Interaction forces of particles dispersed in an electrolyte

Expressions are derived for the forces acting in a disperse medium in the presence of interaction of the double layers surrounding particles or drops of the dispersed phase when the potential of the dispersed particles is small. It is found that the force produced by the presence of double layers is proportional to the concentration gradient of the dispersed particles. It is shown that this force is comparable with the force produced by Brownian motion of the particles and may even exceed it. The equations of motion for the dispersed phase are derived with allowance for the convective terms, the pressure gradient, and the forces caused by Brownian motion and the presence of the double layers. A generalized Fick's law is obtained with effective diffusion coefficient. The equilibrium distribution of the particle concentration in a uniformly rotating cylinder is found with allowance for the interaction of the double layers.Translated from Izvestiya Akademii Nauk SSSR, Meklianika Zhidkosti i Gaza, No. 5, pp. 98–102, September–October, 1984.     

Exponential vs Algebraic Growth and Transition Prediction in Boundary Layer Flow

For applications regarding transition prediction, wing design andcontrol of boundary layers, the fundamental understanding of disturbancegrowth in the flat-plate boundary layer is an important issue. In thepresent work we investigate the energy growth of eigenmodes andnon-modal optimal disturbances. We present a set of linear governingequations for the parabolic evolution of wavelike disturbances validboth for the exponential and algebraic growth scenario. The base flow istaken as the Falkner–Skan similarity solution with favorable, adverseand zero pressure gradients. The optimization is carried out over theinitial streamwise position as well as the spanwise wave number andfrequency. The exponential growth is maximized in the sense that theenvelope of the most amplified eigenmode is calculated. In the case ofalgebraic growth, an adjoint-based optimization technique is used. Wefind that the optimal algebraic disturbance introduced at a certaindownstream position gives rise to a larger growth than for the optimaldisturbance introduced at the leading edge. The exponential andalgebraic growth is compared and a unified transition-predictionmethod based on available experimental data is suggested.     

Motion and heat and mass transfer in a disperse admixture in turbulent nonisothermal jets of a gas and a low-temperature plasma

The motion and heat and mass transfer of particles of a disperse admixture in nonisothermal jets of a gas and a low-temperature plasma are simulated with allowance for the migration mechanism of particle motion actuated by the turbophoresis force and the influence of turbulent fluctuations of the jet flow velocity on heat and mass transfer of the particle. The temperature distribution inside the particle at each time step is found by solving the equation of unsteady heat conduction. The laws of scattering of the admixture and the laws of melting and evaporation of an individual particle are studied, depending on the injection velocity and on the method of particle insertion into the jet flow. The calculated results are compared with data obtained with ignored influence of turbulent fluctuations on the motion and heat and mass transfer of the disperse phase. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 95–108, May–June, 2008.     

Optimal disturbances in suction boundary layers

A well-known optimization procedure is used to find the optimal disturbances in two different suction boundary layers within the spatial framework. The maximum algebraic growth in the asymptotic suction boundary layer is presented and compared to previous temporal results. Furthermore, the spatial approach allows a study of a developing boundary layer in which a region at the leading edge is left free from suction. This new flow, which emulates the base flow of a recent wind-tunnel experiment, is herein denoted a semi-suction boundary layer. It is found that the optimal disturbances for these two suction boundary layers consist of streamwise vortices that develop into streamwise streaks, as previously found for a number of shear flows. It is shown that the maximum energy growth in the semi-suction boundary layer is obtained over the upstream region where no suction is applied. The result indicates that the spanwise scale of the streaks is set in this region, which is in agreement with previous experimental findings.     

Statistical characteristics of the motion of a fluid particle in a medium with random porosity

In the study of flow of a neutral admixture in a porous medium, it is most often assumed in the stochastic formulation that the porosity is constant and a determinate quantity, and the velocity is a random function [1–4]. The velocity distribution is usually regarded as known. Flow in a porous medium with random porosity has been studied to a far lesser extent. We note [5], which studies the averaged equations obtained within the framework of the correlation approximation. We consider the model problem of one-dimensional motion of a fluid particle (position of the front for flow of a neutral admixture in a porous medium) in a medium with random porosity. For a particular form of random porosity field, expressions are obtained for the one- and two-point densities of the distribution of the position of the particle. A study is made of the dependences of the first four moments and the correlation function of the position of the particle as functions of the time. It is shown that for large values of the time the motion of the particle is asymptotically similar to Brownian motion. It is shown by means of numerical modeling that the results obtained transfer to the case of an arbitrary random porosity field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 59–65, November–December, 1986.     

