A modified drag model for power-law fluid-particle flow used in computational fluid dynamics simulation

A modified drag model for the power-law fluid-particle flow considering effects of rheological properties was proposed. At high particle concentrations (ε_s ≥ 0.2), based on the Ergun equation, the cross-sectional shape and the tortuosity of the pore channel are considered, and the apparent flow behavior index and consistency coefficient of the power-law fluid at the surface of the particles are corrected. At low particle concentrations (ε_s < 0.2), based on the Wen-Yu drag model, the modified Reynolds number for power-law fluid and the relational expression between drag coefficient for single particle and Reynolds number that considers the effect of the flow behavior index are adopted. Numerical simulations for the power-law fluid-particle flow in the fluidized bed were carried out using the non-Newtonian drag model. The effects of rheological parameters on the drag coefficient were analyzed. The comparisons of simulation and experiment show that the modified drag model predicts reasonable void fraction under different rheological parameters, particle diameters, and liquid velocities in both low particle concentrations and high particle concentrations. The increase in flow behavior index and consistency coefficient increases the drag coefficient between the two phases and decreases the average particle concentration within the bed.     

Laminar flow of a non-Newtonian fluid in channels with wall suction or injection

The one-dimensional approximate equation in the rectangular Cartesian coordinates governing flow of a non-Newtonian fluid confined in two large plates separated by a small distance of h, with the upper plate stationary while the lower plate is uniformly porous and moving in the x-direction with constant velocity, is derived by accounting for the order of magnitude of terms as well as the accompanying approximations to the full-blown three-dimensional equations by using scaling arguments, asymptotic techniques and assuming the cross-flow velocity is much less than the axial velocity. The one-dimensional governing equation for a power-law fluid flow confined between parallel plates, with the upper plate is stationary and the bottom plate subjected to sudden acceleration with a constant velocity in the x-direction and uniformly porous, is solved analytically for a Newtonian fluid case (n = 1) and numerically for various values of power-law index to determine the transient velocity and thus the overall transient velocity distribution. The effects of mass suction/injection at the porous bottom plate on the flow of non-Newtonian fluids are examined for various values of time and power-law index. The results obtained from the present analysis are compared with the data available in the literature.     

Large eddy simulation of particle deposition and resuspension in turbulent duct flows

Particle deposition and resuspension in a horizontal, fully developed turbulent square duct flow at four flow bulk Reynolds numbers (10,320, 83k, 215k and 250k) is simulated by applying large eddy simulation coupled with Lagrangian particle tracking technique. Forces acting on particles includes drag, lift, buoyancy and gravity. Four particle sizes are considered with the diameters of 5?μm, 50?μm, 100?μm and 500?μm. Results obtained for the fluid phase are in good agreement with the available experimental and numerical data. Predictions for particles show that particle size, flow Reynolds number and the duct (celling, floor and vertical) walls play important roles in near-wall particle deposition and resuspension. For the smallest particle (5?μm), the particle deposition rates in duct ceiling, floor and vertical walls are found to be similar with each other and all increase with the flow Reynolds number, while the particle resuspension tends to occur in the middle wall regions and corners of the duct with less influenced by the flow Reynolds number. The ceiling deposition rate gradually decreases with particle size while the floor and vertical wall deposition rates both increase with particle size. The ceiling particle deposition rate increases with Reynolds number while the floor deposition rate decreases with it. The vertical deposition rate for the small particles (5–50?μm) increases with the flow Reynolds number obviously, while for the large particles (100–500?μm) that becomes insensitive. In addition, the flow Reynolds number is found to have an obvious effect on particle resuspension while the effect of particle size on particle resuspension decreases with Reynolds number. Eventually, a dynamic analysis was conducted for particles deposition and resuspension in turbulent duct flows.     

