Steady Couette Flow Research Articles

Analysis and accurate numerical solutions of the integral equation derived from the linearized BGKW equation for the steady Couette flow

The integral equation for the flow velocity u ( x ; k ) in the steady Couette flow derived from the linearized Bhatnagar–Gross–Krook–Welander kinetic equation is studied in detail both theoretically and numerically in a wide range of the Knudsen number k between 0.003 and 100.0. First, it is shown that the integral equation is a Fredholm equation of the second kind in which the norm of the compact integral operator is less than 1 on L p for any 1 ≤ p ≤ ∞ and thus there exists a unique solution to the integral equation via the Neumann series. Second, it is shown that the solution is logarithmically singular at the endpoints. More precisely, if x = 0 is an endpoint, then the solution can be expanded as a double power series of the form ∑ n = 0 ∞ ∑ m = 0 ∞ c n , m x n ( x ln ⁡ x ) m about x = 0 on a small interval x ∈ ( 0 , a ) for some a > 0 . And third, a high-order adaptive numerical algorithm is designed to compute the solution numerically to high precision. The solutions for the flow velocity u ( x ; k ) , the stress P x y ( k ) , and the half-channel mass flow rate Q ( k ) are obtained in a wide range of the Knudsen number 0.003 ≤ k ≤ 100.0 ; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.

Pulsed Taylor-Couette flow in a viscoelastic fluid under inner cylinder modulation

The influence of elasticity on pulsed Taylor-Couette flow in a linear Maxwell fluid is investigated. We consider the case, in which the inner cylinder is oscillating with a periodic angular velocity, $ \Omega_{0}\cos(\omega t)$ , and the outer cylinder is fixed. Attention is focused on the linear stability analysis which is solved using the Floquet theory and a technique of converting a boundary value problem to an initial value problem. Results obtained in this framework show that, in the high-frequency limit, the Deborah number has a destabilizing effect and the critical Taylor and wave numbers tend toward constant values independently of the frequency number. However, in the low-frequency limit, the Maxwell fluid behaves as a Newtonien one and the Deborah number has no effect on the stability of the basic state which tends to the classical configuration of steady circular Couette flow. These numerical results are in good agreement with the asymptotic analysis performed in the limit of low and high frequencies.

Accurate solution and approximations of the linearized BGK equation for steady Couette flow

We present an accurate and efficient high-order collocation method to solve the integral equation with a singular kernel derived from the linearized BGK equation in kinetic theory. In particular, we use a Chebyshev based collocation method to solve the integral equation with singular kernel for the steady Couette flow in a wide range of Knudsen numbers, i.e., 0.003⩽k⩽10.0. We compute the flow velocity u(y,k), the stress Pxy(k), and the half-channel total mass flow rate Q(k). Our results are uniformly accurate to 11 significant digits or better, thus they can serve as benchmark data. We also construct approximations of the velocity, the slip velocity, and the total mass flow rate as functions of the Knudsen number k.

Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates

In this study, steady and unsteady magneto-hydrodynamic (MHD) Couette flows between two parallel infinite plates have been studied through numerical Differential Quadrature Method (DQM) and analytical Differential Transformation Method (DTM), respectively. Coupled equations by taking the viscosity effect of the two phases for fixed and moving plates have been introduced. The precious contribution of the present study is introducing new, fast and efficient numerical and analytical methods in a two-phase MHD Couette fluid flow. Results are compared with those previously obtained by using Finite Difference Method (FDM). The velocity profiles of two phases are presented and a parametric study of physical parameters involved in the problem is conducted. As an outcome, when magnetic source is fixed relative to the moving plate, by increasing the Hartmann number, velocity profiles for both phases increased, but when it is fixed relative to the fluid an inverse treatment is observed.

