Anomalous behavior of hydrodynamic modes in the two dimensional shear flow of a granular material

The growth rates of the hydrodynamic modes in the homogeneous sheared state of a granular material are determined by solving the Boltzmann equation. The steady velocity distribution is considered to be the product of the Maxwell Boltzmann distribution and a Hermite polynomial expansion in the velocity components; this form is inserted into them Boltzmann equation and solved to obtain the coeificients of the terms in the expansion. The solution is obtained using an expansion in the parameter epsilon =(1 - e)(1/2), and terms correct to epsilon(4) are retained to obtain an approximate solution; the error due to the neglect of higher terms is estimated at about 5% for e = 0.7. A small perturbation is placed on the distribution function in the form of a Hermite polynomial expansion for the velocity variations and a Fourier expansion in the spatial coordinates: this is inserted into the Boltzmann equation and the growth rate of the Fourier modes is determined. It is found that in the hydrodynamic limit, the growth rates of the hydrodynamic modes in the flow direction have unusual characteristics. The growth rate of the momentum diffusion mode is positive, indicating that density variations are unstable in the limit k--> 0, and the growth rate increases proportional to kslash} k kslash}(2/3) in the limit k --> 0 (in contrast to the k(2) increase in elastic systems), where k is the wave vector in the flow direction. The real and imaginary parts of the growth rate corresponding to the propagating also increase proportional to kslash k kslash(2/3) (in contrast to the k(2) and k increase in elastic systems). The energy mode is damped due to inelastic collisions between particles. The scaling of the growth rates of the hydrodynamic modes with the wave vector I in the gradient direction is similar to that in elastic systems. (C) 2000 Elsevier Science B.V. All rights reserved.

Freezing of a colloidal liquid subject to shear flow

A nonequilibrium generalization of the density-functional theory of freezing is proposed to investigate the shear-induced first-order phase transition in colloidal suspensions. It is assumed that the main effect of a steady shear is to break the symmetry of the structure factor of the liquid and that for small shear rate, the phenomenon of a shear-induced order-disorder transition may be viewed as an equilibrium phase transition. The theory predicts that the effective density at which freezing takes place increases with shear rate. The solid (which is assumed to be a bcc lattice) formed upon freezing is distorted and specifically there is less order in one plane compared with the order in the other two perpendicular planes. It is shown that there exists a critical shear rate above which the colloidal liquid does not undergo a transition to an ordered (or partially ordered) state no matter how large the density is. Conversely, above the critical shear rate an initially formed bcc solid always melts into an amorphous or liquidlike state. Several of these predictions are in qualitative agreement with the light-scattering experiments of Ackerson and Clark. The limitations as well as possible extensions of the theory are also discussed.

Compressible Boundary-Layer Swirling Flow in a Nozzle and a Diffuser With Highly Cooled Wall

The steady laminar compressible boundary-layer swirling flow with variable gas properties and mass transfer through a conical nozzle, and a diffuser with a highly cooled wall has been studied. The partial differential equations governing the nonsimilar flow have been transformed to a system of coordinates using modified Lees transformation. The resulting equations are transformed into coordinates having finite ranges by means of a transformation which maps an infinite region into a finite region. The ensuing equations are then solved numerically using an implicit finite-difference scheme. The results indicate that the variation of the density-viscosity product across the boundary layer and mass transfer have strong effect on the skin friction and heat transfer. Separationless flow along the entire length of the diffuser can be obtained by applying suction. The results are found to be in good agreement with those of the local nonsimilarity method but they differ appreciably from those of the local similarity method.

Flow and heat transfer in steady laminar compressible boundary layer swirling flow in a nozzle

The axisymmetric steady laminar compressible boundary layer swirling flow of a gas with variable properties in a nozzle has been investigated. The partial differential equations governing the non-similar flow have been transformed into new co-ordinates having finite ranges by means of a transformation which maps an infinite range into a finite one. The resulting equations have been solved numerically using an implicit finite-difference scheme. The computations have been carried out for compressible swirling flow through a convergent conical nozzle. The results indicate that the swirl exerts a strong influence on the longitudinal skin friction, but its effect on the tangential skin friction and heat transfer is comparatively small. The effect of the variation of the density-viscosity product across the boundary layer is appreciable only at low-wall temperature. The results are in good agreement with those of the local-similarity method for small values of the longitudinal distance.

