Creeping flow past a stationary sphere

This notebook has been written in Mathematica by

Mark J. McCready
Professor and Chair of Chemical Engineering
University of Notre Dame
Notre Dame IN 46556
USA

mjm@nd.edu
http://www.nd.edu/~mjm/


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Version:  8/8/00
More recent versions of this notebook should be available at the web site:
http://www.nd.edu/~mjm/creepingsphere.nb

This notebook shows how to solve creeping flow past a stationary sphere (Stokes's Problem)

Reference:  S. MIddleman  (1999) An Introduction to Fluid Dynamics:Principles of Analysis and Design,
Wiley
pp 166-171.

[Graphics:Images/creeping_sphere_gr_1.gif]

Keywords

Problem of interest

An issue that occurs in many situations of interest to chemical engineers is the contacting of solid particles with liquids or the separation of particles from gases and liquids.  In all of these situations, there is a need to either keep the particles suspended, or, to hasten their settling.  To understand this problem, we need to find an expression that gives the steady -state drag on a particle.  There is no simple relation that is valid for all Reynolds numbers.  However, if the particles is a sphere and the Reynolds number sufficiently small, a relation called "Stokes Law" is valid and it can be obtained through solution of the Navier-Stokes equations.

We will derive this relation below and in the process will hopefully learn something about the physics of "creeping" flows.  It is worthwhile to emphasize that even if a particle is not a sphere, the relation will give the qualitative prediction of the drag and should not be to wrong quantiatively.  

Consider the flow past a stationary sphere where the Reynolds number is significantly less than unity.  Inertia can be neglected.  The far away velocity field is a constant straight flow.  To make the problem as simple as possible, we will use spherical coordinates with the φ axis oriented parallel to the flow.  Thus we have φ symmetry and no [Graphics:Images/creeping_sphere_gr_2.gif][Graphics:Images/creeping_sphere_gr_3.gif].  This means that the [Graphics:Images/creeping_sphere_gr_4.gif] equation is not needed.  If we used a different coordinate system, or different orientation, the problem would be more complicated.   

Learning Objectives

Mathematical Formulation

Solution of the equations with the boundary conditions

Examination of the solution

Conclusions

1.  Creeping flow is a term used for a flow that have effectively no inertia.  In this case the inertia terms are neglected  and the solution is obtained from the resulting linear equations.  The Reynolds number is very much smaller than unity.
        
2.  The solution technique involves using a solution form that is deduced from boundaries of the flow field, faraway from the sphere.

3.  Because viscous forces dominate the flow field, the fluid can never accelerate above the free stream value even if an obstacle causes the fluid to be squeezed.  Thus the velocity in the region of the sphere just slows down and then returns to the free stream value.  

4.  Both normal stresses and tangential stresses contribute to the drag on the sphere.  These can be termed form drag and skin drag.     

5.  Consistent with the fluid not accelerating, the pressure never increases above the free stream value.  The fluid has no inertia that would cause a pressure increase as the fluid slows down.

6.  The velocity decays slowly (as  [Graphics:Images/creeping_sphere_gr_212.gif]) and thus the disturbance is felt very far away from the sphere.  This makes it difficult to do a real experiment, in a reasonable size container, that allows that sphere to fall at a speed specified by the drag that is predicted from the analysis here.  The very high Reynolds number case decays much faster.     


Converted by Mathematica      August 7, 2000