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.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Stability of the T ∞FT < T ∞α relation between Vogel temperatures of flow and glass transition against determination variants

Dynamic shear measurements in the frequency range from 10^–4 to 500 rad/s at the flow and main transition of a polydisperse poly(vinyl acetate) and a monodisperse polystyrene sample are presented. For both samples the Vogel temperature of the flow transition T _FT is smaller than the Vogel temperature of the main transition T , independent of the criteria used for data evaluation. The difference between the two Vogel temperatures corresponds to results for samples with other molecular weight and polydispersity from the literature. The T _FT <T relation is discussed in terms of short () and long (FT) dynamic glass transitions in entangled polymers. The relation is explained by preaveraging of the energy landscape for the long flow transition by the short glass transition.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

Existence and regularity for solutions of the Korteweg-de Vries equation

The construction suggested by an inverse-scattering analysis establishes the existence of solutions u(x, t) of the Korteweg-de Vries equation subject to an initial condition u(x, 0)=U(x), where U has certain regularity and decay properties. It is assumed that UC³(), that U is piecewise of class C ⁴, and that U ^(j) decays at an algebraic rate for j4. The faster the decay of U ^(j) the smoother the solution will be for t0. If U and its first four derivatives decay faster than ¦x¦^–n for all n, then the solution will be infinitely differentiable for t0. For t>0, the decay rate of u(x, t) as x + increases with the decay rate of U; but the decay rate as x - depends on the regularity of U. A solution u ₁ of the Korteweg-de Vries equation such that u ₁(·, 0)C() may fail to remain in class C for all time if u ₁(x, 0) does not decay fast enough as ¦x¦.This research was performed in part as a Visiting Member of the Courant Institute of Mathematical Science.     

Numerical Investigation of Off-Design Regimes of Flow Past Power-Law Waveriders Based on the Flows Behind Plane Shocks

The results of a numerical investigation of supersonic off-design flow past waveriders at the freestream Mach numbers M = 4 and 8 are presented. Flow regimes with M both greater and smaller than the design value M _d are analyzed. Configurations based on the flows behind plane shocks and described by power-law functions are considered. The results are obtained by the finite-volume solution of the Euler equations using higher-order TVD Runge-Kutta schemes.     

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 the existence of limit cycles of Liénard equation

In this paper, we have proved several theorems which guarantee that the Liénard equation has at least one or n limit cycles without using the traditional assmuption G(±) =+. Thus some results in [3–5] are extended. The limit cycles can he located by our theorems. Theorems 3 and 4 give sufficient conditions for the existence of n limit cycles having no need of the conditions that the function F(x) is odd or nth order compatible with each other or nth order contained in each other.     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

A general model for the effective viscosity of pseudoplastic and dilatant fluids

Summary A new and very general expression is proposed for correlation of data for the effective viscosity of pseudoplastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expression. A straightforward procedure is outlined for evaluation of the arbitrary constants.

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Zusammenfassung Eine neue und sehr allgemeine Formel wird für die Korrelation der Werte der effektiven Viskosität von strukturviskosen und dilatanten Flüssigkeiten in Abhängigkeit von der Schubspannung vorgeschlagen. Die meisten schon früher vorgeschlagenen Methoden werden hier als Spezialfälle dieser Gleichung gezeigt. Ein einfaches Verfahren für die Auswertung der willkürlichen Konstanten wird beschrieben.

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Nomenclature b arbitrary constant inSisko model (eq. [5]) - n arbitrary exponent in eq. [1] - x independent variable - y(x) dependent variable - y ₀(x) limiting behavior of dependent variable asx 0 - y(x) limiting behavior of dependent variable asx - z original dependent variable - arbitrary constant inSisko model (eq. [5]) andBird-Sisko model (eq. [6]) - arbitrary exponent in eqs. [2] and [8] - effective viscosity = shear stress/rate of shear - _A effective viscosity at = _A - _B empirical constant in eqs. [2] and [8] - ₀ limiting value of effective viscosity as 0 - ₀() limiting behavior of effective viscosity as 0 - limiting value of effective viscosity as - () limiting behavior of effective viscosity as - rate of shear - arbitrary constant inBird-Sisko model (eq.[6]) - shear stress - _A arbitrary constant in eqs. [2] and [8] - ₀ shear stress at inBingham model - _1/2 shear stress at = ( ₀ + )/2 With 8 figures     

