Cylindrical Couette Flow Research Articles

A differential velocities-based study of diffusion effects in shear banding micellar solutions

By using the generalized bracket approach of non-equilibrium thermodynamics, we recently developed a multi-fluid approach to account for the Fickian diffusion and stress-induced migration in wormlike micellar solutions. In this work, we propose a modification to this framework where differential velocities are not treated as intermediate variables anymore. As the differential velocities are now used as state variables, additional boundary conditions arising from the presence of the diffusive derivative terms can be directly imposed with respect to these variables. The advantage of this modification is that no microstructural information is required for the specification of the boundary conditions. Sample calculations on this new model performed for transient cylindrical Couette flow revealed that the diffusional processes greatly affect the temporal evolution of the rheology and microstructure of the wormlike micelles. In the nonlinear viscoelastic regime, the stress-induced migration dominates over the Fickian diffusion. The smaller the diffusivity constant, the more time is required to reach steady state. Furthermore, the steady-state solution was found to be independent of the (nonzero) value of the diffusivity constant. To smooth the kink between the shear bands and to obtain a unique solution, we believe that nonlocal and surface effects would have to be accounted for. The proposed multi-fluid description of diffusion is general and could be applied to other complex fluids such as polymer solutions.

Frequency staircases in narrow-gap spherical Couette flow

Recent studies of plane parallel flows have emphasised the importance of finite-amplitude self-sustaining processes for the existence of alternative non-trivial solutions. The idea behind these mechanisms is that the motion is composed of distinct structures that interact to self-sustain. These solutions are not unique and their totality form a skeleton about which the actual realised motion is attracted. Related features can be found in spherical Couette flow between two rotating spheres in the limit of narrow-gap width. At lowest order the onset of instability is manifested by Taylor vortices localised in the vicinity of the equator. By approximating the spheres by their tangent cylinders at the equator, a critical Taylor number based on the ensuing cylindrical Couette flow problem would appear to provide a lowest order approximation to the true critical Taylor number. At next order, the latitudinal modulation of their amplitude a satisfies the complex Ginzburg-Landau equation (CGLe)where is latitude scaled on the modulation length scale, t is time and is proportional to the excess Taylor number. The amplitude a governed by our CGLe is linearly stable for all but possesses non-decaying nonlinear solutions at finite , directly analogous to plane Couette flow. Furthermore, whereas the important balance suggests that the Taylor vortices ought to propagate as waves towards the equator with frequency proportional to latitude, the realised solutions are found to exist as pulses, each locked to a discrete frequency, of spatially modulated Taylor vortices. Collectively they form a pulse train. Thus the expected continuous spatial variation of the frequency is broken into steps (forming a staircase) on which motion is dominated by the local pulse. A wealth of solutions of our CGLe have been found and some may be stable. Nevertheless, when higher-order terms are reinstated, solutions are modulated on a yet longer length scale and must evolve. So, whereas there is an underlying pulse structure in the small but finite gap limit, motion is likely to be always weakly chaotic. Our CGLe and its solution provides a paradigm for many geophysical and astrophysical flows capturing in minimalistic form interaction of phase mixing , diffusion and nonlinearity .

Analysis of the accuracy and pressure oscillation of the lattice Boltzmann method for fluid–solid interactions

Investigations have been conducted to analyze the accuracy of the ghost fluid immersed boundary lattice Boltzmann method and the conventional interpolation/extrapolation bounce-back schemes. The intrinsic sources of pressure oscillation for numerical simulation of moving boundary flows have also been investigated. An existing bilinear ghost fluid immersed boundary lattice Boltzmann method (BGFM) (Tiwari and Vanka, 2012) and a proposed quadratic ghost fluid immersed boundary lattice Boltzmann method (QGFM) have been compared with Guo's second-order extrapolation bounce-back scheme (Guo et al.) and the linear and quadratic interpolation bounce-back scheme (LIBB, QIBB) (Bouzidi et al., 2001; Lallemand and Luo, 2003). To study the numerical pressure oscillations in the moving boundary problem, (i) three existing refilling techniques and one proposed refilling technique are compared; (ii) three collision models, including the single-relaxation-time model, the multiple-relaxation-time model and the two-relaxation-time model, are investigated; (iii) two force evaluation schemes, a Galilean invariant momentum exchange method Wen et al. and a stress integration method Inamuro et al., are considered. The accuracy and the performance in pressure oscillation suppression for the studied numerical approaches are compared and discussed by five numerical examples: an eccentric cylinder flow, a Cylindrical Couette flow, an impulsively started cylinder in a channel, an oscillation cylinder in calm water and a particle suspension problem. The numerical results indicate that the accuracy of QGFM scheme is comparable to Guo's scheme while the accuracy of BGFM is worse than both of them. The QIBB scheme shows the best performance in the space convergence accuracy among all the schemes. Selection of the collision model, refilling technique and force evaluation scheme affect the pressure oscillation phenomenon in moving boundary simulations remarkably.