Steady compressible flow of granular materials through a wedge-shaped hopper: The smooth wall, radial gravity problem

A continuum model based on the critical state theory of soil mechanics is used to generate stress and density profiles, and to compute discharge velocities for the plane flow of cohesionless materials. Two types of yield loci are employed, namely, a yield locus with a corner, and a smooth yield locus. The yield locus with a corner leads to computational difficulties. For the smooth yield locus, results are found to be relatively insensitive to the shape of the yield locus, the location of the upper traction-free surface and the density specified on this surface. This insensitivity arises from the existence of asymptotic stress and density fields, to which the solution tends to converge on moving down the hopper. Numerical and approximate analytical solutions are obtained for these fields and the latter is used to derive an expression for the discharge velocity. This relation predicts discharge velocities to within 13% of the exact (numerical) values. While the assumption of incompressibility has been frequently used in the literature, it is shown here that in some cases, this leads to discharge velocities which are significantly higher than those obtained by the incorporation of density variation.

Unsteady laminar compressible boundary-layer flow at a three-dimensional stagnation point

Unsteady laminar compressible boundary-layer flow with variable properties at a three-dimensional stagnation point for both cold and hot walls has been studied for the case when the velocity of the incident stream varies arbitrarily with time. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. Computations have been carried out for two particular unsteady free-stream velocity distributions: (i) an accelerating stream and (ii) a fluctuating stream. The results indicate that the variation of the density-viscosity product across the boundary layer, the wall temperature and the nature of stagnation point significantly affect the skin friction and heat transfer.

Stability of the viscous flow of a fluid through a flexible tube

The stability of the Hagen-Poiseuille flow of a Newtonian fluid in a tube of radius R surrounded by an incompressible viscoelastic medium of radius R < r < HR is analysed in the high Reynolds number regime. The dimensionless numbers that affect the fluid flow are the Reynolds number Re = (rho VR/eta), the ratio of the viscosities of the wall and fluid eta(r) = (eta(s)/eta), the ratio of radii H and the dimensionless velocity Gamma = (rho V-2/G)(1/2). Here rho is the density of the fluid, G is the coefficient of elasticity of the wall and V is the maximum fluid velocity at the centre of the tube. In the high Reynolds number regime, an asymptotic expansion in the small parameter epsilon = (1/Re) is employed. In the leading approximation, the viscous effects are neglected and there is a balance between the inertial stresses in the fluid and the elastic stresses in the medium. There are multiple solutions for the leading-order growth rate s((0)), all of which are imaginary, indicating that the fluctuations are neutrally stable, since there is no viscous dissipation of energy or transfer of energy from the mean flow to the fluctuations due to the Reynolds stress. There is an O(epsilon(1/2)) correction to the growth rate, s((1)), due to the presence of a wall layer of thickness epsilon(1/2)R where the viscous stresses are O(epsilon(1/2)) smaller than the inertial stresses. An energy balance analysis indicates that the transfer of energy from the mean flow to the fluctuations due to the Reynolds stress in the wall layer is exactly cancelled by an opposite transfer of equal magnitude due to the deformation work done at the interface, and there is no net transfer from the mean flow to the fluctuations. Consequently, the fluctuations are stabilized by the viscous dissipation in the wall layer, and the real part of s(1) is negative. However, there are certain values of Gamma and wavenumber k where s((1)) = 0. At these points, the wall layer amplitude becomes zero because the tangential velocity boundary condition is identically satisfied by the inviscid flow solution. The real part of the O(epsilon) correction to the growth rate s((2)) turns out to be negative at these points, indicating a small stabilizing effect due to the dissipation in the bulk of the fluid and the wall material. It is found that the minimum value of s((2)) increases proportional to (H-1)(-2) for (H-1) much less than 1 (thickness of wall much less than the tube radius), and decreases proportional to H-4 for H much greater than 1. The damping rate for the inviscid modes is smaller than that for the viscous wall and centre modes in a rigid tube, which have been determined previously using a singular perturbation analysis. Therefore, these are the most unstable modes in the flow through a flexible tube

