Stability of plane Poiseuille–Couette flows of a piezo-viscous fluid

We examine stability of fully developed isothermal unidirectional plane Poiseuille–Couette flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in Hron et al. [J. Hron, J. Málek, K.R. Rajagopal, Simple flows of fluids with pressure-dependent viscosities, Proc. R. Soc. Lond. A 457 (2001) 1603–1622] and Suslov and Tran [S.A. Suslov, T.D. Tran, Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid, J. Non-Newtonian Fluid Mech. 154 (2008) 170–178]. Stability results for a piezo-viscous fluid are compared with those for a Newtonian fluid with constant viscosity. We show that piezo-viscous effects generally lead to stabilisation of a primary flow when the applied pressure gradient is increased. We also show that the flow becomes less stable as the pressure and therefore the fluid viscosity decrease downstream. These features drastically distinguish flows of a piezo-viscous fluid from those of its constant-viscosity counterpart. At the same time the increase in the boundary velocity results in a flow stabilisation which is similar to that observed in Newtonian fluids with constant viscosity.     

On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions

We consider the use of pressure measurements in contraction flows in the determination of the extensional viscosity behaviour of polymer solutions. The experimental data are interpreted on the basis of the recent theory of Binding. The resulting extensional viscosities are compared with those obtained from a commercial Spin Line Rheometer.We conclude that contraction flows provide a convenient means of determining the extensional viscosity of shear-thinning polymer solutions. The case is not so clear for constant viscosity Boger fluids.In the course of the experiments, it is shown that excess pressure losses in the contractions can be brought about by two distinct flow mechanisms in the case of Boger fluids. In the axisymmetric case, both vortex enhancement and excess pressure loss are observed, although there is not a strict one-to-one correlation between these phenomena. In the planar case, vortex enhancement is not conspicuously present, although there is still a substantial excess pressure loss at high flow rates. This excess must be associated with the ‘bulb’ flow field which essentially replaces the vortex-enhancement regime of the axisymmetric case.     

On the flow of fluids through inhomogeneous porous media due to high pressure gradients

Most porous solids are inhomogeneous and anisotropic, and the flows of fluids taking place through such porous solids may show features very different from that of flow through a porous medium with constant porosity and permeability. In this short paper we allow for the possibility that the medium is inhomogeneous and that the viscosity and drag are dependent on the pressure (there is considerable experimental evidence to support the fact that the viscosity of a fluid depends on the pressure). We then investigate the flow through a rectangular slab for two different permeability distributions, considering both the generalized Darcy and Brinkman models. We observe that the solutions using the Darcy and Brinkman models could be drastically different or practically identical, depending on the inhomogeneity, that is, the permeability and hence the Darcy number.     

Flow of fluids with pressure dependent viscosities in an orthogonal rheometer subject to slip boundary conditions

We consider slow steady flows of Navier–Stokes-like fluids with pressure dependent viscosities between rotating infinite parallel plates with Navier slip boundary conditions. We derive exact solutions which correspond to flows in orthogonal and torsional rheometers, and investigate the effect of the slip coefficient and the material parameters on the solutions. We find that even when inertial effects are ignored, vorticity boundary layers develop at the upper boundary due to the pressure dependence of the viscosity. These boundary layers diminish and eventually disappear with increased slippage.     

Existence Theory for the Isentropic Euler Equations

 We establish an existence theorem for entropy solutions to the Euler equations modeling isentropic compressible fluids. We develop a new approach for constructing mathematical entropies for the Euler equations, which are singular near the vacuum. In particular, we identify the optimal assumption required on the singular behavior on the pressure law at the vacuum in order to validate the two-term asymptotic expansion of the entropy kernel we proposed earlier. For more general pressure laws, we introduce a new multiple-term expansion based on the Bessel functions with suitable exponents, and we also identify the optimal assumption needed to validate the multiple-term expansion and to establish the existence theory. Our results cover, as a special example, the density-pressure law where and are arbitrary constants. (Accepted June 10, 2002) Published online December 3, 2002 Communicated by C. M. DAFERMOS     