Torque data on non-isothermal rarefied Cylindrical Couette flow

The torque exerted on the walls of a micro cylindrical Couette gas flow is studied using Direct Simulation Monte Carlo (DSMC) method and analytical solutions. An analytical solution for temperature filed under specified wall heat flux condition is proposed using a previously introduced power-law (PL) wall-scaling approach. The results of cylindrical Couette flow under specified wall heat flux condition are compared with those of isothermal flow. The effects of rarefaction, compressibility, tangential momentum accommodation coefficient (TMAC) and gas-surface heat exchange on the torque values are investigated for slip to free molecular flow regimes. The results indicate that the torque coefficient has a maximum value at the transitional flow regime for TMAC values less than unity. Lower TMAC values and higher Knudsen numbers lead to reduction and deviation of torque magnitudes from those of continuum limit. Thermal simulations of cylindrical Couette flow show that heating and cooling have significant effects on the torque values. Velocity profiles and normalized torque values demonstrate mixed increasing/decreasing behavior under heating/cooling processes for rarefied gas flow.

Cylindrical Couette flow of a rarefied gas: Effect of a boundary condition on the inverted velocity profile.

The cylindrical Couette flow of a rarefied gas between a rotating inner cylinder and a stationary outer cylinder is investigated under the following two kinds of kinetic boundary conditions. One is the modified Maxwell-type boundary condition proposed by Dadzie and Méolans [J. Math. Phys. 45, 1804 (2004)] and the other is the Cercignani-Lampis condition, both of which have separate accommodation coefficients associated with the molecular velocity component normal to the boundary and with the tangential component. An asymptotic analysis of the Boltzmann equation for small Knudsen numbers and a numerical analysis of the Bhatnagar-Gross-Krook model equation for a wide range of the Knudsen number are performed to clarify the effect of each accommodation coefficient as well as of the boundary condition itself on the behavior of the gas, especially on the flow-velocity profile. As a result, the velocity-slip and temperature-jump conditions corresponding to the above kinetic boundary conditions are derived, which are necessary for the fluid-dynamic description of the problem for small Knudsen numbers. The parameter range for the onset of the velocity inversion phenomenon, which is related mainly to the decrease in the tangential momentum accommodation, is also obtained.

On the insertion of a thin gas layer in micro cylindrical Couette flows involving power-law liquids

Consideration is given to the fluid flow and heat transfer repercussions for introducing a thin (micro) gas layer into a cylindrical Couette flow assembly dealing with a power law liquid (lubricant). The trio influence of the thin gas layer on (1) the torque required for activating the lubrication process, (2) on the maximum temperature of the shaft (inner cylinder) and (3) on Nusselt number of the two-phase flow configuration are studied analytically. The results demonstrate that the thin gas layer normally contributes to reduce in the torque to set the fluid in motion and to downscale the maximum temperature at the shaft, especially for the sub-category of shear thickening liquids. However, in the sub-category of shear thinning liquids, the above-mentioned positive roles attributed to the micro gas layer hold true only for a limited number of flow configurations depending on several factors such as the Knudsen number, the accommodation coefficients and the rheological parameters of the liquid. It is further corroborated that the thin gas layer stabilizes or destabilizes the flow, depending on the magnitude of the power index number characterizing the liquid.

Implicit-correction-based immersed boundary-lattice Boltzmann method with two relaxation times.