Analysis of Red Blood Cell Deformation under Fast Shear Flow for Better Estimation of Hemolysis

We examined the deformation behavior of a red blood cell (RBC) in various flow fields to determine whether the extent of RBC deformation is correlated with the shear stress used as a hemolysis index. The RBC model was introduced to a simple shear flow (Couette flow) and to slightly complex flows (unsteady shear flows and stenosed flows). The RBC deformation was assessed by the maximum first principal strain over the RBC membrane and compared with the shear stress. Although the results were consistent under steady Couette flow, this was not the case under unsteady Couette flow or stenosed flow due to the viscoelastic nature of the RBC deformation caused by fluid forces. These results suggest that there is a limitation in accurately estimating the mechanical damage of RBCs solely from a macroscopic flow field, indicating the necessity of taking into account the dynamic deformation of RBCs to provide a better estimation of hemolysis.

Investigation of multiscale fluid flow characteristics based on a hybrid atomistic–continuum method

A computer program based on a molecular dynamics–continuum hybrid method has been developed in which the Navier–Stokes equations are solved in the continuum region and the molecular dynamics in the atomistic region. The coupling between the atomistic and continuum is constructed through constrained dynamics within an overlap region where both molecular and continuum equations are solved simultaneously. The simulation geometries are solved in three dimensions and an overlap region is introduced in two directions to improve the choice of using the molecular region in smaller areas. The proposed method is used to simulate steady and start-up Couette flow showing quantitative agreement with results from analytical solutions and full molecular dynamics simulations. The prepared algorithm and the computer code are capable of modeling fluid flows in micro and nano-scale geometries.

Shear flows of a new class of power-law fluids

We consider the flow of a class of incompressible fluids which are constitutively defined by the symmetric part of the velocity gradient being a function, which can be non-monotone, of the deviator of the stress tensor. These models are generalizations of the stress power-law models introduced and studied by J. Malek, V. Průsa, K.R. Rajagopal: Generalizations of the Navier-Stokes fluid from a new perspective. Int. J. Eng. Sci. 48 (2010), 1907–1924. We discuss a potential application of the new models and then consider some simple boundary-value problems, namely steady planar Couette and Poiseuille flows with no-slip and slip boundary conditions. We show that these problems can have more than one solution and that the multiplicity of the solutions depends on the values of the model parameters as well as the choice of boundary conditions.

Hydromagnetic flows of a mixture of two Newtonian fluids between two parallel plates

The paper aims to study the flow of a binary mixture of electrically conducting, incompressible and viscous fluids between two parallel plates in the presence of a transverse uniform magnetic field. The solution of such a flow model has many applications in magnetohydrodynamic (MHD) power generators, MHD pumps, MHD accelerators, and MHD flowmeters. Exact solutions have been obtained for the following four problems: (1) steady hydromagnetic Couette flow, (2) unsteady hydromagnetic Couette flow, (3) steady hydromagnetic Poiseuille flow, (4) unsteady hydromagnetic Poiseuille flow. The mean velocity of the mixture is drawn for different values of magnetic parameters and results are interpreted with the aid of graphs. The previous solutions involving single Newtonian fluid appear as the special cases of the present analysis.

Combined Effects of Hall Currents and Rotation on Steady Hydromagnetic Couette Flow

We study a steady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system between two infinitely long parallel plates in the presence of a uniform transverse magnetic field on taking Hall Current into account. The governing equations describing the flow are solved analytically. It is observed that the Hall currents accelerate the primary velocity whereas they retard the secondary velocity. The induced magnetic field is significantly affected by the Hall currents. An increase in Hall currents leads to fall in the fluid temperature. The heat transfer characteristics have also been studied. The rate of heat transfer at the lower plate decreases whereas the rate of heat transfer at the upper plate increases with an increase in Hall parameter. The asymptotic behavior of the solutions are discussed for small and large values of magnetic parameter and rotation parameter. It is interesting to note that either for strong magnetic field or for large rotation there exists a single-deck boundary layer in the region near the stationary plate. The thickness of this boundary layer first decreases, reaches a minimum and then increases with an increase in Hall parameter.

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.

