Laminar plane jet in a gas medium

The effects of compressibility on a laminar plane jet are investigated. First, assuming that the viscosity is directly proportional to the enthalpy it is found that the similarity solutions both for momentum and energy equations exist. By a proper transformation of the similarity variable, the energy equation, in the case of viscous heating, is reduced to a hypergeometric equation and solutions for arbitrary Prandtl number are obtained. Secondly, taking the Prandtl number of the fluid as unity and a power law for the dependence of viscosity on enthalpy the velocity distribution is obtained with the help of Crocco's integral.

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Zusammenfassung Der Einfluss der Kompressibilität auf einer ebenen laminaren Strahl wird untersucht. Zunächst wird unter der Annahme, dass Zähigkeit und Enthalpie einander proportional sind, gefunden, dass sowohl die Impulsgleichung als auch die Energiegleichung Aehnlichkeitslösungen haben. Im Falle der Erwärmung durch Dissipation wird die Energiegleichung durch eine geeignete Transformation der Aehnlichkeitsvariablen auf eine hypergeometrische Differentialgleichung reduziert, und Lösungen für beliebige Prandtl-Zahlen werden bestimmt. Im Falle, dass zwischen Zähigkeit und Enthalpie ein Potenzgesetz herrscht, wird für die Prandtl-Zahl1 die Geschwindigkeitsverteilung mit Hilfe von Crocco's Integral erhalten.

     

Laminar magnetohydrodynamic duct flow in the presence of a magnetic dipole

We study magnetohydrodynamic flow of a liquid metal in a straight duct. The magnetic field is produced by an exterior magnetic dipole. This basic configuration is of fundamental interest for Lorentz force velocimetry (LFV), where the Lorentz force opposing the relative motion of conducting medium and magnetic field is measured to determine the flow velocity. The Lorentz force acts in equal strength but opposite direction on the flow as well as on the dipole. We are interested in the dependence of the velocity on the flow rate and on strength of the magnetic field as well as on geometric parameters such as distance and position of the dipole relative to the duct. To this end, we perform numerical simulations with an accurate finite-difference method in the limit of small magnetic Reynolds number, whereby the induced magnetic field is assumed to be small compared with the external applied field. The hydrodynamic Reynolds number is also assumed to be small so that the flow remains laminar. The simulations allow us to quantify the magnetic obstacle effect as a potential complication for local flow measurement with LFV. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical approximation for laminar film condensation of saturated stream on an isothermal vertical plate

In this paper, the laminar film condensation of saturated stream on an isothermal vertical plate is studied. The boundary layer equations of momentum and thermal energy are reduced to two ordinary differential equations by means of a set of similarity transformations. The problem is then solved analytically using the homotopy analysis method (HAM). The dual solutions are obtained for a range of values of the parameter η_δ. However, it should be noted that the second branch solution of the considered problem has only mathematical meanings. The present work shows the validity and the great potentiality of the proposed technique for the nonlinear problems with multiple solutions.     

Bose-Einstein condensation of a nonideal gas

The canonical distribution with a constraint is applied to the Bogoliubov model of a nonideal Bose gas. Two approximate solutions of the variational equations for the Bose condensate density and the constraint parameter are found. They are treated as nontrivial and trivial parts of the process of Bose condensation as far as they contribute to the thermodynamic properties at temperatures T above and below the critical T_c, respectively. The corresponding branches of the spectrum are investigated and the specific heat C_V is considered for all temperatures with the help of the derived solutions. The low-temperature behavior C_V T^3/2 and C_V T³ due to the contributions of free boson and phonon excitations is derived. The asymptotic behavior of C_V for temperatures approaching the critical temperature T_c from below and above are approximately calculated for values of the parameters ofHe ⁴ that are in qualitative agreement with the experiment.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 295–306, November, 1996.     

