Heat transfer between infinite parallel plates in a rarefied gas

The problem of heat transfer between two infinite parallel plates is investigated on the basis of equations obtained by averaging the Boltzmann kinetic equation with respect to the transverse velocity. A numerical solution of the problem is accomplished for a temperature ratio between the plates of T₀/T₁=1/4 and for various Knudsen numbers.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–91, January–February, 1972.     

平行平板间Couette流起动过程的运算微积解

求解平行平板间Couette流的起动过程通常使用分离变量法,得到的速度分布在平板间距趋于无穷时难以逼近Stokes第一问题的解,而两者在物理上却是完全一致的. 本文利用运算微积法和Heaviside算子进行求解,不但解决了这一问题,而且具有运算简单的特点.     

Subsonic rarefied gas flow over a rack of flat transverse plates

Parameters of a rarefied gas flow through a rack of flat plates aligned across the flow are studied by means of the joint numerical solution of the Boltzmann and Navier-Stokes equations. A subsonic flow regime is considered. The changes in flow characteristics are calculated as functions of the free-stream velocity and plate temperature. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 59–67, January–February, 2008.     

Rarefied gas shear flow between two movable segments of parallel plates

A study is made of the two-dimensional steady-state rarefied gas flow observed between two parallel plane surfaces of finite and different length when one of the surfaces is fixed and the other moves parallel to itself at a constant velocity, while remaining within the bounds of a given segment with fixed ends (the motion is similar to that of a conveyer belt). This flow can be regarded as a twodimensional counterpart of the classical one-dimensional Couette flow. The corresponding problem is formulated in a rectangular domain for the nonlinear kinetic equation with a model collision operator and is solved by a finite-difference method for various boundary conditions. For simplicity's sake, the flow was studied under conditions such that it can be considered near-isothermal. The gas pressures on each side of the gap formed by the plates may be the same or different. If the pressures on both sides of the gap are equal, then a near-zero-gradient flow develops between the plates. In this case, the greater the plate length, the nearer the flow in the middle of the gap to one-dimensional Couette flow. The end effects are examined, together with the conditions in which the flow in the middle of the domain can be assumed to be practically one-dimensional. In the zero-gradient regime, the system operates, in general, as a pump transferring gas from one side of the gap the other. The ability to pump gas also remains if a small counterpressure exists.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 150–155, May–June, 1995.The work was financially supported by the Russian Foundation for Fundamental Research (project No. 93-013-17928).     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Non-isothermal couette flow of a rarefied gas between two rotating cylinders

An accurate numerical solution of the momentum and the heat transfer through a rarefied gas confined between two cylinders rotating with different angular velocities and having different temperatures has been obtained over a wide range of the Knudsen number on the basis of the Bhatnagar, Gross, Krook model equation. The viscous stress tensor, heat flux, and the fields of density, temperature and velocity are found. An analysis of the influence of the angular velocities and the temperature ratio on these quantities is given.     

Strong recondensation in a one-component and a two-component rarefied gas for an arbitrary value of Knudsen number

As is known, surface phenomena such as evaporation, absorption, and reflection of molecules from the surface of a body depend strongly on its temperature [1–5]. This leads to the establishment of a flow of a substance between two surfaces maintained at different temperatures (recondensation). The phenomenon of recondensation was studied in kinetic theory comparatively long ago. However, up to the present, only the case of small mass flows in a onecomponent gas has been investigated completely [3,4]. Meanwhile it is clear that by the creation of appropriate conditions we can obtain considerable flows of the recondensing substance, so that the mass-transfer rate will be of the order of the molecular thermal velocity. Such a numerical solution of the problem with strong mass flows along the normal to the surface for small Knudsen numbers for a model Boltzmann kinetic equation was obtained in [7]. In this study we numerically solve the problem of strong recondensation between two infinite parallel plates over a wide range of Knudsen numbers for a one-component and a two-component gas, on the basis of the model Boltzmann kinetic equation [6] for a one-component gas and the model Boltzmann kinetic equation for a binary mixture in the form assumed by Hamel [8], for a ratio of the plate temperatures equal to ten. We also investigate the effect of the relative plate motion on the recondensation flow.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1972.     