In the present paper, we verify the effectiveness of the two-relaxation-time (TRT) collision operator in reducing boundary slip computed by the immersed boundary-lattice Boltzmann method (IB-LBM). In the linear collision operator of the TRT, we decompose the distribution function into symmetric and antisymmetric components and define the relaxation parameters for each part. The Chapman-Enskog expansion indicates that one relaxation time for the symmetric component is related to the kinematic viscosity. Rigorous analysis of the symmetric shear flows reveals that the relaxation time for the antisymmetric part controls the velocity gradient, the boundary velocity, and the boundary slip velocity computed by the IB-LBM. Simulation of the symmetric shear flows, the symmetric Poiseuille flows, and the cylindrical Couette flows indicates that the profiles of the numerical velocity calculated by the TRT collision operator under the IB-LBM framework exactly agree with those of the multirelaxation time (MRT). The TRT is as effective in removing the boundary slip as the MRT. We demonstrate analytically and numerically that the error of the boundary velocity is caused by the smoothing technique using the δ function used in the interpolation method. In the simulation of the flow past a circular cylinder, the IB-LBM based on the implicit correction method with the TRT succeeds in preventing the flow penetration through the solid surface as well as unphysical velocity distortion. The drag coefficient, the wake length, and the separation points calculated by the present IB-LBM agree well with previous studies at Re = 10, 20, and 40.

Generalized second-order slip boundary condition for nonequilibrium gas flows

It is a challenging task to model nonequilibrium gas flows within a continuum-fluid framework. Recently some extended hydrodynamic models in the Navier-Stokes formulation have been developed for such flows. A key problem in the application of such models is that suitable boundary conditions must be specified. In the present work, a generalized second-order slip boundary condition is developed in which an effective mean-free path considering the wall effect is used. By combining this slip scheme with certain extended Navier-Stokes constitutive relation models, we obtained a method for nonequilibrium gas flows with solid boundaries. The method is applied to several rarefied gas flows involving planar or curved walls, including the Kramers' problem, the planar Poiseuille flow, the cylindrical Couette flow, and the low speed flow over a sphere. The results show that the proposed method is able to give satisfied predictions, indicating the good potential of the method for nonequilibrium flows.

MHD Couette Flow in Cylindrical Porous Annulus with Perfectly Conducting Walls

This work obtained an analytical solution for a steady cylindrical MHD Couette flow in a porous medium between two perfectly conducting rotating cylinders under the influence of a non-uniform radial magnetic field. Since part of the analytical solution is expressed in terms of the integral of the Modified Bessel function of the first and second kinds of variable order, numerical integration was performed. Current results indicate that the flow may become more uniform when the strength of the external magnetic field is increased. The magnetic fluid tends to slow down if the permeability of the porous medium decreases. If the porous annulus is thick, the momentum of the flow is more difficult to propagate from the outer cylinder into the inner part of the annulus. If both the inner and outer cylinders rotate, the shear effect the inner cylinder imposes is only relatively influential in the region close to it. A decrease in Da no less than 10-2 may increase the amount of magnetic field induced. The transfer of momentum across the annular space is easier in a thin porous annulus than a thick one and hence induces a stronger magnetic field. If the inner cylinder rotates in the direction opposite of the outer one, the magnetic field in the clockwise direction will be induced in some region.

Pinch instability of a cylindrical couette flow in a nonuniform axial magnetic field

The paper addresses the linear stability to axisymmetric perturbations of an incompressible nonideal fluid between two rotating coaxial infinitely long cylinders in a nonuniform axial magnetic field. For conducting cylinders, the results for uniform and nonuniform magnetic fields are qualitatively identical. This is also observed for nonconducting cylinders in a magnetic field with a constant direction. Instability appears for nonconducting cylinders in a magnetic field with a varying direction, whose magnitude exceeds a certain critical value. This new instability also exists in the absence of rotation and, hence, is independent of its parameters. In addition, the critical magnetic field is independent of the magnetic Prandtl number, which facilitates experimental observation of the new instability.

The effect of Knudsen layers on rarefied cylindrical Couette gas flows

We investigate a power-law probability distri- bution function to describe the mean free path of rarefied gas molecules in non-planar geometries. A new curvature- dependent model is derived by taking into account the boundary-limiting effects on the molecular mean free path for surfaces with both convex and concave curvatures. The Navier-Stokes constitutive relations and the velocity-slip boundary conditions are then modified based on this power-law scaling through the transport property expres- sions in terms of the mean free path. Velocity profiles for isothermal cylindrical Couette flow are obtained using this power-law model and compared with direct simulation Monte Carlo (DSMC) data. We demonstrate that our model is more accurate than the classical slip solution, and we are able to capture important non-linear trends associated with the non-equilibrium physics of the Knudsen layer. In addition, we establish a new criterion for the critical accommodation coefficient that leads to the non-intuitive phenomenon of velocity inversion. The power-law model predicts that the critical accommodation coefficient is significantly lower than that calculated using the classical slip solution, and is in good agreement with available DSMC data. Our proposed constitutive scaling for non- planar surfaces is based on simple physical arguments and can be readily implemented in conventional fluid dynamics codes for arbitrary geometric configurations of microfluidic systems.