Mixed convection turbulent film condensation on a sphere

The present paper reports a research on condensation heat transfer of an isothermal sphere with an external flow of vapor. The high tangential velocity of the vapor flow is determined from potential flow theory. The transition criterion of the onset turbulence has been given in the local film Reynolds number (Re_Γ). An eddy diffusivity model along with an expression by [H. Kato, N.N. Shiwaki, M. Hirota, On the turbulent heat transfer by free convection from a vertical plate, Int. J. Heat Mass Transfer, 11(1968) 1117–1125] is used to model turbulence. And the local liquid–vapor interfacial shear which occurs for high velocity vapor flow across a sphere surface is defined by the Colburn analogy. The paper then presents analytical analysis for the local dimensionless film thickness and heat transfer characteristics for the film condensation. And a comparison with those generated by previous theoretical of laminar condensation is discussed. The comparison shows the heat transfer coefficient of turbulent film condensation is higher than laminar film condensation under the high vapor velocity.     

Laminar two-phase boundary layers in natural convection film boiling

Zusammenfassung Es wird die zweiphasige Grenzschichtströmung behandelt, die beim laminaren Filmsieden an einer vertikalen Platte und an einem langen horizontalen Zylinder unter den Bedingungen natürlicher Konvektion entsteht. In den Rechnungen wird der gesamte Bereich derjenigen Dampfüberhitzung miterfasst, in welchem rein konvektiver Wärmetransport möglich ist unter Einschluss des Impulstransports an der Phasengrenzfläche. Neue Heliummessungen und bekannte Versuchsergebnisse stimmen in befriedigender Weise mit den Rechnungen überein, wenn man berücksichtigt, dass ein vereinfachtes Strömungsmodell mit einer glatten Phasengrenzfläche vorausgesetzt ist.     

Laminar free convection flow of an electrically conducting fluid from a vertical plate with uniform surface heat flux and variable wall temperature in the presence of a magnetic field

Zusammenfassung Das Problem der freien Konvektionsströmung einer elektrisch leitenden Flüssigkeit im Bereich einer senkrechten Platte ist in zwei Fällen gelöst: a) Die Platte hat einen gleichförmigen Oberflächenwärmefluss. b) Die Platte besitzt eine veränderliche Temperatur. Es wird festgestellt, dassähnliche Lösungen dann zustande kommen, wenn das Magnetfeld, das senkrecht zur Platte wirkt, im Falle a) proportionalx ^–1/5 und im Fall b) proportionalx ^(n-1)/4 ist, wox der längs der Platte gemessene Abstand von ihrem niedrigsten Punkt ist. Es wird weiter festgestellt, dass die Nusselt-Zahl mit zunehmender magnetischer eldstärke geringer wird, während die Grenzschtdichtigkeit zunimmt.     

Laminar boundary layers along an infinite porous plate in the presence of a transverse magnetic field

Exact solutions of the Navier-Stokes equations are derived by a Laplace-transform technique for two-dimensional, incompressible flow of an electrically conducting fluid past on infinite porous plate. It is assumed that the flow is independent of the distance parallel to the plate and that the velocity component normal to the plate is constant. A general formula is derived for the velocity distribution in terms of the given external velocity. The skin friction is obtained and some special cases are considered.     

Stability analysis of the phytoplankton vertical steady states in a laboratory test tube

The positive vertical equilibrium profiles of a phytoplankton population growing in a vertical test tube under controlled experimental conditions (temperature, salinity, light intensity at the top surface) for nutrients are discussed with reference to their stability properties for arbitrary positive initial values of the biomass concentration along the tube. Two different approaches are followed. First a stability result is established in the Sobolev norm H² by estimating the norms of the perturbations recursively in successive subintervals of suitably small amplitude. The second approach provides stability in the sense of the uniform convergence as a corollary of a stability theorem for a rather general class of integro-differential equations.     

Liquid flow in a tube with deforming walls in the presence of valves

A numerical solution is presented for the flow of a viscous liquid in a tube of finite length with valves at the ends and elastic properties of the tube walls periodically changing with time. Such conditions may be encountered in the flow of blood through veins equipped with valves to prevent reverse flow.Scientific-Research Institute of Mechanics. M. V. Lomonosov State University, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 903–908, September–October, 1972.     

