Experiments on sedimentation beneath downward-facing inclined walls

It has been known since Boycott (1920) that sedimentation rates can be several times higher within inclined walls than in vertical vessels. Beneath the downward-facing inclined wall a thin boundary-layer of clear liquid is formed with an upward velocity. Mass continuity brings about the observed effect. Ponder (1925), and Nakamura & Kuroda (1937) proposed a global kinematic model without any detail of the flow field to predict the sedimentation rate. Recently, two asymptotic theories for describing the flow field became available. The investigation by Acrivos & Herbolzheimer (1979) requires both Re²/Gr and Re⁴/Gr to be small, with Re being a sedimentation Reynolds number and Gr a sedimentation Grashof number. The analysis due to Schneider (1982) is valid for small values of Re²/Gr and large values of Re⁴/Gr. This paper presents experimental sedimentation data in a²/Gr and large values of symmetrical, roof-shaped vessel. The experiments with spherical glass beads in a variable mixture of glycerine and water cover the whole parameter range from very small to very large values of Re⁴/Gr. Therefore, both theories could be verified. In the case of small values of Re⁴/Gr, strong waves beneath the downward-facing wall were observed and graphically evaluated in order to find a basis for further theoretical approach. Furthermore, an upward flow of particles in a sublayer of the boundary-layer leads te protruding “borns” and an oscillation of the originally horizontal discontinuity separating the suspension from the clear liquid. The measured particle distribution and deviations from a monodispersed suspension seems to be a possible explanation for this effect, which is investigated theoretically in a separate paper (Schaflinger 1984). A further experimental shortcoming, the shallow depth of the settling vessel, that was necessary for an optoelectronic measuring of the actual volume fraction solids within the bulk, may affect the convective motion when the sedimentation Reynols number is small. However, in this case the experimental data are in conformity with the theoretical predictions which do not consider the influence of wall friction on the sedimentation behaviour.     

Detonation waves in polydisperse gas suspensions of unitary fuel having a continuous particle size distribution function

Institute of the Mechanics of Multiphase Systems, Siberian Branch of the Russian Academy of Sciences, 625000 Tyumen'. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 6, pp. 14–24, November–December, 1995.     

Detonation waves in polydisperse gas suspensions of monofuel in tubes with an abrupt expansion

The distinctive features of detonation wave propagation in polydisperse (double-fraction) gas suspensions of a monofuel in tubes with an abrupt expansion are numerically investigated. Numerical calculations are performed for different sizes and relative mass contents of the particles of both fractions. A comparative analysis of the effect of mono- and polydisperse monofuel particles on the detonation wave mitigation is made. The dependences of the critical tube-diameter ratio of a sectional pipeline on the relative mass content and polydispersity of monofuel particles with different sizes are presented.     

Evolution of shock waves in polydisperse gas suspensions with a nonuniform particle concentration distribution

The results of a numerical investigation of the laws of shock wave propagation in polydisperse (two-fraction) gas suspensions with a non-uniform initial particle concentration distribution are presented. Examples of shock wave propagation in extended layers of a gas suspension with linearly increasing, linearly decreasing and sinusoidal laws of variation of the particle concentration are considered. It is shown that when shock waves pass through layers of a gas suspension with increasing and decreasing laws of variation of the particle concentration, respectively, amplification and attenuation of the waves are observed; when shock waves travel through gas suspensions with a periodic law of variation of the particle concentration the pressure distribution behind the wave fronts is nonmonotonic. The solutions corresponding to polydisperse and monodisperse gas suspensions with an effective particle size are examined. The nonequilibrium and thermodynamic-equilibrium solutions are compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 183–190, September–October, 1991.     