Blasius and Sakiadis problems in nanofluids

The classical problems of forced convection boundary layer flow and heat transfer past a semi-infinite static flat plate (Blasius problem) and past a moving semi-infinite flat plate (Sakiadis problem) using nanofluids are theoretically studied. The similarity equations are solved numerically for three types of metallic or nonmetallic nanoparticles such as copper (Cu), alumina (Al₂O₃), and titania (TiO₂) in the base fluid of water with the Prandtl number Pr = 6.2 to investigate the effect of the solid volume fraction parameter ?? of the nanofluids. Also, the case of conventional or regular fluid (?? = 0) with Pr = 0.7 is considered for comparison with known results from the open literature. The comparison shows excellent agreement. The skin friction coefficient, Nusselt number, and the velocity and temperature profiles are presented and discussed in detail. It is found that the solid volume fraction affects the fluid flow and heat transfer characteristics.     

Unsteady oscillatory stagnation-point flow of a viscoelastic fluid

The unsteady stagnation point flow of the Walters B^′ fluid is examined and solutions are obtained. It is assumed that the infinite plate at y=0 is oscillating and the fluid impinges obliquely on the plate.     

See-saw rocking: an in vitro model for mechanotransduction research

In vitro mechanotransduction studies, uncovering the basic science of the response of cells to mechanical forces, are essential for progress in tissue engineering and its clinical application. Many varying investigations have described a multitude of cell responses; however, as the precise nature and magnitude of the stresses applied are infrequently reported and rarely validated, the experiments are often not comparable, limiting research progress. This paper provides physical and biological validation of a widely available fluid stimulation device, a see-saw rocker, as an in vitro model for cyclic fluid shear stress mechanotransduction. This allows linkage between precisely characterized stimuli and cell monolayer response in a convenient six-well plate format. Models of one well were discretized and analysed extensively using computational fluid dynamics to generate convergent, stable and consistent predictions of the cyclic fluid velocity vectors at a rocking frequency of 0.5 Hz, accounting for the free surface. Validation was provided by comparison with flow velocities measured experimentally using particle image velocimetry. Qualitative flow behaviour was matched and quantitative analysis showed agreement at representative locations and time points. Maximum shear stress of 0.22 Pa was estimated near the well edge, and time-average shear stress ranged between 0.029 and 0.068 Pa. Human tenocytes stimulated using the system showed significant increases in collagen and GAG secretion at 2 and 7 day time points. This in vitro model for mechanotransduction provides a versatile, flexible and inexpensive method for the fluid shear stress impact on biological cells to be studied.     

Multi-scale study of particle flow in silos

The flow characteristics of solid particles in a silo were studied experimentally and theoretically. A multi-scale study of the particles flow was performed by means of discrete element method (DEM). The dependence of flow behaviors on the particles diameter distribution and silo geometry was analyzed to establish the spatial and statistical distributions of microdynamic variables related to flow and silo structures such as velocity, porosity, coordination number, and interaction forces between particles. The results show that the distribution of particle diameter has great effects on particles flow, and the mixing of multi-sized particles is propitious to granular flow. The geometry of silos has greater effects on granular flow than particle size distribution, and inserts can improve the flow behaviors of “funnel flow” type to “mass flow”. Linear equations can be used to describe the relationship between discharge rate and orifice size by G^2/5 vs. D_o for the same distribution of particles diameter. The flow structure of particles in the silos is spatially non-uniform, which is illustrated by spatial and statistical distributions of porosity and coordination number. Both porosity and coordination number are affected by the mode of particles packed, which is affected by the geometry of silos and particle size distribution. The distribution of contact forces between particles is spatially non-uniform too. In flat-bottomed silo, there are arched stress chains in the vicinity of the orifice under the “bridging action”, which disappeared in wedge-shaped hopper silo.     

