Stokes’ flow

Consider the weighted residual statement of the equation of motion in a steady state Stokes flow model, expressed as... 

MODELLING OF STEADY STATE STOKES FLOW OF A GENERALIZED NEWTONIAN FLUID... 

In this section the governing Stokes flow equations in Cartesian, polar and axisymmetric coordinate systems are presented. The equations given in two-dimensional Cartesian coordinate systems are used to outline the derivation of the elemental stiffness equations (i.e. the working equations) of various finite element schemes. 

In the absence of body force the equations of continuity and motion representing Stokes flow in a two-dimensional Cartesian system are written, on the basis of Equations (1.1) and (1.4), as... 

Similarly in the absence of body forces the Stokes flow equations for a generalized Newtonian fluid in a two-dimensional (r, 8) coordinate system are written as... 

Similarly the components of the equation of motion for an axisymmetric Stokes flow of a generalized Newtonian fluid are written as... 

Using a procedure similar to the derivation of Equation (4.13) the working equations of the U-V-P scheme for steady-state Stokes flow in a polar (r, 6) coordinate system are obtained on the basis of Equations (4.5) and (4.6) as... 

We start with the governing equations of the Stokes flow of incompressible Newtonian fluids. Using an axisymraetric (r, z) coordinate system the components of the equation of motion are hence obtained by substituting the shear-dependent viscosity in Equations (4.11) with a constant viscosity p, as... 

Step 1 To solve a Stokes flow problem by this program the inertia term in the elemental stiffness matrix should be eliminated. Multiplication of the density variable by zero enforces this conversion (this variable is identified in the program listing). 

Step 2 General structure of stiffness matrices derived for the model equations of Stokes flow in (x, 3O and (r, z) formulations (see Chapter 4) are compared. 

THIS TERM SHOULD BE MULTIPLIED BY ZERO FOR STOKES FLOW CALCULATIONS... 

Plotnikov P.I. (1995) On a class of curves arising in a free boundary problem for Stokes flow. Siberian Math. J. 36 (3), 619-627. 

Assuming spherical particles, the drag coefficient, in the laminar, the Stokes flow regime is... 

The uniform flow at time t = 0 rapidly changed into Stokes flow around the obstacle. By / 100 psec a pair of counter-rotating eddies had developed... 

If the length scales associated with changes in velocity are normalized by Vv (characteristic length scale for Stokes flow), length scales associated with changes in curvature are normalized by Ss (typical striation thickness) and velocities normalized by V (a characteristic velocity), then the normal stress condition becomes,... 

Chaiken, J., Chevray, R., Tabor, M., and Tan, Q. M., Experimental study of Lagrangian turbulence in Stokes flow, Proc. Roy. Soc. Lond. A408, 165-174 (1986). 

Jana, S. C., Metcalfe, G., and Ottino, J. M., Experimental and computational studies of mixing in complex Stokes flow—the vortex mixing flow and the multicellular cavity flow. J. Fluid Mech. 269, 199-246 (1994). 

Kaper, T. J., and Wiggins, S., An analytical study of transport in Stokes flows exhibiting large scale chaos in the eccentric journal bearing. J. Fluid Mech. 253, 211-243 (1993). 

This is known as Stokes flow, and Eq. (11-3) has been found be accurate for flow over a sphere for NRe < 0.1 and to within about 5% for NRe < 1. Note the similarity between Eq. (11-3) and the dimensionless Hagen-Poiseuille equation for laminar tube flow, i.e.,/ = 16/tVRe. 

However, the criterion for Stokes flow (tVRe < 1) cannot be tested until Vt is known, and if it is not valid then Eq. (11-10) will be incorrect. This will be addressed shortly. 

The viscosity of a Newtonian fluid can be determined by measuring the terminal velocity of a sphere of known diameter and density if the fluid density is known. If the Reynolds number is low enough for Stokes flow to apply (fVRe < 0.1), then the viscosity can be determined directly by rearrangement of Eq. (11-10) ... 

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