First-order chemical reaction on flow past an impulsively started vertical plate with uniform heat and mass flux

Summary A finite-difference solution of the transient natural convection flow of an incompressible viscous fluid past an impulsively started semi-infinite plate with uniform heat and mass flux is presented here, taking into account the homogeneous chemical reaction of first order. The velocity profiles are compared with the available theoretical solution and are found to be in good agreement. The steady-state velocity, temperature and concentration profiles are shown graphically. It is observed that due to the presence of first order chemical reaction the velocity decreases with increasing values of the chemical reaction parameter. The local as well as average skin-friction, Nusselt number and Sherwood number are shown graphically.List of symbols C concentration - C species concentration in the fluid far away from the plate - C _w species concentration near the plate - C dimensionless concentration - D mass diffusion coefficient - Gc mass Grashof number - Gr thermal Grashof number - g acceleration due to gravity - j mass flux per unit area at the plate - K dimensionless chemical reaction parameter - K _l chemical reaction parameter - k thermal conductivity - Nu_x dimensionless local Nusselt number - dimensionless average Nusselt number - Pr Prandtl number - q heat flux per unit area at the plate - Sc Schmidt number - Sh_x dimensionless local Sherwood number - dimensionless average Sherwood number - T temperature - T temperature of the fluid far away from the plate - T _w temperature of the plate - T dimensionless temperature - t time - t dimensionless time - u ₀ velocity of the plate - U, V dimensionless velocity components inX,Y-directions, respectively - u, v velocity components inx, y-directions, respectively - X dimensionless spatial coordinate along the plate - x spatial coordinate along the plate - Y dimensionless spatial coordinate normal to the plate - y spatial coordinate normal to the plate - thermal diffusivity - volumetric coefficient of thermal expansion - * volumetric coefficient of expansion with concentration - coefficient of viscosity - kinematic viscosity - _x dimensionless local skin-friction - dimensionless average skin-friction     

Hydrodynamic pressure on an accelerating dam and criteria for cavitation

Summary The effect of finite reservoir on the hydrodynamic pressure due to horizontal as well as vertical ground excitations has been studied. It is found that for horizontal accelerations the hydrodynamic pressure force decreases as the size of the reservoir decreases. The effect of vertical acceleration on the pressure force on a dam is simply to adjust the hydrostatic pressure by replacing the gravitational constant by an effective gravitational acceleration and this is true for any arbitrary shapes of the reservoir. A simple criterion has been presented in this paper which would enable dam engineers to determine whether a given earthquake could cause cavitation at the dam-water interface or not.     

Density correlation function in superfluid helium nearT λ

The previously calculated hydrodynamic form of the density correlation function of liquid helium is written out explicitly forT nearT . This function, which is proportional to the intensity of light scattered from the fluid, has a finite value at =0, which grows asTT from below. In evaluating the precise form of this contribution in terms of transport coefficients, it is important to keep terms of order (C _p /C _v )–1.     

Dynamics of Trapped Bose Gases at Finite Temperatures

Starting from an approximate microscopic model of a trapped Bose-condensed gas at finite temperatures, we derive an equation of motion for the condensate wavefunction and a quantum kinetic equation for the distribution function for the excited atoms. The kinetic equation is a generalization of our earlier work in that collisions between the condensate and non-condensate (C ₁₂ ) are now included, in addition to collisions between the excited atoms as described by the Uehling–Uhlenbeck (C ₂₂ ) collision integral. The continuity equation for the local condensate density contains a source term ₁₂ which is related to the C ₁₂ collision term. If we assume that the C ₂₂ collision rate is sufficiently rapid to ensure that the non-condensate distribution function can be approximated by a local equilibrium Bose distribution, the kinetic equation can be used to derive hydrodynamic equations for the non-condensate. The ₁₂ source terms appearing in these equations play a key role in describing the equilibration of the local chemical potentials associated with the condensate and non-condensate components. We give a detailed study of these hydrodynamic equations and show how the Landau two-fluid equations emerge in the frequency domain is a characteristic relaxation time associated with C ₁₂ collisions. More generally, the lack of complete local equilibrium between the condensate and non-condensate is shown to give rise to a new relaxational mode which is associated with the exchange of atoms between the two components. This new mode provides an additional source of damping in the hydrodynamic regime. Our equations are consistent with the generalized Kohn theorem for the center of mass motion of the trapped gas even in the presence of collisions. Finally, we formulate a variational solution of the equations which provides a very convenient and physical way of estimating normal mode frequencies. In particular, we use relatively simple trial functions within this approach to work out some of the monopole, dipole and quadrupole oscillations for an isotropic trap.     

