An experimental study on vortex breakdown in a differentially-rotating cylindrical container

The vortex breakdown phenomenon in a closed cylindrical container with a rotating endwall disk was reproduced. Visualizations were performed to capture the prominent flow characteristics. The locations of the stagnation points of breakdown bubbles and the attendant global flow features were in excellent agreement with the preceding observations. Experiments were also carried out in a differentially-rotating cylindrical container in which the top endwall rotates at a relatively high angular velocity _t, and the bottom endwall and the sidewall rotate at a low angular velocity _sb. For a fixed cylinder aspect ratio, and for a given relative rotational Reynolds number based on the angular velocity difference _t–_sb, the flow behavior is examined as |_sb/_t| increases. For a co-rotation (_sb/_t>0), the breakdown bubble is located closer to the bottom endwall disk. However, for a counter-rotation (_sb/_t<0), the bubble is seen closer to the top endwall disk. For sufficiently large values of _sb, the bubble ceases to exist for both cases.     

An existence result for a class of shape optimization problems

Given a bounded open subset of R ⁿ, we prove the existence of a minimum point for a functional F defined on the family A() of all quasiopen subsets of , under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A() with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal k ^th eigenvalue (or with minimal capacity) with respect to a given elliptic operator.     

A priori Estimates and Existence for Elliptic Systems via Bootstrap in Weighted Lebesgue Spaces

We present a new general method to obtain regularity and a priori estimates for solutions of semilinear elliptic systems in bounded domains. This method is based on a bootstrap procedure, used alternatively on each component, in the scale of weighted Lebesgue spaces L^p()=L^p((x)dx), where (x) is the distance to the boundary. Using this method, we significantly improve the known existence results for various classes of elliptic systems.     

On the Discretization of Distributed-Parameter Systems with Quadratic and Cubic Nonlinearities

Approximate methods for analyzing the vibrations of an Euler--Bernoulli beam resting on a nonlinear elastic foundation are discussed. The cases of primary resonance ( _n ) and subharmonic resonance of order one-half ( 2 _n ), where is the excitation frequency and _n is the natural frequency of the nth mode of the beam, are investigated. Approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equation and boundary conditions. The amplitude and phase modulation equations show that single-mode discretization leads to erroneous qualitative as well as quantitative predictions. Regions of softening (hardening) behavior of the system, the spatial dependence of the response drift, and frequency-response curves are numerically evaluated and compared using both approaches.     

Instability of stationary solutions for equations of curvature-driven motion of curves

A study is made for equations of evolving curves on a two-dimensional square domain. It is assumed that a curve moves depending on its curvature, normal vector, and position and is orthogonal to at its end points. Under some conditions, instability of stationary solutions is proved through an eigenvalue analysis.     

Injectivity and self-contact in nonlinear elasticity

Let be a bounded open connected subset of ³ with a sufficiently smooth boundary. The additional condition det dx vol () is imposed on the admissible deformations : ¯ of a hyperelastic body whose reference configuration is ¯. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W ^1,p (;³), p > 3, that is injective almost everywhere.     

Parabolic Equations in Reifenberg Domains

Natural Sobolev-type estimates are proved for weak solutions of inhomogeneous parabolic equations in divergence form in a bounded cylinder _*=×(0,T] which is -Reifenberg flat in the space direction. The principal coefficients of the operator are assumed to be in BMO space with their BMO semi-norms small enough.     

Variants of the Stokes Problem: the Case of Anisotropic Potentials

We investigate the smoothness properties of local solutions of the nonlinear Stokes problem$\begin{eqnarray*}-\diverg \{T(\eps(v))\} + \nabla \pi &=& g \msp \mbox{on $\Omega$,}\\\diverg v&\equiv & 0 \msp \mbox{on $\Omega$,}\end{eqnarray*}$where v: ⁿ is the velocity field, $\pi$: $ denotes the pressure function, and g: ⁿ represents a system of volume forces, denoting an open subset of ⁿ . The tensor T is assumed to be the gradient of some potential f acting on symmetric matrices. Our main hypothesis imposed on f is the existence of exponents 1 < p q < \infty such that\lambda (1+|\eps|^{2})^{\frac{p-2}{2}} |\sigma|^{2} \leq D^{2}f(\eps)(\sigma ,\sigma) \leq \Lambda (1+|\eps|^{2})^{\frac{q-2}{2}} |\sigma|^{2}holds with suitable constants , > 0, i.e. the potential f is of anisotropic power growth. Under natural assumptions on p and q we prove that velocity fields from the space W ¹ _{p, loc} (; ⁿ ) are of class C ^1, on an open subset of with full measure. If n = 2, then the set of interior singularities is empty.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

Time discretization schemes for the Stefan problem in a concentrated capacity

We consider two different time discretization algorithms for a nonlinear parabolic PDE arising in heat conduction phenomena with phase changes in two adjoining bodies and , where can be considered as the boundary of . Stability, convergence and error estimate results are given for both algorithms.

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Sommario Si studiano due algoritmi di discretizzazione nel tempo di un sistema di equazioni a derivate parziali non lineari paraboliche che governa la conduzione del calore, in presenza di cambiamento di fase, in due corpi congiunti e , di cui possa essere considerato come la frontiera di , Vengono dati risultati di stabilità, convergenza e maggiorazione dell'errore per entrambi gli algoritmi.