Influence of the walls on overheating instability in a magnetohyduodynamic channel

The mechanism of the overheating instability of magnetohydrodynamic flows is as follows. if the electrical conductivity of the medium depends on the temperature, then a small local increase in temperature may lead, under specific conditions, to an increase in liberation of Joule heat energy, to a further temperature rise, and so to instability.The overheating instability has been studied before (1, 2) for a unifrom unperturbed temperature and without account for the effect of the boundaries of the region. It has been discovered that the growth increment of the disturbances increases as the wavelength increases. However, it is clear that heat conduction through the boundaries of the region occupied by the conducting medium may effect the development of perturbations, primarily of long wavelength perturbations. Below we will examine the simplest problem of the stability of temperature distribution for an electric discharge in a gas between two planes.     

Heat transfer in a three-dimensional rectangular cavity oriented at an angle to the free-stream

We consider a flow of a viscous incompressible heat-conducting fluid over a cubic cavity. The heat transfer on the bottom of the cavity rotated at an angle to the free stream is studied numerically. The numerical algorithm includes a finite-difference approximation in the spatial coordinates, a semi-implicit time integration method, and a modified version of an iterative stabilized method of biconjugate gradients with an algebraic multigrid preconditioner for solving the Poisson equation. The algorithm is designed for the use of multiprocessor computers. Two different inlet flow conditions are considered: a steady-state flow and a steady flow with superimposed periodic perturbations. In the first case, it is shown that the integral heat transfer rate increases monotonically with increase in the cavity rotation angle α. For α = 45°, the increase in the heat flux amounts to 20%. The presence of periodic disturbances may result in up to 3-fold increase in the integral heat transfer rate as compared to the case of the steady-state inlet flow. The enhancement of heat transfer occurs when the frequency of the inlet flow disturbances is close to the frequency of unstable modes of the mixing layer formed at the upper boundary of the cavity.     

Localized disturbances in parallel shear flows

The development of localized disturbances in parallel shear flows is reviewed. The inviscid case is considered, first for a general velocity profile and then in the special case of plane Couette flow so as to bring out the key asymptotic results in an explicit form. In this context, the distinctive differences between the wave-packet associated with the asymptotic behavior of eigenmodes and the non-dispersive (inviscid) continuous spectrum is highlighted. The largest growth is found for three-dimensional disturbances and occurs in the normal vorticity component. It is due to an algebraic instability associated with the lift-up effect. Comparison is also made between the analytical results and some numerical calculations.Next the viscous case is treated, where the complete solution to the initial value problem is presented for bounded flows using eigenfunction expansions. The asymptotic, wave-packet type behaviour is analyzed using the method of steepest descent and kinematic wave theory. For short times, on the other hand, transient growth can be large, particularly for three-dimensional disturbances. This growth is associated with cancelation of non-orthogonal modes and is the viscous equivalent of the algebraic instability. The maximum transient growth possible to obtain from this mechanism is also presented, the so called optimal growth.Lastly the application of the dynamics of three dimensional disturbances in modeling of coherent structures in turbulent flows is discussed.     

The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.     

Dynamical properties of heterogeneous surface layers. capillary wave scattering

The dynamics of a heterogeneous liquid surface constituting a two-dimensional disperse system is considered. One of the surface phases (the dispersed phase) forms circular regions of diameter comparable with the characteristic length of the mechanical disturbances within the continuous disperse medium. Inhomogeneous boundary conditions for the Navier-Stokes equations with a discontinuity on the surface phase contact line are formulated. Special attention is paid to the conditions on this line. An approximate method of solving the surface wave diffraction problem and the results for the case of transverse surface wave scattering are described. It is shown that if the wavelength is close to the dimensions of the two-dimensional dispersed particles and their concentration is sufficiently high, the energy of the scattered waves may exceed that dissipated in the vorticity layer. Thus, a new nonclassical mechanism of surfactant action on capillary wave damping is possible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–137, January–February, 1991.     

