in the pseudoplastic liquids could not be explained in terms of the known rheological properties of the fluids.

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Zusammenfassung Es wurde experimentell das Verhalten von festen und deformierbaren Teilchen untersucht, die bei der Suspension in strukturviskosen und viskoelastischen Flüssigkeiten einer langsamenCouette-Strömung ausgesetzt sind. Die Geschwindigkeitsprofile zeigten gewisse Abweichungen von denenNewtonscher Flüssigkeiten, aber die gemessenen Winkelgeschwindigkeiten der festen Kügelchen ergaben, daß die Drehung des Feldes gleich dem halben Geschwindigkeitsgradienten war. Die gemessenen Winkelgeschwindigkeiten der Stäbchen und Scheiben stimmten mit der Theorie, die auf Newtonsche Flüssigkeiten zutrifft, überein. In beiden nicht-Newtonschen Flüssigkeiten verschob sich jedoch die Kreisbahn stetig zu einem asymptotischen Wert, der einem Minimum der Dissipationsenergie der Strömung entsprach. Scheibchen in viskoelastischen Lösungen von Polyacrylamid richteten sich bei höherer Scherspannung in Strömungsrichtung aus und zeigten keine Drehung mehr.Es wurden auch Wanderungen von festen Teilchen über die Scherebene im Spalt derCouette-Anordnung beobachtet. In strukturviskosen Flüssigkeiten erfolgte die Wanderung in Richtung der höheren Scherung, während auf elastische Flüssigkeiten das Gegenteil zutraf.Die Deformation, Orientierung und das Aufbrechen strukturviskoser Tröpfchen inNewtonschen Flüssigkeiten und das Verhalten von Newtonschen Tröpfchen in strukturviskosen Flüssigkeiten waren den früher in rein-Newtonschen Systemen beobachteten Phänomenen ähnlich. Die Deformation der viskoelastischen Tröpfchen war jedoch kleiner als die von der Theorie vorhergesagt worden war.Zweikörper-Zusammenstöße zwischen festen gleichförmigen Kügelchen in strukturviskosen Flüssigkeiten waren unsymmetritch und irreversibel. Darin unterschieden sie sich von Zusammenstößen inNewtonschen Flüssigkeiten, in denen völlige Umkehrbarkeit beobachtet worden war.Während einige der beobachteten Phänomene in viskoelastischen Suspensionen qualitativ gedeutet werden konnten, ließ sich das Teilchenverhalten in strukturviskosen Flüssigkeiten nicht anhand der bekannten Theologischen Eigenschaften der Flüssigkeiten erklären.

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This work was supported by the Defence Research Board of Canada (DRB Grant 9530-47).     

Hodograph transformation methods in incompressible second grade fluid

Solutions for the equations of motion of a steady plane flow for a second grade fluid are obtained by using hodograph transformation techniques and the results are compared with the corresponding solutions for viscous fluids.     

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.     

New exact solutions of Stokes' second problem for an MHD second grade fluid in a porous space

We investigate a problem describing the oscillating flow of an incompressible magnetohydrodynamic (MHD) second grade fluid in a porous half space. Exact solutions for sine and cosine oscillations are developed by applying the Laplace transform method. The total obtained solution is a sum of steady and transient solutions. Particular attention is given to the effects of magnetic and porous medium parameters on the velocity. It is shown that previous results for a non-porous medium and hydrodynamic fluid are the limiting cases of the present problem. The results for velocity are plotted and discussed carefully.     

Exact solutions of starting flows for second grade fluid in a porous medium

This paper concentrates on the unsteady flows of a magnetohydrodynamic (MHD) second grade fluid filling a porous medium. The flow modeling involves modified Darcy's law. Three problems are considered. They are (i) starting flow due to an oscillating edge, (ii) starting flow in a duct of rectangular cross-section oscillating parallel to its length, and (iii) starting flow due to an oscillating pressure gradient. Analytical expressions of velocity field and corresponding tangential stresses are developed. These expressions are found to be significantly affected by the applied magnetic field, permeability of the porous medium and the material parameter of the fluid. Moreover, the influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn. Finally, a comparison is also made with the existing results in the literature.     

Plane flow of a fluid of second grade through a contraction

We have studied the flow of a fluid of second grade through three types of plane channels with a contraction. The numerical method makes use of non-symmetric first- and third-order derivatives for maximizing the diagonal terms of the iterative systems. We have examined, in particular, the growth and decay of the corner eddies as a function of the Reynolds and Weissenberg numbers.     

Approximate symmetries of creeping flow equations of a second grade fluid

Creeping flow equations of a second grade fluid are considered. Two current approximate symmetry methods and a modified new one are applied to the equations of motion. Approximate symmetries obtained by different methods and the exact symmetries are contrasted. Approximate solutions corresponding to the approximate symmetries are derived for each method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed in detail.     

High-order numerical methods of fractional-order Stokes' first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractionalorder Stokes' first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial...     

Weakly-imposed Dirichlet boundary conditions for non-Newtonian fluid flow

We propose a new formulation for weakly imposing Dirichlet boundary conditions in non-Newtonian fluid flow. It is based on the Gerstenberger–Wall formulation for Newtonian fluids [1], but extended to non-Newtonian fluids. It uses a stabilization term in the weak form that is independent from the actual fluid model used, except for an adjustable parameter κ, having the physical dimension of a viscosity. The new formulation is tested, combined with an extended finite element method, for the flow past a cylinder between two walls using both a generalized Newtonian and a viscoelastic fluid. It is shown that the convergence is optimal for the generalized Newtonian fluid by comparing with a converged boundary-fitted solution using traditional strong boundary conditions. Also the solution of the viscoelastic fluid compares very well with a traditional solution using a boundary-fitted mesh and strong Dirichlet boundary conditions. For both fluid models we also test various values of the κ parameter and it turns out that a value equal to the zero-shear-viscosity gives good results. But, it is also shown that a wide range of κ values can be chosen without sacrificing accuracy.     

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 + , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The 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 Newtonian fluid are also obtained as limiting cases of our general solutions.     

Stokes’ first problem for a Rivlin-Ericksen fluid of second grade in a porous half-space

Stokes’ first problem for a Rivlin-Ericksen fluid of second grade in a porous half-space is considered under isothermal conditions. Laplace transform techniques are used to determine the exact solution, temporal limits, small-time expressions, and displacement thickness. In addition, special/limiting cases are noted, energy aspects are covered, and numerical results are presented graphically. Most significantly, it is shown that the flow suffers a jump discontinuity on start-up, that due to this jump a nonpositive steady-state development time can result, and that for a special case of the material constants the flow instantly attains its steady-state configuration.     

Symmetry considerations for materials of second grade

The symmetry group associated with a material point of second grade is characterized, thereby eludicating the interplay between first-and second-order strain measures in determining its response to deformation.     

Analytical solutions for non-Newtonian fluid flows in pipe-like domains

This paper deals with some unsteady unidirectional transient flows of an Oldroyd-B fluid in unbounded domains which geometrically are axisymmetric pipe-like. An expansion theorem of Steklov is used to obtain exact solutions for flows satisfying no-slip boundary conditions. The well known solutions for a Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. The steady state solutions are also obtained for t→∞.