The structure and rheology of sheared model swimmer suspensions

We analyze the rheological response of a suspension of simple model swimmers subject to a steady Couette flow. We consider the squirmer model as a means to control systematically the interplay between self-propulsion and active stress generation and analyze their relative impact both on the effective viscosity of a suspension and the microstructure the squirmers develop. We show how self-propulsion introduces an intrinsic contribution to the effective viscosity of the active suspension. Accordingly, apolar squirmers show shear thickening while polar ones develop a shear thinning response to the applied shear. We show that the detailed coupling of the squirmers to the bounding walls has a strong influence on the structure of the suspension and the development of shear bands.

High accuracy numerical solutions of the Boltzmann Bhatnagar-Gross-Krook equation for steady and oscillatory Couette flows

Modeling gas flows generated by micro- and nano-devices often requires the use of kinetic theory. To facilitate implementation, various approximate formulations have been proposed based on the Bhatnagar-Gross-Krook (BGK) kinetic model, including most recently, the lattice Boltzmann (LB) method. While there exists a comprehensive numerical data set for the hard sphere linearized Boltzmann equation for steady Couette flow, no such set of data is available for the Boltzmann-BGK equation. The purpose of this article is to present a high accuracy data set for the linearized Boltzmann-BGK equation over the full range of Knudsen numbers and normalized oscillation frequencies – this encompasses both steady and unsteady Couette flows. This data set is expected to be of particular value in the benchmarking and validation of computational methods such as the LB method and other approaches based on the Boltzmann-BGK equation.

Flow of a binary mixture of fluids in a semicircular duct

The purpose of this work is to examine the flow of a binary mixture including chemically inert incompressible Newtonian fluids in a duct of semicircular cross-section. Such a flow model has great significance not only of its own theoretical interest, but also for application to various engineering processes. The governing equations have been solved analytically using the finite Fourier sine and Hankel transforms for the following four problems: (1) steady Couette flow in a semicircular duct, (2) unsteady Couette flow in a semicircular duct, (3) steady Poiseuille flow in a semicircular duct, (4) unsteady Poiseuille flow in a semicircular duct. The previous solutions corresponding to pure Newtonian fluid appear as the special cases of the present analysis. Key words: Binary mixture, Newtonian fluid, Couette flow, Poiseuille flow, Fourier sine transform, Hankel transform.

Interaction of toroidal focal conic defects with shear flow

Toroidal focal conic defects form in smectic liquid crystals due to antagonistic surface anchoring conditions. The resulting layer structure, defect size, and the formation of ordered patterns of defects have been studied in detail. Here, we investigate the effect of shear flow on toroidal focal conic defects. The defects are formed and subjected to a steady linear Couette flow in a microscale shear cell. In situ visualization of the evolution of the polarization texture of individual defects reveals the impact of three distinct flow regimes on the layer structure. At low Ericksen numbers, individual defects exhibit an early time elastic regime in which the optical intensity of the layer structure does not change in the presence of the applied shear flow. At increasing applied strains, the defect layer structure deforms, resulting in a monotonically increasing optical intensity within its original footprint. At even larger applied strains, satellite defects are emitted from the anchored defect. Cessation of shear flow at moderate strains, prior to the formation of satellite defects, leads to an exponential decay of the polarization intensity of the defect, suggesting that relaxation of the layer structure occurs with a time constant that is consistent with that predicted by a permeation flow mechanism. This study establishes the relevant length and time scales involved in the interaction of shear stresses and elastic stresses in smectic liquid crystalline samples whose structure deviates significantly from an idealized aligned lamellar configuration.

Steady Couette flows of elastoviscoplastic fluids are nonunique

The Herschel–Bulkley rheological fluid model includes terms representing viscosity and plasticity. In this classical model, below the yield stress the material is strictly rigid. Complementing this model by including elastic behavior below the yield stress leads to a description of an elastoviscoplastic (EVP) material such as an emulsion or a liquid foam. We include this modification in a completely tensorial description of cylindrical Couette shear flows. Both the EVP model parameters, at the scale of a representative volume element, and the predictions (velocity, strain and stress fields) can be readily compared with experiments. We perform a detailed study of the effect of the main parameters, especially the yield strain. We discuss the role of the curvature of the cylindrical Couette geometry in the appearance of localization; we determine the value of the localization length and provide an approximate analytical expression. We then show that, in this tensorial EVP model of cylindrical Couette shear f...

