Stokes Problems

Hughes, T. J. R., Franca, L. P. and Balestra, M., 1986. A new finite-element formulation for computational fluid dynamics. 5. Circumventing the Babuska-Brezzi condition - a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolations. Cornput. Methods Appl. Meek Eng. 59, 85-99. 

Freed, KF Muthukumar, M, On the Stokes Problem for a Suspension of Spheres at Finite Concentrations, Journal of Chemical Physics 68, 2088, 1978. 

As in the case of Stokes problem we represent the total potential W at each point of the earth s surface as a sum of the normal and disturbing potentials ... 

Thus, the determination of heights of the quasi-geoid N requires knowledge of the disturbing potential T on the physical surface of the earth. As in the case of the Stokes problem, in order to calculate N we have to determine the disturbing potential, which obeys some boundary condition on the physical surface of the earth instead of the surface of a geoid. This is the main advantage of a new approach. 

Thus, we have derived the integral equation with respect to the function a. As in the case of Stokes problem it is possible to apply the spherical approximation, that is, the magnitude of the normal field at points of the surface S is... 

In this section we describe the spreadsheets used to solve the Stokes problem between a cylindrical shell and an inner rod that rotates with fixed rotation rate, Section 4.8. Both explicit and implicit solution procedures are illustrated. This problem has boundary conditions that are fixed in time, and solves the transient problem to the steady-state solution. Other problems discussed in Chapter 4 have time-varying boundary conditions or time-varying forcing functions. Solving these problems requires only very straightforward modification of the following examples. 

Investigation of the velocity profiles obtained for the first case (crossection (x, y = 70 nm)) indicates a strong decrease of the maximum fluid velocity with increasing electrolyte concentration (see Figure 4). Furthermore, a transition of the velocity curves from a parabolic curve classically obtained from the Navier-Stokes problem (i.e., c = 0 mol/m3) to a very flattend curve for high electrolyte concentration (c = 1000 mol/m3) can be seen. 

To obtain Sf, it is further necessary to solve the Stokes problem for two spheres. In the case of a purely straining motion, where only open trajectories obtain the result of the complete analysis, as outlined above, eventually yields (Batchelor and Green, 1972b)... 

We assume that the spaces X and Qh satisfy the usual inf sup condition for the Stokes problem ... 

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 ... 

For White-Metzner-type models, the associated Stokes problem obtained for We = 0 is then nonlinear. This is related to quasi-Newtonian models and is studied in [111]. Numerical auialysis for We > 0 has not been done yet. 

The main difficulty is to conveniently satisfy the incompressibility condition. In the following we will first recall the continuous mathematical formulation of the Stokes problem. Then it is recalled that a compatibility condition between pressure and velocity elements is necessary to prove convergence. Finally several possible strategies to solve the discretized system are developed. 

Let us consider the following Stokes problem on domain 2 with boundary conditions on dQ = Fiur2 (with rinr2=0)... 

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). 

Let us now consider the same flow domain (represented in Fig. l.a) with the boundary conditions of vanishing velocity on F] and F2 (the fluid is sticking at the wall on Fi and F2). This problem too has been largely studied for a Newtonian fluid. In this case, singular solutions of the homogeneous Stokes problem exist if a is a solution of the following equation (14) 5in(am) 2 sin((0) 2 aof) 0)... 

The non-homogeneous Stokes problem (18)-(20) in velocity and pressure is mathematically coupled to transport equations (16) through Ty. In this case the elimination of the tensor Ty is not possible, it has to be considered as a primitive variable. Two basic ideas (introduced by Marchal and Crochet) guide these developments. 

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). 

An element for the stress components composed of 16 sub-elements (4x4) on which bilinear (continuous) polynomials are used, was introduced by Marchal and Crochet in [28]. This leads to a continuous C° approximation of the three variables. The velocity is approximated by biquadratic polynomials while the pressure is linear. Fortin and Pierre ([17]) made a mathematical analysis of the Stokes problem for this three-field formulation. They conclude that the polynomial approximations of the different variables should satisfy the generalized inf-sup (Brezzi-Babuska) condition introduced by Marchal and Crochet and they proved it was the case for the Marchal and Crochet element. In order to take into account the hyperbolic character of the constitutive equation, Marchal and Crochet have implemented and compared two different methods. The first is the Streamline-Upwind/Petrov-Galerkin (SUPG). Thus a so-called non-consistent Streamline-Upwind (SU) is also considered (already used in [13]). As a test problem, they selected the "stick-slip" flow. With SUPG method applied to this problem, wiggles in the stress and the velocity field were obtained. In the SU method, the modified weighting function only applies to the convective terms in the constitutive equations. 

Let us notice Uq and ao (resp. U i and Oi) velocity and stress tensor at X=0 (resp. derivative according to X of velocity and stress tensor at X.=0 ). The velocity field Uo is the solution of an homogeneous Stokes problem with a "stick-slip singularity ( = jt) and a singular solution (Uo(r,8)= r IJo(e)) exists. As it is easily verified Ui is the solution of the following inhomogeneous Stokes problem ... 

From equation (57b), it is possible to express AT, as a fimction of Au. Substituting in equation (57a) leads then to a non-symmetric system (58) with the velocity as unknown, and for which the storage reqmrement is equivalent to that for a generahzed Stokes problem ... 

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... 

Figure 7-11. A schematic representation of the domain for a uniform flow past an arbitrary, axisymmetric body. For the case of a solid sphere, this is Stokes problem.

As an example of the application of (7-131), we consider creeping flow past an arbitrary axisymmetric body with a uniform streaming motion at infinity. For the case of a solid sphere, this is known as Stokes problem. In the present case, we begin by allowing the geometry of the body to be arbitrary (and unspecified) except for the requirement that the symmetry axis be parallel to the direction of the uniform flow at infinity so that the velocity field will be axisymmetric. A sketch of the flow configuration is shown in Fig. 7 11. We measure the polar angle 9 from the axis of symmetry on the downstream side of the body. Thus ij = I on this axis, and ij = — 1 on the axis of symmetry upstream of the body. 

Here, we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition,... 

Figure 7-12. The streamlines and contours of constant vorticity for uniform streaming flow past a solid sphere (Stokes problem). The streamfunction and vorticity values are calculated from Eqs. (7-158) and (7-162). Contour values plotted for the streamfunction are in increments of 1/16, starting from zero at the sphere surface, whereas the vorticity is plotted at equal increments equal to 0.04125.

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... 

In summary then, the leading-order problem is just the translation of a spherical drop through a quiescent fluid. The solution of this problem is straightforward and can again be approached by means of the eigenfunction expansion for the Stokes equations in spherical coordinates that was used in section F to solve Stokes problem. Because the flow both inside and outside the drop will be axisymmetric, we can employ the equations of motion and continuity, (7-198) and (7-199), in terms of the streamfunctions f<(>> and < l)), that is,... 

Note In solving the preceding problems, be sure to exploit the linearity of the Stokes problem in U and the electrostatic problem in E°°.)... 

From the description of the Stokes problem, we conclude that the incompressible surfactant film causes an oscillating Stokes-type boundary layer to develop, which is the predominant damping mechanism. By the same argument as before, the amplitude of the wave solution satisfying Laplace s equation is modified by an exponential attenuation exp(-)3 Here, however, we take the attenuation length to be given by the viscous boundary layer thickness of Eq. (10.5.34), from which, by comparison with Eq. (10.5.32),... 

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