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 ||||.     

Uniaxial extension of an isothermal fluid cylinder in the presence of inertia,surface tension and gravity

Effects of inertia, surface tension and gravity in the constant force stretching of isothermal cylindrical filaments of Newtonian, power-law and Maxwell-type fluids were analysed in Lagrangian coordinates. Solution for the purely gravitational extension of Newtonian fluid cylinder was found to be as simple as = 1 – C ₃ (1 – ) where designates the cross sectional area, the Lagrangian distance and the time. Analytical solutions were also available for the case of inertialess Newtonian and power-law fluids.A first-order backward differencing scheme and minimal computer time were sufficient to numerically analyse the constant force extension of Maxwell-type fluids in the presence of inertia, gravity and surface tension. Effects of inertia, surface tension and gravity on the severity of neck down occurring at either end of the filament are summarized in diagrams. The present approach is valid on any other constitutive model as far as there is a numerical scheme to analyse thehomogeneous extension of a cylinder of that particular fluid.     

Abkühlungscharakteristik einer gut wärmeleitenden Platte,die bei der Bewegung durch ein Fluid gestreckt wird

Zusammenfassung Das Abkühlungsverhalten einer gut wärmeleitenden Platte, die durch ein Fluid bewegt und dabei nach einem beliebigen Potenzgesetz gestreckt wird, wurde untersucht. Der Impulstransport wird dabei durch eine Gleichung beschrieben, die der Falkner-Skan-Gleichung mit einem Parameter für die Dehnung ähnelt. Für das thermische Problem wurden bezüglich des Ortes ähnliche Lösungen gefunden, die sowohl für kleine als auch große Werte des Fließabstandes gültig sind. Die Konvergenz der Reihen für kleine-Werte konnte durch die Euler-Transformation verbessert werden, so daß die Ergebnisse für alle-Werte Gültigkeit haben.

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The cooling of a low heat resistance stretching sheet moving through a fluid

The cooling characteristics of a low-heat-resistance sheet moving through a fluid have been studied for arbitrary power law stretching. The momentum transfer is governed by an equation that resembles the Falkner-Skan equation with stretching as a parameter. For the thermal problem, locally similar solutions have been obtained for small and large values of the streamwise distance. The convergence of low series has been improved by Euler transformation to yield results valid for all values of.

     

An example of a quasiconvex function that is not polyconvex in two dimensions

We study the different notions of convexity for the function f () = ||² (||² – 2 det ) where ^2×2, introduced by Dacorogna & Marcellini. We show that f is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦¦ 2/3 2, 1, 1+ (for some >0), 2/3, respectively.     

Contributions to the theory of the method of steepest descent

Applying the method of steepest descent to F(x ₁,..., x _n ) one obtains a sequence of points _v . To obtain conditions for convergence of _v , the derived set H of the _v in the case of divergence is studied. In this case H is a continuum on which not only grad F vanishes everywhere, but also the rank of the Hessian of F is everywhere less than n-1.     

Analysis of slip and temperature jump coefficients in a binary gas mixture

The possibility of simplifying the formulas obtained by the Maxwell-Loyalka method for the velocity _u, temperature _T and diffusion _d slip coefficients and the temperature jump coefficient in a binary gas mixture with frozen internal degrees of freedom of the molecules is considered. Special attention is paid to gases not having sharply different physicochemical properties. The formulas are written in a form convenient for use without linearization in the thermal diffusion coefficient. They are systematically analyzed for mixtures of inert gases, N₂, O₂, CO₂, and H₂ at temperatures extending from room temperature to 2500°K. It is shown that for the molecular weight ratios m^* = m₂/m₁ considered the expressions for _u and can be radically simplified. With an error acceptable for practical purposes (up to 10%) it is possible to employ expressions of the same structural form as for a single-component gas: for _u if 1 m^* 6, and for if 1 m^* 3. When 1 m^* 2 the expression for _T can be simplified with a maximum error of 5%. Within the limits of accuracy of the method the expression for _t can be linearized in the thermal diffusion coefficient. An approximate expression convenient for practical calculations is proposed for _d Finally, the , _u, and _T for a single-component polyatomic gas with easy excitation of the internal degrees of freedom of the molecules are similarly analyzed; it is shown that these expressions can be considerably simplified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–159, November–December, 1990.     

