Unsteady Convection in a Binary Mixture in the Neighborhood of a Plane Vertical Surface

The problem of the development of convection in a binary mixture in the neighborhood of an infinite vertical plate, on which a constant (after initial switch-on) heat flux and zero admixture flow are given, is solved. In particular, the cases of neutral and stable density stratification of the medium are considered. It is found that heat transfer to the medium can lead not to an increase but to a decrease in its temperature. This can be interpreted as the effective negative heat capacity of the stratified binary mixture.     

Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel

The evolution of steady-state viscous incompressible fluid flows in a plane divergent channel is investigated. For the classical formulation of the Jeffery-Hamel problem a complete solution is given as a function of the determining parameters. For a fixed angle of divergence the behavior of the main unimodal flow is determined as a function of the Reynolds number. Critical values at which the flow pattern bifurcates and the steady-state unimodal flow ceases to exist are found. The mechanism of bifurcation is established and its diagram is constructed. This mechanism and the diagram were not previously known in the scientific literature in connection with the investigation of the Jeffery-Hamel problem. The critical Reynolds number at which bifurcation occurs is given as a function of the channel divergence angle. The results may be of interest for hydromechanical, technological, and geophysical applications.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 25–36.Original Russian Text Copyright © 2005 by Akulenko and Kumakshev.     

Unsteady Self-Similar Flow of a Viscous Incompressible Fluid in a Plane Divergent Channel

The problem of two-dimensional unsteady flow of a viscous incompressible fluid in a sector-like domain is considered. Initially a strictly radial flow is imposed, which makes it possible to seek solutions within the class of self-similar flows. A numerical method based on mixed finite-difference and spectral spatial discretization is developed, making it possible to find the self-similar solution efficiently. The process of development and establishment of the steady Hamel-Jeffery and Moffatt flows is modeled mathematically.     

平面粘性流体扰动与哈密顿体系

通过变分原理,将哈密顿体系的理论引入到平面粘性流体扰动的问题中,导出一套哈密顿算子矩阵的本征函数向量展开求解问题的方法。基于直接法求解流体力学基本方程,导出流场一般特征关系,通过本征值的求解及本征向量的叠加,得到波扰动解,继可分析流场端部效应。从而在该领域用在哈密顿体系下辛几何空间中研究问题的方法代替了传统在拉格朗日体系欧几里德空间分析问题的方法。为流体力学的研究提供一条新途径。     

Effects of Viscous Dissipation and Flow Work on Forced Convection in a Channel Filled by a Saturated Porous Medium

Fully developed forced convection in a parallel plate channel filled by a saturated porous medium, with walls held either at uniform temperature or at uniform heat flux, with the effects of viscous dissipation and flow work included, is treated analytically. The Brinkman model is employed. The analysis leads to expressions for the Nusselt number, as a function of the Darcy number and Brinkman number.     

Invariant Solutions of the Thermal-Diffusion Equations for a Binary Mixture in the Case of Plane Motion

The group properties of the thermal-diffusion equations for a binary mixture in plane flow are studied. Optimal systems of first-and second-order subalgebras are constructed for the admissible Lie operator algebra, which is infinite-dimensional. Examples of the exact invariant solutions are given, which are found by solving ordinary differential equations. Exact solutions are found that describe thermal diffusion in an inclined layer with a free boundary and in a vertical layer in the presence of longitudinal temperature and concentration gradients. The effect of the thermal-diffusion parameter on the flow regime is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 95–108, January–February, 2006.     