Velocity distribution function for a dilute granular material in shear flow

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle-wall collisions is large compared to particle-particle collisions. An asymptotic analysis is used in the small parameter epsilon, which is naL in two dimensions and na(2)L in three dimensions, where; n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle-wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution e(t) and e(n) in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (u(x), u(y)) = (+/-V, O) after repeated collisions with the wall, where u(x) and u(y) are the velocities tangential and normal to the wall, V = (1 - e(t))V-w/(1 + e(t)), and V-w and -V-w, are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions :in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to epsilon(1/2) in the limit epsilon --> 0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to epsilon(1/2) in the limit epsilon --> 0. The moments of the velocity distribution function are evaluated, and it is found that [u(x)(2)] --> V-2, [u(y)(2)] similar to V-2 epsilon and -[u(x)u(y)] similar to V-2 epsilon log(epsilon(-1)) in the limit epsilon --> 0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.

Oscillatory flow and temperature fields in an open tube with temperature difference across the ends

The oscillating flow and temperature field in an open tube subjected to cryogenic temperature at the cold end and ambient temperature at the hot end is studied numerically. The flow is driven by a time-wise sinusoidally varying pressure at the cold end. The conjugate problem takes into account the interaction of oscillatory flow with the heat conduction in the tube wall. The full set of compressible flow equations with axisymmetry assumption are solved with a pressure correction algorithm. Parametric studies are conducted with frequencies of 5-15 Hz, with one end maintained at 100 K and other end at 300 K. The flow and temperature distributions and the cooldown characteristics are obtained. The frequency and pressure amplitude have negligible effect on the time averaged Nusselt number. Pressure amplitude is an important factor determining the enthalpy flow through the solid wall. The frequency of operation has considerable effect on penetration of temperature into the tube. The density variation has strong influence on property profiles during cooldown. The present study is expected to be of interest in applications such as pulse tube refrigerators and other cryocoolers, where oscillatory flows occur in open tubes. (C) 2011 Elsevier Ltd. All rights reserved.

Studies on the effects of test gas on the flow field around large angle blunt cone flying at hypersonic Mach number

The effect of the test gas on the flow field around a 120degrees apex angle blunt cone has been investigated in a shock tunnel at a nominal Mach number of 5.75. The shock standoff distance around the blunt cone was measured by an electrical discharge technique using both carbon dioxide and air as test gases. The forebody laminar convective heat transfer to the blunt cone was measured with platinum thin-film sensors in both air and carbon dioxide environments. An increase of 10 to 15% in the measured heat transfer values was observed with carbon dioxide as the test gas in comparison to air. The measured thickness of the shock layer along the stagnation streamline was 3.57 +/- 0.17 mm in air and 3.29 +/- 0.26 mm in carbon dioxide. The computed thickness of the shock layer for air and carbon dioxide were 3.98 mm and 3.02 mm, respectively. The observed increase in the measured heat transfer rates in carbon dioxide compared to air was due to the higher density ratio across the bow shock wave and the reduced shock layer thickness.

