In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. A fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function.
In C. POZRIKIDIS's Boundary Integral
and Singularity Methods for Linearized
Viscous Flow,The Green's functions of Stokes flow
represent solutions of the continuity
equation $\nabla\cdot {\bf u}=0$ and the
singularly forced Stokes equation
$$-\nabla P+\mu
\nabla^2{\bf u}+{\bf g}\delta({\bf x-x_0})=0 $$where ${\bf g}$ is an arbitrary constant,
${\bf x_0}$ is an arbitrary point, and
$\delta$ is the three-dimensional
delta function. Introducing the Green's function ${\bf G}$, we write the solution of (2.1.1)
in the form
$$u_i({\bf x})=\frac{1}{8\pi\mu}G_{ij}({\bf x,x_0})g_j$$
I am confused with the Green's function in this text.
Here are my questions:
-
Is $P({\bf x})$ supposed to be the unknown
in the Stokes equations:$$ \begin{align}
-\nabla P+\mu \nabla^2 u+\rho b&=0\\
\nabla \cdot u &=0
\end{align}
$$ -
What does the Green's function mean here? (Is it "with respect to" $u$?)
Why is it of that strange form?Why is the solution of this kind of form? -
How can one get $\frac{1}{8\pi\mu}$?What is the relation between ${\bf G}$ and $G_{ij}$? As I understand, $G_{ij}$ are the components and ${\bf G}:{\mathbb R}^3\to{\mathbb R}^3$. Then one should write:
$${\bf G}({\bf x})=\begin{bmatrix}
G_1({\bf x})\\ G_2({\bf x})\\ G_3({\bf x})\end{bmatrix}$$
where $G_i:{\mathbb R}^3\to{\mathbb R}$. What is $G_{ij}$? - What's the Green's function in the most general case?
Best Answer
I moved this question to math.SE a month ago. This is indeed the problem I got from the research, though it may not very appropriate here.
@Willie Wong gave a very nice answer to the question. Instead of closing or deleting the question, I think it's worth putting the link here.