Wave Equation for Second Sound in Superfluid He-4

I am trying to derive the wave equation for 'Second Sound in superfluid Helium-4 using the basic tenets of the two-fluid model. I am following the derivation in a book which has intermediate steps along the way - I am trying to fill in the gaps. I am almost there - there is only one step that I cannot do:

I cannot get from:

(dVn/dt) - (dVs/dt) = (-p/pn).S.grad(T) ........... (1)

to

(pn/ps).d/dt(p.Div(Vn)) = -p.S.Laplacian(T) ........... (2)

where S = entropy, T = temperature, ps = density of superfluid component, pn = density of normal component, p = pn + ps = overall density of helium, Div = divergence operation, grad = grad operation, Laplacian = Laplacian operator = div(grad(_)), Vn = velocity of normal component of Helium, Vs = velocity of superfluid component of Helium.

The book says to take the divergence (Del) of both sides of eqn 1, and then use eqn 3 below to 'replace the superfluid velocity in the result', where upon eqn 2 apparently pops out after a little wrangling.

Div(j) = -(dp/dt) ........... (3)

Where j = pnVn + psVs ...........(4)

It doesn't happen for me though!

Setting dp/dt = 0 (a fair approximation in this case) in (3) and subbing in (4), I get:

Div(Vs) = -(pn/ps).Div(Vn) ..........(5)

Subbing (5) into (1) after taking the Div of both sides, I get:

d/dt[Div(Vn) + (pn/ps).Div(Vn)] = -(p/pn).S.Laplacian(T) ............... (6)

I can't work out how to go further or whether I've committed some mathematical howler! I need to get (6) to look like (2), unless I've stuffed up in arriving at (6)...

Any help would be much appreciated!

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gabbagabbahey
Homework Helper
Gold Member
You're almost there;

$$\vec{\nabla}\cdot\vec{v}_n+\left(\frac{p_n}{p_s}\right)\vec{\nabla}\cdot\vec{v}_n=\left(1+\frac{p_n}{p_s}\right)\vec{\nabla}\cdot\vec{v}_n$$

And

$$1+\frac{p_n}{p_s}=\frac{p_s+p_n}{p_s}=\frac{p}{p_s}$$

Champion - I appreciate it a great deal - that has been driving me bloody well nuts!