# Can Navier Stokes equations explain pressure on a stationary body's surface?

• schettel
In summary, the Navier Stokes equations can be used to calculate pressure on the surface of a stationary body within a fluid flow. This involves taking into account impulse and wall shear stress. However, if the body is not moving, only shear stress needs to be considered. The equations are Galilean invariant, meaning the pressure on the surface will be the same regardless of whether the body or fluid is moving. The local temporal derivative must not be neglected in the equations to accurately describe the dynamics at a field point. This can be seen by considering the distance from the field point to the body.
schettel
Consider a stationary body within the flow of some fluid. I want to calculate pressure on the surface of the body. From the Navier Stokes (incompressible, stationary, no volume forces) equations, you would get something like:
dp/dx=-rho(u du/dx+v du/dy+w du/dz)+eta(d²u/dx²+d²u/dy²+d²u/dz²)
...likewise for the other coordinates.
This means that, when calculating pressure on the surface of my body, impulse (rho(..) on the right hand side of the equation) and wall shear stress (eta(..)) come into play. That seams quite logical. But my body is not moving, so, assuming a no-slip condition for its surface, u=v=w=0, which leaves me only with shear stress.
If the body is moving and the fluid is not, you should still get the same pressure on the surface. But in this case, the first term on the right hand side of the equation (impulse) is not zero.
There must be a mistake in my reasoning. Can anybody tell me what it is?
Maybe it has something to do Euler and Lagrange?

You're asking if N-S is Galilean invariant, right? It is.
The solution lies in that you cannot neglect the local temporal derivative if you shift to a description in which the body has a non-zero velocity.

This makes sense, because roughly, the dynamics in the fluid at a field point should primarily depend upon the distance from the field point to the body.

arildno said:
You're asking if N-S is Galilean invariant, right? It is.
The solution lies in that you cannot neglect the local temporal derivative if you shift to a description in which the body has a non-zero velocity.

This makes sense, because roughly, the dynamics in the fluid at a field point should primarily depend upon the distance from the field point to the body.

Welcome to Mech&Aero forum. Glad to see you here, Arildno! :rofl:

I do agree with your answer. The local variation of the velocity $$\partial u/\partial t$$ is non zero if the body is in motion. The problem is transformed into an unsteady one.

I do poke my nose into here on occasion..
Thanks for the welcome, though..

## What is Navier Stokes equation?

Navier Stokes equation is a set of equations that describe the motion of fluids such as liquids and gases. It takes into account factors like velocity, pressure, and density to predict the behavior of a fluid in a given situation.

## Why is Navier Stokes equation important?

Navier Stokes equation is important because it helps us understand and predict the behavior of fluids in real-world scenarios. It has various applications in fields such as aerodynamics, weather forecasting, and oceanography.

## What are the limitations of Navier Stokes equation?

Navier Stokes equation is a simplified model of fluid dynamics and has some limitations. It assumes that fluids are incompressible, which is not always the case. It also does not take into account certain factors like turbulence and viscosity, which can significantly affect fluid behavior.

## What are the practical applications of understanding Navier Stokes equation?

Understanding Navier Stokes equation has numerous practical applications. It helps in designing efficient and safe structures such as buildings, bridges, and airplanes. It is also used in developing weather prediction models and optimizing industrial processes.

## What are some common methods used to solve Navier Stokes equation?

There are various methods used to solve Navier Stokes equation, including finite difference, finite volume, and finite element methods. These methods involve breaking down the equations into smaller, more manageable parts and using numerical techniques to solve them.

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