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.     

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.     

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.

     

Instability of a plane axisymmetric flow with a shear discontinuity in the velocity profile

It is proposed to investigate the stability of a plane axisymmetric flow with an angular velocity profile (r) such that the angular velocity is constant when r < r_O – L and r > r_O + L but varies monotonically from ₁ to ₂ near the point r_O, the thickness of the transition zone being small L r_O, whereas the change in velocity is not small ¦₂ – ₁¦ ₂, ₁. Obviously, as L O short-wave disturbances with respect to the azimuthal coordinate (k=m/r_O 1/r_O) will be unstable with a growth rate-close to the Kelvin—Helmholtz growth rate. In the case L=O (i.e., for a profile with a shear-discontinuity) we find the instability growth rate _O and show that where the thickness of the discontinuity L is finite (but small) the growth rate does not differ from _O up to terms proportional to kL 1 and 1/m 1. Using this example it is possible to investigate the effect of rotation on the flow stability. It is important to note that stabilization (or destabilization) of the flow in question by rotation occurs only for three-dimensional or axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–114, January–February, 1985.     

Existence theorem for a minimum problem with free discontinuity set

We study the variational problem Where is an open set in ⁿ ,n2gL ^q () L (), 1q<+, O<, <+ andH _n–1 is the (n–1)-dimensional Hausdorff Measure.     

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.     

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.     

The Hele-Shaw problem and area-preserving curve-shortening motions

We prove existence, locally in time, of a solution of the following Hele-Shaw problem: Given a simply connected curve contained in a smooth bounded domain, find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature. We show that this motion is a curve-shortening motion which does not change the area of the region enclosed by the curve. In case is the whole plane ², we also show that if the initial curve is close to an equilibrium curve, i.e., to a circle, then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity.     

Unsteady viscous shock layer in the case of supersonic motion through an inhomogeneity

The model of a perfect gas is used in a numerical simulation of unsteady effects in a viscous shock layer near the stagnation streamline near the front part of a rotating blunt body in an inhomogeneous external flow with pressure difference P _t8, temperature difference T _t8, and vorticity difference . The evolution of nonlinear disturbances due to the passage of a heated region and a change of the injection regime is followed. A divergence finite-difference scheme of second order of approximation across the shock layer, realized by vector sweeps with allowance for the boundary conditions on the surface of the body and behind the separated bow shock wave, is used.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 138–145, January–February, 1992.     

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.     

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.     

Bestimmung der Schubspannungsfunktion des Blutes mit dem Couette-Rheometer unter Berücksichtigung des Wandverhaltens

Zusammenfassung Bei Blutergeben sich aus Messungen an verschiedenen Couette-Systemen verschiedene Verläufe für die Schubspannungsfunktion, wenn man Wandhaften annimmt. Es wird daher ein Wandeffekt angenommen, bei dem die unentmischte Blutsuspension unmittelbar auf der Wand gleitet. Die Wandgleitgeschwindigkeit wird als Funktion der Wandschubspannung angesetzt und aus Messungen an drei verschiedenen Couette-Systemen bestimmt.Aus der so ermittelten Wandgleitgeschwindigkeit kann die Dicke eines Blutplasmafilmes an der Wand abgeschätzt werden. Sie ergibt sich zu einigen µm. Dadurch wird die angenommene Modellvorstellung für den Wandeffekt bestätigt.Bei Berücksichtigung der ermittelten Wandgleitgeschwindigkeit ergibt sich für die Schubspannungsfunktion aus Messungen an verschiedenen Couette-Systemen derselbe Verlauf. Bei Annahme von Wandhaften ergeben sich dagegen deutlich zu hohe Werte für die Schubspannungsfunktion.

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Summary Measurements with blood in different Couette-Systems are resulting in different shear functionsf() if no wall-slip effect is assumed.Therefore we use the model that the homogeneous blood suspension is sliding directly on the wall. The wall-slip velocity is introduced as a function of the wall shear stress. This wall-slip function can be determined from measurements with three different Couette-Systems.After the wall-slip function is determined the thickness of a plasma film on the wall can be estimated. One gets a thickness of a few µms. Thus the assumed model for the wall effect is confirmed. Measurements with different Couette-Systems evaluated according to the wall-slip model, are leading to the same shear functionf().

