Fluid Mechanics question,velocity potential

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I am new in this place, am studying civil engineering in Spain,Madrid, and is something I do not understand in a theoretical exposition of the velocity/force potential.

They suppose that the external force that it acts in each point of the fluid and the speed, derive from scalar , so they admit a potential :

[tex]\frac{\vec{F}}{m}= - \vec{\nabla} U[/tex]

[tex]\vec{V}= \vec{\nabla} \Omega[/tex]


If the acceleration depends on the coordinates of the point and the time [tex]\vec{a} ( u',v',w') = f(x,y,z,t)[/tex] :

[tex]u'= \frac{du}{dt}= \frac{\partial u}{\partial x} \frac{\partial x}{\partial t} +\frac{\partial u}{\partial y } \frac{\partial y}{\partial t} + \frac{\partial u}{\partial z}\frac{\partial z}{\partial t} + \frac{\partial u}{\partial t}[/tex] and thus with the other coordinates of the acceleration


And here my doubt comes, I do not understand as they obtain to this expression:

[tex]u'= \frac{\partial^2 \Omega}{\partial x^2} \frac{\partial \Omega}{\partial x} +\frac{\partial^2 \Omega}{\partial x \partial y} \frac{\partial \Omega}{\partial y} + \frac{\partial^2 \Omega}{\partial x \partial z}\frac{\partial \Omega}{\partial z} + \frac{\partial^2 \Omega}{\partial x \partial t}[/tex]




if somebody can help to understand it me, would be thanked for. Thank you very much
 

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  • #2
arildno
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First of all, don't use partials where they don't belong!

Now, we have
[tex]\frac{dx}{dt}=u,\frac{dy}{dt}=v,\frac{dz}{dt}=w[/tex]
Thus, we may write the expression for the acceleration in the x-direction, i.e, u' as:
[tex]u'=u\frac{\partial{u}}{\partial{x}}+v\frac{\partial{u}}{\partial{y}}+w\frac{\partial{u}}{\partial{z}}+\frac{\partial{u}}{\partial{t}}[/tex]

Now, insert:
[tex]u=\frac{\partial\Omega}{\partial{x}},v=\frac{\partial\Omega}{\partial{y}},w=\frac{\partial\Omega}{\partial{z}}[/tex]

See if you get it right now!
 
  • #3
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Compound functions always was my nightmare


[tex]\frac{\partial}{\partial t} \left(\frac{\partial \Omega}{\partial x}\right)=\frac{\partial}{\partial x} \left(\frac{\partial \Omega}{\partial x}\right) \underbrace{\frac{dx}{dt}}_{u}+\frac{\partial}{\partial y} \left(\frac{\partial \Omega}{\partial x}\right) \underbrace{\frac{dy}{dt}}_{v} + \frac{\partial}{\partial z} \left(\frac{\partial \Omega}{\partial x}\right) \underbrace{ \frac{dz}{dt}}_{w}+ \frac{\partial}{\partial t} \left(\frac{\partial \Omega}{\partial x}\right) \frac{dt}{dt}[/tex]


that is what it did not see, thank you very much to solve the doubt to me
 

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