An equation of motion for an incompressible Newtonian fluid in a packed bed

The application of a volume average Navier-Stokes equation for the prediction of pressure drop in packed beds consisting of uniform spherical particles is presented. The development of the bed permeability from an assumed porous microstructure model is given. The final model is quasi-empirical in nature, and is able to correlate a wide variety of literature data over a large Reynolds number range. In beds with wall effects present the model correlates experimental data with an error of less than 10%. Numerical solutions of the volume averaged equation are obtained using a penalty finite element method.Nomenclatures d length of a representative unit cell - d _e flow length in Representative Unit Cell - d _p characteristic pore size - D _T column diameter - D _P equivalent particle diameter - e _v energy loss coefficient for elbow - f _app apparent friction factor - f _v packed bed friction factor, defined by Equation (30) - F term representing impermeability of the porous medium - I integral defined by Equation (3) - L length of packed column - N Number of RUC in model microstructure - P pressure - P interstitial pressure - P pressure deviation - Re_p Reynolds number,v _p d _p/ - Re_s Reynolds number,v _s d/gm - Re_b Reynolds number,v _s D _p/ - S _fs fluid solid contact area - T tortuosity - v fluid velocity - v velocity deviation - v _p velocity in a pore - v _s superficial velocity in the medium - v interstitial velocity - V _o total volume of representative unit cell - V pore volume of representative unit cell - change in indicated property - u normal vector onS _fs - porosity - viscosity - density - coefficient in unconsolidated permeability model     

Interactions between two spheres falling along their line of centers in a viscoelastic fluid

An experimental investigation is presented in which the distance between two identical spheres falling along their line of centers in a viscoelastic fluid is determined as a function of time. It is found, for all five experimental fluids, that for small initial separations of the spheres, the two spheres eventually converge. However, for large initial separations, the two spheres eventually diverge. This leads to the definition of a critical initial separation distance. From this quantity a characteristic time can be derived for the two sphere fluid system which depends solely upon fluid properties. The implications of this study for suspension rheology are discussed.     

An experimental investigation of negative wakes behind spheres settling in a shear-thinning viscoelastic fluid

We present detailed experimental results examining “negative wakes” behind spheres settling along the centerline of a tube containing a viscoelastic aqueous polyacrylamide solution. Negative wakes are found for all Deborah numbers (2.43≤De(˙γ)≤8.75) and sphere-to-tube aspect ratios (0.060≤a/R≤0.396) examined. The wake structures are investigated using laser-Doppler velocimetry (LDV) to examine the centerline fluid velocity around the sphere and digital particle image velocimetry (DPIV) for full-field velocity profiles. For a fixed aspect ratio, the magnitude of the most negative velocity, U _min , in the wake is seen to increase with increasing De. Additionally, as the Deborah number becomes larger, the location of this minimum velocity shifts farther downstream. When normalized with the sphere radius and the steady state velocity of the sphere, the axial velocity profiles become self-similar to the point of the minimum velocity. Beyond this point, the wake structure varies weakly with aspect ratio and De, and it extends more than 20 radii downstream. Inertial effects at high Reynolds numbers are observed to shift the entire negative wake farther downstream. Using DPIV to investigate the transient kinematic response of the fluid to the initial acceleration of the sphere from rest, it is seen that the wake develops from the nonlinear fluid response at large strains. Measurements of the transient uniaxial extensional viscosity of this weakly strain-hardening fluid using a filament stretching rheometer show that the existence of a negative wake is consistent with theoretical arguments based on the opposing roles of extensional stresses and shearing stresses in the wake of the sphere. Received: 10 November 1997 Accepted: 1 May 1998     

An experimental study of the effective thermal conductivity of a sheared suspension of rigid spheres

An experimental study was conducted to measure the effective thermal conductivity of a sheared suspension of rigid spherical particles. The objective was to verify the theoretical prediction of Leal (1973) for a dilute suspension undergoing shear at low particle Peclet number, and to extend the range of the experiments to conditions beyond the scope of the theory. Surprisingly, reasonable agreement with the theoretical prediction was observed even for suspensions of moderate concentrations (volume fraction ≤ 0.25) and higher Peclet numbers [Pe 0(1)]. The trend of the data, however, verifies the obvious fact that the theory does not completely describe the transport behavior at higher concentrations and Peclet numbers. The range of quantitative applicability of Leal's result is apparently only for Pe < 0.01 and < 0.01, but the changes in the effective thermal conductivity in this domain were too small to be measured in our apparatus.     

