Negatively isotone optimal policies for random walk type Markov decision processes

Summary This paper considers a random walk type Markov decision process in which the state spaceI is an integer subset of IR ^m , and the action spaceK is independent ofi I. The natural order, overI, and a quasi order,, overK, is assumed, together with aconditional convexity assumption on the returns {r _i ^k }, and certain other assumptions about these rewards and the transition probabilities in relationship to the orders and.A negatively isotone policy is one for whichi i(i))(i) (i.e.(i) (i) or(i) i)). It is shown that, under specified conditions, a negatively isotone optimal policy exists. Some consideration is given to computational implications in particular relationship to Howard's policy space method.

---

Zusammenfassung Wir betrachten einen Markovschen Entscheidungsprozeß vom random walk Typ. Der ZustandsraumI sei eine Teilmenge des IR^m, wobeii I ganzzahlige Komponenten habe. Die MengeK der zulässigen Aktionen ini I sei unabhängig voni I. Sei die natürliche Ordnung aufI und sei eine Quasiordnung aufK. Die Erträge {r _i ^k }seienbedingt konvex, darüberhinaus seien weitere Voraussetzungen über diese Erträge und die Übergangswahrscheinlichkeiten in Bezug auf die Ordnungen und erfüllt. Eine Politik heißt negativ isoton, falls ausi i folgti(i) (d. h.(i) (i) oder(i)(i)). Wir zeigen, daß unter gewissen Voraussetzungen einenegativ isotone optimale Politik existiert: Auch diskutieren wir einige Folgerungen für die Numerik, insbesondere hinsichtlich Howards Politikiteration.

     

Breakup of an anomolously viscous liquid film in a centrifugal force field

An equation is obtained for the breakup radius with consideration of tipping moments and Laplacian pressure forces acting on the liquid ridge at the critical point.Notation K, n rhenological constants - density - surface tension - r current cup radius - R maximum cup radius - r_c critical radius for film breakup - ¯r=¯r=r/R dimensionless current radius - ¯r_c=r_c/R dimensionless critical radius - ₀, _c actual and critical film thicknesses - current thickness - R_r ridge radius - h₀ ridge height - h current ridge height - ₀ limiting wetting angle - current angle of tangent to ridge surface - angle between axis of rotation and tangent to cup surface - angular velocity of rotation - q volume liquid flow rate - v₁ and v meridional and tangential velocities - =4vv _lm/r,=4v_m/r dimensionless velocities - M moments of surface and centrifugal forces - M_v moment from velocity head - p_r pressure within ridge - P_vm pressure from velocity head - p_m, p_pm pressures from centrifugal force components tangent and normal to cup surface - deviation range of breakup radius from calculated value - ¯r_max, ¯r_min limiting deviations of breakup radius - _c angle of tangent to curve _c/°₀=f(¯r) at critical point - t random oscillation of ratio _c/_c Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 51–56, July, 1980.     

Effect of the weight of the admixture on the turbulence structure of a two-phase jet

The effect of gravity on the turbulence structure of an inclined two-phase jet is evaluated according to the Prandtl theory of mixing length.Notation C_x drag coefficient for a particle - D_p particle diameter - g_i components of the acceleration g due to gravity acting on a particle in the direction of jet flow (g_i=g sin ) and in the direction normal to it (g_i=g cos ) - V_poi ^±, V_goi ^± fluctuation components of the velocities of the particles and gas, respectively, at the end of a mole formation - V_fi free-fall velocity of a particle - l _u mixing length - m_p particle mass - t _p length of time of particle-mole interaction - V_pi ^±, V_gi ^± positive and negative fluctuation velocities of particles and of the gas respectively, with the components u_p ^±, u_g ^±, v_p ^±, v_g ^±, k=V_goi/V_fi - V_i ^± relative velocity of the gas - jet inclination angle relative to the earth's surface - empirical constant - _u, jet boundaries in terms of velocity and concentration - _u=y/ _u dimensionless velocity ordinate - =y/ dimensionless concentration ordinate - admixture concentration - u_m, _m velocity and the concentration of the admixture at the jet axis - _g dynamic viscosity of the gas - _s, _g densities of the particle material and of the gas - _g, _p shearing stresses in the gas and in the gas of particles - _m, ₀ shearing stresses in the mixture and in pure gas, respectively Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 422–426, March, 1981.     