Non-orthogonal stagnation-point flow of a micropolar fluid

This paper considers the problem of steady two-dimensional flow of a micropolar fluid impinging obliquely on a flat plate. The flow under consideration is a generalization of the classical modified Hiemenz flow for a micropolar fluid which occurs in the boundary layer near an orthogonal stagnation point. A coordinate decomposition transforms the full governing equations into a primary equation describing the modified Hiemenz flow for a micropolar fluid and an equation for the tangential flow coupled to the primary solution. The solution to the boundary-value problem is governed by two non-dimensional parameters: the material parameter K and the ratio of the microrotation to skin friction parameter n. The obtained ordinary differential equations are solved numerically for some values of the governing parameters. The primary consequence of the free stream obliqueness is the shift of the stagnation point toward the incoming flow.     

Separation characteristics of fluid flow inside two parallel plates with wavy surface

Separation characteristics of fluid flow inside two parallel wavy plates for steady-laminar flow is investigated numerically in the present study. Governing equations are discretized using control volume based finite-volume method with collocated variable arrangement. SIMPLE algorithm is used and SIP solver is applied for solution of system of equations. Effect of surface waviness (defined by amplitude to average interwall spacing ratio, a/H) and aspect ratio (defined by wavelength to average interwall spacing ratio, w/H) on separation characteristics of fluid flow is presented. The present work has been carried out for surface waviness a/H=0-0.3, aspect ratio w/H=1.5-2.25. A critical Reynolds number (Re_c) is used to identify the appearance of first separation of fluid flow in the channel. Critical Reynolds (Re_c) number is calculated for wide range of surface waviness and aspect ratio. The structure of separation bubble depends strongly on waviness of the surface and aspect ratio for a particular Reynolds number and changes little with wave number (n). Finally pressure drop characteristics is presented in terms of average friction factor as a function of Reynolds number.     

Stability of a laminar premixed supersonic free shear layer with chemical reactions

We discuss the stability of a 2-dimensional compressible supersonic flow in the wake of a flat plate. The fluid is a multi-species mixture which is undergoing finite rate chemical reactions. We consider the spatial stability of an infinitesimal disturbance in the fluid. Numerical solutions of the eigenvalue stability equations for both reactive and non-reactive supersonic flows are presented and discussed. The chemical reactions have significant influence on the stability behavior. For instance, a neutral eigenvalue is observed near the freestream Mach Number M_∞ ～- 2.375 for non-reactive case, but disappears when the reaction is turned on. For reactive flows, the eigenvalues are not very dependant on the free stream Mach number. The disturbance amplification rate is higher and the wave speed, c_r is lower for most of the reactive cases when compared with the non-reactive cases.     

��Note on ��Flows induced by a plate moving normal to stagnation-point flow�� by P. D. Weidman and M. A. Sprague (Acta Mech. 219, 219�C229, 2011)��

In a recent paper of Weidman and Sprague (Acta Mech., 2011), the unsteady flows generated by an impermeable infinite flat plate advancing with constant velocity V toward, or receding from an orthogonal (plane or axisymmetric) stagnation-point flow, have been investigated by an exact similarity reduction of the Navier?CStokes equations. It has been shown that in the co-moving reference frame of the plate, the induced flow appears as a steady flow, with an additional term R f??? in the governing equation of the similar stream function f (??). The Reynolds number R involved in this additional term is proportional to the plate velocity V. The present paper shows, however, that with the aid of a simple transformation, the additional term R f??? can be removed from the governing equation, its effect being transferred in the boundary condition for f (??). As a consequence, the unsteady flow problems of Weidman and Sprague reduce to the classical steady stagnation-point flow problems for permeable surfaces with a uniform lateral suction or injection of the fluid, so that the transpiration parameter f (0) coincides with R for the plane and with R/2 for the axisymmetric flow, respectively. The main benefit of this approach is that all the results of the latter well-investigated problems can simply be transcribed for the problems formulated by Weidman and Sprague (Acta Mech, 2011).     

