Multicomponent turbulent boundary layer on a chemically active surface

When a gaseous mixture flows past chemically active surfaces the boundary layer formed on the wetted body may contain a large number of components with different diffusion properties. This leads to the necessity for studying the diffusion of the components in the multicomponent boundary layer.The use of thebinary boundary layer concept in the general case cannot yield satisfactory results, since replacement of the mutual diffusion coefficients D_ij of the various pairs of components by a single diffusion coefficient D in many cases is a rough approximation.In the general case the number of different diffusion coefficients is equal to N(N–1)/2 (N is the number of components). Usually it is possible to identify groups of components with similar molecular weights. Then the number of different diffusion coefficients may be reduced without large error. However, even in the comparatively simple case when it is possible to divide all the components into two groups with similar molecular weights we must take account of three different diffusion coefficients (one diffusion coefficient in each group and also the diffusion coefficient for the components of one group relative to the components of the other group). Only in particular cases when the gaseous mixture consists of only two components with arbitrary molecular weights, or if all the components of the gaseous mixture have similar molecular weights, can we with justification introduce a single diffusion coefficient (if in this case there are no limitations on the direction of the diffusion).Studies have been published covering the laminar multicomponent boundary layer. An analytic method for solving the equations of the laminar multicomponent boundary layer was developed by Tirskii [1]. There are also studies in which concrete results were obtained by numerical methods with the use of computers (for example, [2, 3]).As far as the author knows, for turbulent flow there are studies (for example, [4, 5]) covering flow with chemical reactions only in the case when all the diffusion coefficients are equal (D_ij=D).The present paper presents a method for calculating the turbulent multicomponent boundary layer with account for several different diffusion coefficients.Notation x, y coordinates - u, v velocity components - density - T temperature - h heat content - H enthalpy - c_i mass concentration of the i-th component - c ₁ ⁽¹⁾ element concentrations in solid body - J_i diffusion flux of the i-th component - m molecular weight - dynamic viscosity coefficient - kinematic viscosity coefficient - heat conduction coefficient - c_p specific heat - adiabatic index - D_ij binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - S_t Stanton number - M Mach number - friction - q radiant thermal flux - boundary layer thickness - D rate of displacement of gas-solid interface - degree of gasification - r_ij weight fraction of element i in component j - _ij stoichiometric coefficients - K_i reaction equilibrium constants - l number of components for which I_i0 Indices i, j component number - w quantities for y=0 - * quantities on the edge of the laminar sublayer - ⁽¹⁾ quantities at the solid body - quantities at the outer edge of the boundary layer - molar transport coefficients     

Couette flow of a multicomponent ionized gas

We consider equilibrium flow of a multicomponent ionized gas between two catalytic plates of infinite length, one of which moves parallel to the other with constant velocity. The results of [1] are generalized for ionized gaseous mixtures which are in local thermodynamic equilibrium. Formulas are presented for calculating the thermal flux and the effective thermal conductivity for ambipolar diffusion.Then a special ionization case is discussed.Notation A_i chemical symbol of the i-th component - W_i projection of the molar diffusive flux vector of the i-th component on the y-axis - x_i molar concentration - H_i enthalpy - m_i molecular weight - Q_s heat of the s-th reaction - K_ps(T) equilibrium constant of the s-th reaction - W_i mass formation rate of the i-th component per unit volume - Z_i charge number - e unit charge (electron charge) - E electric field intensity - distance between the plates - N number of components - v _sl stoichiometric coefficients - density - T temperature - p pressure - u projection of average velocity on y-axis - viscosity - thermal conductivity - D_ij binary diffusion coefficient - R universal gas constant - k Boltzmann constant In conclusion, the author wishes to thank G. A. Tirskii for proposing the study and for suggestions made in the course of the investigation.     

A class of hypo-elastic non-elastic materials and their thermodynamics

Thermodynamics is developed for a class of thermo-hypo-elastic materials. It is shown that materials of this class obey the laws of thermodynamics, but are not elastic.

