Effects of radial temperature gradient on the stability of flow of a viscous incompressible fluid between two rotating cylinders

Stability of the flow of a viscous incompressible fluid in a narrow gap between two concentric cylinders rotating in opposite directions is studied. Exact values of critical Taylor number for the first and second modes are derived by a numerical procedure and these are compared with the known results. It is observed that the value of the critical Taylor number T_crit decreases with increasing the values of a parameter $\text{N (N =}\text{Ra}\text{T}\text{,}\text{Ra = Rayleigh number, T = Taylor number}\text{)}$, and the rate of decrease in the value of T_crit is more when the two cylinders rotate in the opposite directions.     

On the flow of a viscous incompressible fluid between two coaxial rotating porous cylinders

Exact solution of the Navier-Stokes equations for the laminar flow of a viscous incompressible fluid between two coaxial rotating porous cylinders, kept at constant temperatures, has been studied. The rate of injection at one cylinder is taken to be the same as the rate of suction at the other. Expressions for the velocity and temperature distributions and for the torque required to turn the outer cylinder are obtained. The effects of λ (injection parameter), σ (the ratio of the radii of the cylinders) and Pé (Péclet number = λPr) on them are shown graphically.     

Exact solutions for the rotational flow of a generalized Maxwell fluid between two circular cylinders

Unsteady flow of an incompressible generalized Maxwell fluid between two coaxial circular cylinders is studied by means of the Laplace and finite Hankel transforms. The motion of the fluid is produced by the rotation of cylinders around their common axis. The solutions that have been obtained, written in integral and series form in terms of the generalized G_a,b,c(·, t)-functions, are presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions. They satisfy all imposed initial and boundary conditions and for λ → 0 reduce to the solutions corresponding to the Newtonian fluids performing the same solution. Furthermore, the corresponding solutions for ordinary Maxwell fluids are also obtained for β = 1. Finally, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity field ω(r, t) have been depicted against r and t for different values of the material and fractional parameters.     

Effect of porous lining on the flow between two concentric rotating cylinders

The effect of non-erodible porous lining on the flow between two concentric rotating cylinders is investigated using Beavers and Joseph slip boundary condition. It is shown that the shearing stress at the walls increases with the porous lining thickness parameter ε.     

Steady-state flow of an incompressible viscoelastic polymer fluid between two coaxial cylinders

A boundary value problem for a quasi-linear equation determining the velocity profile of a flow of a polymer fluid in a pipe formed by two coaxial cylinders is considered. On the basis of methods of approximation without saturation, a computational algorithm of increased accuracy is developed, making it possible to solve the problem in a wide range of parameters, including record-low values of r ₀, the radius of the inner cylinder.     

Vibrational mixing of a fluid between two quasiconcentric cylinders

We consider the possibility of intense mixing of a viscous fluid in the gap between two quasiconcentric cylinders, with one of the cylinders performing high-frequency vibrations about its axis. The motion of the fluid is described by Navier-Stokes equations for the axisymmetric case. The stream function is represented by a generalized Fourier series. The small parameter is the ratio of the vibration amplitude to the radius of the external cylinder. Calculations carried out in the zeroth approximation produced the pattern of stream lines for various Reynolds numbers, vibration amplitudes, and ratios of external and internal radii. The mixing intensity was found to increase substantially with the reduction of the gap between the cylinders, whereas variation of the ratio of the vibration amplitude to the Reynolds number did not produce marked qualitative changes. The fluid flow in this system generates a contraction semigroup, which makes it possible to derive the ergodicity criterion for the stream function.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 35–39, 1986.     

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 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.     

Helical flow of a Bingham fluid arising from axial Poiseuille flow between coaxial cylinders

The Bingham fluid model represents viscoplastic materials that display yielding, that is, behave as a solid body at low stresses, but flow as a Newtonian fluid at high stresses. In any Bingham flow, there may be regions of solid material separated from regions of Newtonian flow by so-called yield boundaries. Such materials arise in a range of industrial applications. Here, we consider the helical flow of a Bingham fluid between infinitely long coaxial cylinders, where the flow arises from the imposition of a steady rotation of the inner cylinder (annular Coutte flow) on a steady axial pressure driven flow (Poiseuille flow), where the ratio of the rotational flow compared to the axial flow is small. We apply a perturbation procedure to obtain approximate analytic expressions for the fluid velocity field and such related quantities as the stress and viscosity profiles in the flow. In particular, we examine the location of yield boundaries in the flow and how these vary with the rotation speed of the inner cylinder and other flow parameters. These analytic results are shown to agree very well with the results of numerical computations.     

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 modes of instability for flow between rotating cylinders with a transverse pressure gradient

Résumé L'analyse de l'effet du gradient transversal de la pression sur la stabilité de l'écoulement de Couette d'un fluide newtonien était basée sur la supposition que l'apparition de l'instabilité est due à la présence d'un mouvement secondaire, stationnaire et axialement symmétrique. Récemment on a demontré (Hughes et Reid) que, pour un tel écoulement, il existe des intervals de nombres d'ondes longitudinales de perturbation, pour lesquels les deux modes stationnaires inferieurs n'existent pas. Comme d'aprés le critère de Rayleigh, des couches stables et instables du fluide existent dans l'écoulement de base, il paraît raisonnable de poser la question si les modes d'oscilla tions neutres, du type étudié par Chang et Sartory pour un écoulement hydromagnetique de Couette, peuvent apparaître à ces nombres d'onde. On prouve dans ce travail qu'un tel type de mode oscillatoire des perturbations peut exister et que l'apparition de l'instabilité peut dépendre de l'apparition d'un tel mode.     

