On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

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.     

On properties of solutions of functional differential equations continuously differentiable on (0, +∞)

We establish the saddle-point property of the system of functional differential equations (t) = Ax(t) + Bx((t)) + C ((t)) + f (x(t), x((t))), (0) = 0.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 302–310, July–September, 2004.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

On the effect of magnetic field on viscous lifting and drainage of conducting fluid

This paper concerns with obtaining the solution of the problem of viscous lifting and drainage of a thin liquid film clinging to a vertical plane surface moving with a velocityf(t) in the presence of a transverse magnetic field. Specializing to the case when the surface moves with a constant acceleration, it has been found that the film thickness, for large magnetic fields, increases with the increase in magnetic field.Nomenclature a acceleration of the plate - A non-dimensional acceleration, =a/g - B magnetic induction vector - B ₀ applied magnetic field - f(t) any function oft - Laplace transform off(t) - g gravitational acceleration - h film thickness - H non-dimensional film thickness, =h(g/ ²)^1/3 - J current density vector - k (/)^1/2 B ₀ - M k( ²/g)^1/3 - n summation index - q mass flow rate - Q non-dimensional mass flow rate, =q/ - t time - T non-dimensional time, =t(g ²/)^1/3 - Laplace transform ofv(x, t) - V fluid velocity vector, =[0,v(x, t), 0] - (x, y, z) space coordinates - Y non-dimensionaly-coordinate, =y(g/ ²)^1/3 Greek symbols _n (n+1/2) - conductivity - density - kinematic viscosity     

Asymptotic analysis of equilibrium states for rotating turbulent flows

The equilibrium states of homogeneous turbulence simultaneously subjected to a mean velocity gradient and a rotation are examined by using asymptotic analysis. The present work is concerned with the asymptotic behavior of quantities such as the turbulent kinetic energy and its dissipation rate associated with the fixed point (/kS)=0, whereS is the shear rate. The classical form of the model transport equation for (Hanjalic and Launder, 1972) is used. The present analysis shows that, asymptotically, the turbulent kinetic energy (a) undergoes a power-law decay with time for (P/)<1, (b) is independent of time for (P/)=1, (c) undergoes a power-law growth with time for 1<(P/)<(C ₂–1), and (d) is represented by an exponential law versus time for (P/)=(C ₂–1)/(C ₁–1) and (/kS)>0 whereP is the production rate. For the commonly used second-order models the equilibrium solutions forP/,II, andIII (whereII andIII are respectively the second and third invariants of the anisotropy tensor) depend on the rotation number when (P/kS)=(/kS)=0. The variation of (P/kS) andII versusR given by the second-order model of Yakhot and Orzag are compared with results of Rapid Distortion Theory corrected for decay (Townsend, 1970).     

Anwendung des Übertragungsverfahrens bei stark abklingenden Lösungsfunktionen

Übersicht Bei stark abklingenden Funktionen wird die Übertragungsmatrix U() aufgespalten in die Anteilc U ₁() e^– und U ₂() e. Der zweite Term spielt am Rand = 0 keinc Rolle. Die unbekannten Anfangswerte sind über die Matrix U ₁(0) an die bekannten gebunden und eindeutig bestimmbar.

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Summary For strongly decaying solution functions the transfer matrix U() is splitted into the parts U ₁() e^– and U ₂() e. The second term does not influence at the boundary = 0. The unknown initial values are related by the matrix U ₁(0) to the known values and they can be uniquely determined.

     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

An asymptotic treatment of the transient development of axisymmetric surface waves

A linearized theory is developed for the derivation of an asymptotic solution of the initial value problem of axisymmetric surface waves in an infinitely deep fluid produced by an arbitrary oscillating pressure distribution. An asymptotic treatment of the problem is presented in detail to obtain the solution for the surface elevation for sufficiently large time. Finally, the main prediction of this analysis for some particular pressure distributions of physical interest is exhibited.Nomenclature R, , Y cylindrical polar coordinates - frequency - g acceleration due to gravity - density of fluid - T time - (R, Y; T) velocity potential - E(R, T) vertical surface elevation - P(R, T) applied surface pressure - r, y nondimensional cylindrical polar coordinates, - p(r, t) nondimensional surface pressure - t nondimensional time, T - (r, y; t) nondimensional velocity potential, - (r, t) nondimensional vertical surface elevation, - (k) Hankel transform of a function p(r) with respect to r - I ₁ transient wave integral - I ₂ steady state wave integral     

