Stability analysis of the plane Couette flow for a model kinetic equation

The stability of the plane Couette flow is studied using the simplified Boltzmann equation (the BGK equation) in which the high modes in the space of velocities and coordinates are truncated. The solution to the Navier-Stokes equation with small additional terms depending on the Knudsen number is used as the stationary solution. We assume that the perturbations depend only on the coordinate that is orthogonal to the flow. The density perturbations are assumed to be nonzero. In this approximation, the problem is found to be unstable in the case of small Knudsen numbers.     

Couette flow of dipolar fluids

An exact solution for the problem of steady flow of an incompressible, isothermal and homogeneous dipolar fluid in the annular space between two concentric circular cylinders rotating with uniform angular velocities is obtained. The effect of the material constants on the torque acting on the cylinders is studied particular cases.     

Couette flow between corrugated cylinders

This note reports some analytic Stokes solutions to the flows outside a rotating corrugated cylinder and between two rotating corrugated cylinders of either epitrochoidal or hypotrochoidal cross-sections. The results are obtained by using conformal mappings and general biharmonic solutions inside and outside of a unit circle.     

Couette flow of a non-homogeneous fluid

The two dimensional Couette flow of a non-homogeneous viscous fluid is studied. The plane boundaries of the channel are maintained at different temperatures. The upper plane moves with a uniform horizontal velocity and the lower plane is at rest. The fluid is subjected to suction and injection at the boundaries. Thesteady equations are solved by introducing similarity variables which are expanded in series of powers of a small stratification parameter. The non-linear theory predicts that the temperature depends on the distancex from the throat section, an observation which is not predicted by the linear theory. The non-linear effects on velocity and temperature are studied. The rate of heat transfer is discussed.     

Non-isothermal spherical Couette flow of Oldroyd-B fluid

The present paper is concerned with non-isothermal spherical Couette flow of Oldroyd-B fluid in the annular region between two concentric spheres. The inner sphere rotates with a constant angular velocity while the outer sphere is kept at rest. The viscoelasticity of the fluid is assumed to dominate the inertia such that the latter can be neglected in the momentum and energy equations. An approximate analytical solution is obtained through the expansion of the dynamical variable fields in power series of Nahme number. Non-homogeneous, harmonic for axial- velocity and temperature equations and biharmonic for stream function equations, have been solved up to second order approximation. In comparison of the present work with isothermal case; [1,2], two additional terms; a first order velocity and a second order stream function are stem as a result of the interaction between the fluid viscoelasticity and temperature profile. These contributions prove to be the most important results for rheology in this work.     

Non-isothermal spherical Couette flow of Oldroyd-B fluid

The present paper is concerned with non-isothermal spherical Couette flow of Oldroyd-B fluid in the annular region between two concentric spheres. The inner sphere rotates with a constant angular velocity while the outer sphere is kept at rest. The viscoelasticity of the fluid is assumed to dominate the inertia such that the latter can be neglected in the momentum and energy equations. An approximate analytical solution is obtained through the expansion of the dynamical variable fields in power series of Nahme number. Non-homogeneous, harmonic for axial- velocity and temperature equations and biharmonic for stream function equations, have been solved up to second order approximation. In comparison of the present work with isothermal case; [1,2], two additional terms; a first order velocity and a second order stream function are stem as a result of the interaction between the fluid viscoelasticity and temperature profile. These contributions prove to be the most important results for rheology in this work.     

Periodic waves in rotating plane Couette flow

A class of nonlinear equations of Navier-Stokes type of the form (**) $$\frac{{dw}}{{dt}} + L_0 w + (\lambda - \lambda _0 )L_1 w + \gamma \lambda (M_1 w + M_2 w) + \gamma ^2 L_3 (\lambda ,\gamma )w + B(w) = 0$$ is investigated, where λ is a “load” parameter (i.e., a Reynolds number), γ is a “structure” parameter,L _o L ₁,M ₁,M ₂ andL ₃(λ, γ) are linear operators, andB is a quadratic operator. An equation of the form (**) describes a variety of spiral flow problems including rotating plane Couette flow which is studied here in detail. Under suitable hypotheses on the operators in (**), it is shown that Hopf bifurcation occurs for γ sufficiently small. In the problem of rotating plane Couette flow, by determining the sign of the real part of a certain “cubic” coefficient, it is shown, in addition, that the bifurcating periodic orbits are supercritical and asymptotically stable, and correspond to periodic waves.     

Radiative magnetogasdynamic Couette flow with variable parameters

Couette flow of an electrically conducting compressible fluid with uniform magnetic field applied transversely between flat walls of arbitrary electrical conductivity, radiative emissivity and temperature is analysed. The equations of steady flow are formulated with the introduction of temperature dependent coefficients of viscosity and thermal and electrical conductivity, together with absorption coefficient in addition dependent upon the density. Profiles of velocity, induced magnetic field, radiative flux and temperature are derived numerically under a one-dimensional modelling. The parameter dependence, expressed as a power law, is found to have a marked effect upon the velocity and magnetic field but to be of little import for the thermal profiles. Similar influence is found for the effect of wall electrical conductivity and emissivity. On the other hand, changes in the strength of the fluid thermal conductivity significantly affect the temperature profile in the presence of electromagnetic interaction.