Thermal diffusivity of inhomogeneous systems

The possibility of analyzing the nonsteady temperature fields of inhomogeneous systems using the quasi-homogeneous-body model is investigated.Notation t, t_I, t_i temperature of quasi-homogeneous body inhomogeneous system, and i-th component of system - a, , c thermal diffusivity and conductivity and volume specific heat of quasi-homogeneous body - a_{i i}, c_i same quantities for the i-th component - q heat flux - S, V system surface and volume - x, y coordinates - macrodimension of system - dimensionless temperature Fo=a/² - Bi=/ Fourier and Biot numbers - N number of plates - =h/ ratio of micro- and macrodimensions - _V, volumeaveraged and mean-square error of dimensionless-temperature determination - time - m_i i-th component concentration Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 126–133, July, 1980.     

Analytical Theory of DC SQUIDS Operating in the Presence of Thermal Fluctuations

A comprehensive analytical theory of symmetric DC SQUIDs is presented taking into account the effects of thermal fluctuations. The SQUID has a reduced inductance < 1/ where = 2LI_c/₀, L is the loop inductance, ₀ is the flux quantum, and I_c is the critical current of the identical Josephson junctions which are assumed to be overdamped. The analysis, based on the two dimensional Fokker–Planck equation, has been successfully performed in first order approximation with considered a small parameter. All important SQUID characteristics (circulating current, current-voltage curves, transfer function, and energy sensitivity) are obtained. In the limit 1( = 2k_BT/I_c0 is the noise parameter, k_B is the Boltzmann constant, and T is the absolute temperature) the theory reproduces the results of numerical simulations performed for the case of small thermal fluctuations. It was found that for < 1 the SQUID energy sensitivity is optimum when is higher than 1/, i.e., outside the range for which the present analysis is valid. However, for 1 the energy sensitivity has a minimum at L = L_F , where L_F = ( ₀ /2) ²/k_B , and therefore, in this case, the optimal reduced DC SQUID inductance is _opt = 1/, i.e., within the range for which the present analysis is valid. In contrast to the case of an RF SQUID, for a DC SQUID the transfer function decreases not only with increasing L/L_F but also with increasing (as 1/). As a consequence, the energy sensitivity of a DC SQUID with < 1/ degrades more rapidly (as ⁴ ) with the increase of than that of an RF SQUID does (as ² ).     

Heat and mass transfer effects on flow past an impulsively started vertical plate

Summary An exact solution to the problem of lfow past an impulsively started infinite vertical plate in the presence of uniform heat and mass flux at the plate is presented by the Laplace-transform technique. The velocity, the temperature and the concentration profiles are shown graphically. The rate of heat transfer, the skin-friction, and the Sherwood number are also shown on graphs. The effect of different parameters like Grashof number, mass Grashof number, Prandtl number, and Schmidt number are discussed.List of symbols C species concentration near the plate - C species concentration in the fluid far away from the plate - C dimensionless concentration - C _p specific heat at constant pressure - D mass diffusion coefficient - g acceleration due to gravity - Gr thermal Grashof number - Gc mass Grashof number - j mass flux per unit area at the plate - K thermal conductivity of the fluid - Nu Nusselt number - Pr Prandtl number - q heat flux per unit area at the plate - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the plate - T temperature of the fluid far away from the plate - T _w temperature of the plate - u velocity of the fluid in thex-direction - u ₀ velocity of the plate - u dimensionless velocity - x coordinate axis along the plate - y coordinate axis normal to the plate - y dimensionless coordinate axis normal to the plate - volumetric coefficient of thermal expansion - ^* volumetric coefficient of expansion with concentration - coefficient of viscosity - kinematic viscosity - density - skin-friction - dimensionless skin-friction - dimensionless temperature - er fc complementary error function - similarity parameter     

