Stokes expansion

Assume that the solutions of u and p can be expressed by the Stokes expansion in the form... 

It is noted that the Stokes expansion is not uniformly valid in the neighborhood of infinity. Therefore, another expansion, i.e., the Oseen expansion, is introduced to satisfy the boundary conditions at infinity. By using the matching technique, the final results can be obtained. 

Similar to the Stokes expansion, the Oseen expansion is of the form... 

The matching principle requires that the Stokes expansion and the Oseen expansion must be asymptotically equal in their common domain of validity. The Stokes expansion is valid from the sphere out to some large distance, whereas the Oseen expansion is valid from infinity to some small radii in the stretched variables. Hence, the common domain is a spherical shell within which the unstretched radius is large but the stretched radius is small. In this domain, the matching principle yields ... 

Fe m F M Fs Lorentz force vector Magnus force vector Saffman force vector Pi Pi Stokes expansion Pressure of fluid Dipole moment vector... 

Incompressible Limit In order to obtain the more familiar form of the Navier-Stokes equations (9.16), we take the low-velocity (i,e. low Mach number M = u I /cs) limit of equation 9,104, We also take a cue from the continuous case, where, if the incompressible Navier-Stokes equations are derived via a Mach-number expansion of the full compressible equations, density variations become negligible everywhere except in the pressure term [frisch87]. Thus setting p = peq + p and allowing density fluctuations only in the pressure term, the low-velocity limit of equation 9,104 becomes... 

Yu. A. Simonov, Cluster Expansion, Non-Abelian Stokes Theorem and Magnetic Monopoles,... 

The above self-similar velocity profiles exists only for a Re number smaller than a critical value (e.g. 4.6 for a circular pipe). The self-similar velocity profiles must be found from the solution of the Navier-Stokes equations. Then they have to be substituted in Eq. (25) which must be solved to compute the local Nusselt number Nu z). The asymptotic Nusselt number 7Vm is for a pipe flow and constant temperature boundary condition is given by Kinney (1968) as a function of Rew and Prandtl (Pr) numbers. The complete Nu(z) curve for the pipe and slit geometries and constant temperature or constant flux boundary conditions were given by Raithby (1971). This author gave /Vm is as a function of Rew and fluid thermal Peclet (PeT) number. Both authors solved Eq. (25) via an eigenfunction expansion. 

The next task is to substitute the Mach-number expansions into the Navier-Stokes equations. The following equations show the term-by-term substitution. By collecting all terms of like order, differential equations can be formed for each order of Mach number ... 

The purpose of this appendix is to spell out explicitly the Navier-Stokes and mass-continuity equations in different coordinate systems. Although the equations can be expanded from the general vector forms, dealing with the stress tensor T usually makes the expansion tedious. Expansion of the scalar equations (e.g., species or energy) are much less trouble. 

Step 1 is purely hydrodynamic and relates the perturbation Q to the velocity near the wall which is the only relevant quantity for the mass transfer response. at are either wall velocity gradients or coefficients involved in the velocity expansion near the wall. This step requires the use of Navier-Stokes equations and will be treated in Chapter 2. 

Stokes vector will contain components of the form sin ( 4 sinOr) and cos (AsinQt). Consequently, the temporal response is rather complicated. To complete the analysis of the Fourier content of the signal the following expansions are required ... 

Stokes, R. H. The Molar Volumes and Thermal Expansion Coefficients of Solid and Liquid Potassium from 0—85°C. J. Phys. Chem. Solids, 27, 51—56 (1966). 

Block the laser beam or spectrometer entrance slit and adjust the spectrometer to an anti-Stokes shift of 1000 cm Caution Exposure of the sensitive phototube to the intense Rayleigh scattering line can seriously damage the detector. Scan the anti-Stokes spectrum from 1000 to 150 cm in the parallel polarization configuration and, using appropriate sensitivity expansion, j measure the ratio of anti-Stokes to Stokes peak heights for each band. 

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