A class of nonlinear,completely integrable abstract wave equations

IfL is a positive self-adjoint operator on a Hubert spaceH, with compact inverse, the second-order evolution equation int,u+Lu+u _H ² u=0 has an infinite number of first integrals, pairwise in involution. It follows from this that no nontrivial solution tends weakly to 0 inH ast. Under an additional separation assumption on the eigenvalues ofL, all trajectories (u,u) are relatively compact inD(L ^1/2)×H. Finally, if all the eigenvalues are simple, the set of initial values of quasi-periodic solutions is dense in the ball B=(u ₀,u ₀ )D(L ^1/2)×H; L^1/2 u _{0 H} ² +u ² < for sufficiently small.     

A direct formulation of correlations for wall-proximity corrections of hot-wire measurements

Correlations for corrections to hot-wire data for the effects of wall proximity within the viscous sublayer are usually presented in the form u/u = F (y u /). The application of such correlations requires a prior knowledge of the wall shear stress; alternatively, the correlation must be used in an iterative fashion. It is shown in the present note that any such correlation may be recast with no loss of generality in the explicit form u/u _m = f (y u _m/), which is more convenient for use.List of symbols u difference between measured and actual velocities, u _m – u - u _m measured velocity - u shear velocity, - u ⁺ on-dimensional velocity, u/u - y distance from wall - y ⁺ non-dimensional distance from wall, y u / - fluid density - fluid kinematic viscosity - _s wall shear stress     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

On nonlinear problems of mixed type: A qualitative theory using infinite-dimensional center manifolds

We consider the equation a(y)u_xx+div_y(b(y)_yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L₂() of the operatorAu= (1/a) div_y(b_yu)+(c/a) in the case ofb changing sign. This leads to the abstract problem u_xx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, u_x) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x) Ce ^-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear.     

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.     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

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.     

Bursting and structure of the turbulence in an internal flow manipulated by riblets

The buffer layer of an internal flow manipulated by riblets is investigated. The distributions of the ejection and bursting frequency from the beginning to the middle part of the buffer layer, together with high moments of the fluctuating streamwise velocity,u, and its time derivative are reported. The profiles of the ejection and bursting frequency are determined and compared using three single point detection schemes. The effect of the riblets on the bursting mechanism is found confined in a localized region in the buffer layer. The multiple ejection bursts are more affected than the single ejection bursts. The skewness and flatness factors of theu signal are larger in the manipulated layer than in the standard boundary layer. That, also holds true for the flatness factor of the time derivative, but the Taylor and Liepman scales are not affected. The spectrum of theu signal is altered at the beginning part of the viscous sublayer.Nomenclature u Friction velocity - Viscosity - l _v ;f _v wall scalesv/u ;u ² /v - y Vertical distance to the wall - z Spanwise extent - (⁺) Variable normalized with wall scales - u Velocity;u=Turbulence intensity - h, s Height and width of the riblets - f _e Ejection frequency - f _b Bursting frequency - f _BME Frequency of the Bursts with Multiple Ejection - f _BSE Frequency of Single Ejection Bursts - S andS _du/dt Skewness factor ofu and its time derivative - F _u andF _du/dt Flatness factor ofu and its time derivative Abbreviations SBL Standard (non-manipulated) Boundary Layer - MBL Manipulated Boundary Layer - BME Bursts with Multiple Ejections - BSE Bursts with Single Ejections - VITA Variable Interval Time Averaging technique - u–l u-level technique - mu Modifiedu-level technique     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

Take-off threshold velocity of solid particles lying under a turbulent boundary layer

The take-off of free solid particles by wind erosion has been investigated. A preliminary bibliographic study (Foucaut and Stanislas 1995) has enabled an analysis of the Bagnold criterion (1941) to be made and its accordance with the results obtained by White (1982) to be shown, thus leading to a semi-empirical model. This initial study led to a more judicious choice of the primary parameters, allowing a more physical representation of the threshold velocity. In the present study, the criterion validation was carried out in a specific boundary layer wind tunnel, by means of a direct measurement of the threshold velocity. The basic idea was to increase the wind tunnel velocity slowly and linearly and to perform an optical detection of the first take-offs.List of symbols D _p particle diameter - D _Pref reference diameter - _p D _p /D _pref - g gravity - H shape factor = (/) - h ⁺ roughness parameter ( =u D _p /2) - R _f =u D _p / - R _x u _e x/v - û u _e /v - u ⁺ flow velocity ( =u/u ) - u _e external velocity - u _elim external threshold velocity - û _e u _e /u _elim - u friction velocity - u ^* threshold friction velocity - u _tref reference velocity - ^* u ^*/u ^ref - y ⁺ yu / - _p ( _p –)/ - boundary layer thickness - displacement thickness - dynamic viscosity - v kinematic viscosity - momentum thickness - fluid density - _p particle density - standard deviation of particle diameter - shear stress Research carried out at Ecole des Mines de Douai     