Taylor-Couette Instability in Anisotropic Clay Suspensions Measured Using Small-Angle X-ray Scattering

In this Letter, we propose an original and novel experimental method to characterize both the onset and morphology of Taylor-Couette instability occurring in a non-Newtonian cylindrical Couette flow. Using synchrotron-based rheological small angle x-ray scattering experiments, we jointly investigate the shear-thinning behavior of natural swelling clays suspensions and the associated anisotropy developing in such media. Combined with a linear stability analysis for power law fluids, a destabilizing effect is shown both numerically and experimentally and the vortices morphology is found to be dependent on the fluid index. Still, the strong destabilizing effect and large vortex size can not be assigned to shear-thinning only, which clearly evidences the impact of medium anisotropy on Taylor-Couette instability.

Velocity Inversion In Cylindrical Couette Gas Flows

We investigate a power-law probability distribution function to describe the mean free path of rarefied gas molecules in non-planar geometries. A new curvature-dependent model is derived by taking into account the boundary-limiting effects on the molecular mean free path for surfaces with both convex and concave curvatures. In comparison to a planar wall, we find that the mean free path for a convex surface is higher at the wall and exhibits a sharper gradient within the Knudsen layer. In contrast, a concave wall exhibits a lower mean free path near the surface and the gradients in the Knudsen layer are shallower. The Navier-Stokes constitutive relations and velocity-slip boundary conditions are modified based on a power-law scaling to describe the mean free path, in accordance with the kinetic theory of gases, i.e. transport properties can be described in terms of the mean free path.Velocity profiles for isothermal cylindrical Couette flow are obtained using the power-law model. We demonstrate that our model is more accurate than the classical slip solution, especially in the transition regime, and we are able to capture important non-linear trends associated with the non-equilibrium physics of the Knudsen layer. In addition, we establish a new criterion for the critical accommodation coefficient that leads to the non-intuitive phenomena of velocity-inversion. Our results are compared with conventional hydrodynamic models and direct simulation Monte Carlo data. The power-law model predicts that the critical accommodation coefficient is significantly lower than that calculated using the classical slip solution and is in good agreement with available DSMC data. Our proposed constitutive scaling for non-planar surfaces is based on simple physical arguments and can be readily implemented in conventional fluid dynamics codes for arbitrary geometric configurations.

Molecular dynamics study of slip behaviors: Role of a soft attractive tail

The role of soft attractive tails in nano-scale cylindrical Couette flows is investigated by using molecular dynamics computer simulations. The cylindrical Couette flow is generated by rotating an inner wall with a fixed outer wall. By varying the cutoff distance of the Lennard-Jones potential, we can easily control the inclusion or exclusion of an attractive component. We found that the number density and the tangential velocity profiles along the radial direction show rather strikingly different behaviors that can be classified into four distinctive flow types of slip behaviors due to the soft attractive tail.

Effects of rarefaction, viscous dissipation and rotation mode on the first and second law analyses of rarefied gaseous slip flows confined between a rotating shaft and its concentric housing

A parametric analytical study is carried out to scrutinize the mechanism of fluid flow, heat transfer and entropy generation in a low-speed rarefied gaseous flow confined between a shaft and its concentric housing, i.e., the cylindrical Couette flow. In the first law analysis, closed form solutions for the radial temperature profiles are obtained by incorporating the calculated velocity distribution into the energy equation. The derivations for three thermal cases, which are founded on imposing different thermal conditions, namely, the Uniform Heat Flux (UHF) and the Constant Wall Temperature (CWT) boundary conditions, are presented. In the second law analysis, the contributions of thermal diffusion and fluid friction irreversibility to the total entropy generation in the micro domain are illustrated, and the relevant expressions for the Bejan number and the entropy generation number as well as the average entropy generation rate are derived. Finally, the variations of major variables with influential parameters such as the Knudsen number, the Brinkman number and rotation mode are investigated to elucidate the associated effects of rarefaction phenomenon, viscous dissipation and geometric condition on the characteristics of the flow.