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D Rohrdurchmesser - f() Schubspannungsfunktion - f() mittlerer Fehler vonf() - h Länge des Couette-Spaltes - M in der Couette-Strömung übertragenes Drehmoment - M _ij Meßwerte fürM - Mittelwert derM _ij - mittlerer Fehler von - r _a ,r _i äußerer bzw. innerer Begrenzungsradius eines Couette-Systems - r ₁,r ₂,r ₃ Begrenzungsradien der Couette-Systeme I, II, III - s Spaltweite im Couette-System - v _w Wandgleitgeschwindigkeit - Schergeschwindigkeit - Dicke des Wandfilms - ( _w) Funktion des Wandgleitens - Viskosität - Schubspannungsfeld im Couette-System - ₁, ₂, ₃ Schubspannungen an den Stellenr ₁,r ₂,r ₃ in den Couette-Systemen I, II, III bei gleichem Moment - _w Wandschubspannung - Winkelgeschwindigkeitsdifferenz zwischen äußerem und innerem Zylinder im Couette-System - _i vorgegebene Werte für - mittlerer Fehler von - _I, _II, _III für die Couette-Systeme I, II, III - _{I, II, III} mittlere Fehler von _I, _II, _III Vortrag, gehalten auf der Jahrestagung der Deutschen Rheologischen Gesellschaft in Aachen vom 5.–7. März 1979.Mit 5 Abbildungen     

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.     

Steady axisymmetric flow with vorticity of an inviscid fluid in multistage turbomachines

The direct problem of steady axisymmetric flow of a gas with vorticity through a multistage turbomachine is formulated precisely and a generalized solution is constructed by a variational-difference method. The turbomachine is represented schematically by an annular channel in which there are fixed (₁) and rotating (₂) three-dimensional cascades and channels free of them (₀).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–15, October–December, 1981.     

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.

     

Equivalent Conditions for Ω-Inverse Limit Stability

Recently we obtained sufficient conditions for an endomorphism to be -inverse limit stable. That is, if an endomorphism f satisfies weak Axiom A and the no-cycles condition, then f is -inverse limit stable. In this paper we give alternative conditions for -inverse limit stability. The following are equivalent: (a) f satisfies weak Axiom A and the no-cycles condition; (b) the chain recurrent set is prehyperbolic; and (c) the closure of the set of -limit points of f, L ⁺(f), is prehyperbolic with no cycles.     

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.     

Analysis of suddenly started laminar flow in the entrance region of a circular tube

Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U ², dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u ₁ velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - _w wall shear stress     

On motions which preserve ellipsoidal holes

Let be an ellipsoid in ³ contained in a region . Suppose one body occupies the region – in a certain stress-free reference configuration while a second body, the inclusion, occupies the region in a stress-free reference configuration. Assume the inclusion is free to slip at . Now suppose that by changing some variable such as the temperature, pressure, humidity, etc., we cause the trivial deformation y(x)=x of the inclusion to become unstable relative to some other deformation. For example, the inclusion may be made out of such a material that if it were removed from the body, it would suddenly change shape to another stress-free configuration specified by a deformation y=Fx, F F=C, C being a fixed tensor characteristic of the material, at a certain temperature. However, with an appropriate material model for the surrounding body, we expect it will resist the transformation, and both body and inclusion will end up stressed.In a recent paper, Mura and Furuhashi [1] find the following unexpected result within the context of infinitesimal deformations: certain homogeneous deformations of the ellipsoid which take it to a stress-free configuration also leave the surrounding body stress-free. These are essentially homogeneous, infinitesimal deformations which preserve ellipsoidal holes. In this paper, we find all finite homogeneous deformations and motions which preserve ellipsoidal holes.     

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].