Eulerian–Lagrangian method for three-dimensional simulation of fluidized bed coal gasification

A comprehensive three-dimensional numerical model has been developed to simulate the coal gasification in a fluidized bed gasifier. The methodology is based on the multiphase particle-in-cell (MP-PIC) model, which uses an Eulerian method for fluid phase and a discrete particle method for particle phase. Dense particulate flow, mass and heat transfer, homogeneous and heterogeneous chemistry between phases and within the fluid mixture are considered. The dynamics of the particle phase is calculated by solving a transport equation for the particle distribution function (PDF) f. Particle collisions and chemical reactions are solved on a grid cell with particle properties mapped from discrete particles to the grid. Solid mass consumed or produced in reactions changes the size of particles. Simulations were carried out in a coal gasifier with a height of 2.0 m and a diameter of 0.22 m at atmosphere. The calculated product gas compositions compare well with the experimental data. The formation of flow patterns, profiles of particle species and gas compositions, distributions of reaction rates and consumption of carbon mass were investigated under different operating conditions.     

Hydrodynamic braking

A viscous fluid lies between two parallel plates. The bottom plate, moving laterally in its own plane, is affected by increasing viscous resistance when the top plate squeezes downward. The problem is solved by a power series expansion in a squeeze number S. Braking characteristics are obtained for several different states: the top plate moves with constant velocity, constant force, or constant power; the bottom moves with constant velocity or when it is free.     

Probing fluid flow using the force measurement capability of optical trapping

Interest in microfluidics is rapidly expanding and the use of microchips as miniature chemical reactors is increasingly common. Microfluidic channels are now complex and combine several functions on a single chip. Fluid flow details are important but relatively few experimental methods are available to probe the flow in confined geometry. We use optical trapping of a small dielectric particle to probe the fluid flow. A highly focused laser beam attracts particles suspended in a liquid to its focal point. A particle can be trapped and then repositioned. From the displacement of the trapped particle away from its equilibrium position one estimates the external force acting on the particle. The stiffness (spring constant) of the optical trap is low thus making it a sensitive force measuring device. Rather than using the optical trap to position and release a particle for independent velocimetry measurement, we map the fluid flow by measuring the hydrodynamic force acting on a trapped particle. The flow rate of a dilute aqueous electrolyte flowing through a plastic microchannel (W × H × L = 5 mm × 0.4 mm × 50 mm) was mapped using a small silica particle (1 μm diameter). The fluid velocity profile obtained experimentally is in very good agreement with the theoretical prediction. Our flow mapping approach is time efficient, reliable and can be used in low-opacity suspensions flowing in microchannels of various geometries.     

Similarity solutions for laminar free convection flow of a thermomicropolar fluid past a non-isothermal vertical flat plate

Laminar free convection boundary layer flow of a thermomicropolar fluid past a non-isothermal vertical flat plate has been studied in detail. It has been established that the flow problem has similarity solutions when the variation in the temperature of the plate is a linear function of the distance from the leading edge measured along the plate. The resulting system of the nonlinear ordinary differential equations has been solved numerically by “Shooting Method” for various values of the material parameters. The effects of these parameters has been studied on the velocity and microrotation fields graphically. Also “Tables” have been given for the values of temperature, skin-friction parameter, microrotation gradient on the wall and Nusselt number. Two types of boundary conditions are prescribed for the microrotation on the wall.     

Thermal radiation effects on magnetohydrodynamic mixed convection flow of a micropolar fluid past a continuously moving semi-infinite plate for high temperature differences

Summary An analysis is presented to study the effect of radiation on magnetohydrodynamic mixed convective steady laminar boundary layer flow of an optically thick electrically conducting viscous micropolar fluid past a moving semi-infinite vertical plate for high temperature differences. A uniform magnetic field is applied perpendicular to the moving plate. The density of the micropolar fluid is assumed to reduce exponentially with temperature. The usual Boussinesq approximation is neglected because of the high temperature differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting governing equations are transformed using a similarity transformation and then solved numerically by applying an efficient technique. The effects of radiation parameter R, magnetic parameter M, couple parameter Δ and density/temperature parameter n on the velocity, angular velocity and temperature profiles as well as the local skin friction coefficient, wall couple stress and the local Nusselt number are presented graphically and in tabular form.     

