Particle-phase Viscosity Research Articles

Numerical solutions for unsteady boundary layer flow of a dusty fluid past a permeable stretching/shrinking surface with particulate viscous effect

PurposeThis present aims to present the numerical study of the unsteady stretching/shrinking flow of a fluid-particle suspension in the presence of the constant suction and dust particle slip on the surface.Design/methodology/approachThe governing partial differential equations for the two phases flows of the fluid and the dust particles are reduced to the pertinent ordinary differential equations using a similarity transformation. The numerical results are obtained using the bvp4c function in the Matlab software.FindingsThe results revealed that in the decelerating shrinking flow, the wall skin friction is higher in the dusty fluid when compared to the clean fluid. In addition, the effect of the fluid-particle interaction parameter to the fluid-phase can be seen more clearly in the shrinking flow. Other non-dimensional physical parameters such as the unsteadiness parameter, the mass suction parameter, the viscosity ratio parameter, the particle slip parameter and the particle loading parameter are also considered and presented in figures. Further, the second solution is discovered in this problem and the solution expanded with higher unsteadiness and suction values. Hence, the stability analysis is performed, and it is confirmed that the second solution is unstable.Practical implicationsIn practice, the flow conditions are commonly varying with time; thus, the study of the unsteady flow is very crucial and useful. The problem of unsteady flow of a dusty fluid has a wide range of possible applications such as in the centrifugal separation of particles, sedimentation and underground disposable of radioactive waste materials.Originality/valueEven though the problem of dusty fluid has been broadly investigated, limited discoveries can be found over an unsteady shrinking flow. Indeed, this paper managed to obtain the second (dual) solutions, and stability analysis is performed. Furthermore, the authors also considered the artificial particle-phase viscosity, which is an important term to study the particle-particle and particle-wall interactions. With the addition of this term, the effects of the particle slip and suction parameters can be investigated. Very few studies in the dusty fluid embedded this parameter in their problems.

Stability of concentrated suspensions under Couette and Poiseuille flow

The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. A linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated, and the connections to the stability problem for the related classical Bingham flow problem are discussed.

Unsteady flow of a dusty Bingham fluid through a porous medium in a circular pipe

A time-varying flow through a porous medium of a dusty viscous incompressible Bingham fluid in a circular pipe is studied. A constant pressure gradient is applied in the axial direction, whereas the particle phase is assumed to behave as a viscous fluid. The effect of the medium porosity, the non-Newtonian fluid characteristics, and the particle phase viscosity on the transient behavior of the velocity, volumetric flow rates, and skin friction coefficients of both the fluid and particle phases is investigated. A numerical solution is obtained for the governing nonlinear momentum equations by using the method of finite differences.

Ion slip effect on a steady flow through a circular pipe of a dusty conducting Oldroyd 8-constant fluid

In this paper, a steady magnetohydrodynamic (MHD) flow of a dusty incompressible electrically conducting Oldroyd 8-constant fluid through a circular pipe is examined with considering the ion slip effect. A constant pressure gradient in the axial direction and an external uniform magnetic field in the perpendicular direction are applied. A numerical solution is obtained for the governing nonlinear momentum equations by using finite differences. The effect of the ion slip, the non-Newtonian fluid characteristics, and the particle-phase viscosity on the velocity, volumetric flow rates, and skin friction coefficients of both the fluid and particle phases is reported.

Steady MHD Flow of a Dusty Incompressible Non-Newtonian Oldroyd 8-Constant Fluid in a Circular Pipe

The steady magneto-hydrodynamic flow of a dusty incompressible electrically conducting and non- Newtonian Oldroyd 8-constant fluid through a circular pipe is examined considering the Hall effect. A constant pressure gradient is imposed in the axial direction of the pipe while an external uniform magnetic field is applied in the perpendicular direction. A numerical solution for the governing nonlinear momentum equations is obtained using the method of finite differences. The effect of the Hall current, the non-Newtonian fluid characteristics and the particle-phase viscosity on the velocity, the volumetric flow rates, and the skin friction coefficients of both the fluid and the particle phases is investigated.

Comparative analysis of CFD models for jetting fluidized beds: Effect of particle-phase viscosity

Under the Eulerian–Eulerian framework of simulating gas–solid two-phase flow, the accuracy of the hydrodynamic prediction is strongly affected by the selection of rheology of the particulate phase, for which a detailed assessment is still absent. Using a jetting fluidized bed as an example, this work investigates the influence of solid rheology on the hydrodynamic behavior by employing different particle-phase viscosity models. Both constant particle-phase viscosity model (CVM) with different viscosity values and a simple two-fluid model without particle-phase viscosity (NVM) are incorporated into the classical two-fluid model and compared with the experimental measurements. Qualitative and quantitative results show that the jet penetration depth, jet frequency and averaged bed pressure drop are not a strong function of the particle-phase viscosity. Compared to CVM, the NVM exhibits better predictions on the jet behaviors, which is more suitable for investigating the hydrodynamics of gas–solid fluidized bed with a central jet.

