Fundamental solutions of the stokes equation.

[funcosed]

Homework Statement

Hi, I going through my class notes for a fluids class, specifically fundamental solutions of the Stokes equations. To derive the stresslet and rotlet involves solving the following

u_i = (1/8*pi*μ)*(∂G_ik/∂x_j)*F_k*A_j

G_ik(x) = δ_ij*(1/r)+(x_ix_j/r³)

We looked at it in a lecture (skipping all the "easy parts"! of course) and I am trying to fill in the gaps. The first step is to taylor expand it at A/2 wit |A/2| << |x|.

This should give u_i(x) = (1/8*pi*μ)*(G_ik(x-0)F_k + (A_i/2)*∂/∂x_j(G_ik)(x - 0)*F_k + ... (Mix of vector and suffix notation here very confusing)

Then do the same at -A/2 (with -F) and add the solutions

Homework Equations

Taylor series: f(a) + f'(a)*(x-a) + (f''(a)/2!)*(x-a)² + ...

The Attempt at a Solution

My problem is the mix of notation, I am not sure how to apply the taylor series here. Any help appreciated.

 

[jbsteele]

Since Taylor's theorem is a local expansion, you can expand the function Gik(x) around the point x-0. To do this, start by expanding Gik(x) as a Taylor series around the point x-0. Since Gik(x) is a function of two variables (x and j) we need to differentiate with respect to both variables. We also need to keep in mind that the derivatives are evaluated at the point x-0, so for example the second derivative with respect to x is ∂2Gik/∂x2(x-0). The Taylor series can then be written as: Gik(x) = Gik(x-0) + (Ai/2)*∂Gik/∂xj(x-0) + (Ai/2)2*∂2Gik/∂xj2(x-0) + ...Plugging this into the equation for ui gives: ui = (1/8*pi*μ)*(Gik(x-0)Fk + (Ai/2)*∂Gik/∂xj(x-0)*Fk + (Ai/2)2*∂2Gik/∂xj2(x-0)*Fk + ... )Then do the same at -A/2 (with -F): ui = (1/8*pi*μ)*(-Gik(x-0)Fk - (Ai/2)*∂Gik/∂xj(x-0)*Fk - (Ai/2)2*∂2Gik/∂xj2(x-0)*Fk + ... )Adding the two solutions gives the final result: ui = (1/8*pi*μ)*[Gik(x-0)(Fk - Fk) + (Ai/2)*∂Gik/∂xj(x-0)*(Fk - Fk) + (Ai/2)2*∂2Gik/∂xj2(x-0)*(Fk - Fk) + ... ]