Material Derivative

1. May 29, 2010

coverband

A fluid moves so that its velocity is $$\vec {u} \equiv (2xt,-yt,-zt)$$, written in rectangular Cartesian coordinates. Show that the surface F(x,y,z,t) = $$x^2exp (-2t^2)+(y^2+2z^2)exp(t^2)=constant$$ moves with the fluid (so that it always contains the same fluid particles; that is, DF/Dt=0)

2. Relevant equations
I got this from http://en.wikibooks.org/wiki/Marine..._Wave_Theory#Kinematic_Free_Surface_Condition

$$\frac{DF(x,y,z,t)}{Dt}=w.$$ If the surface is defined by z=A(x,y,t), then $$w = \frac {\partial A}{\partial t} + u \frac {\partial A}{\partial x}+v \frac{\partial A}{\partial y}$$ but I'm confused. If A refers to the surface, my surface has z's in it. Also what does the F stand for in the wiki equations

3. The attempt at a solution

2. May 29, 2010

jdwood983

I believe $F$ is also the surface. If you look at one of the prior sections, it says,

I would also assume that if your surface has a component of $z$ in it, you cannot use that form of the derivative and must use the total derivative.

3. May 29, 2010

coverband

Thanks very much