This document describes the method of regularized Stokeslets for modeling Stokes flow. It introduces Stokes flow equations and the Green's function solution of a Stokeslet. It then describes how the method of regularized Stokeslets assumes forces are spread over a small region rather than a Dirac delta function. It provides examples of regularized shape functions in 2D and 3D and derives the corresponding regularized Green's functions Gฮต and Bฮต. The document concludes by describing an algorithm to solve for forces given velocities and provides examples comparing results to analytical solutions for 2D and 3D flows.
1. THE METHOD OF REGULARIZED STOKESLETS
Mingjie Zhu
14th, August, 2015
Beijing Computational Science Research Center
2. STOKES FLOW
In the common case of an incompressible Newtonian fluid, the Stokes equations
take the (vectorized) form:
๐๐ป2
โ ๐ป๐ + ๐ = ๐
๐ป โ ๐ = ๐
3. GREEN'S FUNCTION SOLUTION: THE STOKESLET
We first consider the generic situation in which the force are spread over a small ball
centered at the points ๐0. The force is given by
๐ญ ๐ = ๐0 ๐ฟ(๐ โ ๐0)
Then we define a function G(๐ฅ)called Greenโs function
๐ป2G ๐ฅ = ๐ฟ(๐ฅ)
And another function
๐ป2
B ๐ฅ = G ๐ฅ
4. GREEN'S FUNCTION SOLUTION: THE STOKESLET
Take the divergence of Stokes equations, we find that the pressure satisfies
๐ป2
๐ = ๐ป โ ๐
Which gives the particular solution
๐ = ๐0 โ ๐ปG
Now we use this expression to rewrite the equation for ๐ as ๐๐ป2
๐ = ๐0 โ ๐ป ๐ปG โ ๐0 ๐ฟ,
whose particular solution is
๐๐ ๐ฅ = ๐0 โ ๐ป ๐ปB ๐ โ ๐0 โ ๐0G ๐ โ ๐0
6. REGULARIZED STOKESLETS
The method of regularized Stokeslets assumes the force is not a Dirac delta applied at a
single point but uses a shape function. We use ๐ ๐(๐) to denote this function, which is a
radially symmetric smooth function with the property that ๐ ๐(๐) d๐ = 1. The
parameter ๐ determines the spread of the function.
7. EXAMPLES OF THESE FUNCTIONS
In ๐2: ๐ ๐ ๐ =
3๐3
2๐( ๐ 2+๐2)5/2
In ๐3
: ๐ ๐ ๐ =
15๐4
8๐(๐2+๐2)7/2
8. DERIVING G ๐ AND B ๐
Since ๐ ๐ is radially symmetric and ๐ป2
G ๐ = ๐ ๐, we have that
๐ป2
G ๐ =
1
๐
๐G ๐
โฒ
๐ โฒ
= ๐ ๐(๐)
And
G ๐
โฒ
๐ =
1
๐ 0
๐
๐ ๐ ๐ ๐ d๐
After integrating once more, we obtain G ๐(๐). Then, we can get B ๐(๐) in the same way.
10. FINDING FORCES FROM VELOCITIES
We can write last equations into
๐ข ๐๐ =
๐=1
๐
๐๐๐(๐1, ๐2, โฆ , ๐ ๐) ๐ ๐
Or, as a matrix equation:
๐ผ = ๐ด๐ญ
Which is to solve F using U and F. However, it is
impossible to calculate the inverse of ๐ด
directly. We use GMRES with zero initial guess.
Radius=1, velocity=(0.4, 0)
11. ALGORITHM
1. Create all Stokeslets: define their coordinates and initial velocities;
2. Calculate each element in the matrix ๐ด;
3. Solve the matrix equation using GMRES to obtain forces in each Stokeslet; (Normally,
GMRES can iterate converging to the tolerance 1e-8)
4. Calculate velocity of other points in the Stokes flow field;
5. Plot the result in the screen or output these result.
12. RESULTS COMPARE: A 2D CIRCLE
Radius=0.25, velocity=(1, 0)
Up: the x component of velocity
Down: the y component of velocity
13. RESULTS COMPARE: A 3D SPHERE
Radius=1.2, velocity=(0.5, 0, 0)
When z=0.
Up: the x component of velocity
Down: the y component of velocity