- #1
Mr. Cosmos
- 9
- 1
I am not entirely sure how to convert the conservation of mass and momentum equations into the Lagrangian form using the mass coordinate h. The one dimensional Euler equations given by,
[tex] \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0 [/tex]
[tex] \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + \frac{1}{\rho}\frac{\partial p}{\partial x} = 0 [/tex]
need to be converted to,
[tex] \frac{\partial v}{\partial t} - \frac{\partial u}{\partial h} = 0 [/tex]
[tex] \frac{\partial u}{\partial t} + \frac{\partial p}{\partial h} = 0 [/tex]
where the Lagrangian mass coordinate has the relation,
[tex] \frac{\partial h}{\partial x} = \rho [/tex]
and
[tex] v = \frac{1}{\rho} [/tex]
Thanks.
[tex] \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0 [/tex]
[tex] \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + \frac{1}{\rho}\frac{\partial p}{\partial x} = 0 [/tex]
need to be converted to,
[tex] \frac{\partial v}{\partial t} - \frac{\partial u}{\partial h} = 0 [/tex]
[tex] \frac{\partial u}{\partial t} + \frac{\partial p}{\partial h} = 0 [/tex]
where the Lagrangian mass coordinate has the relation,
[tex] \frac{\partial h}{\partial x} = \rho [/tex]
and
[tex] v = \frac{1}{\rho} [/tex]
Thanks.
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