Flow of a viscous fluid in a deformable valved tube in the presence of gravity

The problem of the flow of a viscous incompressible fluid in a tube of finite length with valves at the ends is solved numerically with allowance for gravity and elastic wall properties that vary periodically with time.Paper presented at the First All-Union Conference on Engineering and Medical Biomechanics, Riga, October, 1975.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 4, pp. 756–760, July–August, 1975.     

Laminar flow separation in an axi-symmetric sudden smooth expanded circular tube

This paper concerns with the investigation of laminar flow separation and its consequences in a tube over a smooth expansion under the axi-symmetric approximations. A co-ordinate stretching has been made to map the expanded tube into a straight tube. The two-dimensional unsteady Navier-Stokes equations are solved approximately by using primitive variables in staggered grid. A thorough quantitative analysis is performed through numerical simulations of the desired quantities such as wall shear stress, axial velocity, pressure distribution etc. These quantities are presented graphically and their consequences in the flow field are analysed in details. The dependence of the flow field on the physical parameter like expansion height d and on the Reynolds number has been investigated in details. It is interesting to note that the peak value of wall shear stress decreases with increasing height of expansion and also with the increasing Reynolds number.     

Laminar two-phase boundary layers in natural convection film boiling of subcooled liquids

Zusammenfassung Es wird die zweiphasige Grenzschichtströmung behandelt, die in unterkühlter Flüssigkeit entsteht, wenn die wärmeabgebende (vertikale) Wand eine Temperatur oberhalb des Leidenfrost-Punktes hat. Unter Einschluss des Impulstransportes an der Phasengrenzfläche werden asymptotische Lösungen ermittelt, die für sehr geringe und sehr hohe Flüssigkeitsunterkühlung gelten. Von den beiden Grenzfällen ausgehend lässt sich eine Interpolationsformel angeben, welche mit Wärmeübergangsmessungen verglichen wird, die das thermische Verhalten des organisch gekühlten Reaktors simulieren. In erster Näherung stimmen Rechnung und Experiment überein, wenn man die Streuung der Messwerte und vereinfachende Rechnungsannahmen berücksichtigt.     

Oscillations of a gas in an open-ended tube near resonance

The pressure as function of time was measured near resonance in different axial locations of an open-ended tube. Flow visualisation showed that transition to turbulence was not influenced by the strong disturbance of the open end, except in a region near the open end which had a length of about three particle displacements. The pressure readings were decomposed into the first, second and third harmonic and compared with two different theories. In one case, the linearized theory for the oscillating flow in a tube was fitted to the boundary conditions, the obvious one at the piston and a model at the open end. In the second case, the nonlinear theory of Chester [1] was used. Both theories assume a relation between pressure and velocity at the open end that contains two free constants. The constants were determined by comparing the amplitude of the first and the second harmonic ofone pressure measurement with the theoretical predictions. Once the constants are fixed, the pressurep(ωt, x/L) is completely determined. For weak nonlinear effects, the pressure is essentially determined by one constantα(=k ₂) and the second constantβ(=k ₁) loses its significance. For the range of parameters given there isα=0.825±0.015. A very good approximation of the pressure near resonance can therefore be calculated with the following simple boundary condition at the open end $$p_E = \frac{{4\alpha }}{{3\pi }}\rho \hat u_E u_E = 0.350 \rho \hat u_E u_E .$$ Both theories predict a resonance frequency slightly above the experimental one. Changing Levine and Schwingers [2], end correction from 0.6133R to 1R eliminates the discrepancy for all tube lengths. For the first harmonic the variation of the amplitude and the phase of the pressure signal withω andx is very well predicted by both theories. The nonlinear theory describes also the small second and third harmonics fairly well while the linear theory predicts only the correct order of magnitude of these higher harmonics. The constantα that determines the energy loss at the open end shows an apparent increase if the boundary layer on the tube wall becomes turbulent. This occurs for \(A = 2\hat u/\sqrt {v\omega } \geqq 550\) to 750 which is close to the value observed in a tube with a closed end.