Structural instability in the sedimentation of particulate suspensions through viscoelastic fluids

A theory is developed to describe a structural instability that has been observed during the sedimentation of particulate suspensions through viscoelastic fluids. The theory is based on the assumption that the influence of hydrodynamic interactions in viscoelastic fluids, which tend to cause particles to aggregate, is in competition with hydrodynamic dispersion, which acts to maintain a homogeneous microstructure. In keeping with the experimental observations, it predicts that the suspension structure will stratify into vertical columns when a dimensionless stability parameter exceeds a critical value. The column-to-column separation, measured in particle radii, is predicted to be proportional to the square root of the ratio of the dimensionless dispersion coefficient to the product of the particle volume fraction and the Deborah number. The time for the formation of the columns is predicted to scale with the inverse of the average volume fraction. These predictions are in agreement with experimental data reported in the literature.     

Influence of nonuniform particle size on settling beneath downward-facing inclined walls

Experimental observations of the settling process in a symmetrical, roof-shaped vessel show a deformation of the initially horizontal interface between the suspension and the clear fluid, if Re²/Gr is small and Re⁴/Gr is large. This phenomenon is the result of a particle stream moving upward with a rclatively high vclocity within a sub-boundary-layer between the bulk of the suspension and a clear liquid layer that is formed underneath the inclined walls of the vessel. This particle stream propagates into the clear liquid above the suspension, thus giving rise to the observed deformities with the shape of protruding “horns” and, as a consequence, to an oscillation of the interface. A possible reason for these effects could be the nonuniform particle size. This assumption is confirmed by a theoretical analysis, which proves the existence of the abovementioned sub-boundary layer for the special case of two different particle sizes.     

Natural convection in an inclined quadrantal cavity heated and cooled on adjacent walls

Natural convective flow and heat transfer in an inclined quadrantal cavity is studied experimentally and numerically. The particle tracing method is used to visualize the fluid motion in the enclosure. Numerical solutions are obtained via a commercial CFD package, Fluent. The working fluid is distilled water. The effects of the inclination angle, ? and the Rayleigh number, Ra on fluid flow and heat transfer are investigated for the range of angle of inclination between 0° ? ? ? 360°, and Ra from 10⁵ to 10⁷. It is disclosed that heat transfer changes dramatically according to the inclination angle which affects convection currents inside, i.e. flow physics inside. A fairly good agreement is observed between the experimental and numerical results.     

The viscosity of polydisperse emulsions

Zusammenfassung Nach der gewöhnlichenEinstein-Taylorschen Formulierung der Fließeigenschaften disperser Systeme bleibt die Intrinsic-Viskosität unabhängig von der Teilchengröße oder der Teilchengrößenverteilung, während die Experimente eine geringe aber deutliche Abhängigkeit zeigen. Dieses Phänomen ist auf der Basis eines Modelles erklärbar, bei dem an der Teilchenoberfläche partielle Gleitung stattfindet. Eine solche Gleitung wird stattfinden in Gegenwart von oberflächenadsorbierbaren Substanzen. Auf einfache Weise, mit Hilfe derFröhlich-Sackschen Methode, wird eine quantitative Formel dieses Effektes abgeleitet. Emulsionen, die die monodispersen kleinen Teilchen enthalten, sind zäher als Systeme mit polydispersen großen Teilchen. Die gute Übereinstimmung mit den experimentellen Beobachtungen wird kritisch diskutiert.     

Analysis of the dissolution of a polydisperse system of particles having the shape of a parallelepiped

A mathematical model based on a crystal size distribution function is proposed for the continuous dissolution of particles having the shape of a parallelepiped. An evolution equation for the undersaturation of the solution is derived. Results of calculations using this equation are presented. A stability analysis of the steadystate solution obtained is carried out.     

Mixed convection with viscous dissipation in an inclined porous channel with isoflux impermeable walls

Combined forced and free convection flow in a fluid saturated inclined plane channel is investigated by taking into account the effect of viscous dissipation. Steady parallel flow is considered assuming that the temperature gradient in the parallel flow direction is constant, and the channel walls are subject to uniform symmetric heat fluxes. Two possible formulations of the Darcy–Boussinesq scheme are considered, based on two different choices of the reference temperature for modelling buoyancy. The first choice is a constant temperature, while the second is a streamwise changing temperature. It is shown that both approaches substantially agree in the formulation of the balance equations for the range of values of the Darcy–Rayleigh number such that viscous dissipation is important. The boundary value problem is solved analytically for any tilt angle, revealing that it admits dual solutions for assigned values of the governing parameters. The rather important effect of viscous dissipation in the special case of adiabatic channel walls is outlined. E. Magyari is on leave from Institute of Building Technology, ETH—Zürich     

