Help with Navier-Stokes Equation: Symbols & Meaning

In summary, the Navier-Stokes equations describe fluid motion and are used to model the behavior of fluids. They include terms for the convective derivative, pressure gradient, stress tensor, and external forces. They can be thought of as the continuous version of Newton's second law and are important in understanding fluid dynamics.
  • #1
iasc
17
0
I was wondering if someone could help me this Navier-Stokes Equation.

f[(δv/δt) + v.Dv] = -DP + Dt + f

Could someone maybe explain the symbols and what it means.
I'm not sure but I think Navier-Stokes equations describe fluid motion.

(The P could be ρ. I'm not too sure)

Thanks
 
Physics news on Phys.org
  • #2
You are correct that the equations describe fluid motion. I have a beautiful proof that solutions always exist in three dimensions, but unfortunately it is too big for this marginal comment.
 
  • #3
iasc said:
I was wondering if someone could help me this Navier-Stokes Equation.

f[(δv/δt) + v.Dv] = -DP + Dt + f

Could someone maybe explain the symbols and what it means.
I'm not sure but I think Navier-Stokes equations describe fluid motion.

(The P could be ρ. I'm not too sure)

Thanks

I can't completely parse what you wrote, but some of it I can decipher:

The term in [] (not sure what that 'f' is doing there), is the "total" or "convective" derivative. It simply means that a spatial quantity (the velocity field vector 'v') is allowed to vary both in time and in space. I am assuming 'D' is a nabla (del) operator.

The term DP could be the gradient of pressure term, but the Dt term is normally the divergence of the stress tensor, so those two terms are a little ambiguous. The stress tensor consists of both an isotropic part (the pressure) and the off-diagonal antisymmetric components (the shear stress). The final 'f' is used if there is an external body force: gravity, centripetal forces, electromagnetic, etc. etc.

The Navier-Stokes equation you wrote is nothing more than ma=F for a continuous material. If ma(or []) = 0, then you have conservation of momentum.
 

What is the Navier-Stokes equation?

The Navier-Stokes equation is a set of mathematical equations that describe the behavior of fluid flow. It takes into account factors such as pressure, velocity, and viscosity to predict how a fluid will move in a given situation.

What are the symbols used in the Navier-Stokes equation?

The symbols used in the Navier-Stokes equation include ρ (rho) for density, u for velocity, p for pressure, v for viscosity, and τ (tau) for shear stress. These symbols may vary slightly depending on the specific form of the equation being used.

What does the Navier-Stokes equation help us understand?

The Navier-Stokes equation helps us understand the behavior of fluids in a variety of situations, such as in pipes and around objects. It is used in many fields of science and engineering, including aerodynamics, meteorology, and oceanography.

Is the Navier-Stokes equation difficult to solve?

The Navier-Stokes equation can be difficult to solve, especially for complex fluid flows. In fact, it is one of the seven unsolved Millennium Prize Problems in mathematics, with a prize of $1 million for anyone who can provide a complete solution.

Are there any applications of the Navier-Stokes equation in real life?

Yes, the Navier-Stokes equation has many practical applications in real life. It is used to design airplanes and cars, predict weather patterns, and develop efficient water and air filtration systems. It is also used in computer simulations to study fluid dynamics in a virtual environment.

Similar threads

Replies
18
Views
920
  • Classical Physics
Replies
4
Views
1K
  • Classical Physics
Replies
8
Views
2K
  • Classical Physics
Replies
1
Views
4K
Replies
20
Views
5K
  • Classical Physics
Replies
4
Views
8K
  • STEM Academic Advising
Replies
6
Views
973
  • Classical Physics
Replies
22
Views
3K
Replies
9
Views
2K
  • Classical Physics
Replies
7
Views
1K
Back
Top