Similarity analysis of axisymmetric free convection on a horizontal infinite plate subjected to a mixed thermal boundary condition

Summary Similarity analysis of the problem of axisymmetric free convection on a horizontal infinite plate is considered assuming that the plate is subjected to a mixed thermal boundary condition. It is shown that the thermal boundary condition is characterized by a nonnegative parameterm and the two cases ofm=0 andm=1 correspond to prescribed plate temperature and prescribed surface heat flux respectively. If one has to compute the heat transfer coefficient for various values ofm, there is no need to solve the boundary value problem everytime; it is enough to solve a certain polynomial equation provided the solution is known for any particular value ofm.Notation a ₀(r),a ₁(r),a ₂(r) coefficients in Eq. (1) - A transition parameter used in Eq. (22) - C function ofr defined in Eq. (8.1) - f dimensionless stream function - F dimensionless pressure - g acceleration due to gravity - G function ofr defined in Eq. (8.2) - m mixed thermal boundary condition parameter - N normalized heat transfer coefficient in Eq. (26) - Nu _r Nusselt number - p fluid pressure - p _e ambient pressure - P normalized pressure drop at the plate - Pr Prandtl number - r radial coordinate - S normalized stress at the plate - T temperature - T _e ambient temperature - u velocity in the radial direction - w velocity in the axial direction - z axial coordinate - coefficient of thermal expansion - dimensionless similarity variable - dimensionless temperature - exponent inC - kinematic viscosity - ambient fluid density - stream function     

Periodic flow of a second-grade fluid induced by non-torsional oscillations of eccentric rotating disks

This paper deals with the periodic flow of a second-grade fluid caused by non-torsional oscillations of two disks rotating about non-coincident axes. While the two parallel disks are initially rotating with the same angular velocity about distinct axes, they start to execute non-torsional oscillations in their own planes and in the opposite directions. An exact solution is obtained for the components of the horizontal force per unit area exerted by the top and bottom disks on the fluid in the periodic state. The results are graphically displayed and the influence of the second-grade fluid parameter, the ratio of the frequency of oscillation to the angular velocity of the disks, the Reynolds number and the dimensionless velocity amplitudes of oscillation is discussed. It is observed that the change in the \( x \)-component of the mentioned force gets larger when the second-grade fluid parameter increases. However, an opposite effect is seen for the change in the \( y \)-component.     

Free convection effects on the oscillatory flow of a couple stress fluid through a porous medium

