Electroosmosis over non-uniformly charged surfaces: modified Smoluchowski slip velocity for second-order fluids

In the present paper we focus on deriving the modified Smoluchowski slip velocity of second-order fluids, for electroosmotic flows over plane surfaces with arbitrary non-uniform surface potential in the presence of thin electric double layers (EDLs). We employ matched asymptotic expansion to stretch the electric double layer and subsequently apply regular asymptotic expansions taking the Deborah number ($De$) as the gauge function. Modified slip velocities correct up to $O(De^{2})$ are presented. Two sample cases are considered to demonstrate the effects of viscoelasticity on slip velocity: (i) an axially periodic patterned potential and (ii) a step-change-like variation in the surface potential. The central result of our analysis is that, unlike Newtonian fluids, the electroosmotic slip velocity for second-order fluids does not, in general, align with the direction of the applied external electric field. Proceeding further forward, we show that the slip velocity in a given direction may, in fact, depend on the applied electric field strength in a mutually orthogonal direction, considering three dimensionality of the flow structure. In addition, we demonstrate that the modified slip velocity is not proportional to the zeta potential, as in the cases of Newtonian fluids; rather it depends strongly on the gradients of the interfacial potential as well. Our results are likely to have potential implications so far as the design of charge modulated microfluidic devices transporting rheologically complex fluids is concerned, such as for mixing and bio-reactive system analysis in lab-on-a-chip-based micro-total-analysis systems handling bio-fluids.

Non-Newtonian Effects of Second-Order Fluids on the Hydrodynamic Lubrication of Inclined Slider Bearings

Theoretical study of non-Newtonian effects of second-order fluids on the performance characteristics of inclined slider bearings is presented. An approximate method is used for the solution of the highly nonlinear momentum equations for the second-order fluids. The closed form expressions for the fluid film pressure, load carrying capacity, frictional force, coefficient of friction, and centre of pressure are obtained. The non-Newtonian second order fluid model increases the film pressure, load carrying capacity, and frictional force whereas the center of pressure slightly shifts towards exit region. Further, the frictional coefficient decreases with an increase in the bearing velocity as expected for an ideal fluid.

Modeling Die Swell of Second-Order Fluids Using Smoothed Particle Hydrodynamics

This work presents the development of a weakly compressible smoothed particle hydrodynamics (WCSPH) model for simulating two-dimensional transient viscoelastic free surface flow which has extensive applications in polymer processing industries. As an illustration for the capability of the model, the extrudate or die swell behaviors of second-order and Olyroyd-B polymeric fluids are studied. A systematic study has been carried out to compare constitutive models for second-order fluids available in literature in terms of their ability to capture the physics behind the swelling phenomenon. The effects of various process and rheological parameters on the die swell such as the extrusion velocity, normal stress coefficients, and Reynolds and Deborah numbers have also been investigated. The models developed here can predict both swelling and contraction of the extrudate successfully. The die swell of a second-order fluid was solved for a wide range of Deborah numbers and for two different Reynolds numbers. The numerical approach was validated through the solution of fully developed Newtonian and non-Newtonian viscoelastic flows in a two-dimensional channel as well as modeling the die swell of a Newtonian fluid. The results of these three benchmark problems were compared with analytic solutions and numerical results in literature when pertinent, and good agreements were obtained.

The stress in a dilute suspension of liquid spheres in a second-order fluid

We use an ensemble averaging technique to calculate the average stress for a dilute suspension of liquid drops that are instantaneously spherical. The solvent and the drops consist of second-order fluids with differing properties. The suspension is itself a second-order fluid and its viscosity and normal stress coefficients are determined. For the special case of a rigid sphere suspension the results agree with Koch & Subramanian (J. Non-Newtonian Fluid Mech., vol. 138, 2006, p. 87, and vol. 153, 2008, p. 202). Differences from other results in the literature are discussed.

Numerical Simulation of Die Swell of Second-Order Fluids Using Smoothed Particle Hydrodynamics

This work presents the development of both weakly compressible and incompressible Smoothed Particle Hydrodynamics (SPH) models for simulating two-dimensional transient viscoelastic free surface flow which has extensive applications in polymer processing industries. As an illustration with industrial significance, we have chosen to model the extrudate swell of a second-order polymeric fluid. The extrudate or die swell is a phenomenon that takes place during the extrusion of polymeric fluids. When a polymeric fluid is forced through a die to give a polymer its desired shape, due to its viscoelastic non-Newtonian nature, it shows a tendency to swell or contract at the die exit depending on its rheological parameters. The die swell phenomenon is a typical example of a free surface problem where the free surface is formed at the die exit after the polymeric fluid has been extruded. The swelling process leads to an undesired increase in the dimensions of the extrudate. To be able to obtain a near-net shape product, the flow in the extrusion process should be well-understood to shed some light on the important process parameters behind the swelling phenomenon. To this end, a systematic study has been carried out to compare constitutive models proposed in literature for second-order fluids in terms of their ability to capture the physics behind the swelling phenomenon. The effect of various process and rheological parameters on the die swell such as the extrusion velocity, normal stress coefficients, and Reynolds and Deborah numbers have also been investigated. The models developed here can predict both swelling and contraction of the extrudate successfully. The die swell problem was solved for a wide range of Deborah numbers and for two different Re numbers. The numerical model was validated through the solution of fully developed Newtonian and Non-Newtonian viscoelastic flows in a two-dimensional channel, and the results of these two benchmark problems were compared with analytic solutions, and good agreements were obtained.

EXISTENCE AND UNIQUENESS OF STEADY, FULLY DEVELOPED FLOWS OF SECOND ORDER FLUIDS IN CURVED PIPES

Steady, fully developed flows of second order fluids in curved pipes of circular cross-section have previously been studied using regular perturbation methods.2,3,12,20 These perturbation solutions are applicable for pipes with small curvature ratio: The cross sectional radius of the pipe divided by the radius of curvature of the pipe centerline. It was shown by Jitchote and Robertson12 that perturbation equations could be ill-posed when the second normal stress coefficient is nonzero. Motivated by the singular nature of the perturbation equations, here, we study the full governing equations without introducing assumptions inherent in perturbation methods. In particular, we examine the existence and uniqueness of solutions to the full governing equations for second order fluids. We show rigorously that a solution to the full problem exists and is locally unique for small non-dimensional pressure drop, in agreement with earlier results obtained using a formal expansion in the curvature ratio.12 The results obtained here are valid for arbitrarily shaped cross-section (sufficiently smooth) and for all curvature ratios. An operator splitting method has been employed which may be useful for numerical studies of steady and unsteady flows of second order fluids in curved pipes.