Stokes flow in a junction of two-dimensional orthogonal channels

The purpose of this study is to calculate Stokes flow structures in relation to flow rate distribution in a junction of four orthogonal channels. Particular attention was paid to the existence of recirculation flow inside the channels for low Reynolds numbers. The presence of this recirculation flow may be observed, but only when the flow rate is relatively low in at least one of the branches.     

Stokes flow past a two-dimensional lens

Summary Two problems involving uniform Stokes flow past a two-dimensional lens are considered in an attempt to determine those geometric characteristics which cause separation and theresultant formation of large Stokes eddies. Criteria are derived which identify lenses having this property. It is found that a necessary condition for separation is that the lens be convex-concave.

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Zusammenfassung Zwei Probleme werden untersucht, die die gleichförmige Stokes-Strömung über eine zweidimensionale Linse betreffen, als ein Versuch, die geometrischen Eigenschaften fetzustellen, die die Ablösung und die resultierende Formation von grossen Stokes-Wirbeln verursachen. Kriterien werden abgeleitet, welche Linsen, die diese Chharakteristik haben, identifizieren. Es zeigte sich, dass die konvex-konkave Form der Linse eine notwendige Bedingung ist.

     

Stokes flow solutions in infinite and semi-infinite porous channels

We derive a class of exact solutions for Stokes flow in infinite and semi-infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no-slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi-infinite channel domain, the ground-state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane-based water purification, heat and mass transfer, and fuel cells.     

Stokes flow through a staggered array of rectangular cylinders and the junction resistance

The micro-fluid mechanical problem of viscous flow across an array of staggered rectangular cylinders is studied by eigenfunction expansions and matching. Streamlines and permeabilities are determined for (mean) flows normal and parallel to the stagger. A variety of approximate formulas are deduced. The resistance of a T junction is determined.     

The completed double layer boundary integral equation method for two-dimensional Stokes flow

Power and Miranda (1987) explained how integral equations ofthe second kind can be obtained for general exterior three-dimensionalStokes flows. They observed that, although the double layerrepresentation that leads to an integral equation of the secondkind coming from the jump property of its velocity field acrossthe density carrying surface can represent only those flow fieldsthat correspond to a force and torque free surface, the representationmay be completed by adding terms that give arbitrary total forceand torque in suitable linear combinations, precisely a Stokesletand a Rotlet located in the interior of the three-dimensionalparticles. Karrila and Kim (1989) called Power and Miranda'snew method the completed double layer boundary integral equationmethod, since it involves the idea of completing the deficientrange of the double layer operator. The main objective of thispaper is to extend Power and Miranda's completed method to theproblem of multiple cylinders in twodimensional bounded andunbounded domains. This extension is not trivial, owing to theunbounded behaviour at infinity of the fundamental solutionof the Stokes equation in two dimensions and the associatedparadoxes arising from this unbounded behaviour.     

Smooth interface in a two-component Stokes flow

The problem considered is that of evolution of the free boundary Γ(t) separating two immiscible viscous fluids with different constant densities and viscosities. The motion is described by the Stokes equations driven by the gravity force. We prove the existence of classical solutions for small timet and establish that the free boundary Γ(t)∈C ^l+2 (l>0 is an arbitrary non-integer number)     

On mass flux through a torus in Stokes flow

The symmetrical flow of a viscous fluid past a torus is considered with particular reference to the mass flux through the central hole. Asymptotic expressions are obtained when the hole is small and when it is relatively large, and are compared with the results of a numerical computation.     

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.     

Stokes flow past a slightly deformed fluid sphere

Summary The Stokes flow past a fluid spheroid whose shape deviates slightly from that of a sphere, is examined. To the first order in the small parameter characterizing the deformation, an exact solution is obtained. As an application, the case of a fluid oblate spheroid is considered and the drag experienced by it is evaluated. Special well-known cases are then deduced.

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Zusammenfassung Die Stokes-Strömung um einen Flüssigkeitstropfen, der nur leicht von einer perfekten Kugel abweicht, wird untersucht. Eine Lösung wird gefunden, die exakt ist bis zur ersten Ordnung im Deformations-Parameter. Als Beispiel wird der Strömungswiderstand eines abgeplatteten Flüssigkeits-Sphäroids berechnet. Die Methode liefert für bekannte Spezialfälle korrekte Lösungen.

     

Stokes flow response to a localised change in interfacial tension

The interfacial tension contribution to the normal stress balance leads to a rather complicated low-Reynolds number hydrodynamic response to the imposed surface tension variation. It is shown that the assumptions dictated by the model prevent the steady state of zero response from ever being attained and thereby demand that the forcing be switched off, after which the interface displacement decays algebraically in time with a small amplitude.     

Homogenization of a non-stationary Stokes flow in porous medium including a layer

We study the behavior of the solution to the non-stationary Stokes equations in a porous medium with characteristic size of the pores ε and containing a thin fissure {0?xn?η} of width η. The limit when ε and η tend to zero gives the homogenized behavior of the flow, which depends on the comparison between ε and η.     

Shape optimization for Stokes flow

This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations. The structures of continuous shape gradients with respect to the shape of the variable domain for some given cost functionals are established by introducing the Piola transformation and then deriving the state derivative and its associated adjoint state. Finally we give the finite element approximation of the problem and a gradient type algorithm is effectively used for our problem.     

Transient Stokes flow over a plane with a moving sleeve

The transient and stationary Stokes flow in the half space over a plane boundary containing a moving sleeve is determined in the case when the sleeve starts att=0 and moves with a constant velocity. Stream line plots showing the convergence towards a stationary state are presented.

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Zusammenfassung Die transiente und die stationäre Stokesschen Strömungen in einem Halbraum über einer, mit einem konstant bewegten Schieber versehener Grenzfläche, wird bestimmt. Stromlinienfiguren, die die Konvergenz gegen einen stationären Zustand zeigen, werden vorgestellt.

     

Motivic orthogonal two-dimensional representations of Gal(ℚ/ℚ)

Given a two-dimensional compatible family of ℓ-adic representations which is motivic and which respects an orthogonal form up to similitudes, we show how to express itsL-function in terms of a Hecke character. We give several examples and in particular we analyze a representation associated to a certainK3 surface which arose in the study of Kloosterman sums.     

A hybrid genetic algorithm-heuristic for a two-dimensional orthogonal packing problem

In this paper we address a two-dimensional (2D) orthogonal packing problem, where a fixed set of small rectangles has to be placed on a larger stock rectangle in such a way that the amount of trim loss is minimized. The algorithm we propose hybridizes a placement procedure with a genetic algorithm based on random keys. The approach is tested on a set of instances taken from the literature and compared with other approaches. The computation results validate the quality of the solutions and the effectiveness of the proposed algorithm.