Fluids Bernoulli and a pressure field

[member 428835]

I have a text that writes a pressure balance for a cylindrical shape of fluid, where the linearized Bernoulli gives the pressure field $p = p_0+\rho\partial_t \phi : \vec{v} = -\nabla \phi$ where $\vec{v}$ is the velocity vector. $p_0$ is the static pressure required to maintain the fluid's static interface shape.

Evidently gravity and kinetic energy are neglected. My question is, how is it simply $p$ equates to both the transient and $p_0$ quantity? Wouldn't there have to be a transient quantity corresponding to $p$ (wherever it's located?)

 

[Nidum]

p₀ is the static pressure required to maintain the fluid's static interface shape.

Could you explain that to me please ?

 

[member 428835]

Could you explain that to me please ?

There is a volume of liquid shaped as a cylinder, and a wire runs streamwise, attaching tangentially to the volume of liquid. The surrounding media is a fluid with negligible density compared to the cylindrical liquid. In equilibrium the liquids shape will be pure cylindrical. Disturb the volume slightly and it will start to resonate and note always be cylindrical (the cross-sectional circles won't be circles, but have small waves). Given this, is my post #1 clear now?

 

[Nidum]

Given this, is my post #1 clear now?

Sorry but I still don't entirely understand what you are doing . I can see now what the general idea of the problem is but I can't link that to your equations or the related question .

I'll let this one go I think . Thank you anyway for replying .