On the global regularity of shear thinning flows in smooth domains

In some recent papers we have been pursuing regularity results up to the boundary, in W2,l(Ω) spaces for the velocity, and in W1,l(Ω) spaces for the pressure, for fluid flows with shear dependent viscosity. To fix ideas, we assume the classical non-slip boundary condition. From the mathematical point of view it is appropriate to distinguish between the shear thickening case, p>2, and the shear thinning case, p<2, and between flat-boundaries and smooth, arbitrary, boundaries. The p<2 non-flat boundary case is still open. The aim of this work is to extend to smooth boundaries the results proved in reference [H. Beirão da Veiga, On non-Newtonian p-fluids. The pseudo-plastic case, J. Math. Anal. Appl. 344 (1) (2008) 175-185]. This is done here by appealing to a quite general method, introduced in reference [H. Beirão da Veiga, On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc., in press], suitable for considering non-flat boundaries.     

Corner effects in free shear layers in rotating flows

Correct treatment of the 0(E ^1/2)×0(E ^1/2) corner regions leads to better agreement between the theory and observations of profiles of axial velocity in free shear layers.

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Sommaire Le traitement correct des 0(E ^1/2)×0(E ^1/2) des coins, permet un meilleur accord entre la théorie et les observations des profils de vitesse axial les couches libres de cisaillement (couches de Stewartson).

     

Stability of stratified shear flows

For a steady plane parallel flow of an inviscid, incompressible fluid of variable density under gravity, it is shown that the complex wave velocity for any unstable mode lies in a semiellipse-type region whose major axis coincides with the diameter of Howard's semicircle, while its minor axis depends on the stratification. If kc_i denotes the complex part of wave frequency and J₀ the minimum of the local Richardson number over the flow domain, it is further established that kc_i → 0+ as $\text{J}{}_0\text{→}\text{1}\text{4}\text{?}$. The case of free upper surface and conditional reduction dependent on the curvature of the basic velocity of the unstable region is also studied.     

Barotropic instability of shear flows

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm-Liouville theory and hypergeometric functions.     

Remarks on shear flows of the Euler equations

The gradient estimates for renormalization solutions to the Euler equations are derived. The proof is based on the boundedness of the solutions to the linear transport equation, component-wisely. A shear flow is a unique globally-in-time strong solution in certain class due to the argument of renormalization solutions. Shear flows give lower bounds for gradient estimates as well as the analyticity rate. The threshold between locally well-posedness and ill-posedness of the Euler equations is clarified in terms of functions spaces of initial data.     

Embedding smooth diffeomorphisms in flows

In this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces. First we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain a class of local Ck diffeomorphisms for k∈N∪{∞,ω} which admit embedding vector fields with some smoothness. Finally we prove that for any k∈N∪{∞} under the coefficient topology the subset of local Ck diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local Ck diffeomorphisms.     

Boundary optimal control of MHD flows

This paper deals with some optimal control problems associated with the equations of steady-state, incompressible magnetohydrodynamics. These problems have direct applications to nuclear reactor technology, magnetic propulsion devices, and design of electromagnetic pumps. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations is derived. The theory is applied to an example.The work of L. S. Hou was supported in part by the Natural Science and Engineering Research Council of Canada under Grant Number OGP-0137436 and by a Simon Fraser University President's Research grant.     

Slow entropy for some smooth flows on surfaces

We study slow entropy in some classes of smooth mixing flows on surfaces. The flows we study can be represented as special flows over irrational rotations and under roof functions which are C² everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol’d flows), we show that in the scale a_n(t) = n(log n)^t slow entropy equals 1 (the speed of orbit growth is n log n) for a.e. irrational α. If the singularity is of power type (x^?γ, γ ∈ (0, 1)) (Kochergin flows), we show that in the scale a_n(t) = n^t slow entropy equals 1 + γ for a.e. α.We show moreover that for local rank one flows, slow entropy equals 0 in the n(log n)^t scale and is at most 1 for scale n^t. As a consequence we get that a.e. Arnol’d and a.e Kochergin flow is never of local rank one.     

Elliptical flows perturbed by shear waves

We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin–Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier–Stokes fluid in an orthogonal rheometer).     

