The Euler Equation and Incompressible Fluid Theorems

AI Thread Summary
The discussion revolves around the application of the Euler equation for incompressible fluids, specifically using suffix notation to analyze the equation. Participants explore the implications of the divergence theorem and how to manipulate the equation through integration by parts. A key point is the realization that the divergence of an incompressible fluid leads to the condition that the density remains constant, which simplifies the analysis. The conversation highlights the importance of correctly applying factors such as density in the equations. Ultimately, the conclusion is reached that the constant K in the derived equation must be zero, confirming the correctness of the approach taken.
aclaret
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Homework Statement
Let ##u: R^3 \times R \rightarrow R^3## be flow velocity of incompressible fluid. Let fluid be subject to potential force ##-\nabla \chi##. To prove
$$\frac{d}{dt} \int_{V} \frac{1}{2} \rho \langle u, u \rangle dV + \int_{\partial V} H \langle u, n \rangle dA = 0$$where ##H := \frac{1}{2}\rho \langle u, u \rangle + p + \chi##, and notation ##\langle x, y \rangle## denote standard inner product on ##R^3##.
Relevant Equations
fluid dynamic, euler's equation of the motion
$$\frac{Du}{Dt} = -\frac{\nabla p}{\rho} - \nabla \chi$$I re-write the Euler equation for incompressible fluid using suffix notation
$$\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} + \frac{\partial}{\partial x_i} \left(\frac{p}{\rho} + \chi \right) = 0$$what theorems applies to the problem?
 
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You have integrals involving a volume and a surface. That would suggest the divergence theorem.
 
divergence theorem I can write$$\int_{\partial V} \langle u, n \rangle dA = \int_V \frac{\partial u_j}{\partial x_j} dV$$try multiply equation by ##u_i## and then do implicit summation also over ##i##

$$u_i \frac{\partial u_i}{\partial t} + u_i u_j \frac{\partial u_i}{\partial x_j} + u_i \frac{\partial}{\partial x_i} \left(\frac{p}{\rho} + \chi \right) = 0$$get for second term$$u_i u_j \frac{\partial u_i}{\partial x_j} = \frac{1}{2}u_j \frac{\partial}{\partial x_j} \langle u, u \rangle$$first term also

$$u_i \frac{\partial u_i}{\partial t} = \frac{1}{2} \frac{\partial}{\partial t} \langle u,u \rangle$$do I effect the volume integral to both sides?

$$\int_V \frac{1}{2} \frac{\partial}{\partial t} \langle u,u \rangle dV + \int_V \frac{1}{2}u_j \frac{\partial}{\partial x_j} \langle u, u \rangle dV + \int_V u_i \frac{\partial}{\partial x_i} \left(\frac{p}{\rho} + \chi \right) dV = 0$$can I make use of divergence theorem here?
 
$$\frac{d}{dt} \int_V \frac{1}{2} \rho \langle u,u \rangle dV + \int_V u_i \frac{\partial}{\partial x_i} \left[ \frac{1}{2} \rho \langle u, u \rangle + \left(p + \rho \chi \right) \right] dV = K, \quad K \, \mathrm{= constant}$$I can try integrate second term by parts,$$\int_V u_i \frac{\partial}{\partial x_i} \left[ \frac{1}{2} \rho \langle u, u \rangle + \left(p + \rho \chi \right) \right] dV = \int_{\partial V} \left[ \frac{1}{2} \rho \langle u, u \rangle + \left(p + \rho \chi \right) \right] \langle u, n \rangle dA - \int_{V} \left[ \frac{1}{2} \rho \langle u, u \rangle + \left(p + \rho \chi \right) \right] \frac{\partial u_i}{\partial x_i} dV$$does second term vanish? even if, the first term still incorrect. I wonder if you can tell second hint @pasmith :), I'm in a little confusion ;)
 
What is the divergence of an incompressible fluid?
 
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let me see... clearly from equation of continuity it follow that if ##(\forall x,t) \, \rho(x,t) = \text{constant}##, then ##\nabla \cdot u = 0##. hence, second integral vanish, and get$$\frac{d}{dt} \int_V \frac{1}{2} \rho \langle u,u \rangle dV + \int_{\partial V} \left[ \frac{1}{2} \rho \langle u, u \rangle + \left(p + \rho \chi \right) \right] \langle u, n \rangle dA = K, \quad K \, \mathrm{= constant}$$but this would give me ##H := \frac{1}{2} \rho \langle u, u \rangle + p + \rho \chi##, which tiny bit different to problem statement. would you agree my work is correct, or did I do a mistake somewhere...
 
The problem statement says that the fluid is subject ot a potential force of -\nabla\chi, not -\rho\nabla\chi. Other than that your work is correct.

The constant K is zero; it were not there would be some point where <br /> \rho \frac{Du}{Dt} + \nabla p + \nabla \chi \neq 0.
 
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parfait! yes I see, thank for point out my mistake with the factor of ##\rho##. I also was not certain how to "convince myself" that ##K## indeed vanish.

i think necessary for me do lot more of these problem to learn the tricks of the trade :), the simple realize that ##\nabla \cdot u = 0## was enough here to reveal how to finish the solution!

thank for your assistance :)
 
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