Numerical simulation of the motion of neutrally buoyant particles in the plane Poiseuille flow of a Newtonian fluid

In this note we discuss the generalization of a Lagrange multiplier based fictitious domain method to the simulation of migration of neutrally buoyant particles in plane Poiseuille flow of a Newtonian fluid. The migration away from the center of the channel is believed to be an effect of the curvature of the velocity profile. We found this effect is not weakened by the presence of many particles, but instead by the collisions among the particles. Experiments and simulations show that the particles concentrate in the central region where the shear rate is low.     

Postcritical regimes in the nonlinear problem of vortex motion under the free surface of a weighable fluid

An improved Levi-Civita method in which the singularities of the desired function are taken into account by introducing terms containing power singularities is proposed. Results of numerical analysis of the nonlinear problem of a vortex in a bounded flow of an ideal weighable fluid (Fr>1) are given. The following limiting flow regimes are studied: the Stokes waves with one and two crests, emergence of a critical point on the surface, and the detachment of a vortex from a soliton and a uniform flow. It is shown that nonperiodic waves can form in a local zone in the vicinity of the critical point. Ufa State Technical Aviation University, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 70–76, January–February, 2000.     

An experimental study of the behavior of transient isolated convection in a rigidly rotating fluid

The temperature field of starting thermal plumes were measured in a rotating annulus with various rotation rates and buoyancies. The experiments revealed many details of the internal structure of these convective phenomena and also significant horizontal displacements from their source. Measurements show an increase in the maximum temperature observed in the thermal caps with increasing rotation and a more rapid cooling of the buoyancy source.List of symbols D angle relating inward centripetal acceleration to buoyant acceleration, defined by tan D = R/g - g gravitational acceleration - P total pressure of ambient fluid - R radial coordinate measured from rotation axis - R ₀ distance from rotation axis to buoyancy source - u velocity of fluid parcel along the radial direction - velocity of fluid parcel along the azimuthal direction - w velocity of fluid parcel along the axial direction - z axial coordinate, measured upward from the plane containing the buoyancy source - density of a buoyant parcel of fluid - ₀ density of the ambient fluid - azimuthal angle measured from the radial line passing through the buoyancy source - rotation rate of the R––z coordinate system in radians/second     

Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

An experimental study of electrorheological fluid flow through a packed bed of glass beads

This paper shows that pressure drop-flow rate performance of an electrorheological (ER) fluid flowing through a packed bed of glass beads is consistent with a modified Ergun equation for yield stress flow through a packed bed. ER fluids are of scientific and engineering interest due to the sensitivity of their rheological properties on the applied electric field. As far as we know ER fluids have not been studied for flows through porous media. In this work a silica particle–silicone oil suspension is pumped through a rectangular packed bed of glass beads with applied electric fields. The silica particles are observed to form fibrous structures parallel to the electric field that stretch between the beads and extend between the electrodes. The pressure drop-flow rate performance agrees well with the expected performance calculated from a modified Ergun equation for a yield stress fluid flow through the packed bed with the viscosity and yield stress as functions of the applied electric field.     

Unsteady stagnation-point flow of a Newtonian fluid in the presence of a magnetic field

The unsteady stagnation-point flow of a viscous fluid impinging on an infinite plate in the presence of a transverse magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at y=0 is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann's number.     

An experimental study of fluid flow and heat transfer in decaying swirl through a heated annulus

The mean and turbulent structures of turbulent swirling flow in a heated annulus have been measured. Both forced and free vortex swirling flows were generated, and the outer wall of the test section was heated uniformly. The maximum swirl number was 1.39, Reynolds numbers were up to 200000, and heat input was 10.5 kW. Mean and turbulent velocity components, air and wall temperatures, and wall static pressures were all measured. Hot-film techniques were developed to measure turbulence. From these parameters, the flow and temperature fields, pressure distribution, and heat transfer coefficients were determined. The mechanisms of heat transfer were identified.     

An experimental investigation of the flow about a sphere rotating in a rivlin-ericksen fluid

Tangential and radial velocity profiles were measured for the flow about a sphere rotating slowly in a Newtonian fluid, contained in a rectangular tank. Velocities were determined from enlarged streak photographs of aluminium particles moving in a collimated “sheet” of light, at several planes through the flow field. Similar velocity profiles were measured for the flow of a 1.50% Natrosol 250 H solution about two spheres of different diameters rotating in two different sized rectangular tanks. A set of velocity distributions were also measured for a sphere rotating in a 0.9% Natrosol 250 H solution. A dye tracer study of the flow about a sphere rotating in this liquid is presented as well. Both Natrosol solutions exhibited viscoelastic behaviour. The Newtonian fluid study was carried out at a Reynolds number of 1.2 and the viscoelastic fluid studies were within the Reynolds number range of 0.05–1.24.The zero shear viscosities of the Natrosol solutions were measured using the falling-sphere method. The non-Newtonian material parameters were obtained by fitting the theoretical curves to the measured velocity data. The values of the elastic and shear thinning parameters for the two liquids obtained in the different geometrical and dynamical situations are compared.     