Response of a spinning liquid column to pitch excitation

Summary The response of a solidly rotating liquid bridge consisting of inviscid liquid is determined for pitch excitation about its undisturbed center of mass. Free liquid surface displacement and velocity distribution has been determined in the elliptic (>2₀) and hyperbolic (<2₀) excitation frequency range.List of symbols a radius of liquid column - h length of column - I ₁ modified Besselfunction of first kind and first order - J ₁ Besselfunction of first kind and first order - r, ,z cylindrical coordinates - t time - u, v, w velocity distribution in radial-, circumferential-and axial direction resp. - mass density of liquid - free surface displacement - velocity potential - ₀ rotational excitation angle - ₀ velocity of spin - forcing frequency - _1n natural frequency - surface tension - acceleration potential - for elliptic range >2₀ - for hyperbolic range >2₀     

Effect of the elastic factor on the hydrodynamic stability of a structurally viscous medium

The effect of relaxation phenomena on the hydrodynamic stability of the plane gradient flow of a structurally viscous medium is investigated using linear theory.Notation _ij stress tensor deviator - U_i components of the velocity vector - x_i coordinates - t time - P pressure - =₀L/^*V plasticity parameter - _o limiting shear stress - andc dimensionless wave number and the perturbation frequency - Re=VL/* Reynolds number - density - F_ij deformation rate tensor Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 5, pp. 868–871, November, 1978.     

A nonlinear composite shell element with continuous interlaminar shear stresses

A numerical model for layered composite structures based on a geometrical nonlinear shell theory is presented. The kinematic is based on a multi-director theory, thus the in-plane displacements of each layer are described by independent director vectors. Using the isoparametric apporach a finite element formulation for quadrilaterals is developed. Continuity of the interlaminar shear stresses is obtained within the nonlinear solution process. Several examples are presented to illustrate the performance of the developed numerical model.List of symbols reference surface - convected coordinates of the shell middle surface - ⁱ coordinate in thickness direction - ⁱ h thickness of layer i - X_o position vector of the reference surface - ⁱX_o position vector of midsurface of layer i - t _k orthonormal basis system in the reference configuration - ⁱ a _k orthonormal basis system of layer i - ⁱW axial vector - R_o orthonormal tensor in the reference configuration - ⁱ R orthonormal tensor of layer i - ⁱ Cauchy stress tensor - ⁱ P First Piola-Kirchhoff stress tensor - ⁱ q vector of interlaminar stresses - ⁱ n, ⁱ m vector of stress resultants and stress couple resultants - v _x components of the normal vector of boundary - ⁱ N, ⁱ Q, ⁱ M stress resultants and stress couple resultants of First Piola-Kirchhoff tensor - stress resultants and stress couple resultants of Second Piola-Kirchhoff tensor - ⁱ , ⁱ , ⁱ strains of layer i - _K transformation matrix - u_o displacement vector of layer 1 - ⁱ local rotational degrees of freedom of layer i     

Nonsteady thermocapillary mass transfer in the laser alloying of metals

This article examines convective mass transfer of an impurity in a shallow bath of molten metal with allowance for the motion of the fusion front during the laser alloying of metals.Notation r, z, cylindrical coordinates - t time - T_i temperature of the liquid (i=1) and solid (i=2) phases - q(r) absorbed energy flux - k concentration factor - T_m melting point - L heat of fusion - density - _i, _i thermal conductivity and diffusivity - T₀ initial temperature - , absolute and kinematic viscosities of the melt - v_r, v_z projections of the melt velocity on the coordinate axes r and z - p pressure - surface tension Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 5, pp. 799–805, May, 1989.     

A boundary layer theory for the end problem of a circular cylinder

Summary This paper discusses the nature of an approximate solution for the hollow circular cylinder whose fixed ends are given a uniform relative axial displacement and whose cylindrical surfaces are free from traction. We shall take the solution of this problem to be given by a super-position of the following two problems: problem I considers a finite length cylinder whose ends are given a relative axial displacement, but are no longer fixed; problem II removes the radial displacement at the end of the cylinder obtained in problem I.Nomenclature a mid-surface radius of cylinder - c half-height of cylinder - E, in-plane elastic moduli - E_t, _t, G_t transverse elastic moduli - _z, , _r axial, circumferential, and normal strain - _rz transverse shear strain - h cylinder thickness - _z, , _r axial, circumferential, and normal stress - _rz transverse shear stress - z, r axial and radial coordinates - u_z, u_r axial and normal displacements     

Smoothing splines applied in thermal experiments

Cubic smoothing splines are applied to heat transfer in a region of detached flow on a plate with a cylindrical obstacle.Notation M Mach number of incoming flow - t temperature - q specific heat flux - thermal conductivity - c specific heat - p density - x longitudinal coordinate - x relative longitudinal coordinate - z transverse coordinate - r, cylindrical coordinates - r₀ radius of curvature of isothermal lines - current time - transient-state duration - d cylinder diameter - wall thickness Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 2, pp. 246–249, August, 1980.     