On the unsteady flow and temperature distribution of a viscous incompressible fluid between two parallel flat plates

The unsteady flow and temperature distribution of a viscous incompressible fluid between two parallel flat plates has been investigated. The pressure gradient is varying linearly with time. The velocity and temperature profiles for various values oft have been shown graphically. Fort = 0, the velocity and temperature profiles correspond to the plane Poiseuille flow. The velocity and temperature both increase with time and they are maximum on the central plane of the channel.     

Modeling the join curve between two co-axial carbon nanotubes

Making use of an applied mathematical model, we employ a calculus of variations technique to join two co-axial nanotubes. Due to the axial symmetry of the tubes, the three-dimensional problem can be reduced to a problem in two dimensions. The curvature squared for the join region is minimized for a prescribed join length and given tube radii. In this model, a certain non-dimensional parameter B arises, which approximately has the same numerical value when compared with the standard method for the joining between any two carbon nanotubes of different radii. This value occurs in consequence of adopting an angle of inclination of 9.594°, which occurs in the conventional method for joining two carbon nanotubes of different radii and which is necessary to accommodate a single pentagon. The simple calculus of variations model described here provides a general framework to connect nanotubes or other nanostructures.     

Occurrence of Taylor vortices in the flow between two rotating conical cylinders

The behavior of the flow between two coaxial conical cylinders with the inner one rotating and the outer one stationary is studied numerically. Special attention is paid to the occurrence of Taylor vortices in basic flow and unsteady helical vortices. It is found that, in basic flow, the vortices occur in the direction toward smaller radius, while toward bigger radius in unsteady helical vortices; moreover, the unsteady helical vortices can coexist with unstable steady Taylor vortices. The results suggest that the behavior of conical flow is dominated by a competition between the meridional flow and radial flow. The effect of meridional flow is most significant at small apex angle or in basic flow and helical vortices, while the radial flow dominates the structure at larger apex angle or in steady vortical flow. In order to get better understanding the competition and the transition of Taylor–Couette flow to conical flow, a velocity angle related to velocity components is defined, and the pattern evolution of velocity, streamlines and the velocity angle are examined with respect to apex angle, as well as Reynolds number. Finally, the statistical properties of turbulent conical flow are investigated.     

On an extremal property of the Poiseuille flow between two coaxial cylinders

Some extremal property is established of the axial stationary flow of a Newtonian fluid between two concentric cylinders. It is shown that the ratio of the flow rate through the cross-section of an eccentric gap to the pressure drop (along the axis) reaches its minimum when the eccentricity equals zero; i.e., the minimum of the ratio is realized for the Poiseuille flow.     

A note on the flow of a fluid with pressure-dependent viscosity in the annulus of two infinitely long coaxial cylinders

The dependence of the viscosity of fluids on pressure has been well established by experiments and it needs to be taken into consideration in problems where there is a large variation of pressure in the flow domain. In this paper we consider the flow of a fluid in the annulus between two cylinders whose viscosity depends on the pressure. First we consider the steady flow in the annulus due to the rotation of one cylinder with respect to the other. Then we study the problem of flow in the annular region due to torsional and longitudinal oscillations of one cylinder with respect to the other. In both the problems considered the flow is found to be markedly different from that for the incompressible Navier–Stokes fluid with constant viscosity.     

Hall effects on hydromagnetic flow of an Oldroyd 6-constant fluid between concentric cylinders

The hydromagnetic flow of an electrically conducting, incompressible Oldroyd 6-constant fluid between two concentric cylinders is investigated. The flow is generated by moving inner cylinder and/or application of the constant pressure gradient. Two non-linear boundary value problems are solved numerically. The effects of material parameters, pressure gradient, magnetic field and Hall parameter on the velocity are studied. The graphical representation of velocity reveals that characteristics for shear thinning/shear thickening behaviour of a fluid is dependent upon the rheological properties.     

On the steady flow of a Newtonian fluid between two parallel disk

Summary We show that the steady flow of a viscous fluid of moderate Reynolds number between two parallel disks has an exact solution which takes the form of a power series. Employing this exact solution the two-point boundary value problem for this class of flow is reduced to a nonlinear algebraic system which is then solved by a standard optimization method. Results are given for two particular cases, the continuous squeezing flow and the coaxial-disk flow.

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Zusammenfassung Wir zeigen, daß die ständige Strömung einer viskosen Flüssigkeit von mäßiger Reynoldszahl zwischen zwei parallelen Scheiben eine exakte Lösung hat, die die Form einer Potenzreihe annimmt. Wenn wir diese exakte Lösung anwenden, wird das Zwei-Punkte-Grenzwertproblem für diese Art von Strömung reduziert auf ein nichtlineares algebraisches System, das dann durch eine Optimierungsmethode gelöst werden kann. Angegeben werden Resultate für zwei Einzelfälle: die kontinuierliche Quetschströmung und die Strömung für Koaxialscheiben.