Effect of nose shape on hypersonic flow past slender blunt bodies at an angle of attack

We examine some characteristics of hypersonic flow past slender blunt bodies of revolution at a small angle of attack 1, where is the relative body thickness. It is shown that, within the framework of hypersonic theory, for a correct-consideration of the effect of the conditions in the transitional section between the nose and the lateral surface it is necessary, in the general case, to specify the circumferential distribution of the force effect for the nose and the mass of the gas. For small , the effect of the nose, just as in two-dimensional flows [1–4], shows up only through its drag coefficient c_x, for =0. On this basis, the similarity law [1–4] for flow past such bodies, with arbitrary form of the lateral surface and differing in the shape of the nose blunting, which is valid over the entire disturbed region, with the exception of a small vicinity of the nose, is extended to the case in question.The notation r₀ and L maximum nose radius and characteristic body length - V, M, and density, velocity, Mach number, and adiabatic exponent of the gas in the approaching stream - , V²i, and V²p density, enthalpy, and pressure - x, r, and coordinate system of the cylindrical body with its center at the transitional section between the nose and the side surface - Vu, Vv, and Vw corresponding velocity components     

Hyperbolic phenomena in a strongly degenerate parabolic equation

We consider the equation u _t =((u) (u _x )) _x , where >0 and where is a strictly increasing function with lim _s = <. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u _t = ((u)) _x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.     

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.     

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).     

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.     

Theoretische Untersuchung der laminaren Zweistoff-grenzschichtströmung längs eines verdunstenden Flüssigkeitsfilms bei nichtadiabater Verdunstung

Zusammenfassung Zur Klärung der physikalischen Vorgänge im Verdampferteil einer Filmverdampfungsbrennkammer wird in Erweiterung der adiabaten Verdunstung der Fall der einseitig benetzten ebenen Platte behandelt, die sowohl im Gleichals auch im Gegenstrom von der heißen Außenluft umströmt wird. Die für beide Strömungsfälle maßgebenden Grenzschichtgleichungen werden simultan unter Berücksichtigung temperatur- und konzentrationsabhängiger Stoffwerte mit einem impliziten Differenzenverfahren gelöst. Dabei ergeben sich für den Gleichstrom ähnliche Lösungen des gekoppelten Gleichungssystems, die mit den ähnlichen, für die adiabate Verdunstung geltenden Lösungen verglichen werden. Die Berechnung der durch den Stoffübergang beeinflußten Grenzschicht parameter zeigt, daß das Modell der Gegenstromanordnung, bei der sich nichtähnliche Profile entlang der Filmoberfl äche einstellen, für einen möglichen Einsatz in einer Filmverdampfungsbrennkammer am besten geeignet ist.

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Theoretical investigation on the binary laminar boundary-layer flow along a vaporizing liquid layer at non-adiabatic evaporation

For clarification the physical process in the evaporating part of a film-evaporation combustion-chamber in addition to the adiabatic evaporation the case of a one-sided wet plate in co- and counter-current hot air flow is presented. The boundary-layer equations for both streams are solved simultaneously with an implicit finite-difference method taking into account variable fluid properties. Thereby the similar solutions obtained for the co-current flow are compared with the corresponding similar solutions for the case of the adiabatic evaporation. Contrary to the co-current flow the counter-current flow yields non-similar solutions and the computation of the boundary-layer parameters influenced by the evaporation mass-flow shows, that the model of counter-current flow is best suitable for application in a film-evaporation combustion-chamber.