Vibrations of axially compressed laminated orthotropic cylindrical shells,including transverse shear deformation

Summary The free vibrations, buckling and the effect of initial prestress upon the frequency spectrum of orthotropic composite cylindrical shells are examined in the context of a theory that includes transverse shear deformation. Results obtained are compared with the predictions of refined Love-type theory and simplified Donnell-type theory that do not consider shear deformation.The calculated examples indicate that transverse shear deformation can be significant not only for short composite shells but even for longer shells possessing low shear moduli.

---

Schwingungen axial gedrückter, laminierter, orthotroper zylindrischer Schalen einschließlich Querschubverformung

Zusammenfassung Es werden die freien Schwingungen, das Beulen und der Einfluß einer Anfangsvorspannung auf das Frequenzspektrum orthotroper, kompositer, zylindrischer Schalen im Rahmen einer Theorie, die die Querschubverformung mitberücksichtigt, untersucht. Die erhaltenen Resultate werden mit den Vorhersagen der verbesserten Loveschen Theorie und der vereinfachten Donnellschen Theorie, welche keine Querschubverformung in Betrach ziehen, verglichen.Die durchgerechneten Beispiele zeigen, daß die Querschubverformung nicht nur für kurze, komposite Schalen sondern sogar für längere Schalen mit niederem Schubmodul Bedeutung gewinnen kann.

---

Nomenclature a radius of reference surface of cylindrical shell - A _i amplitudes of displacements - A _ij elastic area - B _ij elastic statical moment - D _ij elastic moment of inertia - E _ij elastic stiffness modulus - h shell thickness - k ₄₄,k ₅₅ shear correction factors - l length of cylindrical shell - L _ij functional operator - m number of half-waves in axial direction - M ,M ,M axial, circumferential and twisting moments, respectively - n number of circumferential waves - N ⁰ axial compression force - N ,N ,N axial, circumferential and shear forces, respectively - Q ,Q transverse shear forces - R ₀,R ₁,R ₂ inertia terms - t time - T matrix defined in Eq. (25) - u, v, w displacements in axial, circumferential and radial directions, respectively - U _i time-independent displacement or rotation - nondimensional axial wave parameter - , , axial, circumferential and shear strains, respectively - radial coordinate, taken positive inward - polar angle - , , curvature changes - circular frequency - nondimensional axial coordinate - density - , , , , stress components - , rotations of normal to undeformed midsurface With 6 Figures     

Stress intensity factors in a fully interacting,multicracked, isotropic plate

An essential part of describing the damage state and predicting the damage growth in a multicracked plate is the accurate calculation of stress intensity factors (SIF's). Analytical derivations of these SIF's for a multicraked plate can be complex and tedious. Recent advances, however, in intelligent application of symbolic computation can overcome these difficulties and provide the means to rigorously and efficiently analyze this class of problems. Here, the symbolic algorithm required to implement the methodology for the rigorous solution of the system of singular integral equations for SIF's is presented. The special problem-oriented symbolic functions to derive the fundamental kernels are described, and the associated automatically generated FORTRAN subroutines are given. As a result, a symbolic/FORTRAN package named SYMFRAC, capable of providing accurate SIF's at each crack tip, has been developed and validated.Simple illustrative examples using SYMFRAC show the potential of the present approach for predicting the macrocrack propagation path due to existing microcracks in the vicinity of a macrocrack tip, when the influence of the microcracks' location, orientation, size, and interaction are accounted for.List of symbols offset angle between inner tips of two parallel cracks - _lr, _mz direction cosines between two local coordinate systems - _jl strain tensor - offset of notch-microcracks system with respect to Y axis - _k four roots of the characteristic equation - v Poisson's ratio - _jl ^o , _jl ^T far-field and total stress field, respectively - _XX ^o , _YY ^o , _XY ^o components of stress in global coordinate system - _j angle defining local frame orientation - , normalized real variable - (s, y) Fourier transform of Airy stress function with respect to x variable - a _j half crack length - a ₁₁, a₁₂, a₂₂ coefficients of strain-stress relationship - d normalized radial (tip) distance - f _nj auxiliary functions - k ₁, k₂ mode-I and mode-II stress intensity factors - ker _inf ^sup Fredholm kernels - p _j normal traction at crack surface - q _j shear traction at crack surface - s Fourier variable - r _j, r_kX, r_kY position vector and its components, for an origin of local frame - u, v x, y component of displacement, respectively - w weight function - x _j y_jand X, Y local and global coordinates - [A] matrix of coefficients - C' _j, C_j functions of s in Fourier space (i.e., constrants in x, y-real space) - E Young modulus - F _j(x_j, y_j) Airy stress function - G_nj(_p) discrete auxiliary function - {} loading function vector     

