Laminar flow of power-law fluids past a rotating cylinder

In this work, the continuity and momentum equations have been solved numerically to investigate the flow of power-law fluids over a rotating cylinder. In particular, consideration has been given to the prediction of drag and lift coefficients as functions of the pertinent governing dimensionless parameters, namely, power-law index (1 ≥ n ≥ 0.2), dimensionless rotational velocity (0 ≤ α ≤ 6) and the Reynolds number (0.1 ≤ Re ≤ 40). Over the range of Reynolds number, the flow is known to be steady. Detailed streamline and vorticity contours adjacent to the rotating cylinder and surface pressure profiles provide further insights into the nature of flow. Finally, the paper is concluded by comparing the present numerical results with the scant experimental data on velocity profiles in the vicinity of a rotating cylinder available in the literature. The correspondence is seen to be excellent for Newtonian and inelastic fluids.     

Variational solutions for the two-stream mixing of power-law fluids

The variational principle of Lebon-Lambermont, originally proposed for Newtonian fluids, is seen to be applicable to generalized Newtonian fluids. As an example, it is applied to obtain approximate solutions of the laminar boundary-layer equations for the two-stream mixing of power-law fluids. The flow along a flat plate is obtained as a particular case when the consistency of one of the fluids diverges.     

Gravity-driven laminar film flow of power-law fluids along vertical walls

A theoretical analysis is presented which brings steady laminar film flow of power-law fluids within the framework of classical boundary layer theory. The upper part of the film, which consists of a developing viscous boundary layer and an external inviscid freestream, is treated separately from the viscous dominated part of the flow, thereby taking advantage of the distinguishing features of each flow region. It is demonstrated that the film boundary layer developing along a vertical wall can be described by a generalized Falkner-Skan type equation originally developed for wedge flow. An exact similarity solution for the velocity field in the film boundary layer is thus made available.Downstream of the boundary layer flow regime the fluid flow is completely dominated by the action of viscous shear, and fairly accurate solutions are obtained by the Von Karman integral method approach. A new form of the velocity profile is assumed, which reduces to the exact analytic solution for the fully-developed film. By matching the downstream integral method solution to the upstream generalized Falkner-Skan similarity solution, accurate estimates for the hydrodynamic entrance length are obtained. It is also shown that the flow development in the upstream region predicted by the approximate integral method closely corresponds to the exact similarity solution for that flow regime. An analytical solution of the resulting integral equation for the Newtonian case is compared with previously published results.     

Non-Newtonian fluids v frictional resistance of discs and cones rotating in power-law non-Newtonian fluids

Summary Relations have been derived for the frictional resistance of finite discs and cones rotating in Ostwald-de Waele (power-law) type non-Newtonian fluids. The obtained equations can be formulated as dimensionless relations between the dimensionless moment coefficient and the generalized Reynolds number; the flow-behaviour index n enters the equations as a parameter. The relations derived for cones contain the apex angle 2₀ as an additional parameter in the form of A=sin ₀. The validity of the theoretically derived relations has been verified by measurements of the torque of discs and cones for a number of pseudoplastic power-law fluids.Nomenclature A sin ₀ parameter - b exponent in regression equation (16) - C coefficient in regression equation (16) - c _Mi dimensionless moment coefficient, for bodies wetted on one side (i=1) and for completely wetted bodies (i=2), equations (8) and (9b) - d diameter of turntable - F, G velocity functions of exact solution, equation (4) - K consistency coefficient of non-Newtonian fluids - M _Ki torque of rotating bodies, i=1 for bodies wetted on one side, i=2 for completely wetted bodies - n flow-behaviour index of non-Newtonian fluids - N=K/ kinematic consistency coefficient - P tangential force - r(y) perpendicular distance of point on cone surface from axis - R radius of disc or of base of cone - modified Reynolds number defined by equation (14) - Re _ow generalized Reynolds number defined by equation (10) - S, S area - u, v components of velocity vector - x, y, z coordinates according to fig. 1 - ₀ half the apex angle of cone - coefficient of frictional resistance defined by equation (11) - thickness of boundary layer - independent variable in exact solution, defined by equation (5) - density of fluid - _zx, _zy tangential stresses - angular velocity of rotation Indices T theoretical value - E experimental value - 0 refers to surface of rotating body     

The effect of side walls on particles mixing in rotating drums

In this paper, we study the effects of the presence and shape of side walls and of the overall length of rotating cylindrical drums on the mixing of particles with differing sizes by application of the discrete element method (DEM). By varying the semi-axis of the spheroidally shaped side walls and the length of the overall drum, we observe the formation of circulation patterns near the side walls. Although there is a vast amount of literature studying mixing regimes in rotating drums, little is known about the effect of the side walls of the drum on particle mixing. The results of our study demonstrate that introducing curved side walls induces a strong circulation pattern near these side walls, but has, paradoxically, a negative impact on mixing and actually promotes segregation. The cause for this segregation is the difference in velocity of differently sized particles near the curved side walls. Large particles accumulate at the curved side walls, whereas small particles move away from the curved side walls. When the length of the drum is increased, the overall effect of the side walls is decreased, although it does remain observable, even in very large drums.     

MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

DEM investigation of mixing indices in a ribbon mixer

Mixing index is an important parameter to understand and assess the mixing state in various mixers including ribbon mixers, the typical food processing devices. Many mixing indices based on either sample variance methods or non-sample variance methods have been proposed and used in the past, however, they were not well compared in the literature to evaluate their accuracy of assessing the final mixing state. In this study, discrete element method (DEM) modelling is used to investigate and compare the accuracy of these mixing indices for mixing of uniform particles in a horizontal cylindrical ribbon mixer. The sample variance methods for mixing indices are first compared both at particle- and macro-scale levels. In addition, non-sample variance methods, namely entropy and non-sampling indices are compared against the results from the sample variance methods. The simulation results indicate that, among the indices considered in this study, Lacey index shows the most accurate results. The Lacey index is regarded to be the most suitable mixing index to evaluate the steady-state mixing state of the ribbon mixer in the real-time (or without stopping the impeller) at both the particle- and macro-scale levels. The study is useful for the selection of a proper mixing index for a specific mixture in a given mixer.     

