Can an Expanding Ball in an Ideal Fluid Affect Distant Areas?

[asheg]

Hi every one,
consider ideal fluid claiming the space all over, and a ball in it which is becoming larger and larger with radius of zero at the begin. Is such a situation possible? Doesn't it in contradiction with ideality (in-compressibility of the fluid). If it's not so, Do places far away from the ball surface sense any change? Is there any wave propagation kind equation for ideal fluids.
Are Div(v)=0 and Curl(v)=0 sufficient for this problem.
(Please give me hints and not the result or solution, since it's on me)
Thank you.

 

[olivermsun]

Well if the space is infinite, then there should be no problem with incompressibility. On the other hand, if you don't specify incompressibility (this depends on what you consider an "ideal" fluid) then of course a finite volume could accommodate a growing ball.

If the fluid is truly incompressible, then places far away from the ball would instantly "feel" the divergence. If there is some compressibility, then you might have acoustic waves radiating from the ball. If the fluid is stratified, then the disturbance might excite internal gravity waves as well.

Lots of possible results. What are you envisioning?

 

[asheg]

Thanks for your reply.
I think if the incompressible be what you say (changes propagates instantly all over space)
then the velocity distribution would be:
V(R,t) = t^2/R^2
t is time
R is Radius from center

Because the radius of ball at time t would be t and the velocity of fluid there would be 1 same as velocity of ball's surface.

But according to navier stokes it is different:
Div (v) = 0
d/dt(v) + v.Grad(v) = Grad(P)/rho

How one can get the answer from above equation? How to calculate P?

 

[olivermsun]

The zero divergence (solenoidal) condition is for the fluid parcels themselves. If you postulate a growing "ball" of something in the middle of the fluid, then of course you can't include its volume in your fluid mass conservation. The fluid on all sides of the "ball" continues to satisfy div v = 0.

 

[PJC01]

Hello,

Thank you for your question. I can provide some insights on this scenario with an ideal fluid and a growing ball.

Firstly, it is important to note that the concept of an ideal fluid is an idealization and does not exist in reality. It is a simplified model used in fluid mechanics to study the behavior of fluids. In an ideal fluid, there is no viscosity, no turbulence, and no compressibility. Therefore, the statement that the fluid is "claiming the space all over" is not entirely accurate. In reality, the fluid would be contained in some sort of container or boundary.

Now, regarding the growing ball in the ideal fluid, it is not in contradiction with the ideal fluid assumption of incompressibility. In an ideal fluid, the density is assumed to be constant, and therefore, any change in volume will not affect the fluid's density. The ball can grow without causing any change in the fluid's properties.

As for the question of whether places far away from the ball's surface sense any change, the answer is yes. This is because any disturbance in a fluid will cause a wave to propagate through it. However, in an ideal fluid, the wave propagation is governed by the Euler equations, which do not have a wave equation like in the case of a compressible fluid.

Finally, regarding the equations Div(v)=0 and Curl(v)=0, they are not sufficient to solve this problem as they only describe the fluid's kinematics (velocity and vorticity), but do not take into account the growing ball's dynamics. To solve this problem, you would need to incorporate the equations of motion for the ball, along with the Euler equations for the fluid.

I hope this provides some helpful hints for your problem. Best of luck!