Transformation from Eulerian to Lagrangian

hunt_mat · October 10, 2023, 10:19

Suppose I have the system:
$\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)=0$
$\frac{\partial}{\partial t}(\rho u)+\frac{\partial}{\partial x}(\rho u^{2})-\frac{\partial}{\partial x}\left(\nu\frac{\partial u}{\partial x}\right)=0$
written in Eulerian co-ordinates, and I wish to write it in Lagrangian co-ordinates. I would normally use the Piola transformation to obtain the equations of motion. However, when I have a secopnd order equation like the one I stated, would I have to use the chain rule to convert the first order equations into Lagrangian co-ordinates to complete the transformation?
$\frac{\partial}{\partial x}=\frac{\partial X}{\partial x}\frac{\partial}{\partial X}+\frac{\partial Y}{\partial x}\frac{\partial}{\partial Y}$
$\frac{\partial}{\partial y}=\frac{\partial X}{\partial y}\frac{\partial}{\partial y}+\frac{\partial Y}{\partial y}\frac{\partial}{\partial Y}$

Is there a compact notation for this?

LuckyTran · October 10, 2023, 16:39

it's the Jacobian. What am I missing?

FMDenaro · October 11, 2023, 04:31

Quote:

Originally Posted by hunt_mat

Suppose I have the system:
$\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)=0$
$\frac{\partial}{\partial t}(\rho u)+\frac{\partial}{\partial x}(\rho u^{2})-\frac{\partial}{\partial x}\left(\nu\frac{\partial u}{\partial x}\right)=0$
written in Eulerian co-ordinates, and I wish to write it in Lagrangian co-ordinates. I would normally use the Piola transformation to obtain the equations of motion. However, when I have a secopnd order equation like the one I stated, would I have to use the chain rule to convert the first order equations into Lagrangian co-ordinates to complete the transformation?
$\frac{\partial}{\partial x}=\frac{\partial X}{\partial x}\frac{\partial}{\partial X}+\frac{\partial Y}{\partial x}\frac{\partial}{\partial Y}$
$\frac{\partial}{\partial y}=\frac{\partial X}{\partial y}\frac{\partial}{\partial y}+\frac{\partial Y}{\partial y}\frac{\partial}{\partial Y}$

Is there a compact notation for this?

Matrix notation

d = J.d'

you can see the trasformation in the book of Chorin &Marsden in the description of the Reynolds transport theorem.

hunt_mat · October 13, 2023, 12:45

I managed to work it out in the end.

October 10, 2023, 10:19	Transformation from Eulerian to Lagrangian	#1
hunt_mat Senior Member Matthew Join Date: Mar 2022 Location: United Kingdom Posts: 184 Rep Power: 4	Suppose I have the system: $\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)=0$ $\frac{\partial}{\partial t}(\rho u)+\frac{\partial}{\partial x}(\rho u^{2})-\frac{\partial}{\partial x}\left(\nu\frac{\partial u}{\partial x}\right)=0$ written in Eulerian co-ordinates, and I wish to write it in Lagrangian co-ordinates. I would normally use the Piola transformation to obtain the equations of motion. However, when I have a secopnd order equation like the one I stated, would I have to use the chain rule to convert the first order equations into Lagrangian co-ordinates to complete the transformation? $\frac{\partial}{\partial x}=\frac{\partial X}{\partial x}\frac{\partial}{\partial X}+\frac{\partial Y}{\partial x}\frac{\partial}{\partial Y}$ $\frac{\partial}{\partial y}=\frac{\partial X}{\partial y}\frac{\partial}{\partial y}+\frac{\partial Y}{\partial y}\frac{\partial}{\partial Y}$ Is there a compact notation for this?

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October 10, 2023, 16:39		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	it's the Jacobian. What am I missing?

October 13, 2023, 12:45		#4
hunt_mat Senior Member Matthew Join Date: Mar 2022 Location: United Kingdom Posts: 184 Rep Power: 4	I managed to work it out in the end.