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Bezeichnungen A_j, B_j Abkürzungen in der allg. Differenzen - C_j gleichung (36) - c Massenkonzentration, bezogen auf Gemischmasse - c_f Dimensionsloser örtlicher Reibungsbeiwert - c_p Spezifische Wärmekapazität - D₁₂ Diffusionskoeffizient - h Enthalpie des Gasgemisches - K₁, K₂ Abkürzungen in der Gl. (5) - K₅, K₆ Abkürzungen in der Gl.(22) - L Plattenlänge - M Molmasse - m₁ Massenstromdichte, verdunstende Masse je Flächen- und Zeiteinheit - m^* Dimensionslose Massenstromdichte, Verdunstungsparameter nach Gl.(32) - m^** Örtliche dimensionslose Massenstromdichte nach Gl. (33) - P_Gr Stellvertretende Größe für die Grenzschicht parameter c_f, St_T und St_m nach Gl. (34) - p Statischer Druck (=Summe der Partialdrücke) - p_1w Sättigungsdruck an der Filmoberfläche - q Wärmestromdichte - r Verdampfungsenthalpie - r _1w ^* Dimensionslose Verdampfungsenthalpie nachGl.(25) - u Geschwindigkeit in x-Richtung - v Geschwindigkeit in y-Richtung - x Längskoordinate - ¯x Längskoordinate für den Gegenstrom s. Bild 14 - x_A Wärmeisolierte Anlaufstrecke s. Bild 14 - x^* Dimensionslose Längskoordinate für das Dreipunkt-Differenzenverfahren x^*=x/_s - y Querkoordinate - y^* Normierte Querkoordinate für das Drei punkt-Differenzenverfahren y^*=y/_s - ₁ Dimensionslose Verdrängungsdicke nach Gl.(27) - ₂ Dimensionslose Impulsverlustdicke nach Gl.(28) - _c Konzentrationsgrenzschichtdicke (y-Wert für =0.99) - _s Strömungsgrenzschichtdicke (y-Wert für u/u=0.99) - _T Temperaturgrenzschichtdicke (y-Wert für = 0.99) - _T Dimensionsloser Wandabstand nach Gl.(37) - Normierte absolute Temperatur (= (T – T_w)/(T _{– T w}) - Wärmeleitfähigkeit - Dynamische Zähigkeit - Kinematische Zähigkeit - Dichte - Schubspannung - Allgemeine abhängige Variable (s. Tabelle 1) Normierte Massenkonzentration (=(c₁–c_1w/(c₁–c_1w)) - Nu Nußelt-Zahl (= L(T/yT/y)_w/(T–T_w)) - Pr Prandtl-Zahl (=c_p/) - Re_x Reynolds-Zahl (=ux/) - Re_L Reynolds-Zahl (=uL/) - Re_s Reynolds-Zahl (= u_s/) - Sc Schmidt-Zahl (=/D₁₂) - St_m Stanton-Zahl des Stoffübergangs nach Gl.(31) - St_T Stanton-Zahl des Wärmeübergangs nach Gl.(30) Indizes 0 Bezogen auf Strömung ohne Stoffübergang - 1 Gas 1 (Benzoldampf) - 2 Gas 2 (Luft) - Ungestörter Anströmzustand der Luft - ad Charakteristische Werte des adiabaten Strömungsfalles - Geg Charakteristische Werte des Gegenstroms - Gl Charakteristische Werte des Gleichstroms - j Diskreter Punkt in y-Richtung - k Diskreter Punkt in x-Richtung - w Werte an der Plattenoberfläche - + Werte an der benetzten Plattenoberseite - – Werte an der trockenen Plattenunterseite Auszug aus der von der Fakultät für Maschinenbau und Elektrotechnik der Technischen Universität Braunschweig zur Erlangung des akademischen Grades eines Doktor-Ingenieurs genehmigten Dissertation über Theoretische Untersuchung der laminaren Zweistoffgrenzschichtströmung längs einer benetzten, ebenen Platte bei nichtadiabater Verdunstung des Diplom-Ingenieurs Klaus Pientka. Berichterstatter: Prof. Dr. phil. Dr.-Ing. E.h. H. Schlichting und Prof. Dr.-Ing. D. Hummel. - Die Dissertation wurde am 14 Juni 1976 bei der Technischen Universität eingereicht. Die mündliche Prüfung fand am 23. November 1976 statt.     

