What happens at the lambda point of superfluid Helium?

[CMJ96]

Homework Statement

2. Homework Equations are given in the above picture
3. The attempt at the solution
In part a) I simply rearranged the other expressions given and subbed them into the relationship for relative velocity and came out with Vns= Qdot/ρsT(1- ρn/ρs).Then in part b) I started by rearranging the equation for Qdot to get an equation for Vn and subbed T=0 in, this then allowed me to come to the conclusion that Vn must blow up to infinity if T -> 0 and hence Vs/Vn will become 0.
However, I have a problem with the second part of b), I have been thinking about it and understand that at the Lambda point there is a discontinuity in the heat capacity and hence it will blow up to infinity too, here is where my problem arises, I'm not sure how that effects the ratio of Vs/Vn, my idea was that if the heat capacity is infinite the helium will simply absorb all of the heat and will store it as internet energy, hence it will not move away from the heat source? Any guidance would be greatly appreciated

 

[CMJ96]

I've figured this out, turns out the ratio of superfluid falls to such an infinitely small number that the velocity of the superfluid would tend to infinity to maintain zero mass flux, so I will mark this as solved