Stokes slip flow between corrugated walls

Stokes flow between corrugated plates in microdomains has been analyzed using a perturbation method. This approach used the incompressible Navier-Stokes equations, but the velocity-slip is present along the solid-fluid interface. For the slip flow regime, if we introduce Knudsen number (K _n) herein, 0.01 K _n 0.1, the total flow rate is increasing as a ratio of 1 + 6K _nto no-slip Stokes flow. If we consider fixedK _ncases, the corrugations still decrease the flow rate, consideringO(²) terms, and the decrease is maximum as the phase shift becomes 180 °.     

The drag on two spheres in contact in the slip flow regime

The drag on two equal spheres in rigid contact falling along their line of centres is calculated in the slip flow regime. Standard viscous flow theory is employed with the necessary slip boundary conditions. The drag is seen to depend on the ratio=l/a wherel is the mean free path of an atom in the surrounding gas and a is the radius of the sphere. Two exact limiting cases are obtained for the drag, viz: small and large, and a useful interpolation formula is derived by means of a variational principle. The equivalent sphere approximation is found to give satisfactory results.

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Zusammenfassung Der aerodynamische Widerstand auf zwei gleiche Kugeln, die aneinander haften und die sich entlang ihrer Verbindungslinie bewegen, wird berechnet. Die Stokes Gleichungen werden mit Gleitgeschwindigkeits-Randbedingungen (slip flow) gelöst. Es zeigt sich, daß der Widerstand vom Verhältnisl/a abhängt, wol die freie Weglänge eines Moleküls im Gas unda der Kugelradius sind. In zwei Grenzfällen kann der Widerstand exakt berechnet werden: Für ganz kleine und ganz große, wo der Parameter proportional zur Knudsen Zahl ist. Für Zwischenwerte wird aus Prinzipien der Variationenrechnung eine nützliche Interpolationsformel hergeleitet. Die Näherungsmethode der aequivalenten Kugel liefert befriedigende Resultate.

     

Flow of a fluid of grade two between two eccentric rotating spheres

The isothermal, stationary and isochoric flow of a fluid of grade two between a pair of rotating eccentric spheres is investigated. The equations of motion of first and second order are formulated and solved for the first order only. However, the equation of second order indicates the presence of secondary flow. The stress distributions are computed and used to determine the resultant forces and torques acting on the stationary outer sphere. An important result for rheometry is that the resultant torques can be used to determine the coefficient of viscosity, while the resultant force in the direction of the axis of symmetry may be employed to determine the second normal stress difference.     

Stokes flow with slip and Kuwabara boundary conditions

The forces experienced by randomly and homogeneously distributed parallel circular cylinder or spheres in uniform viscous flow are investigated with slip boundary condition under Stokes approximation using particle-in-cell model technique and the result compared with the no-slip case. The corresponding problem of streaming flow past spheroidal particles departing but little in shape from a sphere is also investigated. The explicit expression for the stream function is obtained to the first order in the small parameter characterizing the deformation. As a particular case of this we considered an oblate spheroid and evaluate the drag on it.     

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.     

Stokes flow past an assemblage of slip eccentric spherical particle‐in‐cell models

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of slip eccentric spherical particle‐in‐cell models with Happel and Kuwabara boundary conditions is investigated. A linear slip, Basset type, boundary condition on the surface of the spherical particle is used. Under the Stokesian approximation, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based on the particle and fictitious spherical envelope. The boundary conditions on the particle's surface and fictitious spherical envelope are satisfied by a collocation technique. Numerical results for the normalized drag force acting on the particle are obtained with good convergence for various values of the volume fraction, the relative distance between the centers of the particle and fictitious envelope and the slip coefficient of the particle. In the limits of the motions of the spherical particle in the concentric position with cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Heat transfer in a MHD flow between eccentric disks rotating at different speeds

Heat transfer in a MHD flow between two infinite eccentric disks rotating with different speeds is considered when the plates are maintained at different temperatures. The results for the corresponding nonmagnetic case presented wrongly by Banerjee and Borkakati [7] are corrected. It is observed that the eccentric rotation reduces the heat transfer on the disks.     

Heat transfer in a hydrodynamic flow between eccentric disks rotating at different speeds

Heat transfer in the flow between two infinite eccentric disks rotating with different speeds is considered when the plates are maintained at different temperatures.

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Zusammenfassung Es wird die Wärmeübertragung im Fluid zwischen zwei unendlichen Platten mit verschiedenen Temperaturen berechnet, die exzentrisch mit verschiedenen Winkelgeschwindigkeiten rotieren.

     

Asymptotic profile of steady Stokes flow around a rotating obstacle

We analyze the spatial anisotropic profile at infinity of steady Stokes flow around a rotating obstacle. It is shown that the flow is largely concentrated along the axis of rotation in the leading term and that a rotating profile can be found in the second term. The proof relies upon a detailed analysis of the associated fundamental solution tensor.     

