Stokes’ second problem

The cylinder-wall circumferential velocity can be an arbitrary function of time, with the fluid velocity still subject to parallel-flow assumptions. The cylindrical analog of Stokes Second problem is to let the cylinder-wall velocity oscillate in a periodic manner. The wall velocity is specified as... 

Oscillatory motion of aflat plate. Another nonstationary problem that admits an exact solution [427] (known as Stokes second problem) describes the flow near an infinite plate oscillating in its plane. This is a problem without initial data. In this case the boundary conditions are posed as follows ... 

Stokes second problem deals with the behavior of a semi-infinite fluid if the wall bounding the fluid performs harmonic oscillations in its plane. This problem is stated as a problem without initial data [181,482] only the boundary condition... 

Stokes second problem and is specifically referred to as the slip velocity approach in electroosmotic flows. More general discussions of the applicability of such slip velocity approach in electrokinetic flows were provided elsewhere [6]. Using the slip velocity approach, the steady velocity field of a fully developed flow driven by an applied electric field, E, and a pressure gradient, dp/dz, is governed by the Stokes equation, expressed as... 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

Let us, for a moment, consider a single particle in one dimension with a Hamiltonian of the type// = p2/2m + V(x). This is a second-order differential operator, and this means that the general solution to the inhomogeneous Eq. (3.51)—considered as a second-order differential equation—will consist of a linear superposition of two special solutions, where the coefficients will depend on the boundary conditions introduced. As a specific example, one could think of the two solutions to the JWKB problem, their connection formulas, and the Stoke s phenomenon for the coefficients. 

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

When We = 0, the Oldroyd-B model (26) reduces to a three-field version of the Stokes problem. For e < 1, this problem is stable under condition (27). It was proven in [106] that, in the case of the Maxwell-type problem (where = 1), one has to add a second inf sup condition to obtain stability ... 

The linear system obtained by the discretization of equations (5)-(6) can also be solved directly. Notice that this system is symmetric but not definite positive. A three-field version of the Stokes problem was considered in [17] and a second inf-sup condition is then necessary to obtain stability (equation (30) of 6.4). 

The three-field formulation should reduce to a convenient approximation of the Stokes problem when applied to a Newtonian flow. Hence a second inf-sup condition is necessary to obtain stability. If the approximation (Tv)h of the extra-stress tensor is continuous, this supplementary condition can be satisfied by using a sufficient number of interior nodes in each element. On the contrary if this approximation is discontinuous, this can be done by imposing that the derivatives DUh of the approximated velocity field are in the space of (Tv)h- Various possible choices concerning the satisfaction of the inf-sup condition and the introduction of upwinding have been explored since 1987. In the following we will recall the basic steps (see [10], [24] and [38] for details). 

It has been argued that in the higher Knudsen number regime, the Burnett equations will allow continued application of the continuum approach. In practice, many problems have been encountered in the numerical solution and physical properties of the Burnett equations. In particular, it has been demonstrated that these equations violate the second law of thermodynamics. Work on use of the Burnett equations continues, but it appears to be unlikely that this approach will extend our computational capabilities much further into the high Knudsen number regime than that offered by the Navier-Stokes equations. 

The second boundary condition is related to the transformation of the flow profile in the surface layer into the flow profile for the Stokes problem on the flow around a sphere outside the thin surface layer near the particle surface [4] and can be written as... 

A second problem, closely related to Stokes problem, is the steady, buoyancy-driven motion of a bubble or drop through a quiescent fluid. There are many circumstances in which the buoyancy-driven motions of bubbles or drops are of special concern to chemical engineers. Of course, bubble and drop motions may occur over a broad spectrum of Reynolds numbers, not only the creeping-flow limit that is the focus of this chapter. Nevertheless, many problems involving small bubbles or drops in viscous fluids do fall into this class.23... 

A second straightforward example, solved previously by other means, is Stokes original problem of uniform flow past a stationary sphere. To apply the methods of the preceding subsection to this problem, it is convenient to transform to the disturbance flow problem,... 

To calculate the component of Fi in the e direction, we therefore require the solution of the original Stokes flow problem to obtain u0, plus the solution of a second Stokes flow problem for translation through a quiescent fluid to obtain u. However, we do not have to determine to calculate F i, and this represents a substantial simplification of the problem. 

Hence, as Oseen noted, we cannot expect the Stokes solution to provide a uniformly valid first approximation to the solution of (9 75), but instead expect that it will break down for large values of r > 0(Re ). Thus Whitehead s attempt to evaluate the second term in the expansion (9 77) was unsuccessfiil for large r. Indeed, as noted earlier in conjunction with the thermal problem, it is not so much a surprise that we cannot obtain a solution for boundary condition (9 81) for r oo, in spite of the fact that the governing equation (9 80) is not a valid first approximation to the full Navier Stokes equation except for r < 0(1 /Re). 

If we compare (10-12) with the full 2D Navier Stokes equation, expressed in terms of the streamfimction, we note that the latter is fourth order (the viscous terms generate V4i/f), whereas (10-12) is only second order. As a result, it is clear that the velocity field obtained from (10-12) will, at most, be able to satisfy only one of the boundary conditions of the original problem at the body surface. Intuitively, we may anticipate that the kinematic condition on the normal component of velocity,... 

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