Diffuse approximation method for solving natural convection in porous media

The diffuse approximation is presented and applied to natural convection problems in porous media. A comparison with the control volume-based finite-element method shows that, overall, the diffuse approximation appears to be fairly attractive.Nomenclature H height of the cavities - I functional - K permeability - p(M _i ,M) line vector of monomials - p ^T p-transpose - M current point - Nu Nusselt number - Ri inner radius - Ro outer radius - Ra Rayleigh number - x, y cartesian coordinates - u, v velocity components - T temperature - M vector of estimated derivatives - _t thermal diffusivity - coefficient of thermal expansion - practical aperture of the weighting function - scalar field - (M, M _i ) weighting function - streamfunction - kinematic viscosity     

Über Differentialgleichungen für Wärme- und Stoffaustausch von hygroskopischen Stoffen

Zusammenfassung Die beiden Differentialgleichungssysteme vonKrischer undLykow werden miteinander verglichen. Dabei ergibt sich, daß die in der deutschen und russischen Literatur angewandten mathematischen Modelle der Trocknung von kapillarporösen Körpern praktisch übereinstimmen. Es werden die Transformationsgleichungen der dimensionslosen Kenngrößen angegeben, die die Beziehungen zwischen den beiden Systemen herstellen.

---

The differential equations ofKrischer andLuikow for unsteady internal heat and mass transfer in the porous medium are compared. It is shown, that the mathematical models for drying in the German and Russian literature are equivalent. The transform relations of the non-dimensional parameters between the two models are given.

---

Formelzeichen nach Krischer z laufende Koordinate in Strömungsrichtung in m - R kennzeichnende Abmessung des Körpers in m - t Zeit in h - Raumgewicht bei mittlerer Feuchtigkeit in kg/m³ - _w Teilgewicht des Wassers in 1 m³ Trockengut in kg/m³ - _wa Anfangsfeuchtigkeit in kg/m³ - _D Dampfdichte in kg/m³ - _L Luftvolumen je m³ Trocknungsgut in m³/m³ - Temperatur in °C - _u Umgebungstemperatur in °C - _a Anfangstemperatur in °C - r Verdampfungswärme in kcal/kg - q _E Wärmeentwicklung in kcal/m³ h - c spezifische Wärmekapazität des Trockengutes in kcal/kg grd - Wärmeleitfähigkeit in kcal/m h grd - Feuchtigkeitsleitzahl des Trockengutes in m²/h - wirksame Diffusionszahl von Wasserdampf in Luft in m²/h - Diffusionswiderstandszahl des Trockengutes — - Konstante — - Konstante in kg/m³ grd Formelzeichen nach Lykow X=r/R dimensionslose Koordinate des Körpers;r laufende Koordinate in m;R kennzeichnende Abmessung in m; - Fo=a/R ² Fourier-Zahl;a Temperaturleitzahl in m²/h; Zeit in h - T(X, Fo)=t(r, )– 0/t dimensionslose Temperatur des Körpers im Punkt mit KoordinateX für den ZeitpunktFo;t(r, ) Temperatur in °C; 0 mittlere Anfangstemperatur in °C; t ein vorher angenommener Temperaturunterschied in grd - (X, Fo)= ₀–u(r, )/u dimensionsloses Potential des Stoffübergangs im Punkt mit KoordinateX für den ZeitpunktFo;u(r, ) Feuchtigkeitsgehalt des Trockengutes in kg/kg; ₀ mittlerer Anfangsfeuchtigkeitsgehalt in kg/kg; u ein vorher angenommener Unterschied des Feuchtigkeitsgehalts in kg/kg - Ko= u/c t Kosowitsch-Zahl; Verdampfungswärme in kcal/kg;c spezifische Wärmekapazität des Trockengutes in kcal/kg - Ko*=Ko modifizierte Kosowitsch Zhal; Kenngröße der Zustandsänderung - Pn= t/u Posnowsche Zahl;=a _T ^m /a _m Thermogradientkoeffizient in 1/grd;a _T ^m thermische Stoffübergangszahl (charakterisiert den Stoffstrom unter der Einwirkung von Temperaturgradienten) in m²/h grd;a _m Stoffübergangszahl (charakterisiert den Stoffstrom unter der Einwirkung von Feuchtigkeitsgradienten) in m²/h - Lu=a _m/a Lykowsche Zahl - Ki _q=q _q ()·R/ _q t dimensionsloser Wärmestrom (Kirpitschew-Zahl);q _q() Wärmestrom durch die Körperoberfläche in kcal/m²; _q Wärmeleitfähigkeit in kcal/m² h grd - Ki _m=q _m ()·R/a _{m 0} u dimensionsloser Stoffstrom;q _m() Stoffstrom durch die Körperoberfläche in kg/m² h; ₀ Wichte des Trockengutes in kg/m³     

Elastico-viscous properties of deflocculated china clay suspensions — concentration effects

Summary The physical properties of deflocculated china clay suspensions are studied in a combined steady and low-amplitude oscillatory shear flow. Concentration effects are examined and it is shown that, with increasing concentration, an initial shear thinning region is followed by a shear thickening one. Qualitative agreement is obtained between theory and experiment for a range of concentrations of suspensions, all of which exhibit marked elastic properties. The experimental results were obtained using a Weissenberg Rheogoniometer.