Mass transfer with chemical reaction in a laminar falling film

Zusammenfassung Es wird eine analytische Lösung für die Absorption in einem laminaren Rieselfilm mit homogener und heterogener chemischer Reaktion 1. Ordnung vorgestellt, wobei der Stofftransportwiderstand auf der Gasseite liegt. Die Lösung ist eine Funktion von drei dimensionslosen ParameternBi, und, welche die BiotZahl und einen homogenen bzw. heterogenen Reaktionsparameter darstellen. Es wird gezeigt, daß für feste Werte vonBi und die Absorptionsrate (bezogen auf die Breite 1 des Rieselfilms) über eine gewisse Länge (dimensionslos) des Rieselfilms unabhängig von ist, wenn, < 0,6 ist. Die laufende Länge wird von der Stelle aus gemessen, an der die Absorption beginnt. Für b 0,6 nimmt der FlußQ mit zu, erreicht aber einen Sättigungswert bei=10, wonachQ nurmehr sehr langsam anwächst. Jedoch für ein gegebenes und ohne Übergangswiderstand im Film (Bi ) nimmtQ mit für alle 0 zu.

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Mass transfer with chemical reaction in a laminar falling film

An analytical solution is presented for gas absorption in a laminar falling film with first-order homogeneous and heterogeneous chemical reaction and external gas-phase mass transfer resistance. The solution depends on three dimensionless parametersBi, and, wich represent the Biot number, homogeneous and heterogeneous reaction parameters, respectively. It is shown that for fixed values ofBi and, the rate of gas absorption (per unit breadth) over a certain length; (dimensionless) along the falling film measured from the point where surface absorption begins is independent of if < 0.6. For 0.6, this fluxQ increases with but reaches a saturation value at=10 beyond whichQ increases very slowly. But for given and zero gas film resistance (Bi ),Q increases with for all 0.

     

Heat transport in a parabolic tube of circular cross-section

The local temperature has been determined for a viscous liquid flowing through a paraboloidal tube. Wall temperature and inlet temperature have been considered constant. The liquid flow was considered as creeping flow and its velocity distribution was determined by solving the biharmonic differential equation of the stream function. The local temperature was evaluated numerically from the analytical results.

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Wärmetransport im Paraboloidrohr

Zusammenfassung Es wird die lokale Temperatur in einer viskosen Strömung durch ein Paraboloidrohr bestimmt. Dabei wird konstante Wand- und Einlauftemperatur angenommen. Die kriechende Strömungsgeschwindigkeit wurde aus der Lösung der biharmonischen Differentialgleichung der Stromfunktion bestimmt. Die lokale Temperatur wurde aus den analytischen Ergebnissen für einige Paraboloidrohre numerisch bestimmt.

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Nomenclature ₁ F ₁ confluent hypergeometric function - diffusivity - T(, , ) temperature - T _w temperature at the paraboloidal wall - T _i temperature at the inlet - u(, ) flow velocity of viscous liquid in -direction - volumetric flow - eigenvalues of confluent hypergeometric function - streamfunction - _o wall of paraboloidal tube - _o inlet of paraboloidal tube - , , paraboloidal coordinates     

Conditions of simulation of stagnation point heat transfer from a high-enthalpy flow

The problem of local simulation of stagnation point heat transfer to a blunt body is solved within the framework of boundary layer theory on the assumption that the simulation subsonic high-enthalpy flow is in equilibrium outside the boundary layer on the model, while the parameters of the natural flow are in equilibrium at the outer edge of the boundary layer on the body. The parameters of the simulating subsonic flow are expressed in terms of the total enthalpyH ₀, the stagnation point pressurep _w and the velocityV ₁ for the natural free-stream flow in the form of universal functions of the dimensionless modeling coefficients=R _m ^* /R _b ^* ( .<1),=V ₁/2H ₀ ( .<1) whereR _m ^* and R _b ^* are the effective radii of the model and the body at their stagnation points. Approximate conditions for modeling the heat transfer from a high-enthalpy (including hypersonic) flow to the stagnation point on a blunt body by means of hyposonic (M1) flows, corresponding to the case ²1, are obtained. The possibilities of complete local simulation of hypersonic nonequilibrium heat transfer to the stagnation point on a blunt body in the hyposonic dissociated air jets of a VGU-2 100-kilowatt induction plasma generator [4, 5] are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 172–180, January–February, 1993.     