Radiation of internal waves during vertical motion of a body through a nonuniform liquid

Energy losses to radiation of internal waves during the vertical motion of a point dipole in two-dimensional and three-dimensional cases are computed.Notation _o(z), p_o(z) density and pressure of the ground state - z vertical coordinate - v, p, perturbed velocity, pressure, and density - H(d 1n _o/dz)^–1 characteristic length scale for stratification - N=(gH^–1–g²c _o ^–2 )^1/2 Weisel-Brent frequency - g acceleration of gravity - c_o speed of sound - vertical component of the perturbed velocity - V vector operator - k wave vector - frequency - d vector surface element - W magnitude of the energy losses - (t), (r) (x)(y)(z) Dirac functions - v_o velocity of motion of the source of perturbations - d dipole moment of the doublet - _o,l length dimension parameters - _o intensity of the source Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 619–623, October, 1980.     

Transient heat conduction in hollow spheres with a moving inner boundary

The finite integral transform method is used to obtain the solution of unsteady heat conduction problems for a hollow sphere with a moving internal boundary and various boundary conditions at the outer surface. For the solution of the problems of interest integral transform formulas are presented with kernels (16), (20), and (24) and the corresponding inversion formulas (18), (22), (26), (29) and characteristic equations (17), (21), (25), (28), (31), (33).Nomenclature a, thermal diffusivity and conductivity - t temperature of phase transformation - density - heat transfer coefficient - Q total quantity of heat passing through inner boundary - F latent heat of phase transformation - Fo(1,)=a/R ₁ ² , Fo(i,)=/r _i ² , Fo(i, _i)=a _i/r _i ² Fourier numbers - Bi₂=R₂/ Biot number     

Non-linear liquid motion in conical containers

Summary The non-linear behavior of an incompressible and frictionless liquid with a free surface in an annular conical sector frustrum container has been determined. All special cases of conical tanks may be obtained from the results. Some configurations have been evaluated numerically. It was found that the oscillating liquid system exhibits softening behavior, showing for increased free surface amplitudes decreased natural frequencies as compared to the linearized frequencies. This non-linear behavior is more pronounced for larger vertex angles and smaller frustrum heights.Notation a, b radius (describing the free surface,r=a the container bottomr=b) - k=b/a ratio of radii - g gravity constant or acceleration along the axis of the container - h liquid height - P ^m ,Q ^m associated Legendre functions of first and second kind and orderm - r, , spherical coordinates - p liquid pressure - t time - 2, 2 outer- and inner vertex angle resp - 2 sector angle of container - (r, , ,t) velocity potential - liquid density - zeros of associated Legendre function or cross product Legendre functions with respect to the degree - non-linear frequency - _mn ⁽⁰⁾ linearized natural frequencies of liquid - free surface elevation With 7 Figures     

Unsteady motion of a fluid near a disk rotating in a magnetic field

The effect of a magnetic field on the velocity distribution in a fluid close to an unsteadily rotating disk is investigated.Notation r, , and z coordinates in the radial, circular, and axial directions - t time - u, v, and w radial, circular, and axial velocity components - u₀ radial velocity of external potential flux - v₀ circular velocity of the disk - (t) angular velocity of the disk - p pressure - density - v kinematic viscosity - B₀ characteristic of the applied magnetic field - electrical conductivity of fluid - R and Z dimensionless coordinates in the radial and axial directions - =Z/2 dimensionless coordinate - T dimensionless time - U, V, and W radial, circular, and axial components of dimensionless velocity - P dimensionless pressure - a, , and ₀ constants with dimensionality t^–1 - m, n, and positive numbers - k =a constant - = = B ₀ ² / parameter characterizing the magnetic field     

