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

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Discussion Overview

The discussion revolves around the theoretical scenario of an expanding ball within an ideal fluid, exploring the implications of incompressibility and the potential effects on distant areas of the fluid. Participants examine the conditions under which such a situation might be possible and the resulting fluid dynamics, including wave propagation and velocity distributions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether an expanding ball in an ideal fluid contradicts the concept of incompressibility and asks if distant areas would sense any changes.
  • Another participant suggests that if the space is infinite, incompressibility may not pose a problem, but the definition of "ideal" fluid could affect the situation.
  • It is proposed that if the fluid is incompressible, distant areas would feel changes instantly, while some compressibility might lead to acoustic waves radiating from the ball.
  • A participant presents a velocity distribution formula based on the assumption of incompressibility and the relationship between the ball's radius and fluid velocity.
  • Concerns are raised about the application of the Navier-Stokes equations, particularly regarding the divergence condition and how to calculate pressure in this context.
  • Another participant clarifies that the zero divergence condition applies to fluid parcels, indicating that the volume of the expanding ball should not be included in fluid mass conservation calculations.

Areas of Agreement / Disagreement

Participants express differing views on the implications of incompressibility and the behavior of the fluid in response to the expanding ball. There is no consensus on the effects on distant areas or the correct application of fluid dynamics principles.

Contextual Notes

Participants highlight the need for clarity on definitions of ideal fluids and the assumptions regarding incompressibility. The discussion also reflects uncertainty about the mathematical treatment of the scenario, particularly in relation to the Navier-Stokes equations.

asheg
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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.
 
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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?
 
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?
 
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.
 

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