A solitary wave in a porous medium

The one-phase Darcy continuity equation, including the quadratic gradient term, is considered. The exact linearization of the equation is found by a functional transformation for an arbitrary spatial dimension in the limit case where the constant fluid compressibility is much more dominant than the constant compressibilities of the reservoir parameters.The equation permits a solution representing a localized wave travelling through a one-dimensional reservoir without changing its form. This is the actual long-time limit of the transient solution for a constant sandface-rate injection of a compressible fluid with a constant compressibility if the fluid is much more compressible than the matrix. A solitary wave solution is not possible for production.A fully developed solitary wave would appear only for very high pressure increases, but the first signs of the emerging solitary wave are detectable at the sandface for moderate pressure increases which can appear under physical reservoir conditions.Latin symbols a Dimensionless wave propagation velocity - A _N Sandface area (N = 0, 1, 2) - c ₁, c ₂ Sums of compressibilities - c _x Generic (generalized) compressibility - c Fluid compressibility - c _h Reservoir height (i.e. bulk volume) compressibility (N = 0, 1) - c _k , c , c Generalized compressibilities - D Spatial reservoir dimensionality (D = 1, 2, 3) - f Fractional change of p _n1 due to nonlinear effects - h Reservoir height (proportional to bulk volume for N = 0, 1) - Horizontal reservoir width (N = 0) - k Reservoir permeability - K _N Constant with dimension of pressure (N = 0, 1, 2) - n Sum index - N Integer variable (N = D – 1) - p Reservoir pressure - p^* Overburden pressure - p _D Dimensionless (scaled) version of p - p ₀ Initial pressure - q Volumetric flow rate referred to sandface - r Radial (or linear) spatial distance from center of well - r _w Well radius - r _e External reservoir radius (or length) from center of well - t Time variable - t _f Injection/production time corresponding to fraction f - T Cole-Hopf-transformed version of dimensionless pressure y - u Rescaled (dimensionless) version of v _D - v Darcy velocity - v _d Dimensionless (scaled) version of v - x Generic symbol in compressibility expression (also used for auxiliary function and for auxiliary variable) - y Rescaled (dimensionless) version of p _D - z Dimensionless (scaled) version of r Greek symbols Coefficient of inertial resistance - Variable in wave solution for y - p _n1 Absolute change in physical sandface pressure due to production or injection - _p Pressure change over (dimensionless) distance behind and far away from front - _r Physical distance at constant time corresponding to - Characteristic (dimensionless) width of solitary wave - Formation porosity - ₁, ₂ Integration constants - Dimensionless (scaled) length of finite reservoir - Fluid viscosity - Fluid density - Dimensionless (scaled) version of t - Wave solution for dimensionless pressure y - Integer variable (±1) distinguishing between production and injection     

A note on three-dimensional boundary layer growth

Summary This note is an extension of the work of Görtler²) on two-dimensional boundary layer growth to the three-dimensional case. The solutions of three-dimensional boundary layer equations are obtained by considering the potential flow of the body to be governed by the functions At U ₀(, ) and At U ₀(, ) where is any positive number.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

Mass transport in a converging-diverging nozzle

The local concentration has been determined for a viscous liquid flowing through a converging-diverging tube with constant wall- and initial concentration. The liquid flow was considered as creeping flow and its velocity distribution was determined by solving the biharmonic differential equation of the stream function. The mass transport was presented in form of an infinite series of Legendre functions, which rendered with the Galerkin condition a determinant of finite order for the determination of the eigenvalues. The local concentration was evaluated numerically for differently diverging tubes, of which the case of mass transport for viscous (linearized) flow through an orifice of circular cross-section presented a special case.