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

Nucleation of Droplets in a Binary Mixture

We revisit the theory of condensation of a droplet in a vapour with the aim of finding the effect that the surface tension has on the phase diagram of a binary mixture. For that purpose we consider condensation under volume control and rederive the Gibbs phase rule. The Gibbs phase rule is equivalent to the common tangent construction for two free enthalpies corresponding to different pressures: in this case, the pressure in the vapour and the pressure in the droplet. Explicit results are calculated for a droplet mixed from incompressible liquids and for a vapour that is an ideal gas mixture. It turns out that in a certain range of volumes the equilibrium state of droplet and vapour has a higher free energy than the vapour alone. At the lower bound of that range of volumes a stable droplet of finite size will nucleate and consequently the vapour pressure will drop. The usual condensation line in a (p, X ₁)-phase diagram without surface tension is thus replaced by two lines: One for the incipient condensation and one for its completion; both lie above the condensation line without surface tension.     

Linear Stability Analysis of a Viscous Liquid Sheet in a High-Speed Viscous Gas

Air-assisted atomizers in which a thin liquid sheet is deformed under the action of a high-speed air flow are extensively used in industrial applications, e.g., in aircraft turbojet injectors. Primary atomization in these devices is a consequence of the onset and growth of instabilities on the air/liquid interfaces. To better understand this process, a temporal linear instability analysis is applied to a thin planar liquid sheet flowing between two semi-infinite streams of a high-speed viscous gas. This study includes the full viscous effects both in the liquid and gas basic states and perturbations. The relevant dimensionless groups entering the non-dimensional Orr–Sommerfeld equations and boundary conditions are the liquid and gas stream Reynolds numbers, the gas to liquid momentum flux ratio, the gas/liquid velocity ratio, the Weber number and the equivalent gas boundary layer to liquid sheet thickness ratio. Growth rates and temporal frequencies as a function of the wave number, varying the different dimensionless parameters are presented, together with neutral stability curves. From the results of this parametric study it is concluded that when the physical properties of gas and liquid are fixed, the momentum flux ratio is especially relevant to determine the instability conditions. It is also observed that the gas boundary layer thickness strongly influences the wave propagation, and acts by damping sheet oscillation frequency and growth. This is especially important because viscosity in the basic gas velocity profile has always been ignored in instability analysis applied to the geometry under study. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mixed Convection in a Horizontal Channel with Local Heating from Below

Mixed convection induced in the entrance region of a horizontal plane channel by a bottom heat source of finite dimensions is considered. The calculations were performed for the Prandtl number Pr = 1, Grashof numbers ranging from 4 · 10³ to 3.2 · 10⁴, and Reynolds numbers varying from 0 to 10. The dimensions of the heat source and its location were also varied. The results were obtained from a numerical solution of the 2D unsteady Navier-Stokes equations in the Boussinesq approximation, written in vorticity – stream function – temperature variables. The solution was found by the Galerkin finite element method.     

Thermal Convection in a Plane Layer Rotating About a Horizontal Axis

The thermal convection of a fluid in a plane vertical layer with a cylindrical lateral boundary, which rotates uniformly about a horizontal symmetry axis, is investigated experimentally. The structure and excitation limit of the convective flows are studied as functions of the rotation frequency, the temperature difference between the layer boundaries, and the layer thickness. The determining dimensionless parameters are found. It is shown that the period-average gravity action produces convection in the form of hexagonally distributed cells stationary in the reference system tied to the cavity.     

Subcritical Motions of a Binary Mixture with Anomalous Thermodiffusion in a Vertical Layer

The finite difference method is used to analyze the stability of a binary mixture with anomalous thermodiffusion relative to perturbations of finite amplitude. In the region of monotonic instability, subcritical motions are detected. The amplitude curves, the limits of main-flow stability relative to the most dangerous perturbations, and the stream function, temperature and concentration isolines are obtained.     

Asymptotic Theory of the Viscous Interaction Between a Vortex and a Plane

The problem of the viscous interaction between a flow induced by a vortex filament and an orthogonal rigid surface is solved for high Reynolds numbers using the method of matched asymptotic expansions. In view of the impossibility of matching the principal terms of the asymptotic expansions directly for the near-axial boundary layer and the main flow zone, the solution is obtained by introducing two intermediate zones. In this case a logarithmic singularity of the axial velocity arises inevitably on the vortex filament. In the near-axial and intermediate zones the solution is obtained numerically and analytically, respectively, while in the main zone the problem reduces to the problem of the flow induced by a line of weakly swirled vortex-sinks.     