Numerical simulation of solid-liquid mixing characteristics in a stirred tank with fractal impellers

The hydrodynamics of solid-liquid mixing process in a stirred tank with four pitched-blade impellers, fractal 1 impellers, and fractal 2 impellers were investigated using computational fluid dynamics (CFD) simulation. An Eulerian-Eulerian approach, standard k-ε turbulence model, and multiple reference frames (MRF) technique were employed to simulate the solid-liquid two-phase flow, turbulent flow, and impeller rotation, respectively. The effects of impeller speed, impeller type, impeller spacing, impeller blade tilt angle, impeller blade shape, solid particle size and initial solid particle loading on the solid particle suspension quality were investigated. Results showed that the homogenous degree of solid-liquid system increased with the increase of impeller speed. The impeller spacing of T5/6 and T and impeller blade tilt angle of 60° and 45° were appropriate for the solid-liquid suspension process. Fractal shape impeller was more efficient than jagged shape impeller in solid-liquid mixing process. Larger particle diameter and higher initial solid particle loading resulted in less homogenous distribution of solid particles. It was found that fractal impeller could improve the solid particle suspension quality compared with four pitched-blade impeller under the same power consumption, increasingly so with the fractal iteration number of fractal impeller. Moreover, fractal impeller reduced the size of impeller trailing vortex and consumed less power consumption compared with four pitched-blade impeller at the same impeller speed, and the more the number of fractal iteration, the higher the impeller energy utilization rate of fractal impeller.     

On exact solutions of flow problems of a second grade fluid through two parallel porous walls

Exact analytical solutions of two problems of laminar flow of a second grade fluid through two parallel porous walls are obtained. For each problem the rate of injection at one wall is assumed equal to the rate of suction at the other wall. Two geometries are considered: rectangular - when the flow takes place between two parallel flat walls, and cylindrical - when the flow takes place through an annulus. For the exact solution no assumption is made on the size of K, the viscoelastic fluid parameter, or the cross-flow Reynolds number. However, it is assumed that a Taylor series expansion of the solution exists near K=0. Also assuming K small, perturbation solutions are developed for both the geometries. The exact solutions reveal that the viscoelasticity of the fluid tends to destroy the formation of the boundary layer at the wall where the suction takes place for large values of the cross-flow Reynolds number. It is further shown that the commonly used perturbation technique does not give satisfactory results even for moderate values of the cross-flow Reynolds number.     

Flows induced by a plate moving normal to stagnation-point flow

The flow generated by an infinite flat plate advancing toward or receding from a normal stagnation-point flow is obtained as an exact reduction of the Navier?CStokes equations for the case when the plate moves at constant velocity V. Both Hiemenz (planar) and Homann (axisymmetric) stagnation flows are considered. In each case, the problem is governed by a Reynolds number R proportional to V. Small and large R behaviors of the shear stress parameters are found for both advancing and receding plates. Numerical solutions determined over an intermediate range of R accurately match onto the small and large R asymptotic behaviors. As a side note, we report an interesting exact solution for plates advancing toward or receding from an exact rotational stagnation-point flow discovered by Agrawal (1957).     

Effects of axial conduction in the fluid on cryogenic regenerator performance

Although axial conduction in the matrix has been recognized as a major source of irreversibility in cryogenic regenerators, axial conduction in the fluid phase has largely been neglected. However, in spite of the negligible intrinsic thermal conductivity of most gases the effective conductivity of the gaseous medium in a porous bed may be quite significant, due to eddy diffusion and the consequent mixing of sections of gas at different temperatures. The governing equations of a thermal regenerator have been written in terms of the reduced length, A, reduced period, II, and an axial conduction parameter, λ, which depends only on the void fraction and the bed length to particle diameter ratio for a flow Reynolds number Re > 2. Numerical solutions, using the finite difference technique developed by Willmott and co-workers, have been obtained for several values of the three parameters. It has been established that axial conduction in the fluid phase is important, particularly when the design reduced length Λ > 1 /λ.