An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates

An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G).     

Analytical solution of the poiseuille flow problem using the ellipsoidal-statistical model of the Boltzmann kinetic equation

An analytical solution (in the form of a Neumann series) of the problem of rarefied gas flow in a plane channel with infinite walls in the presence of a pressure gradient (Poiseuille flow) parallel to them is constructed within the framework of the kinetic approach in an isothermal approximation. The ellipsoidal-statistical model of the Boltzmann kinetic equation and the diffuse reflection model are used as the basic equation and the boundary condition, respectively. Using the resulting distribution function, the mass and heat flux densities in the direction of the pressure gradient per unit channel length in the y′ direction are calculated, and profiles of the gas mass velocity and heat flux in the channel are constructed. The results obtained for the continuum and free-molecular flow models are analyzed and compared with similar results obtained by numerical methods.     

Stability of rarefied dusty gas and suspension flows in a plane channel

The stability of two-dimensional dispersed Poiseuille flow is analyzed within the framework of the linear theory. A numerical solution of the corresponding Orr-Sommerfeld equation is constructed. The effect of the particle mass concentration, dimensions, and relaxation time on the flow stability is considered.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 79–85, September–October, 1995.     

Buoyancy modelling with incompressible SPH for laminar and turbulent flows

This work aims to model buoyant, laminar or turbulent flows, using a two‐dimensional incompressible smoothed particle hydrodynamics model with accurate wall boundary conditions. The buoyancy effects are modelled through the Boussinesq approximation coupled to a heat equation, which makes it possible to apply an incompressible algorithm to compute the pressure field from a Poisson equation. Based on our previous work [1], we extend the unified semi‐analytical wall boundary conditions to the present model. The latter is also combined to a Reynolds‐averaged Navier–Stokes approach to treat turbulent flows. The k ? ? turbulence model is used, where buoyancy is modelled through an additional term in the k ? ? equations like in mesh‐based methods. We propose a unified framework to prescribe isothermal (Dirichlet) or to impose heat flux (Neumann) wall boundary conditions in incompressible smoothed particle hydrodynamics. To illustrate this, a theoretical case is presented (laminar heated Poiseuille flow), where excellent agreement with the theoretical solution is obtained. Several benchmark cases are then proposed: a lock‐exchange flow, two laminar and one turbulent flow in differentially heated cavities, and finally a turbulent heated Poiseuille flow. Comparisons are provided with a finite volume approach using an open‐source industrial code. Copyright © 2015 John Wiley & Sons, Ltd.     

Empirical slip and viscosity model performance for microscale gas flow

For the simple geometries of Couette and Poiseuille flows, the velocity profile maintains a similar shape from continuum to free molecular flow. Therefore, modifications to the fluid viscosity and slip boundary conditions can improve the continuum based Navier–Stokes solution in the non‐continuum non‐equilibrium regime. In this investigation, the optimal modifications are found by a linear least‐squares fit of the Navier–Stokes solution to the non‐equilibrium solution obtained using the direct simulation Monte Carlo (DSMC) method. Models are then constructed for the Knudsen number dependence of the viscosity correction and the slip model from a database of DSMC solutions for Couette and Poiseuille flows of argon and nitrogen gas, with Knudsen numbers ranging from 0.01 to 10. Finally, the accuracy of the models is measured for non‐equilibrium cases both in and outside the DSMC database. Flows outside the database include: combined Couette and Poiseuille flow, partial wall accommodation, helium gas, and non‐zero convective acceleration. The models reproduce the velocity profiles in the DSMC database within an L₂ error norm of 3% for Couette flows and 7% for Poiseuille flows. However, the errors in the model predictions outside the database are up to five times larger. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability of wave forms of motion of a viscous fluid layer under tangential stress

We study the stability of wave flow of a viscous incompressible fluid layer subjected to tangential stress and an inclined gravity force with respect to long-wave disturbances.An asymptotic solution is constructed for the equations of the disturbed motion and the problem is reduced to the study of a second-order ordinary differential equation. It is shown that after loss of stability by a Poiseuille flow the laminar nature of the flow is not destroyed, but the form of the free surface acquires a wave-like profile. The Poiseuille regime is stable for low Reynolds numbers. The critical Reynolds number for wave flow is found, and the stability and instability regions are determined.     