Combined natural convection heat and mass transfer in an enclosure having finite thickness walls

Unsteady three-dimensional conjugate heat and mass transfer in an enclosure having finite thickness heat-conducting walls has been analyzed numerically. The governing unsteady, three-dimensional flow, energy and contaminant transport equations for the gas cavity and unsteady heat conduction equation for solid walls, written in dimensionless terms of the vector potential functions, the vorticity vector, the temperature and the concentration, have been solved using an iterative implicit finite-difference method. Main attention was paid to the effects of the Rayleigh number, buoyancy ratio and the dimensionless time on the flow structure and heat and mass transfer regimes. It should be noted that the dominant cause of the oscillations in the dimensionless time dependences of the average Nusselt number on the heat source surface and the average Sherwood number on the contaminant source surface at Ra>5?10⁵ is the mutual influence of the analyzed object geometry and the thermo-diffusivity impact on the flow. The change in the buoyancy ratio can lead to the essential modifications of the flow, temperature and concentration fields owing to the significant influence of the concentration gradient.     

Gas flow from plane vessels with large angles of inclination of the walls

Numerical solutions are obtained for problems of steady supersonic gas flow from infinite plane vessels with angles of inclination of the walls to the plane of symmetry _c on the interval 90° < _c 180°. The problems are posed and solved in the hodograph plane. It is shown that starting from a certain _c* the flow choking mechanism is determined not by the arrival of the limiting characteristic at the edge of the opening (classical choking mechanism) but by the interaction of the jet with the outside surface of the vessel wall. The effect of _c and the ambient pressure on the local and integral flow characteristics is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 144–151, January–February, 1991.The author wishes to thank A. N. Kraiko for discussing the results and N. I. Tillyaev for assisting with the work.     

Natural convection flow under a magnetic field in an inclined square enclosure differentialy heated on adjacent walls

Steady, laminar, natural-convection flow in the presence of a magnetic field in an inclined square enclosure differentially heated along the bottom and left vertical walls while the other walls are kept isothermal was considered. The governing equations were solved numerically for the stream function, vorticity and temperature ratio using the differential quadrature method for various Grashof and Hartmann numbers, inclination angle of the enclosure and direction of the magnetic field. The orientation of the enclosure changes the temperature gradient inside and has a significant effect on the flow pattern. Magnetic field suppresses the convective flow and its direction also influences the flow pattern, causing the appearance of inner loops and multiple eddies. The surface heat flux along the bottom wall is slightly increased by clockwise inclination and reduced by half by the counterclockwise inclination. The surface heat flux along the upper portion of the left side wall is reversed by the rise of warmer fluids due to the convection currents for no inclination and clockwise inclination of the enclosure.     

Propagation of small perturbations in a polydisperse vapor-gas-dropletmixture with polydisperse particles in the presence of phase transitions

The propagation of small disturbances in two-component gas-vapor mixtures with poly-disperse particles and droplets in the presence of phase transitions is studied. A mathematical model is proposed, the dispersion relation is obtained, high- and low-frequency asymptotics are derived, the equilibrium and frozen sonic speeds are found, and dispersion curves are calculated. The effect of the droplet and particle polydispersity on the dispersion and dissipation of small disturbances is analyzed.     

Stationary flow of a conducting liquid in a rectangular channel with two nonconducting walls and two walls having an arbitrary conductivity parallel to an external magnetic field

The flow of a conducting liquid in a channel of rectangular cross section with two walls (parallel to the external magnetic field) having an arbitrary conductivity, the other two being insulators, is considered. The solution of the problem is presented in the form of infinite series. The relationships obtained are used for numerical calculations of the velocity distribution and the distribution of the induced magnetic field over the cross section for several modes of flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostt i Gaza, No. 5, pp. 46–52, September–October, 1970.