leoflindall
Dec2-10, 12:06 PM
1. The problem statement, all variables and given/known data
Consider plane-parrallel, non magnetic single fluid flow in the z direction. Assume that the realtionship between pressure, velocity and density is constrained by equation (3), shown below.
Show that
\frac{P_{1}}{\rho_{1}} = \frac{5P_{0}}{3\rho_{0}}
Where
P_{0} and \rho_{0}
are the pressure and density of the uniform background on which a wave with small pressure and density fluctiations of
P_{1} and \rho_{1} propagates in the z direction.
2. Relevant equations
(1) \frac{d\rho}{dt} + \frac{d}{dz} (\rho v) = 0
(Mass Conversation)
(2) v = vz^
(3) \frac{d}{dt} [ \frac{1}{2} \rhov^{2} + \frac{3}{2}P ] + \frac{d}{dz} [v(\frac{1}{2} \rho v^{2} + \frac{5}{2} P )] = 0
(The Constraining equation)
Please note where the notation \frac{d}{dt/dz} has been used to indicate the partial differential (I'm not the best at Latex!)
3. The attempt at a solution
I'm not looking for an answer to this problem. This a set problem that I have been set, and am at a loss at how to tackle it. I can the see there is a realtionship between the conservation of mass equation and the constraining equation, however I can't see how to tackle it to show what is required.
Any thoughts of hints inthe right direction would be greatly appreciated.
Many Thanks
Leo
Consider plane-parrallel, non magnetic single fluid flow in the z direction. Assume that the realtionship between pressure, velocity and density is constrained by equation (3), shown below.
Show that
\frac{P_{1}}{\rho_{1}} = \frac{5P_{0}}{3\rho_{0}}
Where
P_{0} and \rho_{0}
are the pressure and density of the uniform background on which a wave with small pressure and density fluctiations of
P_{1} and \rho_{1} propagates in the z direction.
2. Relevant equations
(1) \frac{d\rho}{dt} + \frac{d}{dz} (\rho v) = 0
(Mass Conversation)
(2) v = vz^
(3) \frac{d}{dt} [ \frac{1}{2} \rhov^{2} + \frac{3}{2}P ] + \frac{d}{dz} [v(\frac{1}{2} \rho v^{2} + \frac{5}{2} P )] = 0
(The Constraining equation)
Please note where the notation \frac{d}{dt/dz} has been used to indicate the partial differential (I'm not the best at Latex!)
3. The attempt at a solution
I'm not looking for an answer to this problem. This a set problem that I have been set, and am at a loss at how to tackle it. I can the see there is a realtionship between the conservation of mass equation and the constraining equation, however I can't see how to tackle it to show what is required.
Any thoughts of hints inthe right direction would be greatly appreciated.
Many Thanks
Leo