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Stofftransport in einem konvergierenden-divergierenden Rohr

Znsammenfassung Es wird die lokale Konzentration eines in einer viskosen Strömung durch ein Venturirohr befindlichen Stoffes bei konstanter Wand-und Einlaßkonzentration bestimmt. Die viskose (kriechende) Strömungsgeschwindigkeit wurde aus der Lösung der biharmonischen Differentialgleichung der Stromfunktion bestimmt. Die Bestimmung des Massentransportes im Rohr wird mit einer Lösung in Form einer unendlichen Reihe von Legendrefunktionen bewältigt, wobei die Eigenwertgleichung eine Determinante unendlicher Ordnung ergibt. Die lokale Konzentration wurde für verschieden divergierende Rohre numerisch ausgewertet, wobei der Massentransport bei Strömung durch eine kreisförmige Öffnung als Sonderfall auftrat.

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Nomenclature a radius of throat area - c (, ) concentration - c _w wall concentration - c _i initial concentration - D diffusion coefficient - P _n ⁰ Legendre function - u (, ) flow velocity of liquid (in-direction) - flow volume per time unit - _n roots ofP ⁰ ( ₀)=0 - stream function - , , oblate spheroidal coordinates - ₀ wall of the nozzle     

The swirling radial free jet

Summary A general similarity solution suggested by Watson for the problem of the laminar, radial, free-jet with swirl has been previously discussed by Riley who also calculated the order to which the solution was valid. That problem is considered in more detail here and higher order terms are given. It is shown that a perturbation scheme for the stream function consisting of a series of inverse powers of and which uses the asymptotic similarity solution as the basic solution is inadequate, and a modification to the series so as to include terms like ⁿ (ln ) ^m must be adopted in order to satisfy the boundary conditions. It is also shown that the general similarity solution may be obtained from the asymptotic series representing the general case with swirl for certain special values of the free constants and also for the no-swirl or free-jet problem. The asymptotic series is given to order ^–13 for the case of swirl and to order ^–29 when there is no swirl.     

Wave angle for oblique detonation waves

The flow field associated with a steady, planar, oblique detonation wave is discussed. A revision is provided for- diagrams, where is the wave angle and is the ramp angle. A new solution is proposed for weak underdriven detonation waves that does not violate the second law. A Taylor wave, encountered in unsteady detonation waves, is required. Uniqueness and hysteresis effects are also discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Flow of second-order fluids over an enclosed rotating disc

Summary This paper is devoted to a study of the flow of a second-order fluid (flowing with a small mass rate of symmetrical radial outflow m, taken negative for a net radial inflow) over a finite rotating disc enclosed within a coaxial cylinderical casing. The effects of the second-order terms are observed to depend upon two dimensionless parameters ₁ and ₂. Maximum values ₁ and ₂ of the dimensionless radial distances at which there is no recirculation, for the cases of net radial outflow (m>0) and net radial inflow (m<0) respectively, decrease with an increase in the second-order effects [represented by T(=₁+₂)]. The velocities at ₁ and ₂ as well as at some other fixed radii have been calculated for different T and the associated phenomena of no-recirculation/recirculation discussed. The change in flow phenomena due to a reversal of the direction of net radial flow has also been studied. The moment on the rotating disc increases with T.Nomenclature , , z coordinates in a cylindrical polar system - z ₀ distance between rotor and stator (gap length) - =/z ₀, dimensionless radial distance - =z/z ₀, dimensionless axial distance - _s = _s/z₀, dimensionless disc radius - V =(u, v, w), velocity vector - dimensionless velocity components - uniform angular velocity of the rotor - , p fluid density and pressure - P =p/(₂ z ₀₂ ² , dimensionless pressure - ₁, ₂, ₃ kinematic coefficients of Newtonian viscosity, elastico-viscosity and cross-viscosity respectively - ₁, _{2 2}/z ₀ ² , resp. ₃/z ₀ ² , dimensionless parameters representing the ratio of second-order and inertial effects - m = , mass rate of symmetrical radial outflow - l a number associated with induced circulatory flow - Rm =m/(z ₀₁), Reynolds number of radial outflow - R _l =l/(z ₀₁), Reynolds number of induced circulatory flow - Rz =z ₀ ² /₁, Reynolds number based on the gap - ₁, ₂ maximum radii at which there is no recirculation for the cases Rm>0 and Rm<0 respectively - ₁(T), ₂(T) ₁ and ₂ for different T - U _1(T) ⁽⁺⁾ = dimensionless radial velocity, Rm>0 - V _1(T) ⁽⁺⁾ = , dimensionless transverse velocity, Rm>0 - U _2(T) ^(–) = , dimensionless radial velocity, Rm=–Rn<0, m=–n - V _2(T) ^(–) = , dimensionless transverse velocity, Rm<0 - C _m moment coefficient     

