# Green's function for Stokes equation

• A
• steve1763
In summary, the conversation discusses the use of Greens functions to solve non-homogenous linear problems, specifically the Stokes equations in Cartesian coordinates. The Green's function for the 3D laplacian is given by -1/(4*pi*r), and can be used to find solutions for a point force. The conversation also touches on converting delta functions and the use of Greens functions to find solutions for a point source. It concludes with the derivation of P and u for the Stokes equations using Greens functions.
steve1763
TL;DR Summary
How do I reach the expression for pressure?
So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force.

$$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$
$$\nabla \cdot \textbf{u}=0$$

We can apply the second relation to the first to get

$$- \nabla^2 \textbf{P} + \textbf{F} \cdot \nabla \delta(\textbf{x})=0$$
$$\nabla^2 \textbf{P} = \textbf{F} \cdot \nabla \delta(\textbf{x})$$

The Greens function for the 3D laplacian, according to Wikipedia, is $$-1 \over{4 \pi r}$$ where $$r=(x^2+y^2+z^2)^{1 \over 2}$$ Generally, with Greens functions

$$u(x)= \int G(x,s) f(s) ds$$

So would we get something like

$$\textbf{P}= {-1\over {4 \pi}} \int {1 \over r} \textbf{F} \cdot \nabla \delta(\textbf{x}) ds$$

I believe you might have to convert $$\delta(\textbf{x})={1\over{4 \pi r^2}}\delta(r)$$ I'm a little confused as to where to go next and how to deal with an integral that might have delta functions in it.

Thank you

Pressure is a scalar quantity; it doesn't need a bold symbol.

The Green's function is used to decompose a non-homogenous linear problem $L(u) = f$ into a superposition of problems $L(G_{x_0}) = \delta(x - x_0)$ from which it follows that $$u(x) = \int f(x_0) G_{x_0}(x)\,dx_0.$$ If we want the solution for a point source of strength $k$ at $x_1$, then we get $$u(x) = \int k\delta(x_0 - x_1)G_{x_0}(x)\,dx_0 = kG_{x_1}(x)$$ and we are looking for a constant multiple of the Green's function $G_{x_1}$.

To solve $L(G_{x_0}) = \delta(x - x_0)$, we treat it as a homogenous problem $L(G_{x_0}) = 0$ subject to appropriate conditions on $G_{x_0}$ and its derivatives.

In this case, we want $P$ and $\mathbf{u}$ to be linear in $\mathbf{F}$ and we want $-\nabla P + \mu\nabla^2 \mathbf{u}$ to be proportional to $\nabla^2(1/r)$ with $$\int_{r \leq a} \nabla P - \mu \nabla^2 \mathbf{u}\,dV = \mathbf{F}.$$ That leads us to $$\begin{split} P &= \frac{1}{4\pi} \frac{\mathbf{F} \cdot \mathbf{r}}{r^3} = -\frac{1}{4\pi}\mathbf{F} \cdot \nabla\left(\frac1r\right),\\ \mathbf{u} &= \frac{1}{8\pi\mu}\left(\frac{\mathbf{F}}{r} + \frac{(\mathbf{F} \cdot \mathbf{r})\mathbf{r}}{r^3}\right) = \frac{1}{8\pi \mu}\left(\frac{\mathbf{F}}{r} - (\mathbf{F} \cdot \mathbf{r}) \nabla\left(\frac1r\right)\right).\end{split}$$

steve1763

## 1. What is the Green's function for Stokes equation?

The Green's function for Stokes equation is a mathematical tool used to solve the Stokes equation, which describes the motion of a viscous fluid. It is a function that represents the response of the fluid to a point force or disturbance at a specific location and time.

## 2. How is the Green's function for Stokes equation used in fluid dynamics?

The Green's function for Stokes equation is used to solve complex fluid flow problems, such as calculating the velocity and pressure fields for a given flow. It can also be used to predict the behavior of a fluid in response to external forces, such as in aerodynamics or hydrodynamics.

## 3. What are the properties of the Green's function for Stokes equation?

The Green's function for Stokes equation has several important properties, including linearity, symmetry, and translation invariance. It also satisfies the boundary conditions of the Stokes equation and has a singularity at the point of disturbance.

## 4. How is the Green's function for Stokes equation calculated?

The Green's function for Stokes equation can be calculated using various techniques, such as the method of images, the method of reflections, or the method of fundamental solutions. These methods involve solving a system of equations or integrals to obtain the Green's function for a specific problem.

## 5. What are some applications of the Green's function for Stokes equation?

The Green's function for Stokes equation has various applications in fluid dynamics, including in the study of laminar flow, boundary layer flow, and flow around objects. It is also used in the development of numerical methods for solving fluid flow problems, such as the boundary element method and the method of fundamental solutions.

• Differential Equations
Replies
0
Views
408
• Differential Equations
Replies
33
Views
5K
• Quantum Physics
Replies
3
Views
1K
• High Energy, Nuclear, Particle Physics
Replies
8
Views
1K
• Differential Equations
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Classical Physics
Replies
6
Views
1K
• General Math
Replies
4
Views
1K
• Quantum Physics
Replies
1
Views
1K
• Electromagnetism
Replies
19
Views
2K