Some Global Stability Results for Shear Flows of Viscoelastic Fluids

We consider plane shear flows of viscoelastic fluids. For a number of constitutive models, we prove stability of the rest state for perturbations of arbitrary size. We also consider stability of plane Poiseuille flow in a few special cases. This research was supported by the National Science Foundation under Grant DMS-0405810.     

Newtonian Limit of Maxwell Fluid Flows

In this paper, we revise Maxwell’s constitutive relation and formulate a system of first-order partial differential equations with two parameters for compressible viscoelastic fluid flows. The system is shown to possess a nice conservation–dissipation (relaxation) structure and therefore is symmetrizable hyperbolic. Moreover, for smooth flows we rigorously verify that the revised Maxwell’s constitutive relations are compatible with Newton’s law of viscosity.     

Unconditional Stability of Stationary Flows of Compressible Heat-Conducting Fluids Driven by Large External Forces

We prove that any solution to the full Navier-Stokes system of equations of heat-conducting compressible fluid stabilizes to an equilibrium when time tends to infinity.     

Visualization of Unsteady Creeping Viscous Flows Using Vector Products

A finite element method for analyzing unsteady incompressible creeping flows is presented. Marker particles are introduced to analyze the flow motions. To determine the marker position in the element, vector products are used. By checking the signs of the product, the marker position during the transient analysis can be determined in a simple manner. A benchmark-type problem for which an analytical solution is available and the filling process of a simple axisymmetrical mould shape are solved to illustrate this method.     

One-Dimensional Dynamics of Jet Flows of Elastic Fluids

Flows of elasticoviscous liquids in free jets and in drawn filaments are investigated in the one-dimensional approximation. The interaction of the flows in the jet and the jet-forming die is taken into account. The structure of the transition zones, which ultimately degenerate into shock waves, is investigated. It is shown that depending on the type of instantaneous elastic response jet cross-section necking or swelling shocks may occur. The steady-state flow regimes are investigated numerically.     

Stationary Flows of Shear Thickening Fluids in 2D

We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on a domain in \mathbbR²{\mathbb{R}^{2}}. We assume that the stress tensor is generated by a potential of the form H = h (|e(u)|){H = h (|\varepsilon (u)|)}, e(u){\varepsilon (u)} denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions h having the property that h′ (t)/t increases (shear thickening case).     

Regularity Results for Two-Dimensional Flows of Multiphase Viscous Fluids

Global regularity results for weak solutions of the Navier-Stokes equations for two-dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density. As an application, we deduce regularity properties for the integral curves of the corresponding velocity field. Finally, we prove regularity results “in the small” for strong solutions. (Accepted October 10, 1995)     

Steady Flows of Shear-Dependent Oldroyd-B Fluids around an Obstacle

This work is concerned with the study of steady flows of an incompressible viscoelastic fluid of Oldroyd type, with viscosity depending on the second invariant of the rate of deformation tensor in an exterior domain. We establish a result of existence and uniqueness of strong solutions for sufficiently small data and give estimates relating these solutions to those of the corresponding generalized Newtonian fluid.     

Stability of a Maxwell fluid in hydromagnetics

The character of equilibrium of a Maxwell fluid in the presence of a magnetic field has been investigated. It is shown that the solution is characterized by a variational principle when the direction of magnetic field is either horizontal or vertical. For both directions, an approximate solution has been developed for a fluid layer of finite depth and exponentially varying density. It is found that the stability criterion remains unaffected by the viscosity parameters, although they influence the rate at which the unstable stratification departs from the equilibrium position. The question of excitation of waves has also been considered in detail.     

Global Analysis of the Flows of Fluids with Pressure-Dependent Viscosities

 To describe the flows of fluids over a wide range of pressures, it is necessary to take into account the fact that the viscosity of the fluid depends on the pressure. That the viscosity depends on the pressure has been verified by numerous careful experiments. While the existence of solutions local-in-time to the equations governing the flows of such fluids are available for small, special data and rather unrealistic dependence of the viscosity on the pressure, no global existence results are in place. Our interest here is to establish the existence of weak solutions for spatially periodic three-dimensional flows that are global in time, for a large class of physically meaningful viscosity-pressure relationships. (Accepted May 1, 2002) Published online November 15, 2002 Communicated by S. S. ANTMAN     

Vortex Formations in Plane Flows of Viscous and Ideal Fluids

An analytic representation of individual vortex formations with different cross-sections is obtained for plane-parallel flows of viscous and ideal fluids. A flow in which one of the streamlines has cusp is also found. In these flows the vorticity is constant; therefore, the solutions obtained for the equations describing the kinematic variables in ideal fluid flows must simultaneously satisfy the Navier-Stokes equations.     

On Flows of Viscoelastic Fluids of Oldroyd Type with Wall Slip

We consider the boundary-value problem for the steady isothermal flow of an incompressible viscoelastic liquid of Oldroyd type in a bounded domain with a Navier type slip boundary condition. We prove that under some restrictions on the material constants and the data, there exists a strong solution which is locally unique. The proof is based on a fixed point argument in which the boundary-value problem is decomposed into a transport equation and a Stokes system.     

