Secondary flow of non-Newtonian liquids between confocal elliptic cylinders in relative motion

Résumé On analyse l'écoulement de certains liquides non-newtoniens dans des tuyaux cylindriques qui contiennent des noyaux cylindriques parallèles se mouvant longitudinalement avec une vitesse constante. Les liquides sont tels qu'un écoulement purement rectiligne n'est pas toujours possible. On donne des calculs explicites pour le cas des cylindres elliptiques qui servent de limites rigides. On fait constater que l'on peut avoir un écoulement rectiligne parallèle aux axes des cylindres sur lesquels se superposent quatre écoulements tourbillonnaires dans des plans qui sont normaux par rapport à l'axe des cylindres.     

The effect of a pressure gradient on the secondary flow of non-Newtonian liquids between non-intersecting cylinders

Résumé On fait une analyse de l'écoulement de certains liquides non-newtoniens dans des tuyaux cylindriques qui contiennent des noyaux cylindriques parallèles, l'écoulement étant celui qu'induit une inclinaison de pression constante. Les liquides sont tels qu'un écoulement purement rectiligne n'est pas toujours possible. On donne des calculs explicites dans les cas où les limites rigides sont (i) des cylindres excentriques et circulaires et (ii) des cylindres confocaux et elliptiques. On constate que l'on peut avoir un écoulement rectiligne parallèle aux axes des cylindres sur lesquels se superposent des écoulements de tourbillon dans des plans qui sont normaux par rapport à l'axe des cylindres.     

Rheological and flow modelling of viscoelastic fluids between eccentric cylinders

Viscoelastic fluid flows within eccentric rotating cylinders are simulated using a finite element based commercial software POLYFLOW, and an approach based on the stream-tube method (STM) and domain decomposition.     

两个非同心旋转圆柱间粘性流动的广义雷诺方程及其基本流

运用张量分析方法及修正双极坐标系,建立了轴承润滑流动所应满足的广义Reynolds方程.应用薄流层中的Navier-Stokes方程的渐近分析方法和张量分析工具,得到了两个非同心旋转圆柱之间粘性流动的基本流所应满足的方程.这个基本流可以表示为两个同心旋转圆柱之间的Taylor流加上一个扰动项,并且给出了数值计算例子.     

The hydromagnetic flow between eccentric cylinders with velocity slip at the cylinder walls

The hydrodynamic flow between two eccentric cylinders is examined for small values of modified Reynolds number, porosity parameter and the non-dimensional slip velocity parameter in the presence of a radial magnetic field.The stream function and the pressure distribution are calculated and the results are presented graphically.     

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.     

Non-Newtonian flow between concentric cylinders

The nonlinear rheological effects of Oldroyd 6-constant fluid between concentric cylinders is addressed. Numerical solution of nonlinear differential equation is given. The nonlinear effects on the velocity is shown and discussed. This reveal that characteristics for shear thickening/shear thinning behavior of a fluid is dependent upon the rheological properties.     

Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized Maxwell fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and after some time both cylinders suddenly begin to oscillate around their common axis with different angular frequencies of their velocities. The solutions that have been obtained are presented under integral and series forms in terms of generalized G and R functions. Moreover, these solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary Maxwell fluid are also obtained as limiting cases of our general solutions. At the end, flows corresponding to the ordinary Maxwell and generalized Maxwell fluids are shown and compared graphically by plotting velocity profiles at different values of time and some important results are remarked.     

The mixing of a viscous fluid in a layer between rotating eccentric cylinders

The motion of an incompressible viscous fluid in a thin layer between two circular cylinders, inserted into one another, with parallel axes is investigated. The cylinders rotate relative to one another about an axis parallel to the axes of the cylinders. The stream function of the unsteady plane-parallel flow that occurs is found by solving the boundary-value problem for the equations of hydrodynamic lubrication theory. The motion of the fluid particles is found from the solution of a non-autonomous time-periodic Hamiltonian system with a Hamiltonian equal to the stream function. The positions of fluid particles over time intervals that are a multiple of the period of rotation (Poincaré points) are calculated. The set of points is investigated using a Poincaré mapping on the phase flow. The observed transition to chaotic motion is related to the mixing of the fluid particles and is investigated both numerically and using a mapping, calculated with an accuracy up to the third power of the small eccentricity. The optimum mode of motion is observed when the area of the mixing (chaos) region reaches its highest value.     

Stability of viscous flow between rotating cylinders

Zusammenfassung Ein Beweis des Rayleighschen Kriteriums der Stabilität einer zähen Strömung zwischen zwei unendlichen, coaxialen, rotierenden Zylindern, der auf einer einfachen Abänderung des Energieintegrals für die Perturbationsgeschwindigkeit begründet ist, wird dargestellt.     

Stability analysis of shear-thinning flow between rotating cylinders

Stability of the shear thinning Taylor–Couette flow is carried out and complete bifurcation diagram is drawn. The fluid is assumed to follow the Carreau–Bird model and mixed boundary conditions are imposed. The low-order dynamical system, resulted from Galerkin projection of the conservation of mass and momentum equations, includes additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. It is observed, that the base flow loses its radial flow stability to the vortex structure at a lower critical Taylor number, as the shear thinning effects increases. The emergence of the vortices corresponds to the onset of a supercritical bifurcation which is also seen in the flow of a linear fluid. However, unlike the Newtonian case, shear-thinning Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Complete flow field together with viscosity maps are given for different scenarios in the bifurcation diagram.     