Comparative analysis of CFD models for jetting fluidized beds: The effect of inter-phase drag force

In the Eulerian–Eulerian approach for simulating gas–solid two-phase flow, a proper selection of an interfacial force to describe the momentum transfer between the two phases has a significant influence on the particle dynamics. Based on a detailed review of the inter-phase drag force models, a simple two-fluid model with zero particle-phase viscosity is used to investigate the effects of inter-phase drag force on the jet characteristics in a two-dimensional jetting fluidized bed. The simulation results from different inter-phase drag force models are compared with a designed experimental study. Both qualitative and quantitative results show that the jet behavior, including the distribution of solid volume fraction, bed pressure drop, jet penetration depth and jet frequency, could be significantly affected by the selection of drag force models. None of the models assessed in this work could predict all these parameters accurately. Some models can predict the averaged value well but are weak in predicting the dynamics and vice versa, which calls for a caution in interpreting data based on a single drag model.

Constitutive Models for Filtered Two-Fluid Models of Fluidized Gas–Particle Flows

Our prior study [Igci et al., AIChE J., 2008, 54, 1431–1448] presented a methodology where computational results obtained through highly resolved simulations of a given microscopic two-fluid model (TFM) for gas-particle flows are filtered to deduce models for the residual correlations appearing in the corresponding filtered TFM equations that are appropriate for coarse-grid simulations of gas-particle flows. In the present study, we have performed a large number of highly resolved simulations of a kinetic theory based TFM in two- and three-dimensional periodic domains using this methodology and filtered the results to generate computational data on the filtered drag coefficient, the filtered particle phase pressure, and the filtered particle phase viscosity. These results have been captured in the form of correlations that can readily be employed in coarse-grid simulations of gas-particle flows.

Flow of gas and particles in a bubbling fluidized bed with a filtered two-fluid model

Numerical simulations of gas-particles flow in a bubble fluidized bed with two large eddy simulations of gas and solid phases are presented. For gas phase and solid phase, the sub-grid scale model for the viscosity is based on the Smagorinsky form. The sub-grid model for the particle pressure proposed by Igci et al. (2008) is modified by replacing the minimum fluidization velocity. The collisional interaction of particles is considered by the kinetic theory of granular flow. Flow behavior of gas and particles is performed by means of these two sub-grid scale models. The subgrid closure for the particle phase viscosity and pressure led to a qualitative change in the simulation results. Predictions are compared with experimental data measured by Yuu et al. (2000) and Taghipour et al. (2005) in the bubbling fluidized beds. The distributions of concentration and velocity of particles are predicted in the bubbling fluidized bed. The predicted filtered particle phase pressure increases and the filtered particle phase viscosity decreases with the increase of particle concentration. The qualitative importance of the model constant c s of particles is demonstrated.

Fluid dynamics simulation of the high shear mixing process

The Eulerian–Eulerian two-fluid approach for modelling multiphase flows is used to simulate the flow in a high shear mixer. The results are compared with experimental velocity profiles for the solids phase at the wall in the mixer obtained using a high speed camera (Darelius et al. Chem. Eng. Sci. 62 (2007) 2366). The governing equations are closed using relations from the Kinetic Theory of Granular Flow (KTGF) combined with a frictional stress model due to Johnson and Jackson and Schaeffer and inter-phase drag due to Wen and Yu. In addition, calculations are presented for a model with a constant particle phase viscosity (CPV). Free slip and partial slip boundary conditions for the solid phase velocity at the vessel wall and the impeller have been utilized. The results show that the bed height could be well predicted by the partial slip model, whereas the free slip model could not capture the experimentally found bed height satisfactorily. For the KTGF model, the swirling motion of the rotating torus that is formed by the moving powder bed was over-predicted and the tangential wall velocity was under-predicted, probably due to the fact that the frictional stress model needs to be further developed, e.g. to tackle cohesive particles in dense flow. The CPV model gave predictions in good agreement with the experiments for a solids viscosity of 0.1 Pa s and a wall slip parameter of 0.005 m/Pa s. However, for a very low or very high value of the particle phase viscosity and for a high value of the wall slip parameter the agreement with experiments was poor. Interestingly, values of the viscosity that are commonly employed for fluidized beds seem applicable also in the present case.