Fully Developed Forced Convection in a Parallel Plate Channel with a Centered Porous Layer

In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu _cl = 4.12 and fRe _cl = 24) are used to define heat transfer increment ratio (e_th = Nu_p/Nu_cl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (e_p = fRe_p/fRe_cl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer and pressure drop increment ratios are used to define an overall performance (e = e_th/e_p)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k _r) when k _r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls.     

具有大负分离比的混合流体局部行进波对流

从流体层底部加热引起的对流运动是研究非平衡对流的时空结构或斑图(Pattern)及非线性动力学特性的典型模型之一.本文通过流体力学基本方程组的数值模拟,探讨了具有强Soret效应的混合流体局部行进波的形成过程,发现当分离比ψ=-0.6时,在局部行进波的存在范围内,向局部行进波过渡的不同过程依赖于相对瑞利数r.进一步,讨论了具有强Soret效应的混合流体局部行进波流速场,温度场, 浓度场的结构和特性,分析了局部行进波的存在区间对分离比ψ的依赖性.发现随着Soret效应的增强或负分离比ψ的绝对值的增加,局部行进波稳定存在的区间Δr也在增加.     

Analytical Solution for Forced Convection in a Semi-Circular Channel Filled with a Porous Medium

Fully developed laminar forced convection inside a semi-circular channel filled with a Brinkman-Darcy porous medium is studied. Analytical solutions for flow and constant flux heat transfer are found using a mixture of Cartesian and cylindrical coordinates. The problem depends on a parameter s, which is proportional to the inverse square of the Darcy number. Velocity boundary layers exist when s is large. Both friction factor-Reynolds number product and Nusselt number are determined. Closed form expressions for the clear fluid () limit are found. Rare analytical solutions not only describe fundamental channel flows, but also serve as a check for more complicated numerical solutions.     

Inertial Motions of a Rigid Body with a Cavity Filled with a Viscous Liquid

We study inertial motions of the coupled system, \({\mathscr{S}}\), constituted by a rigid body containing a cavity entirely filled with a viscous liquid. We show that for arbitrary initial data having only finite kinetic energy, every corresponding weak solution (à la Leray–Hopf) converges, as time goes to infinity, to a uniform rotation, unless two central moments of inertia of \({\mathscr{S}}\) coincide and are strictly greater than the third one. This corroborates a famous “conjecture” of N.Ye. Zhukovskii in several physically relevant cases. Moreover, we show that, in a known range of initial data, this rotation may only occur along the central axis of inertia of \({\mathscr{S}}\) with the larger moment of inertia. We also provide necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations, which improve and/or generalize results previously given by other authors under different types of approximation. Finally, we present results obtained by a targeted numerical simulation that, on the one hand, complement the analytical findings, whereas, on the other hand, point out new features that the analysis is yet not able to catch, and, as such, lay the foundation for interesting and challenging future investigation.     

Forced Convection with Laminar Pulsating Flow in a Saturated Porous Channel or Tube

A perturbation approach is used to obtain analytical expressions for the velocity, temperature distribution, and transient Nusselt number for the problem of forced convection, in a parallel-plates channel or a circular tube occupied by a saturated porous medium modeled by the Brinkman equation, produced by an applied pressure gradient that fluctuates with small amplitude harmonically in time about a non-zero mean. It is shown that the fluctuating part of this Nusselt number alters in magnitude and phase as the dimensionless frequency increases. The magnitude increases from zero, goes through a peak, and then decreases to zero. The height of the peak decreases as the modified Prandtl number increases. The phase (relative to that of the steady component) decreases from π/2 to − π/2. The height of the peak at first increases, goes through a maximum, and then decreases as the Darcy number decreases.