Finite-element formulations of shear localization in granular materials

Displacement and mixed finite element formulations of shear localization in granular materials are presented. The formulations are based on hypoplastic constitutive laws for soils and the mixed-enhanced treatment involving displacement, strain and stress rates as independently varied fields. Included in these formulations are the standard displacement method, the three-field mixed formulation, the method of incompatible modes, the enhanced assumed strain method and the mixed enhanced strain method. Several numerical examples demonstrating the capability and performance of the different finite element formulations are presented. The numerical results are compared with available experimental data of Hostun RF sand and numerical results of Karlsruhe sand on biaxial tests. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Speed-up of combustion fronts in shear flows

This paper is concerned with the analysis of speed-up of reaction-diffusion-advection traveling fronts in infinite cylinders with periodic boundary conditions. The advection is a shear flow with a large amplitude and the reaction is nonnegative, with either positive or zero ignition temperature. The unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity, solving an open problem from Berestycki (Nonlinear PDEs in condensed matter and reactive flows, Kluwer, Doordrecht, 2003). The asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem, and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional Hörmander-type condition on the flow.     

A note on nonlinear stability of plane parallel shear flows

We present a generalized energy functional E for plane parallel shear flows which provides conditional nonlinear stability for Reynolds numbers Re below some value ReE depending on the shear profile. In the case of the experimentally important profiles, viz. combinations of laminar Couette and Poiseuille flow, ReE is shown to be at least 174.     

Exact topological classification of Hamiltonian flows on smooth two-dimensional surfaces

The present paper contains an exact topological classification of all nondegenerate Hamiltonian systems on smooth closed two-dimensional surfaces. Bibliography: 8 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1900, pp. 22–53.     

Hydromagnetic free shear layers in rotating flows

The hydromagnetic vertical free shear layers, which occur in a linearized steady axisymmetric flow of an electrically conducting homogeneous fluid, confined between two electrically insulated differentially rotating identical parallel plates, are analysed.

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Zusammenfassung Die hydromagnetischen vertikalen freien Scherschichten, die in einer linearisierten, stationären, achsensymmetrischen Strömung einer elektrisch leitenden homogenen Flüssigkeit entstehen, werden untersucht, wenn die Flüssigkeit durch zwei differentiell rotierende, identische parallele Platten begrenzt wird.

     

Two dimensional solitary waves in shear flows

In this paper we study existence and asymptotic behavior of solitary-wave solutions for the generalized Shrira equation, a two-dimensional model appearing in shear flows. The method used to show the existence of such special solutions is based on the mountain pass theorem. One of the main difficulties consists in showing the compact embedding of the energy space in the Lebesgue spaces; this is dealt with interpolation theory. Regularity and decay properties of the solitary waves are also established.     

Heat transfer modeling for turbulent shear flows on curved surfaces

This paper examines two models for two-dimensional curved flow heat transfer. Both models are based on the assumption of local-equilibrium turbulence. Yet, one model predicts an increase for the turbulent Prandtl number with convex curvature, while the other gives the opposite behavior. Through examination of the near wall temperature profiles, it is possible to identify the correct behavior for turbulent Prandtl number. It is found that the turbulent Prandtl number for curved flows increases with convex curvature. The discrepancy between the two models is traced to the modeling terms proposed for the pressure-temperature-gradient correlation and for wall corrections. While the introduction of the rapid component and the wall corrections in the modeling of the pressure-strain correlation do not affect the behavior of the resultant shear stress, these modeling terms in the pressure-temperature-gradient correlation cause the turbulent Prandtl number to decrease with convex curvature.

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Zusammenfassung In dieser Arbeit werden zwei Modelle für zweidimensionale Wärmeübertragung in gekrümmten Strömungen untersucht. Beide Modelle basieren auf der Annahme von lokalem Gleichgewicht innerhalb der turbulenten Strömung. Trotzdem resultiert das eine Modell in einer Erhöhung der turbulenten Prandtl-Zahl für konvexe Krümmungen, während das zweite Modell genau das Gegenteil vorhersagt. Es ist möglich, durch Untersuchung des Temperaturprofiles in Wandnähe das korrekte Verhalten der turbulenten Prandtl-Zahl zu identifizieren. Es stellt sich dabei heraus, daß die turbulente Prandtl-Zahl für gekrümmte Strömungen in konvexen Kurven zunimmt. Der Widerspruch zwischen den beiden Modellen läßt sich auf die Abbildungsausdrücke zurückführen, die für die Wechselbeziehungen zwischen den Druck- und Temperaturgradienten und die für die Wandkorrekturen angesetzt worden sind. Während die Einführung der schnellen Komponente und die Wandkorrekturen in der Darstellung der Wechselbeziehung zwischen Druck und Beanspruchung keinen Einfluß auf das Verhalten des resultierenden Schubdruckes haben, verursachen diese Darstellungsausdrücke für die Wechselbeziehung zwischen Druck- und Temperaturgradienten ein Abnehmen der turbulenten Prandtl-Zahl für konvexe Kurven.

     

Exponential mixing for smooth hyperbolic suspension flows

We present some simple examples of exponentially mixing hyperbolic suspension flows. We include some speculations indicating possible applications to suspension flows of algebraic Anosov systems. We conclude with some remarks about generalizations of our methods.