Fully developed incompressible flow between non-coaxial circular cylinders

Exact formulae which relate the rate of mass flow to the constant axial pressure gradient in fully developed flow between non-coaxial circular cylinders are obtained for the first time. Of the two formulae which are presented, that which corresponds to the case of fully eccentric cylinders is particularly simple, being in structure no more complicated than the well known result for coaxial cylinders. Attention is also given to the wall shear stresses and to the drag which the fluid exerts on unit length of each cylinder. Exact formulae provides results which describe the dependence of these quantities on the ratio of the diameters of the cylinders and on the eccentricity.

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Résumé On a trouvé pour la première fois les formules exactes qui mettent en relation le flux de la masse au gradient de la pression pour le mouvement d'un fluide dans l'espace entre deux longs cylindres non coaxiaux. Quant les cylindres se touchent nous trouvons un résultat très simple. Nous dérivons aussi les formules exactes pour la tension de frottement à la paroi et la résistance visqueuse par unité de longueur de chaque paroi.

     

Linear oscillatory hydromagnetic flow between two coaxial cylinders

Hydromagnetic flow between two coaxial circular cylinders is discussed when the inner cylinder oscillates axially under a radial magnetic field. Exact solution is given for the case of a perfectly conducting fluid. Expressions for velocity, induced magnetic field, current density, electric field, viscous drag and energy transfer are derived and expressed in polar forms so as to facilitate the study of magnitude and phase variations. Current sheets are found to exist on the two boundaries.     

Oscillatory motion of a viscous fluid in a gap between vibrating cylinders

We investigate the achievement of intensive periodic vortical motion between two quasiconcentric cylinders of circular cross section, undergoing periodic oscillations with integral frequencies about a motionless axis parallel to the axes of the cylinders.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 74–78, 1990.     

Helical modes of instability in swirling flow between concentric cylinders

Zusammenfassung Die Stabilität einer reibungsfreien und schraubenartigen Strömung zwischen zwei konzentrischen Zylindern wird untersucht. Schranken für die Anfachungsgrösse und Fortpflanzungsgeschwindigkeit von nicht-rotationssymmetrischen Störungen sind abgeleitet und genaue Werte dieser Grössen für die Spezialfälle von Couette-Strömung zwischen einem ruhenden äusseren und einem drehenden inneren Zylinder mit zusätzlicher Axial-Geschwindigkeitskomponente, die lienare oder parabolische Funktionen der Radialrichtung sind, werden erhalten. Es ergibt sich, dass der Zusatz einer linear abhängenden axialen Geschwindigkeit destabilisierend wirkt, weil der Zusatz einer parabolischen Komponente stabilisierend ist. Die Maximalanfachungsgrössen befinden sich bei schraubenförmigen Störungen mit endlicher Steigung.     

Gliding motion of bacterium in a non-Newtonian slime

Gliding bacteria are a taxonomically heterogeneous group of rod-shaped prokaryotes that adhere to and translocate on certain substrata by mechanism(s) that are unknown. Organelles of motility, such as flagella, have not been observed on these bacteria. Furthermore, their rigidified cell walls preclude amoeboid type locomotion. The present investigation analyzes a motility model that involves an undulating cell surface which transmits stresses through a layer of exuded slime to the substratum. The non-Newtonian nature of the slime is taken into account by considering a specific fluid of the differential type, namely an incompressible homogenous fluid of grade three. The non-linear differential equation resulting from the balance of momentum and mass is solved analytically and numerically.     

Stokes flow between eccentric rotating spheres with slip regime

The steady axisymmetric flow problem of a viscous fluid contained between two eccentric spheres that rotate about an axis joining their centers with different angular velocities is considered. A linear slip of Basset-type boundary condition at both surfaces of the spherical particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in the spherical coordinate systems based on the inner solid particle and the spherical container. The boundary conditions on the particle’s surface and spherical container are satisfied by a collocation technique. Numerical results for the coupling coefficient acting on the particle are obtained with good convergence for various values of the ratio of particle-to-container radii, the relative distance between the centers of the particle and container, the slip coefficients and the relative angular velocity. In the limiting cases, the numerical values of the coupling coefficient for the solid sphere in concentric position with the container and when the particle is near the inner surface of the container are obtained, and the results are in good agreement with the available values in the literature. The variation of the coupling coefficient with respect the parameters considered are tabulated and displayed graphically.     

Stability of flow of viscoelastic fluids between arbitrarily spaced cylinders

Résumé On a étudié le problème de la stabilité d'un fluide viscoélastique du troisième ordre contenu entre deux cylindres concentriques. L'approximation habituelle que l'écart entre les cylindres est petit n'est pas faite. On a démontré que pour un fluide viscoélastique le nombre de Taylor critique dépend considerablement de l'écart entre les cylindres.