Optimal Control of Nonmodal Disturbances in Boundary Layers

The optimal control of infinitesimal flow disturbances experiencing the largest transient gain over a fixed time span, commonly termed “optimal perturbations,” is undertaken using a variational technique in two- and three-dimensional boundary layer flows. The cost function employed includes various energy metrics which can be weighted according to their perceived importance, simplifying the task of determining which terms are essential for a “good” control scheme. In the accelerated boundary layers investigated, disturbance kinetic energy can be typically reduced by about one order of magnitude. However, it seems impossible to suppress completely over the entire control interval; “good” control strategies still permit approximately an order magnitude growth over the initial energy at some point in the interval. It is shown that the control effort efficiently targets the physical mechanisms behind transient growth. Received 5 February 2001 and accepted 15 June 2001     

The near wake structure of a square cylinder

The near wake structure of a square cross section cylinder in flow perpendicular to its length was investigated experimentally over a Reynolds number (based on cylinder width) range of 6700–43,000. The wake structure and the characteristics of the instability wave, scaling on θ at separation, were strongly dependent on the incidence angle () of the freestream velocity. The nondimensional frequency (St_θ) of the instability wave varied within the range predicted for laminar instability frequencies for flat plate wakes, jets and shear layers. For = 22.5°, the freestream velocity was accelerated over the side walls and the deflection of the streamlines (from both sides of the cylinder) towards the center line was higher compared to the streamlines for = 0°. This caused the vortices from both sides of the cylinder to merge by x/d 2, giving the mean velocity distribution typical of a wake profile. For = 0°, the vortices shed from both sides of the cylinder did not merge until x/d 4.5. The separation boundary layer for all cases was either transitional or turbulent, yet the results showed good qualitative, and for some cases even quantitative, agreement with linearized stability results for small amplitude disturbances waves in laminar separation layers.     

Stability of a vertical high-concentration fine suspension flow

The neutral stability conditions are found with allowance for the effects associated with the disappearance of the particle free volume on transition to the close-packed state and the characteristics of the disturbances with the maximum growth rate are investigated on the interval of intermediate and high particle concentrations. Results relating to the effect of quasi-viscous stresses and Brownian motion on flow stabilization on this concentration interval are obtained. The stability of a bounded uniform vertical flow of not too small particles is investigated and the well-known scale effect, associated with the phenomenon of increasing instability on transition from laboratory to geometrically similar industrial apparatus, is examined. Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 87–96, July–August, 1994.     

Double-diffusive convection for a heated cylinder submerged in a salt-stratified fluid layer

Double-diffusive convection due to a cylindrical source submerged in a salt-stratified solution is numerically investigated in this study. For proper simulation of the vortex generated around the cylinder, a computational domain with irregular shape is employed. Flow conditions depend strongly on the thermal Rayleigh number, Ra _T , and the buoyancy ratio, R _ρ. There are two types of onset of instability existing in the flow field. Both types are due to either the interaction of the upward temperature gradient and downward salinity gradient or the interaction of the lateral temperature gradient and downward salinity gradient. The onset of layer instability due to plume convection is due to the former, whereas, the onset of layer instability of layers around the cylinder is due to the latter. Both types can be found in the flow field. The transport mechanism of layers at the top of the basic plume belongs to former while that due to basic plume and layer around the cylinder are the latter. The increase in Ra _T reinforces the plume convection and reduces the layer numbers generated around the cylinder for the same buoyancy ratio. For the same Ra _T , the increase of R _ρ suppresses the plume convection but reinforces the layers generated around the cylinder. The profiles of local Nusselt number reflects the heat transfer characteristics of plume convection and layered structure. The profiles of averaged Nusselt number are between the pure conduction and natural convection modes and the variation is due to the evolution of layers. Received on 13 September 1996     

Particle suspension in a simple shear flow

A hydrodynamic model describing the particle distribution over the cross-section of a finely dispersed flow is proposed. The model is constructed on the basis of notions concerning the diffusion of particles induced by their random displacements in the process of relative motion of neighboring layers at constant shear velocity. It is shown that the suspension capacity of the flow is large for small particles due to thermal fluctuations and for relatively large particles due to shear-induced particle pulsations. There are critical particle sizes for which the particles are suspended and transported by the flow less effectively than larger or smaller particles.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–121, January–February, 1995.     