A mixed finite element formulation for shells through a reduced Reissner's principle in elastodynamics

Summary The use of Mixed models based in Reissner's principle in statics has been found to lead to some desirable simplifications in Finite Element formulations, in particular in plates and shells. Reduced formulations of Reissner's principle such as the one used by Prato have proved to be even more successful. In this paper, a reduction similar to that of Prato is attempted on a mixed elastodynamic variational principle by Karnopp.

---

Eine gemischte finite Elemente-Formulierung für Schalen durch ein reduziertes Reissnersches Prinzip der Elastodynamik

Zusammenfassung Die Verwendung von gemischten Modellen basiert auf Reissners Prinzip der Statik führt zu erwünschten Vereinfachungen bei der Formulierung von finiten Elementen im speziellen bei Untersuchungen von Platten und Schalen. Reduzierungen des Reissnerschen Prinzips, wie sie von Prato angewendet worden sind, haben sich sogar als noch erfolgreicher erwiesen. In dieser Untersuchung wird eine Reduktion, ähnlich der von Prato, für ein gemischtes elastodynamisches Variationsprinzip nach Karnopp, vorgenommen.

---

Symbol Table A Domain of integration of the Functional. Also area of the triangle - b Second fundamental form of the shell middle surface - C ^ijkl Elastic Constants - E ₁,E ₁ ^* Strain Energy and Co-Energy density - e _ij Elastic strain tensor - f _i Body force density tensor - I _ks Karnopp's functional, specialized to shells - I _ksc Contracted Karnopp's functional, specialized to shells - i, j, k Index 1, 2, 3 - K ₁,K ₁ ^* Kinetic Energy and Co-Energy density - K ^* Kinetic co-energy density for shell - m Moment tensor defined at the mid-surface - n In-plane stress tensor defined at the middle surface - n Qualifier for the boundary normal - p ,p ³ Boundary forces - Prescribed boundary forces - p Shear force tensor defined at the mid-surface - R Position vector of a point in the volume of the shell - r Position vector of a point on the mid-surface - r _i Net impulse density tensor - S _u Portion of the boundary where displacements are preseribed - S Portion of the boundary where forces are prescribed - s Qualifier for the direction tangent to the boundary - t Time variable - t ^ij Stress tensor - u ,u ₃ Mid-surface displacements - Mid-surface velocities - V Volume - v _i Displacement tensor - , Indices. Range 1, 2 - Shear strain tensor for the middle surface - Variation operator - Mid-surface strain tensor - Mid-surface curvature strain tensor - Direction cosine tensor for boundary normal - Mid-surface rotation tensor - Mid-surface angular velocity tensor - _M Strain energy density - _M * Strain co-energy density - _B * Bending strain co-energy density - _TS ^* Transverse shear strain co-energy density - | Covariant differentiation with respect tox , etc - Partial differentiation with respect tox , etc - .(dot) Time differentiation - -(bar) Prescribed quantities     

Use of adjoint functions in investigations of heat conduction and transfer processes

An examination is made of the use of adjoint functions in heat conduction and convection theory. Formulas of perturbation theory are obtained for steady and unsteady cases, an interpretation of the physical meaning of adjoint temperature is given, and some applications of the theory are discussed.Notation (r,) thermal conductivity - t(r,) temperature - t *(r,) adjoint temperature - q_V(r,) density of heat release sources - p(r,) a parameter of adjoint equation - r generalized coordinate - time - (r_s, ) heat transfer coefficient - I linear functional of temperature - (r,;r₀,₀) and *(r,; r₀,₀) Green's function for t(r, ) and t *(r, ) - C(r,) volume specific heat - W(r, ) vector distribution of flow velocities - V, S volume and surface areas of body - R radius of HRE - r, radial and angular coordinates - F_in, F_out inlet and outlet flow areas of channel