Table of Symbols

Latin Letters A _ijkl tensor-valued function of t _ij appearing in hypo-elastic constitutive relation - B _ijkl another tensor-valued function. See equation (4.2) - B the square of - d _ij rate of deformation tensor - d _ij deviator of rate of deformation - f, k functions of pressure, p - g, h functions of the invariant - p pressure - q _i heat flux vector - s _ij stress deviator - _ij co-rotational derivative of stress deviator - t time - t ₁ t ₂ specific values of time - t _ij stress tensor - t _ij ⁰ a specific value of stress - T Temperature - T ₀ a specific value of temperature - u _i velocity - V(t) a material volume as a function of time, t - V ₀ a material volume at a reference configuration - W work (W = work done in a deformation—section 5) Sript Letters Specific internal energy - Specific Helmholtz free energy - G Specific Gibbs function Greek Letters an invariant of the stress deviator—see eq. (2.4) - _ij kroneker delta - (W = work done in a deformation—section 5) - specific entropy - hypo-elastic potential - hypo-elastic potential - mass density - ₀ mass density in a reference configuration - specific volume = 1/ - a function of p - _ijkl a constant tensor—see eq. (2.5) - –G/ - _ij rate of rotation tensor This work is dedicated to Jerald L. Ericksen, without whose influence it would not have been possible     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

The energy-integral method: application to linear hyperbolic heat-conduction problems

This paper utilizes the energy-integral method to obtain approximate analytic solutions to a linear hyperbolic heat-conduction problem for a semi-infinite one-dimensional medium. As for the mathematical formulation of the problem, a time-dependent relaxation model for the energy flux is assumed, leading to a hyperbolic differential equation which is solved under suitable initial and boundary conditions. In fact, analytical expressions are derived for uniform as well as varying initial conditions along with (a) prescribed surface temperature, or (b) prescribed heat flux at the surface boundary. The case when a heat source (or sink) of certain type takes place has also been discussed. Comparison of the approximate analytic solutions obtained by the energy-integral method with the corresponding available or obtainable exact analytic solutions are made; and the accuracy of the approximate solutions is generally acceptable.Nomenclature A,C constants - a ₀(t),a ₁(t),...,a _n (t) arbitrary time-dependent coefficients, equation (3.2) - b thermal propagation speed - C _p specific heat of solid at constant pressure - g(x) given function, equation (5.1) - h(t) specified function of time - I _n modified Bessel function of the first kind - K thermal conductivity - j,n positive constants - P _n (x,t) polynomial of degreen - q(x,t) heat flux - Q(t),R(t),H(t),E(t) see equations (3.9), (II.d), (4.10), (4.12), respectively - (t) thermal penetration depth - (t,) approximate thermal penetration depth - T(x,t) temperature distribution - t time - y dimensionless time, equation (3.17) - V(y) dimensionless surface heat flux - W(y) dimensionless surface temperature - U-(t) unit-step function - G(x;t,) Green's function - x spatial variable - ()₀ surface value (atx=0) Greek symbols thermal diffusivity - density of solid - parameter, see equations (3.11) and (3.13) - parameter depending onn and - specified parameter, equations (4.5a) and (5.12b) - (t),(t) given functions of time, equations (4.6) and (5.5b) - , dummy variables - relaxation time - energy integral - (y),(y) specified functions ofy; equations (3.22) and (4.19)     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

The effect of ablation products on heat exchange in the disintegration of graphite in a radiating air plasma