Steady hydromagnetic Couette flow in a rotating system with nonconducting walls

Steady hydromagnetic Couette flow of class-II of a viscous incompressible electrically conducting fluid in a rotating system with non-conducting walls is studied. Exact solution of the governing equations is obtained in closed form. Expressions for the shear stress at the lower and upper plates due to primary and secondary flows and mass flow rates in primary and secondary flow directions are also derived. Asymptotic behavior of the solution for fluid velocity and induced magnetic field is analyzed for small and large values of rotation parameter K2 and magnetic parameter M2 to gain further insight into the flow pattern. Heat transfer characteristics of the flow are considered taking viscous and Joule dissipations into account. The numerical solution of energy equation and numerical values of rate of heat transfer at both plates are obtained with the help of MATLAB software. The numerical values of fluid velocity, induced magnetic field and fluid temperature are depicted graphically versus channel width variable c for various values of M2 and K2 while numerical values of primary and secondary shear stress at the lower and upper plates, mass flow rates in primary and secondary flow directions and rate of heat transfer at the lower and upper plates are presented in tabular form for various values of K2 and M2.

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.

Oscillatory Kelvin–Helmholtz instability. Part 2. An experiment in fluids with a large viscosity contrast

The stability of two-layer oscillatory flows was studied experimentally in a cylindrical container with a vertical axis. Two superposed immiscible liquids, differing greatly in viscosity, were set in relative oscillatory motion by alternating container rotation. Waves arising beyond a threshold were observed in detail for small oscillation frequencies ranging from 0.1 to 6 Hz. Measurements were performed on the growth rate and the wavenumber of these waves. The instability threshold was determined from the growth rate data. It was found that the threshold and the wavenumber varied with the frequency. In particular, significantly lower thresholds and longer waves were found than those predicted by the inviscid theory of the oscillatory Kelvin–Helmholtz instability. Favourable agreement with the predictions of an existing viscous theory for small oscillation amplitude flows indicates the important role of viscosity, even at the highest frequency, and suggests a similar mechanism behind the instability as that for the short wave instability in steady Couette flows. A semi-numerical stability determination for finite amplitude flows was also performed to improve the prediction in experiments with a frequency lower than 1 Hz.

Class of dilute granular Couette flows with uniform heat flux

In a recent paper [F. Vega Reyes et al., Phys. Rev. Lett. 104, 028001 (2010)] we presented a preliminary description of a special class of steady Couette flows in dilute granular gases. In all flows of this class the viscous heating is exactly balanced by inelastic cooling. This yields a uniform heat flux and a linear relationship between the local temperature and flow velocity. The class (referred to as the LTu class) includes the Fourier flow of ordinary gases and the simple shear flow of granular gases as special cases. In the present paper we provide further support for this class of Couette flows by following four different routes, two of them being theoretical (Grad's moment method of the Boltzmann equation and exact solution of a kinetic model) and the other two being computational (molecular dynamics and Monte Carlo simulations of the Boltzmann equation). Comparison between theory and simulations shows a very good agreement for the non-Newtonian rheological properties, even for quite strong inelasticity, and a good agreement for the heat flux coefficients in the case of Grad's method, the agreement being only qualitative in the case of the kinetic model.

Clustering and phase separation in dense shear granular flow

We investigate various regimes of steady dense Couette flow of inelastically colliding hard disks in the absence of gravity. The two governing parameters in this two-dimensional system are the inelasticity of particle collisions and the average density of particles. The simplest steady state is the uniform shear flow (USF), where the temperature and the density profiles are homogeneous over the system and the velocity of the flow changes linearly between the two moving walls. The USF becomes unstable when the inelasticity of particle collisions exceeds a certain threshold, which depends on the average density of particles. Then the USF gives a way to a "plug flow" regime, where a solid-like cluster coexists with one or two fluid layers. These regimes are investigated using equations of granular hydrodynamics with constitutive relations that interpolate between low and high densities. The results are tested in event-driven molecular dynamics (MD) simulations, and a good agreement is observed.