DEM investigation of mixing indices in a ribbon mixer

Mixing index is an important parameter to understand and assess the mixing state in various mixers including ribbon mixers,the typical food processing devices.M...     

On sheet-driven motion of power-law fluids

A rigorous analysis of non-Newtonian boundary layer flow of power-law fluids over a stretching sheet is presented. First, a systematic framework for treatment of sheet velocities of the form U(x)=Cx^m is provided. By means of an exact similarity transformation, the non-linear boundary layer momentum equation transforms into an ordinary differential equation with m and the power-law index n as the only parameters. Earlier investigations of a continuously moving surface (m=0) and a linearly stretched sheet (m=1) are recovered as special cases.For the particular parameter value m=1, i.e. linear stretching, numerical solutions covering the parameter range 0.1n2.0 are presented. Particular attention is paid to the most shear-thinning fluids, which exhibit a challenging two-layer structure. Contrary to earlier observations which showed a monotonic decrease of the sheet velocity gradient -f^″(0) with n, the present results exhibit a local minimum of -f^″(0) close to n=1.77. Finally, a series expansion in (n-1) is proved to give good estimates of -f^″(0) both for shear-thinning and shear-thickening fluids.     

Drag forces of interacting spheres in power-law fluids

Drag forces of interacting particles suspended in power-law fluid flows were investigated in this study. The drag forces of interacting spheres were directly measured by using a micro-force measuring system. The tested particles include a pair of interacting spheres in tandem and individual spheres in a cubic matrix of multi-sphere in flows with the particle Reynolds number from 0.7 to 23. Aqueous carboxymethycellulose (CMC) solutions and glycerin solutions were used as the fluid media in which the interacting spheres were suspended. The range of power-law index varied from 0.6 to 1.0. In conjunction to the drag force measurements, the flow patterns and velocity fields of power-law flows over a pair of interacting spheres were also obtained from the laser assisted flow visualization and numerical simulation.

Both experimental and computational results suggest that, while the drag force of an isolated sphere depends on the power-index, the drag coefficient ratio of an interacting sphere is independent from the power-law index but strongly depends on the separation distance and the particle Reynolds number. Our study also shows that the drag force of a particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.     

Flow of power-law fluids over a moving wedge surface with wall mass injection

The boundary layer problem of a power-law fluid flow with fluid injection on a wedge whose surface is moving with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. The free stream velocity, the injection velocity at the surface, moving velocity of the wedge surface, the wedge angle and the power law index of non-Newtonian fluid are assumed variables. The fourth order Runge–Kutta method modified by Gill is used to solve the non-dimensional boundary layer equations for non-Newtonian flow field. Without fluid injection, for every angle of wedge β, a limiting value for velocity ratio λ _cr (velocity of the wedge surface/velocity of the uniform flow) is found for each power-law index n. The value of λ _cr increases with the increasing wedge angle β. The value of wedge angle also restricts the physical characteristics of the fluid to be used. The effects of the different parameters on velocity profile and on skin friction are studied and the drag reduction is discussed. In case of C = 2.5 and velocity ratio λ = 0.2 for wedge angle β = 0.5 with the fluid with power law-index n = 0.5, 48.8% drag reduction is obtained.     

Numerical analysis of turbulent flow in a rotating U-bend with roughened walls

A numerical analysis has been performed for a developing turbulent flow in a rotating U-bend of strong curvature with rib-roughened walls using an anisotropic turbulent model. In this calculation, an algebraic Reynolds stress model is used to precisely predict Reynolds stresses, and a boundary-fitted coordinate system is introduced as a method of coordinate transformation to set the exact boundary conditions along the complicated shape of U-bend with rib-roughened walls. Calculated results for mean velocity and Reynolds stresses are compared to the experimental data in order to validate the proposed numerical method and the algebraic Reynolds stress model. Although agreement is certainly not perfect in all details, the present method can predict characteristic velocity profiles and reproduce the separated flow generated near the outer wall, which is located just downstream of the curved duct. The Reynolds stresses predicted by the proposed turbulent model agree well with the experimental data, except in regions of flow separation.     

Wall effects on the transportation of a cylindrical particle in power-law fluids

The present work deals with the numerical calculation of the Stokes-type drag undergone by a cylindrical particle perpendicularly to its axis in a power-law fluid. In unbounded medium, as all data are not available yet, we provide a numerical solution for the pseudoplastic fluid. Indeed, the Stokes-type solution exists because the Stokes’ paradox does not take place anymore. We show a high sensitivity of the solution to the confinement, and the appearance of the inertia in the proximity of the Newtonian case, where the Stokes’ paradox takes place. For unbounded medium, avoiding these traps, we show that the drag is zero for Newtonian and dilatant fluids. But in the bounded one, the Stokes-type regime is recovered for Newtonian and dilatant fluids. We give also a physical explanation of this effect which is due to the reduction of the hydrodynamic screen length, for pseudoplastic fluids. Once the solution of the unbounded medium has been obtained, we give a solution for the confined medium numerically and asymptotically. We also highlight the consequence of the confinement and the backflow on the settling velocity of a fiber perpendicularly to its axis in a slit. Using the dynamic mesh technique, we give the actual transportation velocity in a power-law “Poiseuille flow”, versus the confinement parameter and the fluidity index, induced by the hydrodynamic interactions.