Summary Effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature, have been studied. The surface absorbs the fluid with a constant suction and the free stream velocity oscillates about a constant mean value. Analytical expressions for the velocity and the angular velocity fields have been obtained, using the regular perturbation technique. The effects of Grashof numberG; material parameters and ; Prandtl numberP; permeability parameterK and frequency parametern on the velocity and the angular velocity are discussed. The effects of cooling and heating of a polar fluid compared to a Newtonian fluid have also been discussed. The velocity of a polar fluid is found to decrease as compared to the Newtonian fluid.List of symbols C _p specific heat at constant pressure - g acceleration due to gravity - G Grashof number - K ⁺ permeability of the porous medium - K dimensionless permeability - P Prandtl number - t ⁺ time - t dimensionless time - T _w ⁺ mean temperature of the surface - T ⁺ temperature of the fluid - T ⁺ temperature of the fluid away from the surface - density of the fluid - viscosity - _r rotational viscosity - C _a ,C _d coefficients of couple stress viscosities - I a scalar constant of dimension equal to that of the moment of inertia of unit mass - x ⁺,y ⁺ coordinate system - u ⁺,v ⁺ velocity components in thex ⁺ andy ⁺ directions - u dimensionless velocity in thex ⁺-direction - ⁺ angular velocity component - dimensionless angular velocity - n ⁺ frequency of oscillations - n dimensionless frequency - perturbation parameter - U a constant velocity - u ₀ mean velocity - u ₁ fluctuating part of the velocity - ⁰ mean angular velocity - ¹ fluctuating part of the angular velocity - T ₀ mean temperature - T ₁ fluctuating part of the temperature - ⁰ coefficient of the volume expansion - kinematic viscosity - ^r rotational kinematic viscosity - , material parameters characterizing the polarity of the fluid - v ₀ suction velocity - density of the fluid far from the surface - y dimensionless coordinate normal to the surface     

Havelock wavemakers,Westergaard dams and the Rayleigh hypothesis

Water of constant finite depth fills a semi-infinite channel, with a wavemaker, W, at one end. The generation of small-amplitude gravity waves by harmonic oscillations of W leads to a linear boundary-value problem for a velocity potential, . For vertical, plane wavemakers, there is a theory due to Havelock in which is represented as a convergent series of eigenfunctions, with coefficients determined by the boundary condition on W. We show that the same representation (with different coefficients) can also be used for some wavemakers with other shapes; the allowable geometries and forcings are determined. This is a hydrodynamic analogue of the so-called Rayleigh hypothesis in the theory of gratings. Similar results obtain for the hydrodynamic loading of dams due to short-duration earthquakes.     

Effect of thermal radiation on natural convection over cylinders of elliptic cross section

Summary The effect of conduction-radiation on natural convection flow of an optically dense viscous incompressible fluid along an isothermal cylinder of elliptic cross section has been investigated. The boundary layer equations governing the flow are shown to be nonsimilar. Full numerical solutions of the governing equations are obtained using the implicit finite difference method. The solutions are expressed in terms of the Nusselt number Nu against the eccentric angle in the range [0, ]. The working fluid is taken to have unit value of the Prandtl number, Pr, and the effects of varying the Planck number,R _d, the surface temperature parameter, _w, and the parameterA _O representing the ratio of the major and minor axes of the cylinder are investigated. From the present analysis it is found that the rate of heat transfer from the slender body is higher than from the blunt body and that these higher values become even higher due to an increase in the effect of radiation in the flow field.Nomenclature a semi-major axis of the cylinder - a _r Rosseland mean absorption coefficient - b semi-minor axis of the cylinder - C _p specific heat at constant pressure - f dimensionless stream function - g acceleration due to gravity - Gr Grashof number - Nu Nusselt number - Q _w surface heat flux - Pr Prandtl number - R _d Planck number (radiation-conduction parameter) - T temperature of the fluid - T _w temperature of the heated surface - T temperature of the ambient fluid - u velocity in thex-direction - v velocity in they-direction - x coordinate measuring distance round the cylinder - y coordinate measuring distance normal to the cylinder Greek symbols eccentric angle - coefficient of cubial expansion - coefficient of thermal diffusivity - v kinematic viscosity     

Free convection about a wedge and a cone subjected to mixed thermal boundary conditions