     

Existence and Uniqueness of Time-Periodic Physically Reasonable Navier-Stokes Flow Past a Body

Let be a three-dimensional exterior domain of class C^2,, 0<<1. Assume that a Navier-Stokes liquid is moving in under the action of a body force F that is time-periodic of period T, and that the velocity of the liquid is zero at spatial infinity. In this paper we show that, if F satisfies suitable conditions, and its norm, in appropriate function spaces, is sufficiently small, there is at least one time-periodic strong solution. Furthermore, the velocity field v of such a solution decays to zero for large |x| as |x|^–1 and its spatial gradient decays as |x|^–2, both uniformly in time. In addition, the pressure p decays like |x|^–2 and its gradient like |x|^–3, for almost all t[0,T]. In the special case where F is time-independent, these solutions are also time-independent and coincide with Finns physically reasonable solutions [4]. Moreover, we show that our time-periodic solutions are unique in a very large class, namely, the class of time-periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in ×[0,T].     

Qualitative behavior of dissipative wave equations on bounded domains

The qualitative behavior of solutions of the mixed problem u_tt = u-a(x)u_t in IR x , u=0 on IR x , is studied in the case when a>0 and IRⁿ is bounded. Roughly speaking, if aa_min>0, then solutions decay at least as fast as exp t( –1/2a_min), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If a_max is sufficiently small, depending on , then no overdamping occurs.Partially supported by NSF grant NSF GP 34260.This work was partially supported by the National Science Foundation under Grant No. GP 34260     

Fluctuating flow in a rotating fluid

Summary Fluctuating flow of a viscous fluid rotating over a disk whose angular velocity oscillates about a nonzero mean is investigated. Initially the disk and the fluid rotate in the same sense with different angular velocities ₁ and ₂ ( ₂> ₁) and at a particular instant of time, the angular velocity of the disk becomes ₁[1+ sin( )]. The problem is solved as an initial boundary value problem and it is found that for small values of the results of analytical and numerical methods are in excellent agreement. The effect of frequency parameter on surface skin frictions has been analysed for various values of angular velocity ratio s and amplitude parameter .

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Fluktuierende Strömung in einer rotierenden Flüssigkeit

Übersicht Untersucht wird die fluktuierende Strömung einer viskosen Flüssigkeit, die über einer Scheibe, deren Winkelgeschwindigkeit um einen von Null verschiedenen Mittelwert schwankt, rotiert. Anfangs drehen sich die Scheibe und die Flüssigkeit gleichsinnig, aber mit verschiedenen Winkelgeschwindigkeiten ₁ und ₂ ( ₂> ₁). Zu einem Anfangszeitpunkt geht die Winkelgeschwindigkeit der Scheibe über in ₁[1+ sin ( )]. Die Aufgabe wird als Anfangs-/Randwertproblem gelöst. Für kleine Werte stimmen die analytischen und numerischen Ergebnisse hervorragend überein. Für verschiedene Werte des Winkelgeschwindigkeitsverhältnisses und des Amplitudenparameters wurde der Einfluß des Frequenzparameters auf die Reibspannungen an der Scheibe untersucht.

     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

On the collapse solution in limit analysis

The existence and regularity of collapse solutions in limit analysis of a plastic continuum is examined. Collapse fields for stresses and velocity exist as a saddle point for the duality between the static and kinematic formulations. The velocity field is defined as a pair u = (u , u ^T), where u is of bounded deformation in , while u^T is the velocity of the surface. A generalized Green's formula for the collapse fields is proved under certain regularity conditions.     

Influence of rotation of the body and exterior vorticity on the heat transfer near the stagnation point of a blunt body in a supersonic stream

A study is made of the influence of longitudinal vorticity on the hypersonic viscous shock layer near an axisymmetric cooled surface that rotates about the longitudinal axis with angular velocity ₁ [1, 2]. The equations of the viscous shock layer for the neighborhood of the stagnation point are simplified on the basis of the theory of a thin three-dimensional shock layer [1, 2]. The results are given of some calculations of the influence of the parameters and ₁ on the heat transfer and the structure of the shock layer in the case of steady flow. An iterative numerical method proposed by the author [1] is used, and a modification to accelerate the convergence of the iterations is proposed. It is noted that the parameters and ₁ have characteristic ranges of variation, which depend on the Mach and Reynolds numbers, for which the distributions of the streamlines in the shock layer are qualitatively different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 179–182, May–June, 1984.I thank V. Ya. Neiland and Yu. D. Shevelev for helpful discussions of this paper.     

The shape of a liquid surface between rotating concentric cylinders

Summary The shape of the free surface of a second order fluid, enclosed between vertical concentric cylinders with the inner cylinder rotating with angular velocity, is calculated. The calculation uses a perturbation procedure in which, for slow flows, the boundary conditions on the free surface are transformed to boundary conditions on the fixed horizontal surface obtained when is zero. The results are tested experimentally.With 3 figures     

Reduced and Generalized Stokes Resolvent Equations in Asymptotically Flat Layers, Part I: Unique Solvability

We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In the first part we prove the unique solvability in L^q-Sobolev spaces, 1 < q < , by extending the known results in the case of an infinite layer ₀ via a perturbation argument to asymptotically flat layers which are sufficiently close to ₀. Combining this result with standard cut-off techniques and the parametrix constructed in the second part, we prove the unique solvability for an arbitrary asymptotically flat layer. Moreover, we show equivalence of unique solvability of the generalized and the reduced Stokes resolvent equations, which is essential for the second part of this contribution.     

Scalar Minimizers with Fractal Singular Sets

Lack of regularity of local minimizers for convex functionals with non-standard growth conditions is considered. It is shown that for every >0 there exists a function aC() such that the functional admits a local minimizer uW^1,p() whose set of non-Lebesgue points is a closed set with dim()>N–p–, and where 1<p<N<N+<q<+.