Transient flow towards a well in an aquifer including the effect of fluid inertia

Transient propagation of weak pressure perturbations in a homogeneous, isotropic, fluid saturated aquifer has been studied. A damped wave equation for the pressure in the aquifer is derived using the macroscopic, volume averaged, mass conservation and momentum equations. The equation is applied to the case of a well in a closed aquifer and analytical solutions are obtained to two different flow cases. It is shown that the radius of influence propagates with a finite velocity. The results show that the effect of fluid inertia could be of importance where transient flow in porous media is studied.List of symbols b Thickness of the aquifer, m - c ₀ Wave velocity, m/s - k Permeability of the porous medium, m² - n Porosity of the porous medium - p( ,t) Pressure, N/m² - Q Volume flux, m³/s - r Radial coordinate, m - r _w Radius of the well, m - s Transform variable - S Storativity of the aquifer - S _d(r, t) Drawdown, m - t Time, s - T Transmissivity of the aquifer, m²/s - ( ,t) Velocity of the fluid, m/s - Coordinate vector, m - z Vertical coordinate, m - Coefficient of compressibility, m²/N - Coefficient of fluid compressibility, m²/N - Relaxation time, s - (r, t) Hydraulic potential, m - Dynamic viscosity of the fluid, Ns/m² - Dimensionless radius - Density of the fluid, Ns²/m⁴ - (, ) Dimensionless drawdown - Dimensionless time - , x Dummy variables - ₀, ₁ Auxilary functions     

Convergence for Semilinear Degenerate Parabolic Equations in Several Space Dimensions

We show that any global nonnegative and bounded solution to the degenerate parabolic problemu_t-u^m+f(u)=0 qquad {\rm on} quad R^N,u|{}=0converges to a single stationary state as time goes to infinity. Here m>0, f is a restriction of a real analytic function defined on a sector containing the half-line [0, ), and f(u ^1/m ) is a continuously differentiable function of u.     

Theory of single well tests in a leaky aquifer. Analytical solution for non steady flow

The evaluation of a pump test or a slug test in a single well that completely penetrates a leaky aquifer does not yield a unique relation between the hydraulic properties of the aquifer, independent of the testing conditions. If the flow is transient, the drawdown is characterized by a single similarity parameter that does not distinguish between the storativity and the leakage factor. If the flow is quasi stationary, the drawdown is characterized by a single similarity parameter that does not distinguish between the transmissivity and the leakage factor. The general non steady solution, which is derived in closed form, is characterized bythree similarity parameters.Nomenclature a e 0.8905 = auxiliary parameter - b thickness of the aquifer - b _c thickness of the semipervious stratum - B() auxiliary function - f(s),g(s) auxiliary functions in the complex plane - F(t),G(t) auxiliary functions of time - h undisturbed level of the phreatic surface - K conductivity of the aquifer - K _c conductivity of the semipervious stratum - m ₀ leakage factor - m dimensionless leakage factor - N(s) auxiliary function in the complex plane - Q _w (t) discharge flux - Q steady discharge flux - Q ₀ constant discharge flux during limited time - Q(t) dimensionless discharge flux - r ₀ radius of the well - r radial coordinate - r dimensionless radial coordinate - s complex variable - s ₀ pole - S storativity of the aquifer - S _n n'th part of an integration contour - t time - t dimensionless time - T transmissivity of the aquifer - ,,,,, dimensionless parameters - Euler's number - dummy variable - ₁(), ₂() auxiliary functions - (r, t) drawdown - ₀(t) drawdown in the well - (r, t) dimensionless drawdown - ₀(t) dimensionless drawdown in the well     