Summary The paper deals with a similarity analysis of free convection about a wedge and a cone which are subjected to mixed thermal boundary conditions. The governing equations and the boundary conditions are reduced to a boundary value problem involving a non-negative parameterm which assumes the values 0,1 and for the cases of prescribed temperature, prescribed heat flux and radiation boundary condition. A numerical solution has been computed for the case of radiation boundary condition. The results for constant temperature and constant heat flux available in literature are deduced with the aid of a simple transformation. Critical cases have been found for which the solution is same for all values ofm.Notation a ₀,a ₁,a ₂ coefficients defined in Eq. (5) - A transition parameter used in Eq. (16) - c coefficient ofC in Eq. (7) - C function ofx defined in Eq. (7) - f dimensionless stream function - g acceleration due to gravity - g ₁ =g cos - G function ofx defined in Eq. (7) - m constant defined in Eq. (12) - n = 0 for wedge = 1 for cone - PHF prescribed heat flux - Pr Prandtl number - PT prescribed temperature - r =x sin - RBC radiation boundary condition - T temperature - T _e ambient temperature - u velocity component inx-direction - v velocity component iny-direction - x distance from the vertex measured along the surface - y distance normal to the surface - coefficient of thermal expansion - dimensionless similarity variable - dimensionless temperature - exponent inC in Eq. (7) - kinematic viscosity - semivertical angle - stream function     

Free convection on a horizontal plate in a saturated porous medium with prescribed heat transfer coefficient

Summary The free convection on a horizontal plate embedded in a saturated porous medium is considered assuming that the plate is subjected to a prescribed temperature or a prescribed heat flux or a prescribed heat transfer coefficient. By similarity transformation the governing equations are reduced to identical coupled equations for all the three cases with three common boundary conditions and one boundary condition depending on the thermal boundary condition imposed. It is shown that there is no need to solve the three boundary value problems independently and that the solution for one case can be used to obtain the solution for any other case by a simple algebraic method.Notation A transition parameter used in Eq. (19) - C function ofx defined in Eq. (11.1) - f dimensionless stream function - g acceleration due to gravity - k thermal conductivity - K permeability of the porous medium - N heat transfer coefficient - p fluid pressure - p _e ambient pressure - PT prescribed temperature - PHF prescribed heat flux - PHTC prescribed heat transfer coefficient - q _w surface heat flux - R function ofx defined in Eq. (11.2) - T temperature - T _w plate temperature - T _e ambient temperature - T _w temperature difference=T _w-T _e - u velocity in thex-direction - v velocity in they-direction - x coordinate along the plate in the upward direction - y coordinate normal to the plate - equivalent thermal diffusivity - coefficient of thermal expansion - dimensionless similarity variable - dimensionless temperature - exponent inC - viscosity - kinematic viscosity - fluid density - _e ambient density - stream function     

Aluminum. II. Derivation of C v0from C p and comparison to C v0 calculated from anharmonic models

The specific heat at constant pressure, C _p, of aluminum measured by Ditmars, Plint, and Shukla has been reduced to the volume V ₀ appropriate for 0 K employing the Murnaghan equation. The C _v0 thus obtained is compared with the theoretical C _v0 calculated in the harmonic and the lowest-order anharmonic approximation from three different pseudopotentials (Harrison, Ashcroft, and Dagens-Rasolt-Taylor) as well as a phenomenological Morse potential. The higher-order ( ⁴) anharmonic contributions are calculated from the same nearest-neighbor Morse potential as in the lowest-order anharmonic theory. The role of the vacancy and the higher-order anharmonic contributions to C _v0 has been examined and we conclude that the ⁴ contributions to C _v0 are much smaller than the vacancy contribution. After removal of the vacancy contribution, the reduced C _v0 is found to be in excellent agreement with the Ashcroft and Harrison pseudopotentials as well as the Morse potential including the ² and ⁴ contributions to C _v0.     