Adaptive finite elements for the steady free fall of a body in a Newtonian fluid

The numerical simulation of the free fall of a solid body in a viscous fluid is a challenging task since it requires computational domains which usually need to be several order of magnitude larger than the solid body in order to avoid the influence of artificial boundaries. Toward an optimal mesh design in that context, we propose a method based on the weighted a posteriori error estimation of the finite element approximation of the fluid/body motion. A key ingredient for the proposed approach is the reformulation of the conservation and kinetic equations in the solid frame as well as the implicit treatment of the hydrodynamic forces and torque acting on the solid body in the weak formulation. Information given by the solution of an adequate dual problem allows one to control the discretization error of given functionals. The analysis encompasses the control of the free fall velocity, the orientation of the body, the hydrodynamic force and torque on the body. Numerical experiments for the two dimensional sedimentation problem validate the method. To cite this article: V. Heuveline, C. R. Mecanique 333 (2005).     

A numerical study of the motion of a spherical drop rising in shear-thinning fluid systems

The dynamical processes of a Newtonian spherical drop rising freely through shear-thinning fluids expressed by the generalized Cross-Carreau (Carreau–Yasuda) model were considered experimentally and computationally. The local effects of shear-thinning on the drop motion, which are hard to evaluate from an experimental approach, are clearly revealed by the numerical results. The relation between the drop motion and the change in viscosity is clearly indicated and considered in detail for the cases in which the non-Newtonian fluids are highly and weakly pseudoplastic. In addition, the modified Reynolds and Morton numbers are discussed in order to describe the drop or bubble free motion in shear-thinning fluids using a more convenient and practical form.     

Numerical and experimental study on the motion characteristics of single bubble in a complex channel

This paper is an extended study from previous work. In this study, the focus is paid to the dynamics of bubble rising and deformation in a complex channel, while the previous work is in straight channel. For this purpose, a three-dimensional lattice Boltzmann method (LBM) is employed to simulate the dynamics behaviour of a bubble rising in a complex channel consisting of three half-round throats. To validate the numerical method, a visual experiment was carried out by means of a high-speed digital camera and computer image processing technology. The behaviour of the rising bubble through glycerine solution in a complex channel was recorded. Some physical parameters such as rising velocities, trajectory and shapes of the bubble were calculated and processed based on the experimental data. In the same conditions, the trajectory, shapes and rising velocities of the bubble were simulated during its rising process by the proposed LBM. The numerical results are in good agreement with the experimental results. It demonstrates that LBM used in this work is feasible for simulating two-phase flow in such a complex channel.     

An Eulerian model for the motion of granular material with a large Stokes number in fluid flow

This study introduced a novel Euler–Euler approach for modeling granular multiphase flow. The motion of particles with a large Stokes number was investigated assuming that granular material has unilateral compressibility. Solid pressure in the momentum equations for granular multiphase flow was determined so that the unilateral incompressibility condition was satisfied. Using the continuity condition of the granular phase, the equation was rewritten in the optimal form to calculate the solid pressure. A discrete formulation of smoothed particle hydrodynamics was applied for the convective terms so that the discrete matrix was positive semidefinite for the convergence and the discretization for an unstructured mesh was allowed. Frictional stress was then determined from solid pressure and, by using the solid pressure and frictional stress, momentum equations for the granular phase were solved. The method was incorporated into ANSYS FLUENT by a UDF (user defined function). Model validation was performed through a comparison with two previous results, and efficacy of the proposed model was confirmed.     

On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid

The complete solution for the pressure and the velocity field up to O(De) of a dilute suspension of neutrally buoyant, non-Brownian rigid spheres suspended in an unbounded, weakly viscoelastic matrix fluid, where is the solid volume fraction and De is the Deborah number of the matrix fluid, is presented. The spheres are subjected to an arbitrary linear velocity profile at infinity. The analytical solution is used for the prediction of the bulk stress, and specifically for the calculation of the first and the second normal stress differences in simple shear and uniaxial elongational flows. A comparison of the results with available values reported in the literature is also offered. The final expressions for the bulk normal stress differences in shear and uniaxial elongational flow fully agree with those reported earlier by Greco et al., J. Non-Newton. Fluid Mech., 147 (2007) 1–10.