The effect of particle size and migration on the formation of flow-induced structures in viscoelastic suspensions

Flow-induced structures in suspensions containing spheres in viscoelastic suspending media were investigated by microscopy and rheo-optical methods. Suspensions of monodisperse polystyrene spheres with diameters ranging from 1.2 to 2.8 μm and dispersed in aqueous solutions of hydroxypropylcellulose were studied in simple shear flows. Optical microscopy observations as well as small-angle light-scattering (SALS) experiments were performed using a parallel plate geometry. In agreement with previous work, necklaces of particles aligned in the flow direction were observed when shearing faster then a critical shear rate, which was found to be independent of particle size. In contrast to earlier work, however, the role of particle migration was found to be of prime importance. Particles were shown to migrate toward the plates where the particles assembled and aligned in strings running in the flow direction. For the smallest particles (1 μm diameter), the formation of particle doublets or short strings along the vorticity direction was observed at low shear rates, which flipped to an orientation into the flow direction and grew into longer strings at higher shear rates. SALS experiments were used to quantify the degree of alignment and its dependence on particle size, shear rate, and gap. For the system under investigation, the degree of alignment was found to increase with increasing shear rate and particle size and with decreasing gap. The present results suggest that, depending on the details of the suspending medium and the size and nature of the suspended particles, the formation of aligned structures is affected by the relative magnitude of the colloidal and hydrodynamic forces and the kinetics of string formation versus the kinetics of migration.     

On the Secondary Instability of Three-Dimensional Boundary Layers

One of the possible transition scenarios in three-dimensional boundary layers, the saturation of stationary crossflow vortices and their secondary instability to high-frequency disturbances, is studied using the Parabolized Stability Equations (PSE) and Floquet theory. Starting from nonlinear PSE solutions, we investigate the region where a purely stationary crossflow disturbance saturates for its secondary instability characteristics utilizing global and local eigenvalue solvers that are based on the Implicitly Restarted Arnoldi Method and a Newton–Raphson technique, respectively. Results are presented for swept Hiemenz flow and the DLR swept flat plate experiment. The main focuses of this study are on the existence of multiple roots in the eigenvalue spectrum that could explain experimental observations of time-dependent occurrences of an explosive growth of traveling disturbances, on the origin of high-frequency disturbances, as well as on gaining more information about threshold amplitudes of primary disturbances necessary for the growth of secondary disturbances. Received 13 July 1998 and accepted 7 July 2000     

Interaction of continuous-spectrum waves with each other and with discrete-spectrum waves

The nonlinear problem of the evolution of an initial perturbation in Couette flow is solved in the quadratic approximation and it is shown that the energy of the initial perturbation is transmitted to the main flow so that its profile is somewhat modified. The evolution of the initial perturbation in a fluid with a very simple model flow profile which, in addition to continuous-spectrum waves, also admits the existence of a single neutral mode of the discrete spectrum is then investigated. It is shown that as a result of the linear resonant interaction of the discrete-spectrum and continuous-spectrum waves disturbances that grow linearly with time may be formed. A flow that does not contain exponentially growing modes will be unstable with respect to certain initial disturbances; this instability is called algebraic [6, 7]. A physical interpretation of this effect is given. From this interpretation it is clear that algebraic instability is possible in a fluid with flow profiles of a more general type, in which there are neutral or weakly damped discrete-spectrum modes having a critical layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 116–123, July–August, 1989.The author is grateful to G. I. Barenblatt, S. Ya. Gertsenshtein, M. A. Mironov, S. A. Rybak, O. S. Ryzhov, and E. D. Terent'ev for their interest and useful comments.     

Flow of a dispersed phase in the Laval nozzle and in the test section of a two-phase hypersonic shock tunnel

An unsteady gas-particle flow in a hypersonic shock tunnel is studied numerically. The study is performed in the period from the instant when the diaphragm between the high-pressure and low-pressure chambers is opened until the end of the transition to a quasi-steady flow in the test section. The dispersed phase concentration is extremely low, and the collisions between the particles and their effect on the carrier gas flow are ignored. The particle size is varied. The time evolution of the particle concentration in the test section is obtained. Patterns of the quasi-steady flow of the dispersed phase in the throat of the Laval nozzle and the flow around a model (sphere) are presented. Particle concentration and particle velocity lag profiles at the test-section entrance are obtained. The particle-phase flow structure and the time needed for it to reach a quasi-steady regime are found to depend substantially on the particle size. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 102–113, September–October, 2008.