Simultaneous convection and radiation in flow between two parallel plates

A numerical solution is described for simultaneous forced convection and radiation in flow between two parallel plates forming ahannel. The front plate is transparent to thermal radiation while the back one is thermally insulated. Analyses for both flow and heat are presented for the case of a non-emitting ‘blackened’ fluid. The governing equations of the stream function and the temperature together with their boundary conditions are presented in non-dimensional expressions. The solution is found to depend on eight dimensionless parameters, namely the ratio of the height of the channel to the distance between the plates, the initial dimensionless temperature, the optical thickness, the absorptivities of both plates, the Reynolds number, the Prandtl number and the heat transfer coefficient from the front plate to the surroundings. The numerical solution is obtained using a finite-difference technique. A study has been made of the effect of the initial temperature of the flow at the channel inlet, the dimensionless loss coefficient from the front plate, the absorptivity of the back plate and the optical thickness, on the temperature distribution in the channel, the heat collection efficiency and the average temperature rise in the channel. Results showed that increasing the optical thickness increases the temperature of the front plate and decreases the temperature of the back plate. Also, increasing the optical thickness increases the efficiency of heat collection, which reaches its maximum asymptotic value at an optical thickness of about 1.5. Moreover, the location of the maximum temperature is found to depend on both the optical thickness and the dimensionless heat loss coefficient from the front plate.     

Analysis of heat conduction in a rarefied gas at rest with a temperature jump at the boundary by consistent-order extended thermodynamics

In the context of the first- and second-order theories of consistent-order extended thermodynamics, a systematic approach is established for analyzing the temperature jump at the boundary through studying one-dimensional stationary heat conduction in a rarefied gas at rest. Thereby an approach to the free boundary-value problem in general is explored. Boundary values of temperature are assumed to be in the form of power expansion with respect to the Knudsen number, based on which analytical expressions of the temperature jump aswell as entropy production at the boundary are derived explicitly. Dependencies of these two boundary quantities on both the Knudsen number and accommodation factor are also extensively discussed. The present analysis is expected to be the basis for the study of higher-order theories of consistent-order extended thermodynamics.      

Drag on a body placed between two parallel plates in a hele-shaw flow

In [1], the drag was found that acts on a circular gas bubble between two parallel plates in a slow viscous flow. In the present paper, the problem considered in [1] is solved for a body of arbitrary shape under the assumption that the conditions of a Hele-Shaw flow are satisfied. An expression is obtained for the drag containing only one coefficient in the expansion of the complex potential in a Laurent series.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–162, September–October, 1979.     

On velocity and temperature distribution in turbulent flow between parallel plates

Summary By means of mixing-length and turbulent Prandtl number hypothesis we solved the problem of parallel turbulent flow at constant density, both from the dynamic and thermal point of view; we then analyzed the fit with experimental data of various mixing-length formulas, and also the dependence of temperature profiles on the value of the turbulent Prandtl number.This critical analysis allowed the choice of the most suitable mixing-length formula and the value for the turbulent Prandtl number. On the basis of these results we extended the study discarding the condition of constant density; in particular we considered the case of liquids whose density was taken dependent on temperature changes across the walls, but independent of the pressure changes in flow direction.The study belongs to the case of fully developed temperature and velocity profiles.

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Sommario Mediante l'introduzione del numero di Prandtl turbolento e della lunghezza di miscelamento nelle equazioni di Reynolds, viene risolto il problema della distribuzione di velocità e temperatura in un fluido a densità costante, in moto turbolento tra due piani paralleli. Le distribuzioni di velocità, ottenute con diverse espressioni della lunghezza di miscelamento, vengono poi confrontate con i dati sperimentali, allo scopo di scegliere la più opportuna di queste lunghezze; infine viene esaminata l'influenza del numero di Prandtl turbolento sulla distribuzione di temperatura.In accordo con le suddette scelte, lo studio è successivamente esteso al caso di densità dipendente dalla sola temperatura, ritenendo trascurabili le variazioni di densità per effetto del gradiente di pressione. In altri termini si limita lo studio ai liquidi.Tutti i risultati ottenuti si riferiscono a moti stabilizzati in velocità e temperatura.

     

Onsager's reciprocal relations in the linear problem of flow of rarefied gas past a heat conducting body

Symmetry of integral transport coefficients is established on the basis of a linear stationary Boltzmann equation for the problem of flow past a heat conducting body. An expression is obtained for the entropy production in the gas-body system, and this determines the thermodynamic fluxes and forces. Bakanov and Roldugin [2] have considered the problem of motion of a heat conducting sphere at small Knusden numbers Kn, using the symmetry of the Onsager coefficients to construct an asymptotic solution as Kn → 0. In the present paper, a general method is proposed for establishing the Onsager relations that does not require the actual construction of a solution to the problem and is applicable for bodies of arbitrary shape and all values of the Knudsen number.     

The growth of the thermal boundary layer in laminar flow between parallel flat plates

Summary The problem considered is that of the heat transfer occurring at the inlet to a parallel plate channel. Instead of separating variables, the energy equation is solved, after transformation, in the form of a power series. This method supplies information concerning the initial growth of the thermal boundary layer which is not obtainable by previous methods using eigen-function expansions. A sufficient number of coefficients of the series is computed to allow the present solution to be joined to the asymptotic eigen-function solution, thus completing the treatment of the problem for all values of the longitudinal variable.