Steady Flows of Viscoelastic Fluids in Domains with Outlets to Infinity

The equations governing the motion of incompressible viscoelastic fluids of Rivlin—Ericksen and Oldroyd type are investigated in domains with cylindrical and paraboloidal outlets to infinity. For sufficiently small fluxes, prescribed in each outlet, existence and uniqueness of solutions are proven in weighted Hölder spaces. In domains with paraboloidal outlets the solution is obtained as a perturbation of the corresponding Navier—Stokes solution and in domains with cylindrical outlets as a perturbation of a flux carrier, constructed by joining together the exact solutions found in each outlet. These exact solutions are shown to be either rectilinear flows of Poiseuille type or flows composed of a rectilinear and of a transverse secondary component.     

Existence and Stability of Solutions for Steady Flows of Fibre Suspension Flows

We establish existence, uniqueness, convergence and stability of solutions to the equations of steady flows of fibre suspension flows. The existence of a unique steady solution is proven by using an iterative scheme. One of the restrictions imposed on the data confirms a well known fact proven in Galdi and Reddy (J Non-Newtonian Fluid Mech 83:205–230, 1999), Munganga and Reddy (Math Models Methods Appl Sci 12:1177–1203, 2002) and Munganga et al. (J Non-Newtonian fluid Mech 92:135–150, 2000) that the particle number N _p must be less than 35/2. Exact solutions are calculated for Couette and Poiseuille flows. Solutions of Poiseuille flows are shown to be more accurate than those of Couette flow when a time perturbation is considered.     

Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids

We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial density is close to a constant in L ²∩L ^∞, the initial temperature is close to a constant in L ², and the initial velocity is small in H ^s ∩L ⁴, where s=0 when n=2 and when n=3. (The L ^p norms must be weighted slightly when n=2.) In particular, the initial data may be discontinuous across a hypersurface of ⁿ . A great deal of qualitative information about the solution is obtained. For example, we show that the velocity, vorticity, and temperature are relatively smooth in positive time, as is the “effective viscous flux”F, which is the divergence of the velocity minus a certain multiple of the pressure. We find that F plays a central role in the entire analysis, particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer. Moreover, F is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself, an effect which is crucial for obtaining time-independent pointwise bounds for the density. (Accepted June 13, 1996)     

Liouville Theorems for Stationary Flows of Shear Thickening Fluids in the Plane

We consider entire solutions of the equations for stationary flows of shear thickening fluids in 2D and prove Liouville results under conditions like global boundedness of the velocity field or finiteness of the energy.     

Liouville-Type Theorems for Steady Flows of Degenerate Power Law Fluids in the Plane

We extend the Liouville-type theorems of Gilbarg and Weinberger and of Koch, Nadirashvili, Seregin and Sverák valid for the stationary variant of the classical Navier–Stokes equations in 2D to the degenerate power law fluid model.     

A Numerical Study of Oscillating Peristaltic Flow of Generalized Maxwell Viscoelastic Fluids Through a Porous Medium

This article presents a numerical study on oscillating peristaltic flow of generalized Maxwell fluids through a porous medium. A sinusoidal model is employed for the oscillating flow regime. A modified Darcy-Brinkman model is utilized to simulate the flow of a generalized Maxwell fluid in a homogenous, isotropic porous medium. The governing equations are simplified by assuming long wavelength and low Reynolds number approximations. The numerical and approximate analytical solutions of the problem are obtained by a semi-numerical technique, namely the homotopy perturbation method. The influence of the dominating physical parameters such as fractional Maxwell parameter, relaxation time, amplitude ratio, and permeability parameter on the flow characteristics are depicted graphically. The size of the trapped bolus is slightly enhanced by increasing the magnitude of permeability parameter whereas it is decreased with increasing amplitude ratio. Furthermore, it is shown that in the entire pumping region and the free pumping region, both volumetric flow rate and pressure decrease with increasing relaxation time, whereas in the co-pumping region, the volumetric flow rate is elevated with rising magnitude of relaxation time.     

Unsteady Rotating Flows of a Viscoelastic Fluid with the Fractional Maxwell Model Between Coaxial Cylinders

The fractional calculus is used in the constitutive relationship model of viscoelastic fluid. A generalized Maxwell model with fractional calculus is considered. Based on the flow conditions described, two flow cases are solved and the exact solutions are obtained by using the Weber transform and the Laplace transform for fractional calculus.The project supported by the National Natural Science Foundation of China (10272067, 10426024), the Doctoral Program Foundation of the Education Ministry of China (20030422046) and the Natural Science Foundation of Shandong University at Weihai. The English text was polished by Keren Wang.     

Stability of Large Ekman Boundary Layers in Rotating Fluids

The aim of this paper is to investigate the stability of Ekman boundary layers for rotating fluids when the Ekman number and the Rossby number go to zero. More precisely, we prove that spectral stability implies linear and nonlinear stabilities of approximate solutions. In particular, we replace the smallness condition obtained with energy methods in [5] by a weaker spectral condition which is sharp.