Fluid Flow Symmetry: Showing $\pi_b G^b$ is Constant

In summary, the conversation discusses how to show that the equation \begin{align*}u^a \nabla_{a} (\pi_b G^b) = 0\end{align*} is true if ##\pi_a = (\mu + TS) u_a##, where ##S## is entropy/baryon, ##T## is temperature, ##u_a## is a one-form field corresponding to a fluid-comoving observer and ##\mu## is chemical potential. The suggestion is to use Killing's equation and to expand out the left hand side, taking into account properties of ##u^a##.
  • #1
ergospherical
1,072
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I don't know where to start with this problem. If ##\pi_a = (\mu + TS) u_a## then show that \begin{align*}
u^a \nabla_{a} (\pi_b G^b) = 0
\end{align*}where the field ##G^a## is a symmetry generator. [##S## is entropy/baryon, ##T## is temperature, ##u_a## is a one-form field corresponding to a fluid-comoving observer and ##\mu## is chemical potential].
 
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  • #3
What have you tried? Killing's equation?
 
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Likes ergospherical
  • #4
How would I use Killing's equation here? The only thing I wrote down so far was ##L_{G} \pi = 0##, the statement of the invariance of ##\pi## under the symmetry transformation corresponding to the Killing vector ##G##, but I don't know how to obtain the target equation.
 
  • #5
Did you expand out the left hand side? I think Killing’s equation will kill off some terms involving [itex] \nabla_a G_b [/itex]. What remains may vanish due to other properties of, say, [itex] u^a[/itex] that are implicit in your problem statement.
 

FAQ: Fluid Flow Symmetry: Showing $\pi_b G^b$ is Constant

What is fluid flow symmetry?

Fluid flow symmetry is the concept that the physical properties of a fluid, such as velocity and pressure, remain constant along a line of symmetry. This means that the fluid behaves in a predictable and consistent manner, allowing for mathematical equations to be used to describe its behavior.

What does $\pi_b G^b$ represent in fluid flow symmetry?

$\pi_b G^b$ is a dimensionless parameter used in fluid flow symmetry equations. It represents the ratio of the fluid's inertial forces to its viscous forces, and is used to determine when the fluid flow can be considered laminar or turbulent.

How is $\pi_b G^b$ calculated?

Calculating $\pi_b G^b$ involves determining the fluid's characteristic length, velocity, and viscosity, and plugging these values into the equation $\pi_b G^b = \frac{L\rho u^2}{\mu}$, where L is the characteristic length, $\rho$ is the fluid density, u is the velocity, and $\mu$ is the viscosity.

Why is it important to show that $\pi_b G^b$ is constant in fluid flow symmetry?

Showing that $\pi_b G^b$ is constant is important because it allows for the simplification of equations used to describe fluid flow. This makes it easier to analyze and predict the behavior of fluids in various situations, such as in pipes or around objects in a flow.

What are some real-world applications of fluid flow symmetry?

Fluid flow symmetry has many practical applications, such as in designing efficient pipes and pumps, predicting the behavior of air and water around vehicles and aircraft, and understanding the flow of blood in the human body. It is also used in industries such as aerospace, automotive, and biomedical engineering.

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