In this article we consider the problem of the stationary hypersonic flow of a viscous, heat-conducting stream of radiating high-temperature air past the forward critical point of a blunt body made of graphite in the region between a passing shock wave and the surface of the body.We investigated the radiative and convective heat exchange taking place at an impenetrable surface, as well as such heat exchange when air is injected.We obtained the characteristics of the graphite ablation on the assumption that there is radiative transfer of heat by the graphite-disintegration products.The diffusion was calculated on the basis of a binary model, i.e., it was assumed that the mixture consists of two components: the advancing air and the disintegration products; the chemical reactions in the boundary layer were assumed to be frozen, and on the outer edge of the boundary layer, reaching out as far as the shock wave, they were assumed to be equilibrium reactions.The condition of the gas on the disintegrating surface was also determined on the assumption that there was chemical equilibrium, and the saturation pressure of the vapors was assumed to be equal to the stagnation pressure.It should be noted that the effect of ablation on heat exchange has been considered in [1–3]. Anfimov and Shari [1] assumed that pure air was injected and their calculations were carried out up to stagnation temperatures of 15·10³° K. Hoshisaki and Zasher [2] gave an analysis of a small number of variants, where the values of the absorption cross sections of the injected components used in their calculations were independent of temperature; and the discussion in [3] was limited to the case of low injection velocities.Notation c_p specific heat at constant pressure - c_i mass concentration of the i-th component - D₁₂ coefficient of binary diffusion - f dimensionless stream function - h static enthalpy of the mixture - h_i enthalpy per unit mass of the i-th component - k thermal conductivity - m_w total rate of mass loss through the surface - p stagnation pressure - q heat flux - T absolute temperature - u velocity component along the x axis - v velocity component along the y axis - optical thickness - coefficient of absorption - Lees-Dorodnitsyn self-similar variable - viscosity coefficient - absorption cross section - density - Stefan-Boltzmann constant - wavelength - angle between the normal to the surface and the incident intensity ray Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–31, January–February, 1971.     

Heat and mass transfer near the stagnation point with injection and suction of various gases through the body surface

Numerical solutions of the equations of the laminar boundary layer in the vicinity of the stagnation point of an axisymmetric blunted body with injection of single-component gases into a homogeneous external stream are obtained and generalized. More than 30 different pairs of gases are investigated. The heat and mass transfer in a multicomponent laminar boundary layer with the injection of a gas mixture, and also with simultaneous injection and suction of different gases through the body surface, is analyzed. An approximate method is proposed for calculating the heat and mass transfer in a laminar boundary layer.Notation density - T temperature - J enthalpy - M molecular weight - c_i mass concentration - x_i molar concentration - viscosity coefficient - heat conductivity - D_ij binary diffusion coefficient - D_i generalized diffusion coefficient - V_i diffusion velocity - q convective heat flux - surface friction - G over-all mass flow rate through the surface - G_i flow rate of the i-th component through the surface - /c_p heat transfer coefficient - _i mass transfer coefficient of the i-th component - _q injection coefficient for heat transfer (2. 7) - injection coefficient for mass transfer (2. 7) - , are the parameters of the intermolecular interaction potential function - /c _p =q/(J _e -J _wo), _i = (pc _i Vi) _w /(c _iw -c _ie ) (pc _i V _f) _w /(c _iw –c _ie ) The author wishes to thank V. S. Dranichkin, M. V. Gusev, and A. I. Noikin, who assisted with the computer calculations and the analysis of the computer results.     

Modeling pressure-strain rate correlations in turbulence theory

On the basis of a spectral representation of the rapid part _ij,2 of the correlation tensor p(u _i /x _j ) using Cramer's theorem the inequality _ij,2(U _j /x _i )0 is obtained. As distinct from the realizability conditions, it can serve as a direct and very rigorous test of the adequacy of model expressions for _ij,2. In particular, it is shown that the best known of such expressions do not satisfy this test.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 42–46, March–April, 1992.     

On forced vibration of anisotropic cylinders

Summary The dynamic response of a circular cylinder with thick walls of transverse curvilinear isotropy subjected to a uniformly distributed pressure varying periodically with time is analyzed by means of the Laplace transformation, and the exact solution is obtained in closed form. The previously obtained solutions for forced vibrations with isotropy, and free vibrations with transverse curvilinear isotropy are included as special cases of the general results reported here.Nomenclature t time - r, , z cylindrical coordinates - _ii components of normal strain - _ii components of normal stress - u radial displacement - c _ij elastic constant - mass density - c ² c ₁₁/ - ² c ₂₂/c ₁₁ - a, b inner, outer radius of the cylinder - , A, B constants - forced angular frequency - function defined by (9) - p, real, complex variables - constant defined by (14) - real number - , Lamé elastic constants - J (x) Bessel function of first kind - Y (x) Bessel function of second kind - I (x) modified Bessel function of first kind - K (x) modified Bessel function of second kind     

Correlation of experimental data on the pulsation velocity intensity for turbulent fluid flow in channels of different form