Time-Periodic Solutions of the Navier-Stokes Equations in Unbounded Cylindrical Domains – Leray's Problem for Periodic Flows

Poiseuille flows in infinite cylindrical pipes, in spite of their enormous simplicity, have a main role in many theoretical and applied problems. As is well known, the Poiseuille flow is a stationary solution of the Stokes and the Navier-Stokes equations with a given constant flux. Time-periodic flows in channels and pipes have a comparable importance. However, the problem of the existence of time-periodic flows in correspondence to any given time-periodic total flux, is still an open problem. A solution is known only in some very particular cases, for instance, the Womersley flows. Our aim is to solve this problem in the general case. The above existence result opens the way to further investigations. As an example of this possibility we consider the extension of the classical Leray's problem for Poiseuille flows to arbitrary time-periodic flows. Dedicated to Louis Nirenberg on the occasion of his 80^th birthday     

On the steady simple shear flows of the one-mode Giesekus fluid

Analytical solutions for the plane Couette flow and the plane Poiseuille flow of the one-mode Giesekus fluid without any retardation time have been obtained by considering the domain of definition for each of the two branch solutions which arise due to the presence of the quadratic stress terms in the constitutive equations. For each fixed value of the mobility parametera, the limiting value of the Weissenberg number for the upper branch solution, i.e., the physically realistic solution is determined in terms of the corresponding dimensionless shear stress for the plane Couette flow and in terms of the corresponding dimensionless pressure gradient for the plane Poiseuille flow. In the case of the plane Couette flow, it is shown that fora falling in the range 0a1/2 only the physically realistic solution exists while for 1/2<a 1 a nonphysical solution coexists with the realistic one. In the case of the plane Poiseuille flow, it is shown that the non-physical solution cannot even exist around the center plane of the channel, and the effects of the mobility parameter and the dimensionless pressure gradient on the flow variables are investigated. Possible extensions of the present approach to other steady simple shear flows with and without the introduction of the retardation time are also discussed.     

Combined electroosmotically and pressure driven flow of power-law fluids in a slit microchannel

Electroosmotic flow of power-law fluids in the presence of pressure gradient through a slit is analyzed. After numerically solving the Poisson–Boltzmann equation, the momentum equation with electroosmotic body force is solved through an iterative numerical procedure for both favorable and adverse pressure gradients. The results reveal that, in case of pressure assisted flow, shear-thinning fluids reach higher velocity magnitudes compared with shear-thickening fluids, whereas the opposite is true when an adverse pressure gradient is applied. The Poiseuille number is found to be an increasing function of the dimensionless Debye–Hückel parameter, the wall zeta potential, and the flow behavior index. Comparison between the exact and the results based on the Debye–Hückel linearization reveals that the simplified solution leads to large errors in evaluating the velocity profile for zeta potentials higher than 25 mV, except for shear-thickening fluids in the presence of favorable pressure gradient.     

Effect of random longitudinal vibrations on the Poiseuille flow of a complex liquid

In this work, the rectilinear Poiseuille flow of a complex liquid flowing in a vibrating pipe is analyzed. The pipe wall performs oscillations of small amplitude that can be adequately represented by a weakly stochastic process, for which a quasi-static perturbation solution scheme is suggested. The flow is analyzed using the Bautista–Manero–Puig constitutive equation, consisting on the upper-convected Maxwell equation coupled to a kinetic equation to account for the breakdown and reformation of the fluid structure. A drastic enhancement of the volumetric flow is predicted in the region where the fluid experiences pronounced shear-thinning. Finally, flow enhancement is predicted using experimental data reported elsewhere for wormlike micellar solutions of cetyl trimethyl ammonium tosilate.     