Analysis of melt spinning transients in Lagrangian coordinates

Transients in melt spinning of isothermal power law and Newtonian fluids were found to be governed by an extremely simple partial differential equation ² ( ^1/n )/() = 0 in Lagrangian coordinates where is the cross-sectional area,n the power law exponent, the time and the the time at which a fluid molecule constituting the spinline left the spinneret. The general integral ^1/n =f() +g () of the above governing equation containing two arbitrary functions represents physically attainable spinline transients. Hitherto unknown analytical transient solutions of the above governing equation were obtained for the response of isothermal constant tension spinlines to a stepwise change in tension, spinneret hole area, extrusion speed or extrusion viscosity and for the starting transient in gravitational spinning. Linearized perturbation solutions and the stability limit of the spinline derived from the above new found nonlinear solutions were in agreement with previous findings and the above nonlinear response of the spinline to a step increase in the spinneret hole area was found to be equivalent to Orowan's tandem cylinder model of dent growth in filament stretching.     

Study of transient and quasi-steady ablation of polytetrafluoroethylene layers

A study of the transient one-dimensional ablation of a PTFE-layer heated by a constant heat flux at the surface and cooled by finite heat transfer at the back is performed using a previously presented analytical model. The influences of various parameters upon the course of ablation are investigated and some limiting cases are discussed. The numerical solution for the quasi-steady ablation is presented by graphs, which are also approximated by correlations for the computation of the coupled boundary layer and PTFE-ablator.

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Instationäre und quasi-stationäre Ablation von PTFE-Schichten

Zusammenfassung Die instationäre, eindimensionale Ablation einer PTFE-Schicht wird untersucht, der ein konstanter Wärmestrom an der Oberfläche zugeführt und ein Kühlstrom durch endlichen Wärmeübergang an der Rückseite abgeführt wird. Dabei wird ein analytisches Modell zugrundegelegt, über das kürzlich berichtet wurde, und verschiedene Einflüsse auf den Ablationsverlauf betrachtet sowie Grenzfälle diskutiert. Die numerische Lösung für quasistationäre Ablation kann Diagrammen entnommen werden. Außerdem werden hierfür Korrelationen angegeben für die simultane Berechnung des PTFE-Ablators mit einer Grenzschicht.

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Nomenclature a thermal diffusivity - c polymer mass fraction: density ratio of the decomposing and the undecomposed material - k coefficient of heat transmission - m ablation rate (ablating mass flux) - q heat flux to the surface - Q dimensionless heat flux - t time - T temperature - x coordinate - x_O initial layer thickness - y₁ (x-)/(-), transformed coordinate - y₂ (x-)/(-), transformed coordinate - penetration bond - phase interface - thermal conductivity of crystalline PTFE at the melting point - layer thickness Indices k coolant - at the phase interface (melting point) - at the surface     

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     

The swirling radial jet

The similarity solution of the radial turbulent jet with weak swirl is discussed and a new solution of the radial turbulent jet with swirl is proposed without restrictions assumed in the weak swirl solution.Nomenclature e swirl parameter - k experimental constant - l non-negative constant - M, M , N, P integral invariants - q velocity component in -direction - q _max maximum velocity component in -direction - u radial velocity component - u _max maximum radial velocity component - v axial velocity component - w peripheral velocity component - w _max maximum peripheral velocity component - x radial coordinate - y transverse coordinate - angle introduced in (28) - characteristic width of a jet - (x, y) similarity variable (scaled x and y coordinate) - molecular kinematic viscosity - _T eddy kinematic viscosity - tangential coordinate - fluid density - turbulent shear stress in -direction - _xy , _y components of turbulent shear stress tensor - (x, y) stream function