An analysis is made of experimental data on the intensity of the velocity pulsations in turbulent fluid flow in channels of different shape. Correlating relations are constructed for the intensity of the velocity pulsation components as a function of the flow regime and coordinates.Notation x, y, z coordinates in the flow direction, along the normal and parallel to the channel wall, respectively - a normal distance from the channel center to the wall - b distance from the channel corner to the point of intersection with the wall of the normal from the channel center - ₁ dimensionless distance along the normal from the channel wall - ₂ dimensionless distance in the direction parallel to the channel wall - U local fluid velocity - U_m maximal fluid velocity - U average fluid velocity across the section - _i intensity (mean square value) of the i-th component of the velocity pulsations (u, v, w are the indices in the directions x, y, z, respectively) - _i0 value of the intensity of the velocity pulsation components at the center of the channel - U_L velocity difference within the limits of the hydrcdynamic macroscale - q² total turbulence energy at a fixed point of the flow     

A turbulent immersed air wall jet on a burning graphite surface

The turbulent boundary layer in an immersed air jet traveling along a burning graphite wall is analyzed. In order to study the heat and mass transfer and the friction in the boundary layer, a method of calculation based on the solution of the integrated energy and momentum relations is employed, allowing for the conservative properties of the turbulent boundary layer at the wall [1].Notation x longitudinal coordinate - y transverse coordinate - s height of slot - thickness of boundary layer - * displacement thickness - ** momentum-loss thickness - velocity - j transverse mass flow (flux) - density - T temperature - i total enthalpy - temperature factor - ₁ enthalpy factor - k reduced concentration of the i-th component - tangential stress - b permeability parameter - C₁ form (shape) parameter - dynamic viscosity coefficient - c _f friction coefficient - relative friction coefficient Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 60–67, January–February, 1971.     

A model of stress dissipation in second-moment closures

The paper presents a modified expression for the dissipation rate tensor _ij in the second-moment closure models, which employs the dissipation flatness parameterE and the turbulenceRe number. The expression reproduced the distribution among the three diagonal components of _ij in agreement with the direct numerical simulation of a plane channel flow ofMansour, Kim and Moin, 1988. Implemented in a low-Re-number differentialRe-stress model the relationship yielded predictions of dissipative components better than other models, albeit spoiled by still unsatisfactory modelling of the equation for the energy dissipation rate . on leave from Mainski Fakultet, University of Sarajevo, Bosnia Hercegovina.     

The contribution of internal degrees of freedom to the non-Newtonian rheology of model polymer fluids

We report non-equilibrium molecular dynamics simulations of rigid and non-rigid dumbbell fluids to determine the contribution of internal degrees of freedom to strain-rate-dependent shear viscosity. The model adopted for non-rigid molecules is a modification of the finitely extensible nonlinear elastic (FENE) dumbbell commonly used in kinetic theories of polymer solutions. We consider model polymer melts — that is, fluids composed of rigid dumbbells and of FENE dumbbells. We report the steady-state stress tensor and the transient stress response to an applied Couerte strain field for several strain rates. We find that the rheological properties of the rigid and FENE dumbbells are qualitatively and quantitatively similar. (The only exception to this is the zero strain rate shear viscosity.) Except at high strain rates, the average conformation of the FENE dumbbells in a Couette strain field is found to be very similar to that of FENE dumbbells in the absence of strain. The theological properties of the two dumbbell fluids are compared to those of a corresponding fluid of spheres which is shown to be the most non-Newtonian of the three fluids considered.Symbol Definition b dimensionless time constant relating vibration to other forms of motion - F force on center of mass of dumbbell - F _i force on bead i of dumbbell - F force between center of masses of dumbbells and - F _ij force between beads i and j - h vector connecting bead to center of mass of dumbbell - H dimensionless spring constant for dumbbells, in units of / ² - I moment of inertia of dumbbell - J general current induced by applied field - k _B Boltzmann's constant - L angular momentum - m mass of bead, (= m/2) - M mass of dumbbell, g - N number of dumbbells in simulation cell - P translational momentum of center of mass of dumbbell - P pressure tensor - P _xy xy component of pressure tensor - Q separation of beads in dumbbell - Q _eq equilibrium extension of FENE dumbbell and fixed extension of rigid dumbbell - Q ₀ maximum extension of dumbbell - r _ij vector connecting beads i and j - r position vector of center of mass dumbbell - R vector connecting centers of mass of two dumbbells - t time - t ^* dimensionless time, in units of m/ - T ^* dimensionless temperature, in units of /k - u potential energy - u velocity vector of flow field - u _x x component of velocity vector - V volume of simulation cell - X general applied field - strain rate, s^–1 - ^* dimensionless shear rate, in units of /m ² - general transport property - Lennard-Jones potential well depth - friction factor for Gaussian thermostat - shear viscosity, g/cms - ^* dimensionless shear viscosity, in units of m/ ² - ^* dimensionless number density, in units of ^–3 - Lennard-Jones separation of minimum energy - relaxation time of a fluid - angular velocity of dumbbell - orientation angle of dumbbell      

