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Flow, etc.
Physics 9HB, Jo e Kiskis
Total, lo cal, and convective derivatives
The uid quantities we are interested in are functions of t and x. Examples are the pressure and the velo city. Let's just use Q(t; x) to stand for any eld. If we want to know how Q changes in time at a xed point in space, that's @ Q(t;x) @ t ,^ which^ is^ called^ the^ lo^ cal^ derivative.^ However,^ if^ we^ want^ to^ know how Q changes as we ride along with the uid, we must think a bit more. The path of an element of the uid x(t) satis es dx dt(t )= v (t; x(t)). Thus, if we are riding with the uid Q(t) = Q(t; x(t)) and
dQ(t) dt
@ Q(t; x) @ t
X^3
i=
@ Q(t; x) @ xi
dxi (t) dt
@ Q(t; x) @ t
X^3
i=
vi
@ Q(t; x) @ xi
@ Q(t; x) @ t
The total derivative is the lo cal derivative plus the convective derivative (second term).
Derivatives of v
We want to get an equation of motion for the uid. Newton's second law has the time derivative of the momentum. So we need the time derivative of the momentum of a little element of the uid in the volume dV. The mass is dV , and the velo city is v , so p(t; x) = dV v (t; x). The time rate of change of the momentum of a given element of uid is the total derivative of p. Since we are restricting ourselves to incompressible ow, we do not need to di erentiate , and the derivative of p is (the derivative of v ). Thus, for each comp onent of v ,
dvi dt
@ vi (t; x) @ t
If we collect the comp onents together, this can b e written
dv dt
@ v (t; x) @ t
If the notation of the last term here is not clear, rememb er that it means just what is in the previous comp onent version.
Equation of motion
The equation of motion for an element of the uid is d dtp = F. Now we know how to write the LHS as
dp dt
= dV [
@ v (t; x) @ t
On the RHS, there must b e the forces on the element of uid. We have already seen what these can b e. There is a force dV g due to gravity, and a force rP dV due to the pressure gradient. There is also a force due to the internal friction of the uid|viscosity. For now, we will consider the case of negligibl e viscosity. Later we will add a viscous force. So, after cancelling the dV , that leaves us with
[
@ v (t; x) @ t
- (v r)v (t; x)] = rP + g (7)
as the equation of motion for inviscous, incompressible ow. Sometimes it is convenient to write this in a slightly di erent form. There an identity in vector calculus which says
(v r)v = (r v ) v +
rv 2 : (8)
You can check this by tediously writing out the terms on each side and comparing them. Also, the curl of the velo city is called the vorticity = r v. So the alternative form for the equation of motion is
[
@ v (t; x) @ t
rv 2 ] = rP + g : (9)
This constant may vary from streamline to streamline. This is Bernoulli's equation. For irrotation al ow, there is a stronger result. In that case, the equation of motion b ecomes simply
r(
v 2 ) = r(P + ) (15)
so
r(
v 2 + P + ) = 0 (16)
and 1 2
v 2 + P + = constant: (17)
Now the constant is the same for all streamlines, i.e. it's indep endent of space and time. However, it do es dep end up on the problem at hand and must b e determined from some given data.