Thermal free convection from a solid sphere

Summary Thermal free convection from a sphere has been studied by melting solid benzene spheres in excess liquid benzene (Pr=8,3; 10⁸<Gr<10⁹). Overall heat transfer as well as local heat transfer were investigated. For the effect of cold liquid produced by the melting a correction has been applied. Results are compared with those obtained by other workers who used alternative experimental methods.Nomenclature coefficient of heat transfer - d characteristic length, here diameter of sphere - thermal conductivity - g acceleration of free fall - cubic expansion coefficient - T temperature difference between wall and fluid at infinity - kinematic viscosity - density - c specific heat capacity - a thermal diffusivity (=/c) - D diffusion coefficient - Nu dimensionless Nusselt number (=d/) - Nu* the analogous number for mass transfer (=kd/D) - mean value of Nusselt number - Gr dimensionless Grashof number (=gd ³T/ ²) - Gr* the analogous number for mass transfer (=gd ³x/ ²) - Pr dimensionless Prandtl number (=/a) - Sc dimensionless Schmidt number (=/D)     

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation u_t=÷ a(u, Du), where a(z,)=f(z,), and f is a convex function of with linear growth as ||||, satisfying other additional assumptions. In particular, this class includes the case where f(z,)=(z)(), >0, and is a convex function with linear growth as ||||.     

The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws

Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u _t+f(u) _x =u _xx on a half-line R⁺, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u)u–u₁ induced by L ¹(R⁺). We prove here that v is asymptotically stable with respect to d: if u ₀–vL ¹, then u(t)–v₁0 as t+. When v is a constant, we show that this property holds if and only if f(v)0. These results complement our study of the Cauchy problem [2].     

Das erweiterte Korrespondenzgesetz für die dynamische Idealviskosität und die Idealwärmeleitfähigkeit reiner Stoffe

Zusammenfassung Zur Berechnung der dynamischen Idealviskosität Ideal (T) und der Idealwärmeleitfähigkeit ideal (T) benötigt man die kritische TemperaturT _kr, das kritische spezifische Volum _kr, die MolmasseM, den kritischen Parameter _kr und die molare isochore WärmekapazitätC _v(T). Sowohl das theoretisch, als auch das empirisch abgeleitete erweiterte Korrespondenzgesetz ergeben eine für praktische Zwecke ausreichende Genauigkeit für die Meßwertwiedergabe, die bei den assoziierenden Stoffen und den Quantenstoffen jedoch geringer ist als bei den Normalstoffen.

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The extended correspondence law for the ideal dynamic viscosity and the ideal thermal conductivity of pure substances

For the calculation of the ideal dynamic viscosity Ideal (T) and the ideal thermal conductivity ideal (T) the critical temperatureT _kr, the critical specific volumev _kr, the molecular massM, the critical parameter _kr, and the molar isochoric heat capacityC _v(T) is needed. Not only the theoretically determined but also the empirically determined extended correspondence law gives for practical use a good representation of the measured data, which for the associating substances and the quantum substances is not so good as for the normal substances.

     

Robustness of Exponential Dichotomies in Infinite-Dimensional Dynamical Systems

In this paper we examine the issue of the robustness, or stability, of an exponential dichotomy, or an exponential trichotomy, in a dynamical system on an Banach space W. These two hyperbolic structures describe long-time dynamical properties of the associated time-varying linearized equation _t +A=B(t) , where the linear operator B(t) is the evaluation of a suitable Fréchet derivative along a given solution in the set K in W. Our main objective is to show, under reasonable conditions, that if B(t)=B(, t) depends continuously on a parameter and there is an exponential dichotomy, or exponential trichotomy, at a value ₀, then there is an exponential dichotomy, or exponential trichotomy, for all near ₀.We present several illustrations indicating the significance of this robustness property.     

The motion of a detaching elastic body

Conclusions The qualitative behavior of the displacement (t) and the radius R(t) during the different phases of the motion is illustrated in the diagram of Fig. 6.1.After the first impact at t = 0 the displacement (t) varies according to (5.2). If the first maximum of (t) is higher than ₁ then at time t ₁ the graph of (t) intersects the straight line = _cand detachment first occurs. In the second phase the dependance of on t is expressed by (5.6). The detachment will end at the instant t ₂ when vanishes.The radius R remains equal to R ₀ until (t) reaches the critical value ₁ = _c that is at t = t ₁. After t ₁, R(t) will decrease according to (4.4) up to its final value ₂.A rather unexpected property of the solution is that the greatest elongation is finite for every non-vanishing value of the ratio .To Jerry Ericksen for his 60^th birthday     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.