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

 The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions. Received on 21 August 2000     

Global Solutions to a Model for Exothermically Reacting,Compressible Flows with Large Discontinuous Initial Data

 We prove the global existence of solutions of the Navier-Stokes equations describing the dynamic combustion of a compressible, exothermically reacting fluid, and we study the large-time behavior of solutions, giving necessary and sufficient conditions for complete combustion in certain cases. The adiabatic constants and specific heats of the burned (product) and unburned (reactant) fluids may differ, and the initial data may be large and discontinuous. (Accepted August 31, 2002) Published online January 9, 2003 Communicated by C. M. Dafermos     

Analysis on creeping channel flows of compressible fluids subject to wall slip

Creeping channel flows of compressible fluids subject to wall slip are widely encountered in industries. This paper analyzes such flows driven by pressure in planar as well as circular channels. The analysis elucidates unsteady flows of Newtonian fluids subject to the Navier slip condition, followed by steady flows of viscoplastic fluids, in particular, Herschel–Bulkley fluids and their simplifications including power law and Newtonian fluids, that slip at wall with a constant coefficient or a coefficient inversely proportional to pressure. Under the lubrication assumption, analytical solutions are derived, validated, and discussed over a wide range of parameters. Analysis based on the derived solutions indicates that unsteadiness alters cross-section velocity profiles. It is demonstrated that compressibility of the fluids gives rise to a concave pressure distribution in the longitudinal direction, whereas wall slip with a slip coefficient that is inversely proportional to pressure leads to a convex pressure distribution. Energy dissipation resulting from slippage can be a significant portion in the total dissipation of such a flow. A distinctive feature of the flow is that, in case of the pressure-dependent slip coefficient, the slip velocity increases rapidly in the flow direction and the flow can evolve into a pure plug flow at the exit.     

Global Existence and Large-Time Behavior of Solutions to the Three-Dimensional Equations of Compressible Magnetohydrodynamic Flows

The three-dimensional equations of compressible magnetohydrodynamic isentropic flows are considered. An initial-boundary value problem is studied in a bounded domain with large data. The existence and large-time behavior of global weak solutions are established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent ${\gamma > \frac 32}${\gamma > \frac 32} and constant viscosity coefficients.     

From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations,Existence of Global Weak Solution for the Quasi-Solutions

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are \({C^{\infty}}\) on \({(0,T)\times \mathbb{R}^{N}}\) for any \({T > 0}\). Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to \({\mu(\rho)=\rho^{\alpha}}\) with \({\alpha > 1}\). Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density \({\rho_{0}}\).     

Some analytical solutions for viscometric flows of power-law fluids with heat generation and temperature dependent viscosity

Exact solutions are given for flows of power-law fluids, with heat generation and temperature dependent viscosity, in three situations, namely pressure flow through a circular tube, shear flow between rotating concentric cylinders and shear flow between parallel plates. The application of the results to the experimental determination of the material parameters is discussed and it is shown that the method usually adopted in practice, does in fact lead to accurate estimates.     

Shear flows of a thixotropic fluid

A thixotropic fluid with a viscosity dependent on a structural parameter, which satisfies a very simple kinetic equation, is examined. The stability and evolution of shear flows of such a fluid are investigated. Some classes of problems for which approximate solutions can be obtained are considered. Solutions are obtained for problems of changes in structure in oscillatory Couette flows. The apparent viscoelasticity of thixotropic fluids is analyzed. Some aspects of the thixotropic behavior of blood are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp.3–12, May–June, 1978.In conclusion, I thank S. A. Regirer for discussion of the work and useful comments.     

Global Existence of Solutions for Shear Flow of Certain Viscoelastic Fluids

We prove the global existence in time of solutions to time-dependent shear flows for certain viscoelastic fluids. The essential point in the proof is an a priori estimate for the shear stress. Positive definiteness constraints for the stress play a crucial role in obtaining such estimates. This research was supported by the National Science Foundation under Grant DMS-0405810.     