Combined heat and mass transfer in natural convection flow from a vertical wavy surface

Summary In the present paper, effects of combined buoyancy forces from mass and thermal diffusion by natural convection flow from a vertical wavy surface have been investigated using the implicit finite difference method. Here we have focused our attention on the evolution of the surface shear stress,f(0), rate of heat transfer,g(0), and surface concentration gradient,h(0) with effect of different values of the governing parameters, such as the Schmidt number Sc ranging from 7 to 1500 which are appropriate for different species concentration in water (Pr=7.0), the amplitude of the waviness of the surface ranging from 0.0 to 0.4 and the buoyancy parameter,w, ranging from 0.0 to 1.Notation C species concentration in the boundary layer - C species concentration of the ambient fluid - C _w species concentration at the surface - D chemical molecular diffusivity - f dimensionless stream function - g acceleration due to gravity - Gr _x local modified Grashof number - N ratio of the buoyancy forces due to the temperature difference and the concentration difference - p pressure of the fluid - T temperature of the fluid in the boundary layer - T temperature of the ambient fluid - T _w temperature at the surface - u, v thex- andy-components of the velocity field - x, y axis in the direction along and normal to the plate Greek symbols thermal diffusivity - _T volumetric coefficient of thermal expansion - _C volumetric coefficient of expansion with concentration - stream function - nondimensional similarity variable - x/L - density of the ambient fluid - v kinematic coefficient of viscosity - stream function - dimensionless skin friction - fluid viscosity     

Boojums on the fermi surface in3He-A and Hamiltonian dynamics of the orbital momentum

Boojums (nodes) on the Fermi surface of³He-A lead to peculiar behavior of the intrinsic orbital momentum L in this superfluid. AtT=0, L has the form L=(/2m ₃)l(–C ₀), whereC ₀ is the prefactor of the anomalous term in the supercurrent l(l rot l). From the algebra of the hydrodynamical Poisson brackets and from microscopic theory it follows thatC ₀ is a dynamical invariant, (/t)C ₀=0. The closed system of nonlinear hydrodynamic equations of³He-A atT=0 is constructed in the framework of the Poisson brackets scheme.     

Optical second harmonic generation in CsLiB6O10 nanocrystallites incorporated to the olygoether photopolymer matrices induced by acustical field

Optical second harmonic generation in CsLiB₆O₁₀ (CLBO) large-sized nanocrystallites (15–40 nm) incorporated into the olygoether photopolymer matrices has been found at liquid helium temperature (LHeT). The second harmonic generation (SHG) has been measured for the source wavelength of the YAG : Nd laser ( = 1.06 m). Increasing acoustical power (up to 18.1 W/cm²) and acoustical frequencies at 14 kHz give maximal values of the output SHG. The measured SHG ₂₂₂ tensor component was comparable with the one for the traditional nonlinear optical crystals such as KH₂PO₄, KTiOPO₄, LiNbO₃. With decreasing temperature below 28 K the acoustically induced SHG signal strongly increases. A correlation between the acoustically induced SHG and low-frequency Raman modes has been found. The maximal acoustically induced SHG has been observed for the nanocrystallite concentration about 3.1% (in weighting units) and the crystallite sizes lying within the 35–40 nm. The SHG tensor coefficients were higher than for the proper CLBO crystals at least on the 13%. Advantageous and drawbacks of the presented model are discussed. The theory of the observed phenomena is explained on the ground of ab initio band energy calculations with taking into account of anharmonic electron-phonon interactions.     

Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective flow and mass transfer over an accelerating surface with a heat source in the presence of suction and blowing in the case of variable viscosity

Summary. An analysis has been carried out to obtain the thermal-diffusion and the diffusion-thermo effects on the mixed free-forced convective and mass transfer steady laminar boundary-layer flow over an accelerating surface with a heat source in the presence of suction and blowing. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The partial differential equations governing the problem under consideration have been transformed by a similarity transformation into a system of ordinary differential equations which is solved numerically by applying the shooting method. The results for an impermeable accelerating surface are discussed. The effects of the variable viscosity parameter _r, the thermal diffusion parameter Sr, the diffusion-thermo parameter Df, suction or blowing parameter m, heat flux parameter s and Schmidt number Sc have been examined on the flow field of a hydrogen-air mixture as a non-chemical reacting fluid pair. The effects of varying these parameters are studied in the case of a surface with prescribed wall temperature and a surface with prescribed wall heat flux.     

