THE METHOD OF REGULARIZED STOKESLETS

Mingj ie Zhu14 t h , August , 2015

Be i j ing Computat iona l Sc ience Research Center

STOKES FLOW In the common case of an incompressible Newtonian fluid, the Stokes equations

take the (vectorized) form:

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 . The force is given by

Then we define a function called Green’s function

And another function

GREEN'S FUNCTION SOLUTION: THE STOKESLET

Take the divergence of Stokes equations, we find that the pressure satisfies

Which gives the particular solution

Now we use this expression to rewrite the equation for as , whose particular solution is

GREEN'S FUNCTION SOLUTION: THE STOKESLET

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 to denote this function, which is a radially

symmetric smooth function with the property that . The parameter determines the spread

of the function.

EXAMPLES OF THESE FUNCTIONS

In :

In :

DERIVING AND Since is radially symmetric and , we have that

And

After integrating once more, we obtain . Then, we can get in the same way.

FINAL RESULTS

FINDING FORCES FROM VELOCITIES We can write last equations into

Or, as a matrix equation:

Which is to solve F using U and F. However, it is impossible to calculate the inverse of directly. We use GMRES with zero initial guess.

Radius=1, velocity=(0.4, 0)

ALGORITHM1. Create all Stokeslets: define their coordinates and initial velocities;

2. Calculate each element in the matrix ;

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.

RESULTS COMPARE: A 2D CIRCLE

Radius=0.25, velocity=(1, 0)

Up: the x component of velocityDown: the y component of velocity

RESULTS COMPARE: A 3D SPHERE

Radius=1.2, velocity=(0.5, 0, 0)When z=0.

Up: the x component of velocityDown: the y component of velocity

RESULTS COMPARE: A 3D SPHEREx y stokes_u stokes_v analytical_u analytical_v error_u error_v1 1 0.346046 0.034137 0.351291 0.0356382 0.014931 0.0421331 1.3 0.29273 0.04787 0.296579 0.0492919 0.012979 0.0288511 1.6 0.245579 0.049912 0.248442 0.0510662 0.011525 0.0226031 1.9 0.207925 0.046623 0.210136 0.0475185 0.010524 0.0188451 2.2 0.178654 0.041589 0.180432 0.042281 0.009856 0.0163661 2.5 0.155852 0.036406 0.157331 0.0369465 0.009398 0.0146421 3.7 0.101882 0.021099 0.102764 0.0213386 0.008585 0.0112071 4 0.093677 0.018602 0.0944791 0.018804 0.008491 0.010768

1.6 4 0.095528 0.026299 0.0963557 0.0265792 0.008586 0.0105351.6 4.3 0.088386 0.023647 0.0891441 0.0238912 0.0085 0.0102341.6 4.6 0.082207 0.021328 0.0829062 0.0215431 0.008432 0.0099841.6 4.9 0.076817 0.0193 0.0774665 0.0194909 0.008378 0.0097741.9 1 0.27197 0.046623 0.275412 0.0475185 0.012498 0.0188451.9 2.2 0.173029 0.050128 0.174771 0.0508224 0.009968 0.013657

RESULTS COMPARE: A 3D SPHERE

APPLICATION: BACTERIA

IMPLEMENTATION USING C++1. Run faster (Native code and OpenMP supported);

2. Better readability;

3. Expandable.