Unsteady flow of a dusty conducting non-Newtonian fluid through a pipe

In this paper, the unsteady flow of a dusty viscous incompressible electrically conducting non-Newtonian power-law fluid through a circular pipe is investigated. A constant pressure gradient in the axial direction and a uniform magnetic field directed perpendicular to the flow direction are applied. The particle phase is assumed to behave as a viscous fluid. A numerical solution is obtained for the governing nonlinear momentum equations using finite differences. The effects of the magnetic-field parameter Ha, the non-Newtonian fluid characteristics (the flow index n), and the particle-phase viscosity β on the transient behavior of the velocity, volumetric flow rates, and skin friction coefficients of both fluid and particle phases are studied. It is found that all the flow parameters for both phases decrease as the magnetic field increases or the flow index decreases. On the other hand, increasing the particle-phase viscosity increases the skin friction of the particle phase, but decreases the other flow parameters. PACS No.: 47.50.+d

Effect of Combined Particle-Phase Diffusivity and Viscosity on the Compressible Boundary Layer of a Particulate Suspension Over a Flat Plate

A mathematical dilute fluid-particle suspension model governing steady, laminar, compressible, boundary layer flow and heat transfer over a semi-infinite flat plate based on the Eulerian or continuum approach is developed. The model accounts for both particulate viscous and diffusive effects. Both the fluid and the particle phases are assumed to have general power-law viscosity-temperature relations. For the case of finite particle-phase viscosity, a general boundary condition borrowed from rarefied gas dynamics is used for the particle phase at the surface. Uniform and nonuniform particle-phase slip coefficients are investigated. Numerical solution of the governing equations is obtained by an implicit, iterative, tridiagonal finite difference method. Graphical results for the displacement thicknesses and skin-friction coefficients of both phases as well as the wall heat transfer are presented for various parametric conditions.

Fully developed travelling wave solutions and bubble formation in fluidized beds

It is well known that most gas fluidized beds of particles bubble, while most liquid fluidized beds do not. It was shown by Anderson, Sundaresan & Jackson (1995), through direct numerical integration of the volume-averaged equations of motion for the fluid and particles, that this distinction is indeed accounted for by these equations, coupled with simple, physically credible closure relations for the stresses and interphase drag. The aim of the present study is to investigate how the model equations afford this distinction and deduce an approximate criterion for separating bubbling and non-bubbling systems. To this end, we have computed, making use of numerical continuation techniques as well as bifurcation theory, the one- and two-dimensional travelling wave solutions of the volume-averaged equations for a wide range of parameter values, and examined the evolution of these travelling wave solutions through direct numerical integration. It is demonstrated that whether bubbles form or not is dictated by the value of Ω = (ρsv3t/Ag) 1/2, where ρs is the density of particles, vt is the terminal settling velocity of an isolated particle, g is acceleration due to gravity and A is a measure of the particle phase viscosity. When Ω is large (> ∼ 30), bubbles develop easily. It is then suggested that a natural scale for A is ρsvtdp so that Ω2 is simply a Froude number.

The flow induced by a rotating disk in a particulate suspension

A typical finite volume fraction two phase flow model is used to investigate the influence of assumptions about the nature of particle phase viscosity and the structure of particle phase boundary conditions at a solid surface on solutions to the Von Karman problem for a particle/fluid suspension. Solutions are obtained numerically by an iterative finite difference method. Comparisons of these solutions with one another and with previously published work reveal that both the particle phase viscosity and solid surface boundary conditions influence the nature of predictions in important ways. Selected numerical results are presented graphically to illustrate these conclusions

The propagation of a voidage disturbance in a uniformly fluidized bed

By considering the evolution of a localized voidage disturbance imposed on an otherwise uniformly fluidized bed we are able to determine the dominant effects of the many terms in the continuum equations of motion governing a fluidized bed. For small perturbations a linearized theory is developed, showing that the stability of the uniform state is critically dependent upon the particle-phase collisional pressure and the flow rate of the uniform state, while the effect of particle phase viscosity is shown to be purely dispersive. When the uniform state is stable, the disturbance is shown to develop into a decaying pulse followed by a decaying wavetrain.For finite-amplitude disturbance, nonlinear effects are considered. These are shown to give rise to the propagation of high voidage gradients through the bed. Having established that such voidage fronts will develop, a detailed study of their structure is made. This gives strong indications that, for flow rates at which the uniform state is unstable, the bed will restabilize into a quasisteady periodic state.

On plane, laminar two-phase jet flow

A plane, laminar two-phase jet flow is considered, where the two-phase medium in the jet consists of small spherical solid particles suspended in an incompressible carrier fluid The development of the jet flow is studied in the presence of a moving free stream, containing no particles and alined with the jet. Diffusion of particles as well as the effect of carrier- and particle-phase viscosity is taken into account. A perturbation analysis is performed based on small particle concentration and small velocity differences between jet- and free-stream speed. Accordingly, the derived linear equations are solved byLaplace transforms. Resutiing expressions for particle velocity profiles contain two parameters. One of these is the ratio of the carrierReynolds number and the particle velocity relaxation parameter, and the other is the ratio of the particle diffusion coefficient and the carrier fluid viscosity. It is shown that both parameters can have a profound influence on the solution.