# Fluid Mechanics question,velocity potential

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 :

$$\frac{\vec{F}}{m}= - \vec{\nabla} U$$

$$\vec{V}= \vec{\nabla} \Omega$$

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

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

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

$$u'= \frac{\partial^2 \Omega}{\partial x^2} \frac{\partial \Omega}{\partial x} +\frac{\partial^2 \Omega}{\partial x \partial y} \frac{\partial \Omega}{\partial y} + \frac{\partial^2 \Omega}{\partial x \partial z}\frac{\partial \Omega}{\partial z} + \frac{\partial^2 \Omega}{\partial x \partial t}$$

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

arildno
Homework Helper
Gold Member
Dearly Missed
First of all, don't use partials where they don't belong!

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

Now, insert:
$$u=\frac{\partial\Omega}{\partial{x}},v=\frac{\partial\Omega}{\partial{y}},w=\frac{\partial\Omega}{\partial{z}}$$

See if you get it right now!

Compound functions always was my nightmare

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

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