Classification of linear viscoelastic solids based on a failure criterion

An isotropic, incompressible linear viscoelastic solid subjected to a step shear displacement fails if the relaxation function G(s) is such that 0<G(0)< and –<G(0)0. In this case, the discontinuity in displacement propagates into the interior of the body. The discontinuity will not propagate however if G(0)= or G(0)=–. In the former case there is a diffusion-like smoothening of discontinuous data characteristic of parabolic equations. The case G(0)= may be achieved by composing the kernel as a sum of a smooth kernel and a delta function at the origin times a viscosity coefficient. If the viscosity is small, the smoothing will take place in a propagating layer which scales with the small viscosity. The case of G(0)=– is interesting in the sense that the solution is C smooth but the boundary of the support of the solution propagates at a constant wave spped. If 0<G(0)< and –<G(0)<0, then the material accomodates stress waves under step traction leading to an elastic steady state.     

The behavior of a spherical bubble near a solid wall in a compressible liquid

Summary The behavior of a spherical bubble near a solid wall is analysed by considering the liquid compressibility. The equation of motion of the bubble with first order correction for the effects of liquid compressibility and solid wall is derived. The equation obtained here coincides with the known result in case of L or C . Further experimental study is made on the motion of bubbles produced by a spark discharge in water. The theoretical results are in good agreement with the experiments.

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Das Verhalten einer kugelförmigen Blase in einer kompressiblen Flüssigkeit in der Nähe einer festen Wand

Übersicht Bei Berücksichtigung der Flüssigkeitskompressibilität wird das Verhalten einer kugelförmigen Blase in der Nähe einer festen Wand analysiert. Die Gleichung der Bewegung der Blase wird mit der Korrektur erster Ordnung für den Einfluß der Flüssigkeitskompressibilität und der festen Wand angegeben. Aus der erhaltenen Gleichung wird für L oder C das bekannte Ergebnis hergeleitet. Darüber hinaus wird eine experimentelle Untersuchung der Blasenbewegung durchgeführt. Die Blase wird mit Hilfe von Funkendurchschlägen zwischen Elektroden in Wasser erzeugt. Die theoretischen Ergebnisse stimmen gut mit den Experimenten überein.

     

Maximum density effects on thermal instability of horizontal laminar boundary layers

Linear stability theory is used to investigate the onset of longitudinal vortices in laminar boundary layers along horizontal semi-infinite flat plates heated or cooled isothermally from below by considering the density inversion effect for water using a cubic temperature-density relationship. The analysis employs non-parallel flow model incorporating the variation of the basic flow and temperature fields with the streamwise coordinate as well as the transverse velocity component in the disturbance equations. Numerical results for the critical Grashof number Gr _L ^* =Gr _X ^* /Re _X< ^{Emphasis>/3/2} are presented for thermal conditions corresponding to –0.5 ₁–2.0 and –0.8 ₂1.2.Nomenclature a wavenumber, 2/ - D operator, d/d - F (f–f)/2 - f dimensionless stream function - g gravitational acceleration - G eigenvalue, Gr _L/Re_L - Gr _L Grashof number based on L - Gr _X Grashof number based on X - L characteristic length, (X/U)^1/2 - M number of divisions in y direction - P pressure - Pr Prandtl number, / - p dimensionless pressure, P/( ² /Re _L) - Re _L, Re_X Reynolds numbers, (U L/)=Re _X< ^1/2 and (U), respectively - T temperature - U, V, W velocity components in X, Y, Z directions - u, v, w dimensionless perturbation velocities, (U, V, W)/U - X, Y, Z rectangular coordinates - x, y, z dimensionless coordinates, (X, Y, Z)/L - thermal diffusivity - coefficient of thermal expansion - ₁, ₂ temperature coefficients for density-temperature relationship - similarity variable, Y/L=y - dimensionless temperature disturbance, /T - dimensionless wavelength of vortex rolls, 2/a - ₁, ₂ thermal parameters defined by equation (12) - kinematic viscosity - density - dimensionless basic temperature, (T _b T )/T - –1 - T temperature difference, (T _w–T ) - * critical value or dimensionless disturbance amplitude - prime, disturbance quantity or differentiation with respect to - b basic flow quantity - max value at a density maximum - w value at wall - free stream condition