Pulsatile flows of Leslie–Ericksen liquid crystals

Capillary pulsatile flows of calamitic (rod-like) and discotic nematic liquid crystals are analyzed using the Leslie–Ericksen equations for low-molar mass liquid crystals, using computational, analytical, and scaling methods. The dependence of flow-enhancement and power requirement on frequency, amplitude, pressure drop wave-form, molecular geometry is characterized. The unique roles of orientation-dependent local viscosity and backflow (orientation-driven flow) on flow-enhancement and power requirement are elucidated. The local viscosity effect is shown to be a significant factor in flow-enhancement at all pressure drops, but only affects power requirement at higher pressure drops. Backflow has weak effects on flow-enhancement and large effects on power requirements at low average pressure drops. Amplitude, frequency, and molecular geometry effects are clearly manifested through viscosity and backflow. A detailed comparison with predictions for power law fluids shows a clear correspondence between these non-Newtonian fluids and nematic liquid crystals. The unique distinguishing feature of pulsatile flows of liquid crystals is found to be backflow, such that power increases with increasing frequency, a featured that does not exist in other non-Newtonian fluids due to lack of a strong flow driven by restructuring/re-orientation processes. Future use of these new results may include measurements of viscoelastic parameters that control backflow.     

Application of differential constraint method to exact solution of second-grade fluid

A differential constraint method is used to obtain analytical solutions of a second-grade fluid flow. By using the first-order differential constraint condition, exact solutions of Poiseuille flows, jet flows and Couette flows subjected to suction or blowing forces, and planar elongational flows are derived. In addition, two new classes of exact solutions for a second-grade fluid flow are found. The obtained exact solutions show that the non-Newtonian second-grade flow behavior depends not only on the material viscosity but also on the material elasticity. Finally, some boundary value problems are discussed.     

Breathers for a Relativistic Nonlinear Wave Equation

 A new class of one-dimensional relativistic nonlinear wave equations with a singular δ-type nonlinear term is considered. The sense of the equations is defined according to the least-action principle. The energy and momentum conservation is established. The main results are the existence of time-periodic finite-energy solutions, the existence of global solutions and soliton-type asymptotics for a class of finite-energy initial data. (Accepted May 28, 2002) Published online November 12, 2002 Communicated by G. FRIESECKE     

Global Existence of Solutions in H4 for a Nonlinear Thermoviscoelastic Equations with Non-monotone Pressure

We consider a one-dimensional continuous model of nutron star, which is described by a compressible thermoviscoelastic system with a non-monotone equation of state, due to the effective Skyrme nuclear interaction between particles. We will prove that, despite a possible destabilizing influence of the pressure, which is non-monotone and not always positive, the presence of viscosity and a sufficient thermal dissipation can yield the global existence of solutions in H ⁴ with a mixed free boundary problem for our model.      

Attractors for Gradient Flows of Nonconvex Functionals and Applications

This paper addresses the long-time behaviour of gradient flows of nonconvex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of nonconvex evolution problems. In particular, we discuss the long-time behaviour of solutions of quasistationary phase field models and prove the existence of a global attractor.     

Effects of a transverse flows with variable viscosity in micropolar fluids

A regular perturbation analysis is presented for three laminar natural convection flows in micropolar fluids in liquids with temperature dependent viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface. While these flows have well-known power-low similarity solutions when the fluid viscosity is taken to be constant, they are non-similar when the viscosity is considered to a function of temperature. A single similar flow, that adjacent to a vertical isothermal surface, is also analysed for comparison in order to estimate the extent of validity of perturbation analysis. The formulation used here provides a unified treatment of variable viscosity effects on those four flows. Computed first-order perturbation quantities are presented for all four flows. Numerical results for velocity, angular velocity and thermal functions has been shown graphically or tabulated for different values of micropolar parameters. Received on 20 October 1997     

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.