Couette flow of a third grade fluid with rotating frame and slip condition

An incompressible third grade fluid occupies the porous space between two rigid infinite plates. The steady rotating flow of this fluid due to a suddenly moved lower plate with partial slip of the fluid on the plate is analysed. The fluid filling the porous space between the two plates is electrically conducting. The flow modeling is developed by employing a modified Darcy’s law. A numerical solution of the governing problem consisting of a non-linear ordinary differential equation and non-linear boundary conditions is obtained and discussed. Several limiting cases of the arising problem can be obtained by choosing suitable parameters.     

Shape optimization for Stokes problem with threshold slip

We study the Stokes problems in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning the smoothness of Ω solutions to the Stokes system with the slip boundary conditions depend continuously on variations of Ω. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero.     

Approach to equilibrium of a rotating sphere in a Stokes flow

We study the unsteady rotary motion of a sphere immersed in a Stokes fluid. The equation of motion for the sphere leads to an integro-differential equation, and we are interested in the asymptotic behavior in time of the solution. Preparing initially the system (sphere + fluid) as a stationary state, we prove that the angular velocity of the sphere slows down with a law t ^−3/2 if no other forces than the one exerted by the fluid act on the sphere, while if the sphere is subject also to an elastic torque the asymptotic behavior of the angular position of the sphere is t ^−γ , with γ = 5/2 if the initial angular velocity is zero, γ = 3/2 otherwise. This behavior is due to the memory effect of the surrounding fluid. We discuss briefly other initial preparations of the system.     

Navier–Stokes equations with slip boundary conditions

In this paper, we consider incompressible viscous fluid flows with slip boundary conditions. We first prove the existence of solutions of the unsteady Navier–Stokes equations in n‐spacial dimensions. Then, we investigate the stability, uniqueness and regularity of solutions in two and three spacial dimensions. In the compactness argument, we construct a special basis fulfilling the incompressibility exactly, which leads to an efficient and convergent spectral method. In particular, we avoid the main difficulty for ensuring the incompressibility of numerical solutions, which occurs in other numerical algorithms. We also derive the vorticity‐stream function form with exact boundary conditions, and establish some results on the existence, stability and uniqueness of its solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

A weighted L q -approach to Stokes flow around a rotating body

Considering time-periodic Stokes flow around a rotating body in or we prove weighted a priori estimates in L ^q -spaces for the whole space problem. After a time-dependent change of coordinates the problem is reduced to a stationary Stokes equation with the additional term in the equation of momentum, where ω denotes the angular velocity. In cylindrical coordinates attached to the rotating body we allow for Muckenhoupt weights which may be anisotropic or even depend on the angular variable and prove weighted L ^q -estimates using the weighted theory of Littlewood-Paley decomposition and of maximal operators. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503, by the Grant Agency of the Academy of Sciences No. IAA100190505, and by the joint research project of DAAD (D/04/25763) and the Academy of Sciences of the Czech Republic (D-CZ 3/05-06).     

Flow between eccentric disks rotating at different speeds: Inertia effects

Summary The problem of flow between infinite eccentric disks rotating at different speeds is considered and a solution to the complete Navier-Stokes equations obtained. An approximate analytic solution is also presented.

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Zusammenfassung Das Strömungsproblem zwischen unendlichen exzentrischen Scheiben, rotierend bei verschiedenen Geschwindigkeiten, wird behandelt und eine Lösung der kompletten Navier-Stokes-Gleichungen erhalten. Eine analytische Näherungslösung wird ebenfalls beschrieben.

     

Qualitative behavior of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres

In this article, the bounds of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres have been studied. Based on Lagrange multiplier method, the function extremum theory and the generalized positive definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and the globally exponentially attractive set for this system. The results that obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension and the Lyapunov dimension of chaotic attractors. © 2016 Wiley Periodicals, Inc. Complexity 21: 67–72, 2016     

同心球间流动三模类Lorenz系统的动力学行为及数值仿真

讨论了同心球间旋转流动的类Lorenz型方程组的动力学行为及其数值模拟问题,求出了该方程组平衡点,并对其稳定性进行了分析,证明了该方程组吸引子的存在性,对类Lorenz方程组的动力学行为进行了数值模拟,数值试验表明此类Lorenz型方程组存在极限环和奇怪吸引子.     

Soret and Dufour effects on MHD slip flow with thermal radiation over a porous rotating infinite disk

In this investigation, thermal radiation effect over an electrically conducting, Newtonian fluid in a steady laminar magnetohydrodynamic convective flow over a porous rotating infinite disk with the consideration of heat and mass transfer in the presence of Soret and Dufour diffusion effects is investigated. The partial differential equations governing the problem under consideration are transformed by a similarity transformation into a system of ordinary differential equations which are solved numerically using fourth order Runge–Kutta based shooting method. The effects of the magnetic interaction parameter, slip flow parameter, Soret number, Dufour number, Schmidt number, radiation parameter, Prandtl number and suction parameter on the fluid velocity, temperature and concentration distributions in the regime are depicted graphically and are analyzed in detail. The corresponding skin-friction coefficients, the Nusselt number and the Sherwood number are also calculated and displayed in tables showing the effects of various parameters on them.     

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.