Viscosity of multicomponent gas mixtures of polar gases

Summary The rigorous binary viscosity expression _mix as transformed to the form originally suggested by Sutherland is studied for mixtures involving polar gases. Any attempt to simplify the _ij of the Sutherland viscosity expression turns out to be only approximately successful. A relation for _ij / _ji is however derived, and the procedure suggested for computing _mix on this basis appears to be very successful. The _ij to a large extent are temperature and composition independent and it has been shown that this fact can be utilised with success for predicting _mix values at high temperatures.     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)     

Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at x 3=0

The three Barnett-Lothe tensors S, H, L and the three associated tensors S(), H(), L() appear frequently in the real form solutions to two-dimensional anisotropic elasticity problems. Explicit expressions of the components of these tensors are derived and presented for monoclinic materials whose plane of material symmetry is at x ₃=0. We use the algebraic formalism for these tensors but the results are derived not by the straight-forward substitution of the complex matrices A and B into the formulae. Instead, we find the product –AB ^-1, whose real and imaginary parts are SL ^-1 and L ^-1, respectively. The tensors S, H, L are then determined from SL ^-1 and L ^-1. For S(), H(), L() we again avoid the direct substitution by employing an alternate approach. The new approaches require minimal algebra and, at the same time, provide simple and concise expressions for the components of these tensors. Although the new approaches can be extended, in principle, to monoclinic materials whose plane of symmetry is not at x ₃=0 and to materials of general anisotropy, the explicit expressions for these materials are too complicated. More studies are needed for these materials.     

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     

The secondary flow induced around a sphere in an oscillating stream of elastico-viscous liquid

The present paper is devoted to the theoretical study of the secondary flow induced around a sphere in an oscillating stream of an elastico-viscous liquid. The boundary layer equations are derived following Wang's method and solved by the method of successive approximations. The effect of elasticity of the liquid is to produce a reverse flow in the region close to the surface of the sphere and to shift the entire flow pattern towards the main flow. The resistance on the surface of the sphere and the steady secondary inflow increase with the elasticity of the liquid.Nomenclature a radius of the sphere - b ^ik contravariant components of a tensor - e contravariant components of the rate of strain tensor - F() see (47) - G total nondimensional resistance on the surface of the sphere - g _ik covariant components of the metric tensor - f, g, h secondary flow components introduced in (34) - k ₀ measure of relaxation time minus retardation time (elastico-viscous parameter) - K =k ₀ ²/V ₀ ² , nondimensional parameter characterizing the elasticity of the liquid - n measure of the ratio of the boundary layer thickness and the oscillation amplitude - N, T defined in (44) - p arbitrary isotropic pressure - p _ik covariant components of the stress tensor - p ^ik contravariant components of the stress tensor associated with the change of shape of the material - R =V ₀ a/v, the Reynolds number - S =a/V ₀, the Strouhall number - r, , spherical polar coordinates - u, v, w r, , component of velocity - t time - V(, t) potential velocity distribution around the sphere - V ₀ characteristic velocity - u, v, t, y, P nondimensional quantities defined in (15) - reciprocal of s - density - defined in (32) - defined in (42) - ₀ limiting viscosity for very small changes in deformation velocity - complex conjugate of - oscillation frequency - = ₀/, the kinematic coefficient of viscosity - , defined in (52) - (, y) stream function defined in (45) - =(NT/2n)^1/2 y - /t convective time derivative (1) _ik