Experimental Surface Profile of Dense Granular Media in Cylindrical Couette Flow

Granular materials have widespread use in the industry. The flow of granular media requires careful studies through experiments to understand the rheology of these complex materials. The present work determines the surface profile of dense granular media subjected to cylindrical Couette flow. A translation stage is used to obtain measurement of depth varying with radial distance. A depth gauge attached to the translation stage is used to measure the surface profile. Glass beads (average diameter 0.8-1 mm) and mustard seeds (average diameter 1.2-1.4 mm) are used as model granular materials. Two different Couette gaps are used (4cm and 3cm). Cylinders with smooth surfaces, as well as coated with emery paper are used. The surface profile varies with material, surface roughness of the cylinders, and the gap between the two cylinders. The comparison of the various cases has been done by graphical representation. The probable reasons for the development of such a surface profile are given.

Analytical solution of axi-symmetrical lattice Boltzmann model for cylindrical Couette flows

Analytical solution for the axi-symmetrical lattice Boltzmann model is obtained for the low-Mach number cylindrical Couette flows. In the hydrodynamic limit, the present solution is in excellent agreement with the result of the Navier–Stokes equation. Since the kinetic boundary condition is used, the present analytical solution using nine discrete velocities can describe flows with the Knudsen number up to 0.1. Meanwhile, the comparison with the simulation data obtained by the direct simulation Monte Carlo method shows that higher-order lattice Boltzmann models with more discrete velocities are needed for highly rarefied flows.

A ghost fluid Lattice Boltzmann method for complex geometries

SUMMARYA ghost fluid Lattice Boltzmann method (GF‐LBM) is developed in this study to represent complex boundaries in Lattice Boltzmann simulations of fluid flows. Velocity and density values at the ghost points are extrapolated from the fluid interior and domain boundary via obtaining image points along the boundary normal inside the fluid domain. A general bilinear interpolation algorithm is used to obtain values at image points which are then extrapolated to ghost nodes thus satisfying hydrodynamic boundary conditions. The method ensures no‐penetration and no‐slip conditions at the boundaries. Equilibrium distribution functions at the ghost points are computed using the extrapolated values of the hydrodynamic variables, while non‐equilibrium distribution functions are extrapolated from the interior nodes. The method developed is general, and is capable of prescribing Dirichlet as well as Neumann boundary conditions for pressure and velocity. Consistency and second‐order accuracy of the method are established by running three test problems including cylindrical Couette flow, flow between eccentric rotating cylinders and flow over a cylinder in a confined channel. Copyright © 2011 John Wiley & Sons, Ltd.

Flow and heat transfer in a micro-cylindrical gas–liquid Couette flow

Analytic solutions for the gas and liquid velocity and temperature distribution are determined for steady state one-dimensional microchannel cylindrical Couette flow between a shaft and a concentric cylinder. The solution is based on the continuum model and takes into consideration the velocity slip and temperature jump in the gaseous phase defined by the Knudsen number range of 0.001 < Kn < 0.1. The two fluids are assumed immiscible. The gas layer is adjacent to the shaft which rotates with angular velocity ω s and is thermally insulated. The outer cylinder rotates with angular velocity ω o and is maintained at uniform temperature. The governing parameters are identified and the effects of the Knudsen number and accommodation coefficients on the velocity and temperature profiles, reduction in the overall temperature rise due to the gas layer, the Nusselt number and shear reduction are examined. It was found that the required torque to rotate the liquid in the annular space is significantly reduced by introducing a thin gas layer adjacent to the shaft. Also, reduction in shaft temperature is enhanced through a combination of high energy accommodation coefficient and low momentum accommodation coefficients. Results also indicate that the gas layer becomes more effective in reducing the shaft temperature when the housing angular velocity is much larger than the shaft angular velocity.

Ginzburg–Landau description of laminar-turbulent oblique band formation in transitional plane Couette flow

Plane Couette flow, the flow between two parallel planes moving in opposite directions, is an example of wall-bounded flow experiencing a transition to turbulence with an ordered coexistence of turbulent and laminar domains in some range of Reynolds numbers [R_g,R_t]. When the aspect-ratio is sufficiently large, this coexistence occurs in the form of alternately turbulent and laminar oblique bands. As R goes up trough the upper threshold R_t, the bands disappear progressively to leave room to a uniform regime of featureless turbulence. This continuous transition is studied here by means of under-resolved numerical simulations understood as a modelling approach adapted to the long time, large aspect-ratio limit. The state of the system is quantitatively characterised using standard observables (turbulent fraction and turbulence intensity inside the bands). A pair of complex order parameters is defined for the pattern which is further analysed within a standard Ginzburg--Landau formalism. Coefficients of the model turn out to be comparable to those experimentally determined for cylindrical Couette flow.