Nonlinear stability theory of channel flow with heat transfer – an asymptotic approach

A fully developed laminar Poiseuille flow subject to constant heat flux across the wall is analysed with respect to its stability behavior by applying a weakly nonlinear stability theory. It is based on an expansion of the disturbance control equations with respect to a perturbation parameter ε. This parameter is the small initial amplitude of the fundamental wave. This fundamental wave which is the solution of the linear (Orr-Sommerfeld) first order equation triggers all higher order effects with respect to ε. Heat transfer is accounted for asymptotically through an expansion with respect to a small heat transfer parameter ε _T . Both perturbation parameters, ε and ε _T , are linked by the assumption ε _T =O(ε²) by which a certain distinguished limit is assumed. The results for a fluid with temperature dependent viscosity show that heat transfer effects in the nonlinear range continue to act in the same way as in the initial linear range. Received on 11 August 1997     

Direct simulation of the motion of neutrally buoyant balls in a three-dimensional Poiseuille flow

In a previous article the authors introduced a Lagrange multiplier based fictitious domain method. Their goal in the present article is to apply a generalization of the above method to: (i) the numerical simulation of the motion of neutrally buoyant particles in a three-dimensional Poiseuille flow; (ii) study – via direct numerical simulations – the migration of neutrally buoyant balls in the tube Poiseuille flow of an incompressible Newtonian viscous fluid. Simulations made with one and several particles show that, as expected, the Segré–Silberberg effect takes place. To cite this article: T.-W. Pan, R. Glowinski, C. R. Mecanique 333 (2005).     

Asymptotic analysis of stability problem of plane poiseuille flow for high Reynolds number

A study of the stability of plane Poiseuille flow at higher Reynolds number is made. Within a “triple-deck” structural framework, the qualitative behavior of the eingenvalue of Orr-Sommerfeld equation is analytically obtained. The corresponding eigenfunction is formulated approximately.     

Inertial-microfluidic radial migration in solid/liquid two-phase flow through a microcapillary: Particle equilibrium position

Although equilibrium of spherical particles under radial migration has been extensively investigated, mostly in macroscale flows with characteristic lengths on the order of centimeters, it is not fully characterized at relatively small Reynolds numbers, 1 ≤ Re ≤ 100. This paper experimentally studies “inertial microfluidic” radial migration of spherical particles in circular Poiseuille flow through a microcapillary. Microparticle tracking experiments are performed to obtain the spatial distribution of the particles by adopting a depth-resolved measurement technique. Through the analysis of the radial distribution of particles, inertial microfluidic circular Poiseuille flow is shown to induce a strong radial migration of particles at substantially small Re, which is quite in contrast to the pipe flows at large Re previously reported. This particle migration phenomenon is so prominent that particle equilibrium positions are formed even at small Re. However, it turns out that there exists a certain critical Re below which particle equilibrium position is almost fixed, but above which it seems to drift toward the channel wall.     

Nonlinear Electrohydrodynamic Stability of a Poiseuille Two–Layer Flow

The long–wave stability of the Poiseuille two–layer flow of homogeneous viscous dielectrics between plate electrodes under a constant potential difference is studied in an electrohydrodynamic approximation. A linear asymptotic stability analysis shows that surface polarization forces are a destabilizing factor, in addition to viscous stratification. The method of many scales is used to obtain the Kuramoto—Sivashinsky equation governing the weakly nonlinear evolution of the interface between the dielectrics. Within the framework of the approaches used, it is shown that nonlinear interactions limit perturbation growth and the interface does not fail even for a rather large potential difference.     

Finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements: plane Poiseuille flow

This paper presents a comprehensive review of the numerical techniques used during the past half century and their accuracy in hydrodynamic stability analysis of plane parallel flows. The paper also describes a finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements. A stability analysis technique is performed by imposing an infinitesimal perturbation to the laminar base flow to determine the thresholds of neutral instabilities or the growth rate of the perturbation for any Reynolds and wave numbers. Validation of the present numerical technique is performed for plane Poiseuille flow. The numerical results, obtained with uniform and nonuniform meshes, show excellent agreement with the most accurate results available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Modeling plane turbulent Couette flow

Among the salient features of shear-driven plane Couette flow is the constancy of the total shear stress (viscous and turbulent) across the flow. This constancy gives rise to a quasi-homogenous core region, which makes the bulk of the flow substantially different from pressure-driven Poiseuille flow. The present second-moment closure study addresses the conflicting hypotheses relating to turbulent Couette flow. The inclusion of a new wall-proximity function in the wall-reflection part of the pressure-strain model seems mandatory, and the greement with recent experimental and direct numerical simulation (DNS) results is encouraging. Analysis of model computations in the range 750 ≤ Re ≤ 35,000 and comparisons with low-Re DNS data suggest that plane Couette flow exhibits a local-equilibrium core region, in which anisotropic, homogeneous turbulence prevails. However, the associated variation of the mean velocity in the core, as obtained by the model, conflicts with the intuitively appealing assumption of homogeneous mean shear. The constancy of the velocity gradient exhibited by the DNS therefore signals a deficiency in the modeled transport equation for the energy dissipation rate.     

Effect of duct velocity profile and buoyancy‐induced flow on efficiency of transient hydrodynamic removal of a contaminant from a cavity

This paper describes a preliminary numerical analysis of the effect of duct velocity profile and buoyancy‐induced flow generated by the heat source on hydrodynamic removal of contaminants contained in cavities. The process of fluid renewal in a cavity is modelled via a numerical solution of the Navier–Stokes equations coupled with the energy equation for transient flows. The foulant has the same density as the fluid in the duct and the duct velocity profile is considered to be Poiseuille flow and Couette flow, respectively. The results show that the change in Grashof number and duct flow velocity profile causes a dramatic difference in the observed flow patterns and cleaning efficiency. From a cleaning perspective, the results suggest that Couette flow at higher value of Grashof number becomes more effective in further purging of contaminated fluid from a cavity. Copyright © 2004 John Wiley & Sons, Ltd.     

Stabilité linéaire d'écoulements symétriques presque parallèles en canalLinear stability of nearly parallel symmetric flows in a channel.

In this Note, we present a temporal linear stability analysis of symmetric developing flows slightly perturbed from Poiseuille flow. The Chebyshev spectral collocation method is used to resolve the Orr–Sommerfeld equation. For the main flow, the solution considered is analytic. The results of the stability study depend essentially on the shape and amplitude of the velocity profiles imposed at the channel entry. To cite this article: A. Hifdi et al., C. R. Mecanique 332 (2004).     

An exact mathematical solution for entrance-region laminar heat transfer with axial conduction

Summary A relatively simple mathematical scheme is proposed by which the entrance-region temperature solution for laminar flow heat transfer with axial conduction can be rigorously obtained.For Poiseuille pipe flow (parabolic velocity profile) with uniform wall heat flux, the accurate first twelve eigenvalues, eigenfunctions, and the coefficients of the series expansion which are required in the temperature solution have been determined for Peclet numbers of 5, 10, 20, 30, 50, and 100. In addition, asymptotic expressions for the eigenvalues and the eigenfunction,R _n (1), are derived. It is possible to use the asymptotic equation to predict, with satisfactory accuracy, even the first few eigenvalues for all the Peclet numbers considered.By employing the computed eigenvalues and the relevant constants, the effect of axial conduction on the entrance-region temperature profile and local Nusselt numbers has been examined and reported.This work was performed under the auspices of the U.S. Atomic Energy Commission.