Ferrofluid lubrication of cylindrical rollers with cavitation

Summary This paper analyses the ferrofluid lubrication of cylindrical rollers under combined rolling and normal motion. The analysis, which takes into account the rotation of magnetic particles, has been made for general cases where the magnetization vectors need not be parallel to the applied magnetic field. Cavitation boundary conditions are used and the applied magnetic field is assumed to be imposed in a direction transverse to the fluid motion. A perturbation scheme in terms of non-dimensional Brownian time relaxation parameter has been used and the effects of various parameters on bearing characteristics have been studied.List of symbols A non-dimensional relaxation time parameter - b length of the cylindrical roller [m] - F frictional force [N] - h film thickness [m] - h _o minimum film thickness [m] - h ₂ film thickness at the point of cavitation [m] - H applied magnetic field [A/m] - H ₀ constant applied magnetic field strength [A/m] - I sum of moments of inertia of particles per unit volume [kg/m] - M magnetization vector (M ₁, 0,M ₂) [A/m] - M ₀ equilibrium magnetization [A/m] - M ₁,M ₂ components of magnetization vector [A/m] - N dimensionless parameter - p pressure [(N/m²)] - p _i pressure ofi-th order,i=0, 1, 2, 3 [(N/m²)] - q nondimensional velocity parameter - R equivalent radius of the cylinders [m] - u, v velocity components [m/s] - u _i velocity component ofi-th order,i=0, 1, 2, 3 [m/s] - U ₀,V ₀ reference velocity components of moving surface [m/s] - W load capacity [N] - x, z coordinates - x ₂ cavitation point [m] - viscosity coefficient [Ns/m²] - ₀ permeability of free space [Kg ms^–2 A^–2] - _B Brownian relaxation parameter [s] - _S relaxation time parameter due to rotations [s] - non-dimensional _B      

Use of the WKB method to investigate connective heat transfer in a micropolar fluid

A general approximate solution is obtained for problems of heat transfer associated with a flow of micropolar fluid in a plane channel with boundary conditions of the first and second kind and its accuracy is determined.Notation T_o and T_w temperatures of entrance section and wall of channel, respectively - dp/dx pressure gradient - x₁, x₂ longitudinal and transverse coordinates, respectively (or x and y) - Pe=2v _m ^N h/a Peclet number - v _m ^N mean velocity of Newtonian fluid with viscosity +/2 in channel of width 2h - boundary condition parameter - 2h width of channel - v_x and v_z nonzero components of velocity and microrotation of micropolar fluid - a and thermal diffusivity and thermal conductivity of fluid - , , and viscosities of micropolar fluid - q_w heat flux density on wall - _n and Y_n(y) eigenvalues and eigenfunctions of Sturm-Liouville problem - C_n constants that can be determined by using orthogonality of eigenfunctions Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 732–738, October, 1980.     

Unsteady MHD-boundary-layer of a source and vortex flow adjacent to a stationary surface

Summary The steady laminar incompressible flow of an electrically conducting fluid over an infinite permeable disk in the presence of an axial magnetic field has been investigated, and a self-similar solution of the boundary-layer equations is obtained numerically. For large values of the suction parameter, a closed form solution is obtained. Also, an asymptotic solution is found for large values of the independent variable. The surface-shear stresses in the radial and tangential directions and the surface heat transfer strongly depend on the suction parameter, the ratio of the source and vortex flow and the magnetic field except the surface heat transfer which weakly depends on the magnetic field. The similarity solution of the boundary-layer equations exists only when a certain minimum suction or magnetic field is applied. The results of the analytical solution are in good agreement with those of the numerical solution for the suction parameterf _w3.