Stability of the flow of a fluid through a flexible tube at high Reynolds number

The stability of the Hagen-Poiseuille flow of a Newtonian fluid in a tube of radius R surrounded by an incompressible viscoelastic medium of radius R < r < HR is analysed in the high Reynolds number regime. The dimensionless numbers that affect the fluid flow are the Reynolds number Re = (ρVR / η), the ratio of the viscosities of the wall and fluid ηr = (ηs/η), the ratio of radii H and the dimensionless velocity Γ = (ρV2/G)1/2. Here ρ is the density of the fluid, G is the coefficient of elasticity of the wall and Vis the maximum fluid velocity at the centre of the tube. In the high Reynolds number regime, an asymptotic expansion in the small parameter ε = (1/Re) is employed. In the leading approximation, the viscous effects are neglected and there is a balance between the inertial stresses in the fluid and the elastic stresses in the medium. There are multiple solutions for the leading-order growth rate do), all of which are imaginary, indicating that the fluctuations are neutrally stable, since there is no viscous dissipation of energy or transfer of energy from the mean flow to the fluctruations due to the Reynolds strees. There is an O(ε1/2) correction to the growth rate, s(1), due to the presence of a wall layer of thickness ε1/2R where the viscous stresses are O(ε1/2) smaller than the inertial stresses. An energy balance analysis indicates that the transfer of energy from the mean flow to the fluctuations due to the Reynolds stress in the wall layer is exactly cancelled by an opposite transfer of equal magnitude due to the deformation work done at the interface, and there is no net transfer from the mean flow to the fluctuations. Consequently, the fluctuations are stabilized by the viscous dissipation in the wall layer, and the real part of s(1) is negative. However, there are certain values of Γ and wavenumber k where s(l) = 0. At these points, the wail layer amplitude becomes zero because the tangential velocity boundary condition is identically satisfied by the inviscid flow solution. The real part of the O(ε) correction to the growth rate s(2) turns out to be negative at these points, indicating a small stabilizing effect due to the dissipation in the bulk of the fluid and the wall material. It is found that the minimum value of s(2) increases [is proportional to] (H − 1)−2 for (H − 1) [double less-than sign] 1 (thickness of wall much less than the tube radius), and decreases [is proportional to] (H−4 for H [dbl greater-than sign] 1. The damping rate for the inviscid modes is smaller than that for the viscous wall and centre modes in a rigid tube, which have been determined previously using a singular perturbation analysis. Therefore, these are the most unstable modes in the flow through a flexible tube.

Steady compressible flow of cohesionless granular materials through a wedge-shaped bunker

A continuum model based on the critical-state theory of soil mechanics is used to generate stress, density, and velocity profiles, and to compute discharge rates for the flow of granular material in a mass flow bunker. The bin–hopper transition region is idealized as a shock across which all the variables change discontinuously. Comparison with the work of Michalowski (1987) shows that his experimentally determined rupture layer lies between his prediction and that of the present theory. However, it resembles the former more closely. The conventional condition involving a traction-free surface at the hopper exit is abandoned in favour of an exit shock below which the material falls vertically with zero frictional stress. The basic equations, which are not classifiable under any of the standard types, require excessive computational time. This problem is alleviated by the introduction of the Mohr–Coulomb approximation (MCA). The stress, density, and velocity profiles obtained by integration of the MCA converge to asymptotic fields on moving down the hopper. Expressions for these fields are derived by a perturbation method. Computational difficulties are encountered for bunkers with wall angles θw [gt-or-equal, slanted] 15° these are overcome by altering the initial conditions. Predicted discharge rates lie significantly below the measured values of Nguyen et al. (1980), ranging from 38% at θw = 15° to 59% at θw = 32°. The poor prediction appears to be largely due to the exit condition used here. Paradoxically, incompressible discharge rates lie closer to the measured values. An approximate semi-analytical expression for the discharge rate is obtained, which predicts values within 9% of the exact (numerical) ones in the compressible case, and 11% in the incompressible case. The approximate analysis also suggests that inclusion of density variation decreases the discharge rate. This is borne out by the exact (numerical) results – for the parameter values investigated, the compressible discharge rate is about 10% lower than the incompressible value. A preliminary comparison of the predicted density profiles with the measurements of Fickie et al. (1989) shows that the material within the hopper dilates more strongly than predicted. Surprisingly, just below the exit slot, there is good agreement between theory and experiment.

Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe

We report an experimental study of a new type of turbulent flow that is driven purely by buoyancy. The flow is due to an unstable density difference, created using brine and water, across the ends of a long (length/diameter=9) vertical pipe. The Schmidt number Sc is 670, and the Rayleigh number (Ra) based on the density gradient and diameter is about 108. Under these conditions the convection is turbulent, and the time-averaged velocity at any point is ‘zero’. The Reynolds number based on the Taylor microscale, Reλ, is about 65. The pipe is long enough for there to be an axially homogeneous region, with a linear density gradient, about 6–7 diameters long in the midlength of the pipe. In the absence of a mean flow and, therefore, mean shear, turbulence is sustained just by buoyancy. The flow can be thus considered to be an axially homogeneous turbulent natural convection driven by a constant (unstable) density gradient. We characterize the flow using flow visualization and particle image velocimetry (PIV). Measurements show that the mean velocities and the Reynolds shear stresses are zero across the cross-section; the root mean squared (r.m.s.) of the vertical velocity is larger than those of the lateral velocities (by about one and half times at the pipe axis). We identify some features of the turbulent flow using velocity correlation maps and the probability density functions of velocities and velocity differences. The flow away from the wall, affected mainly by buoyancy, consists of vertically moving fluid masses continually colliding and interacting, while the flow near the wall appears similar to that in wall-bound shear-free turbulence. The turbulence is anisotropic, with the anisotropy increasing to large values as the wall is approached. A mixing length model with the diameter of the pipe as the length scale predicts well the scalings for velocity fluctuations and the flux. This model implies that the Nusselt number would scale as Ra1/2Sc1/2, and the Reynolds number would scale as Ra1/2Sc−1/2. The velocity and the flux measurements appear to be consistent with the Ra1/2 scaling, although it must be pointed out that the Rayleigh number range was less than 10. The Schmidt number was not varied to check the Sc scaling. The fluxes and the Reynolds numbers obtained in the present configuration are much higher compared to what would be obtained in Rayleigh–Bénard (R–B) convection for similar density differences.

Effect of density variation and non-covalent functionalization on the compressive behavior of carbon nanotube arrays

Arrays of aligned carbon nanotubes (CNTs) have been proposed for different applications, including electrochemical energy storage and shock-absorbing materials. Understanding their mechanical response, in relation to their structural characteristics, is important for tailoring the synthesis method to the different operational conditions of the material. In this paper, we grow vertically aligned CNT arrays using a thermal chemical vapor deposition system, and we study the effects of precursor flow on the structural and mechanical properties of the CNT arrays. We show that the CNT growth process is inhomogeneous along the direction of the precursor flow, resulting in varying bulk density at different points on the growth substrate. We also study the effects of non-covalent functionalization of the CNTs after growth, using surfactant and nanoparticles, to vary the effective bulk density and structural arrangement of the arrays. We find that the stiffness and peak stress of the materials increase approximately linearly with increasing bulk density.

The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

The generalizations of the Onsager model for the radial boundary layer and the Carrier-Maslen model for the end-cap axial boundary layer in a high-speed rotating cylinder are formulated for studying the secondary gas flow due to wall heating and due to insertion of mass, momentum and energy into the cylinder. The generalizations have wider applicability than the original Onsager and Carrier-Maslen models, because they are not restricted to the limit A >> 1, though they are restricted to the limit R e >> 1 and a high-aspect-ratio cylinder whose length/diameter ratio is large. Here, the stratification parameter A = root m Omega(2)R(2)/2k(B)T). This parameter A is the ratio of the peripheral speed, Omega R, to the most probable molecular speed, root 2k(B)T/m, the Reynolds number Re = rho w Omega R(2)/mu, where m is the molecular mass, Omega and R are the rotational speed and radius of the cylinder, k(B) is the Boltzmann constant, T is the gas temperature, rho(w) is the gas density at wall, and mu is the gas viscosity. In the case of wall forcing, analytical solutions are obtained for the sixth-order generalized Onsager equations for the master potential, and for the fourth-order generalized Carrier-Maslen equation for the velocity potential. For the case of mass/momentum/energy insertion into the flow, the separation-of-variables procedure is used, and the appropriate homogeneous boundary conditions are specified so that the linear operators in the axial and radial directions are self-adjoint. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order and second-order in the radial and axial directions for the Onsager equation, and fourth-order and second-order in the axial and radial directions for the Carrier-Maslen equation) are determined. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be matched with care. The commonly used `diffuse reflection' boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a `temperature slip' (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analysis in order to make quantitative predictions. In the case of mass/momentum/energy sources within the flow, it is necessary to ensure that the homogeneous boundary conditions are accurately satisfied in the simulations. When these precautions are taken, there is excellent agreement between analysis and simulations, to within 10 %, even when the stratification parameter is as low as 0.707, the Reynolds number is as low as 100 and the aspect ratio (length/diameter) of the cylinder is as low as 2, and the secondary flow velocity is as high as 0.2 times the maximum base flow velocity. The predictions of the generalized models are also significantly better than those of the original Onsager and Carrier-Maslen models, which are restricted to thin boundary layers in the limit of high stratification parameter.