---

Zusammenfassung Es werden die physikalischen Eigenschaften deflockulierter Suspensionen von Porzellanerde in einer kombinierten stationären und oszillatorischen Scherströmung mit niedriger Amplitude studiert. Der Einfluß der Konzentration wird untersucht, und es wird gezeigt, daß mit wachsender Konzentration sich an den anfänglich allein vorhandenen Bereich mit Scherentzähung ein Bereich mit Scherverzähung anschließt. Zwischen Theorie und Experiment wird eine qualitative Übereinstimmung in einem Konzentrationsbereich gefunden, in dem ausgeprägte viskoelastische Eigenschaften vorhanden sind. Die experimentellen Ergebnisse werden mit Hilfe eines Weissenberg-Rheogoniometers erhalten.

---

c phase lag in oscillatory testing - D(t – t) deformation history - F, G non-dimensional complex functions of - complex conjugate ofF - G dynamic rigidity - i - I % increase in mean couple under superposed shear rates - I ₁ moment of inertia of the top platen (i.e. cone) - J amplitude ratio, ₁/ ₁ - K ₁ restoring constant of the torsion bar - q steady shear rate - r, , spherical polar coordinates - t current time - v _i velocity vector - w/w concentration by weight - W a function of andt - ₁ angular amplitude of the motion of the plate - shear rate - /q - apparent viscosity - dynamic viscosity - ^* complex dynamic viscosity - ₀ limiting viscosity at small rates of shear - ₀ gap angle in cone and plate system - ₁, ₂, ₃, ₄,µ ₀ relaxation time constants - shear stress - ₀ unperturbed shear stress - ₁, ₂ kernel functions - angular frequency of oscillation - steady angular velocity of the plate With 16 figures     

Power spectra of fluid velocities measured by laser Doppler velocimetry

The power spectrum and the correlation of the laser Doppler velocimeter velocity signal obtained by sampling and holding the velocity at each new Doppler burst are studied. Theory valid for low fluctuation intensity flows shows that the measured spectrum is filtered at the mean sample rate and that it contains a filtered white noise spectrum caused by the steps in the sample and hold signal. In the limit of high data density, the step noise vanishes and the sample and hold signal is statistically unbiased for any turbulence intensity.List of symbols A cross-section of the LDV measurement volume, m² - A empirical constant - B bandwidth of velocity spectrum, Hz - C concentration of particles that produce valid signals, number/m³ - d _m diameter of LDV measurement volume, m - f(₁, ₂ | u) probability density of t _i; and t _j given (t) for all t, Hz² - probability density for t _j-t_i, Hz - n (t, t) number of valid bursts in (t, t) = N + n - N (t, t) mean number of valid bursts in (t, t) - N _e mean number of particles in LDV measurement volume - valid signal arrival rate, Hz - mean valid signal arrival rate, Hz - R _uu time delayed autocorrelation of velocity, m²/s² - S _u power spectrum of velocity, m²/s²/Hz - t ₁, t ₂ times at which velocity is correlated, s - t _i, t _j arrival times of the bursts that immediately precede t ₁ and t ₂, respectively, s - t _ij t _j–t _i s - T averaging time for spectral estimator, s - T _u integral time scale of u (t), s - T Taylor's microscale for u (t), s - u velocity vector = U + u, m/s - u fluctuating component of velocity, m/s - U mean velocity, m/s - u _m sampled and held signal, m/s Greek symbols (t) noise signal, m/s - _m (t) sampled and held noise signal, m/s - bandwidth of spectral estimator window, radians/s - time between arrivals in pdf, s - Taylor's microscale of length = UT m - kinematic viscosity - ₁, ₂ arrival times in pdf, s - root mean square of noise signal, m/s - _u root mean square of u, m/s - delay time = t ₂ - t ₁ s - B duration of a Doppler burst, s - circular frequency, radians/s - _c low pass frequency of signal spectrum radians/s Other symbols ensemble average - conditional average - ^ estimate     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

Discrete time semigroup transformations with random perturbations

Let (X, ) and (Y,C) be two measurable spaces withX being a linear space. A system is determined by two functionsf(X): X X and:X×YX, a (small) positive parameter and a homogeneous Markov chain {y _n } in (Y,C) which describes random perturbations. States of the system, say {x _n X, n=0, 1,}, are determined by the iteration relations:x _n+1 =f(x _n )+(x _n ,Y_n+1) forn0, wherex ₀ =x ₀ is given. Here we study the asymptotic behavior of the solutionx _n as 0 andn under various assumptions on the data. General results are applied to some problems in epidemics, genetics and demographics.Supported in part by NSF Grant DMS92-06677.Supported in part by NSF Grant DMS93-12255.