Deriving the Navier-Stokes equation from energy-momentum tensor

In summary: The gamma term arises because the pressure is distributed over a much larger area than the density, and so the pressure field has more contributions from large-scale perturbations.
  • #1
PhyPsy
39
0
The energy-momentum tensor for a perfect fluid is [itex]T^{ab}=(\rho_0+p)u^au^b-pg^{ab}[/itex] (using the +--- Minkowski metric).

Using the conservation law [itex]\partial_bT^{ab}=0[/itex], I'm coming up with [itex](\rho+\gamma^2p) [\frac{\partial\mathbb{u}}{{\partial}t}+ (\mathbb{u}\cdot\mathbb{\nabla})\mathbb{u}]= -{\nabla}p[/itex] instead of [itex]\rho[\frac{\partial\mathbb{u}}{{\partial}t}+ (\mathbb{u}\cdot\mathbb{\nabla})\mathbb{u}]= -{\nabla}p[/itex] (disregarding the body force part of the equation). Why is there a term for the part in brackets multiplied by [itex]\rho[/itex], but not for [itex]\gamma^2p[/itex]?
 
Physics news on Phys.org
  • #2
Where did the gamma come from ? Can you post your derivation/scaned copy of it ?
 
  • #3
[itex]u^a=\gamma(1,\mathbf{u})[/itex]
Keeping b constant and cycling a from x to z, I get [itex]\partial_tT^{at}= \partial_t[\gamma^2(\rho_0+p)\mathbf{u}][/itex]
[itex]\partial_xT^{ax}= \partial_x[\gamma^2(\rho_0+p)u_x\mathbf{u}]+\partial_xp[/itex]
[itex]\partial_yT^{ay}= \partial_y[\gamma^2(\rho_0+p)u_y\mathbf{u}]+\partial_yp[/itex]
[itex]\partial_zT^{az}= \partial_z[\gamma^2(\rho_0+p)u_z\mathbf{u}]+\partial_zp[/itex]
[itex]\rho=\gamma^2\rho_0[/itex], so that is why you see a [itex]\gamma^2[/itex] coefficient for p but not for [itex]\rho[/itex]. When I solve for [itex]\partial_bT^{tb}=0[/itex], I get [itex]\partial_t(\rho+\gamma^2p)+ \partial_x[(\rho+\gamma^2p)u_x]+ \partial_y[(\rho+\gamma^2p)u_y]+ \partial_z[(\rho+\gamma^2p)u_z]=0[/itex], so summing the four equations for [itex]T^{ab}[/itex] above and simplifying using [itex]\partial_bT^{tb}=0[/itex], I get [tex](\rho+\gamma^2p)\partial_t\mathbf{u}+ (\rho+\gamma^2p)u_x\partial_x\mathbf{u}+ (\rho+\gamma^2p)u_y\partial_y\mathbf{u}+ (\rho+\gamma^2p)u_z\partial_z\mathbf{u}+ {\nabla}p=\mathbf{0}[/tex]
This simplifies to what I put in the first post: [itex](\rho+\gamma^2p)[\partial_t\mathbf{u}+ (\mathbf{u}\cdot\mathbf{\nabla})\mathbf{u}]= -\mathbf{\nabla}p[/itex]
 
  • #4
There must be a specially relativistic correction to the Galilei invariant Euler equations. You may wish to check L&L <Fluid Mechanics>, pp. 505 to 508.
 
  • #5
It looks like the reason the [itex]\gamma^2p[/itex] term is not in the Navier-Stokes equation is because a nonrelativistic limit is applied where [itex]p<<\rho[/itex], so [itex]\rho+\gamma^2p\approx\rho[/itex].
 

1. How is the energy-momentum tensor related to the Navier-Stokes equation?

The energy-momentum tensor represents the flow of energy and momentum in a fluid. The Navier-Stokes equation is a set of equations that describe the motion of a fluid in terms of its velocity, density, and pressure. The energy-momentum tensor is used to derive the Navier-Stokes equation by considering the conservation of energy and momentum in the fluid.

2. What assumptions are made when deriving the Navier-Stokes equation from the energy-momentum tensor?

In order to derive the Navier-Stokes equation from the energy-momentum tensor, several assumptions are made. These include assuming that the fluid is incompressible, homogeneous, and isotropic, and that it obeys Newton's second law of motion.

3. What are the limitations of using the Navier-Stokes equation for fluid flow?

The Navier-Stokes equation is a simplified model that is valid for many fluid flow situations. However, it does have certain limitations, such as not being able to accurately describe highly turbulent flows, flows with shock waves, or flows with high Reynolds numbers. In these cases, more complex equations, such as the Euler or Navier-Stokes equations, may be more appropriate.

4. How is the Navier-Stokes equation used in practical applications?

The Navier-Stokes equation is used in a wide range of practical applications, including engineering, meteorology, and oceanography. It is used to model and predict the behavior of fluids in various scenarios, such as the flow of air over an airplane wing, the movement of water in a river, or the circulation of blood in the human body.

5. Are there any known solutions to the Navier-Stokes equation?

The Navier-Stokes equation is a set of partial differential equations that are notoriously difficult to solve. There are only a few known analytical solutions for specific cases, such as laminar flow in a pipe or Couette flow between two parallel plates. In most cases, numerical methods are used to approximate solutions to the Navier-Stokes equation.

Similar threads

  • Advanced Physics Homework Help
Replies
4
Views
784
Replies
10
Views
578
Replies
18
Views
938
  • Classical Physics
Replies
4
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
229
  • Advanced Physics Homework Help
Replies
4
Views
1K
  • Special and General Relativity
Replies
21
Views
2K
Replies
16
Views
2K
  • Special and General Relativity
Replies
7
Views
2K
Replies
2
Views
863
Back
Top