Stokes approximation

While mathematically attractive, this force law is of limited interest physically it represents only the interaction between permanent quadrupoles, and even this with neglect of angles of orientation. However, although the details of the dependence of viscosity upon temperature are affected by the force law used, the general form of the hydrodynamic equation in the Navier-Stokes approximation is not affected. 

Eqs. (1-76) show that these values of the coefficients produce the Navier-Stokes approximations to pzz and qz [see Eq. (1-63)] the other components of p and q may be found from coefficient equations similar to Eq. (1-86) and (1-87) (or, by a rotation of coordinate axes). The first approximation to the distribution function (for this case of Maxwell molecules) is ... 

This is called the Stokes approximation to the equation of motion. At steady-state conditions, this may be written as... 

The zero-order Stokes approximation thus satisfies... 

The mass and momentum equations, that is, the Navier-Stokes approximation expressed in cylindrical coordinates with axisymmetry assumption, are [50, 51] ... 

Analysis of the hydrodynamic interactions of many particles in a laminar flow, carried out by Saito [56] showed that in view of the complexity of the physical picture of interactions in many body systems introduction of Stokes approximations in a theoretical consideration of the flow of dispersions can lead to incorrect results. For the laminar flow Saito proposed a formula containing a power series ... 

For this problem, you should assume that the mass transfer process occurs within the region surrounding the bubble where the Stokes approximation to the velocity field can be used. Thus the solution will be valid provided that Re Pe. Explain the reason for this condition. Your calculation of mass transfer rate should be carried out to include the first correction because of convention. 

Other results about shears flow past spherical particles. The motion of a freely floating solid spherical particle in a simple shear flow was considered in [100]. In this case, all the coefficients Gij except for Gn in the boundary conditions (2.5.1) are zero. The fact that the shear tensor has an antisymmetric component (see Section 1.1) results in the rotation of the particle because of the fluid no-slip condition on the particle boundary. The corresponding three-dimensional hydrodynamic problem was solved in the Stokes approximation. It was discovered that near the particle there is an area in which all streamlines are closed and outside this area, all streamlines are nonclosed. 

Motion of two spheres along a line passing through their centers. In the Stokes approximation, an exact closed-form solution of the axisymmetric problem about the motion of two spheres with the same velocity was obtained in [463]. This solution is practically important and can be used for estimating the accuracy of approximate methods, which are applied for solving more complicated problems on the hydrodynamic interaction of particles. 

For a body of revolution whose axis is inclined at an angle ui to the incoming flow direction, the following formula [166] holds in the Stokes approximation (as Re —> 0) ... 

The solution of hydrodynamic problems for an arbitrary straining linear shear flow (Gkm = Gmk) past a solid particle, drop, or bubble in the Stokes approximation (as Re -> 0) is given in Section 2.5. In the diffusion boundary layer approximation, the corresponding problems of convective mass transfer at high Peclet numbers were considered in [27, 164, 353]. In Table 4.4, the mean Sherwood numbers obtained in these papers are shown. 

In the Stokes approximation, the stream function for the flow (4.9.1) is equal to the sum of the stream functions of the constituent flows. 

By equating the sum of the drag force Fv in (5.10.8) and the capillary force (5.10.11) with zero, one can find the velocity of the capillary drift of the drop (in the Stokes approximation) ... 

Stokes approximation is applicable. For a nonscrew-like body (i.e., Kjj = 0), or one which is not rotating, the hydrodynamic force on any solid particle is (B20)... 

As a special case of some results due to Cox (C17), discussed at length in Section III,C, it appears that the torque on any body of revolution is unaffected by Reynolds number to the first order in R when it rotates about its symmetry axis. Rather, the torque is affected only in the O(R ) approximation. Cox also finds that when the body lacks fore-aft symmetry it experiences a force of 0(R) parallel to the symmetry axis [see Eq. (236)]. No such force appears in the Stokes approximation. 

There are two terms in the right part of this equation. The first one, G(t), is the random force due to the molecules of the liquid. This force changes in magnitude and direction in a random manner, and the characteristic time of this change is very small ( 10 s). The second term F is the force of hydrodynamic resistance to the particle s motion. Using the expression for F in Stokes approximation, we obtain... 

Equations (3.99) and (3.102) are called the Stokes approximation, and are the basis of the hydrodynamic interactions in suspensions and polymer solutions. 

Typically a mass of an organic molecule (or ion) is mj = Mntp 4 x (here M 200 is molecular mass, is proton mass) and, indeed, the estimated relaxation time is very short ti 2 x lO s. Thus, the dispersion of the ionic conductivity can only occur in the range of optical frequencies where the physical sense of the friction force is doubtful. By the way, the Stokes approximation seems to be quite good at lower frequencies. 

Thus, the governing equations (10)-(12) with the first order transport terms describe a flow of reacting mixture of viscous gases with strong non-equilibrium chemical reactions in the Navier-Stokes approximation. Transport properties in the one-temperature approach in reacting gas mixtures are considered in Em Giovangigli (1994) Kustova (2009) Kustova et al. (2008) Nagnibeda Kustova (2009). 

In this Chapter, the theoretical models for non-equilibrium chemical kinetics in multi-component reacting gas flows are proposed on the basis of three approaches of the kinetic theory. In the frame of the one-temperature approximation the chemical kinetics in thermal equilibrium flows or deviating weakly from thermal equilibrium is studied. The coupling of chemical kinetics and fluid dynamics equations is considered in the Euler and Navier-Stokes approximations. Chemical kinetics in vibrationaUy non-equilibrium flows is considered on the basis of the state-to-state and multi-temperature approaches. Different models for vibrational-chemical coupling in the flows of multi-component mixtures are derived. The influence of non-equilibrium distributions on reaction rates in the flows behind shock waves and in nozzle expansion is demonstrated. 

In 1910, Oseen provided the first correction term to the Stokes approximation as well as a mathematically self-consistent first approximation, as shown in Eq. (33) ... 

Such very small Reynolds numbers are typical for the motion of particles, ions and molecules, in a liquid, even under the action of an external field (e.g. in an electrophoresis experiment). They permit to simplify the Navier-Stokes equation by neglecting the non-linear convective term (v.V)v. The resulting relation is known as the Stokes approximation. 

---