---

Résumé Pour un fluide conducteur, rayonnant et compressible sous l'influence d'un champ magnétique constant et transverse l'écoulement de Couette entre deux surfaces planes conductrices et rayonnantes avec des températures arbitraires est analysé. On formule les équations d'un écoulement permanent quand la viscosité, la conductibilité thermique et la conductibilité électrique dépendent de la température et le coffficient d'absorption de la densité. On calcule la vitesse, le champ magnétique induit, le rayonnement et la température pour des écoulements qui ne sont fonction que de la cordonnée transverse.La vitesse et le champ magnétique dépendent de façon significative de ce paramètre, mais pas la température ni le rayonnement. On obtient des résultats similaires pour la conductibilité électrique et l'emissivité de rayonnement des surfaces planes. D'autre part quand la conductiblité électrique du fluide n'est pas nulle, une variation de la conductibilité thermique modifie la température d'une manière significative.

     

Some considerations of radiation in magnetohydrodynamic Couette flow

Zusammenfassung In der Abhandlung wird der Einfluss magnetischer Felder auf den Energietransport in inkompressibler Couette-Strömung zwischen zwei sich relativ zueinander bewegenden ebenen Platten untersucht. Das äussere Magnetfeld wirkt senkrecht zur Strömungsrichtung und senkrecht zur Oberfläche der Platten. Die beiden isothermen Platten emittieren und reflektieren diffus Wärmestrahlung. Das Strömungsmedium ist isotrop und kann Wärme absorbieren und emittieren. Die nicht lineare Integro-Differential-Energiegleichung ist näherungweise gelöst. Die numerischen Ergebnisse zeigen den Einfluss der Hartmannschen Zahl und anderer dimensionsloser Kenngrössen auf die Temperaturverteilung und die Wärmeübertragung.     

DNS of stochastically forced laminar plane Couette flow

Background of three dimensional hydrodynamic/vortical fluctuations in a stochastically forced, laminar, incompressible, plane Couette flow is simulated by direct numerical simulations (DNS). It was found that the fluctuating field has well pronounced peculiarities: (i) The hydrodynamic fluctuations exhibits non–exponential, transient growth; (ii) Streamwise non–constant fluctuations with the characteristic length scale of the order of the channel width are predominant in the fluctuating spectrum; (iii) Existence of coherent structures in the fluctuating background; (iv) Stochastic forcing breaks the spanwise reflection symmetry (inherent to the linear and full Navier–stokes equations in a case of the Couette flow) and inputs an asymmetry on dynamical processes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of quasi-steady acceleration in circular Couette flow

The effect of inner cylinder acceleration in the transition from circular Couette flow to Taylor vortex regime was numerically investigated. The solution of the eigenvalue problem when the Couette flow was perturbed allowed the determination of the relationship between the inner cylinder acceleration rate and the acceleration time duration expressed in terms of viscous diffusion time. The analysis of the numerical results also allowed the concept of quasi-steady acceleration in the Couette system to be qualitatively described.     

Numerical study of multiple periodic flow states in spherical Couette flow

The supercritical flow states of the spherical Couette flow between two concentric spheres with the inner sphere rotating are investigated via direct numerical simulation using a three-dimensional finite difference method. For comparison with experiments of Nakabayashi et al. and Wimmer, a narrow gap and a medium gap with clearance ratio β=0.06 and 0.18 respectively are considered for the Reynolds number range covering the     

Numerical study of multiple periodic flow states in spherical Couette flow

The supercritical flow states of the spherical Couette flow between two concentric spheres with the inner sphere rotating are investigated via direct numerical simulation using a three-dimensional finite difference method. For comparison with experiments of Nakabayashi et al. and Wimmer, a narrow gap and a medium gap with clearance ratio β=0.06 and 0.18 respectively are considered for the Reynolds number range covering the first Hopf bifurcation point. With adequate initial conditions and temporary imposition of small wave-type perturbation, multiple periodic flow states with three different pair numbers of spiral Taylor-Görtler (TG) vortices have been simulated successfully for β=0.06, of which the 1-pair and 2-pair of spiral TG vortices are newly obtained. Three different periodic flow states with shear waves, Stuart vortices or wavy outflow boundary, have been obtained for β=0.18. Analysis of the numerical results reveals these higher flow modes in terms of fundamental frequency, wave number and spatial structure.     

Unsteady MHD Couette flow in a rotating system

The unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system has been considered. An exact solution of the governing equations has been obtained by using a Laplace transform. Solutions for the velocity distributions as well as shear stresses have been obtained for small times as well as for large times. It is found that for large times the primary velocity decreases with increase in the rotation parameter K² while it increases with increase in the magnetic parameter M². It is also found that with increase in K², the secondary velocity v₁ decreases near the stationary plate while it increases near the moving plate. On the other hand, the secondary velocity decreases with increase in the magnetic parameter.     

Time-dependent MHD Couette flow in a porous annulus

This study presents the solution for the MHD transient Couette flow in an annulus formed by two concentric porous cylinders of infinite length. The fluid flow is induced by either the impulsive or the accelerated movements of the outer cylinder. A uniform magnetic field is assumed to be applied perpendicular to the direction of flow. General solution of the governing equations is obtained using a combination of Laplace transform and the Riemann-sum approximation method of Laplace inversion. The expressions for the skin friction at the two walls are obtained in both cases. The variations of the velocity and the skin friction with respect to the Hartmann number and suction/injection parameter have been discussed. It is found out that suction accelerates the flow whereas injection retards the flow.     

A theorem on stability of a planar Couette flow

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 11, pp. 1541–1548, November, 1989.     

Numerical analysis of oscillatory Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for a hard sphere molecular gas

The time-dependent motion of a rarefied gas between two parallel planes caused by an oscillatory motion of the plane is studied based on the linearized Boltzmann equation for a hard sphere molecular gas. With the aid of a deterministic numerical method, an accurate numerical analysis is carried out for a wide range of gas rarfaction and oscillatory frequency. The detailed data of the shear stress acting on the planes is provided in a complete form for a wide range of the parameters. The transition of the solution from low to high frequencies under various degrees of gas rarfaction is discussed.