Three doubts in a paragraph on the equilibrium of Incompressible fluids

[Adesh]

Homework Statement

See the main body.

Relevant Equations

We don’t need them.

These are the images from Sommerfeld’s Lectures on Theoretical Physics, Vol 2 chapter 2, section 6, Equilibrium of Incompressible Fluids.

Image 1

Image 2

Doubt 1 : What does it mean for a force to act on a fluid volume? Force acts on a point, force may act on a surface but I’m unable to understand what it means “external force per unit of fluid volume”. Did Sommerfeld mean that $\mathbf F$ acts on all six faces of that fluid volume element ?

Doubt 2 : I really couldn’t understand what happened after equation 3a. Why we divided by $\Delta \tau$ and how $F_x$ came in equation 3b ?

Doubt 3 : How equation 3b and 4 “obviously” constitute a necessary condition for the equilibrium? I’m unable to understand how $grad~ p = \mathbf F$ is a necessary condition for the equilibrium.

Please explain the above said doubts to me, if my question is too broad please let me know.

Thank you.

 

[Orodruin]

Homework Statement:: See the main body.
Relevant Equations:: We don’t need them.

These are the images from Sommerfeld’s Lectures on Theoretical Physics, Vol 2 chapter 2, section 6, Equilibrium of Incompressible Fluids.
Image 1
Image 2

Doubt 1 : What does it mean for a force to act on a fluid volume? Force acts on a point, force may act on a surface but I’m unable to understand what it means “external force per unit of fluid volume”. Did Sommerfeld mean that $\mathbf F$ acts on all six faces of that fluid volume element ?

No, he meant exactly what he said. It is external force per unit volume. For example, the force of gravity per unit volume would be $\rho \vec g$ where $\rho$ is the density and $\vec g$ the gravitational field.

Doubt 2 : I really couldn’t understand what happened after equation 3a. Why we divided by $\Delta \tau$ and how $F_x$ came in equation 3b ?

We divided in order to get rid of the volume element and obtain a differential equation for p. $F_x$ is the x-component of $\vec F$. Equation 3b is based on equilibrium of the fluid.

Doubt 3 : How equation 3b and 4 “obviously” constitute a necessary condition for the equilibrium? I’m unable to understand how $grad~ p = \mathbf F$ is a necessary condition for the equilibrium.

That equation is a direct consequence of force equilibrium on each fluid element.

 

[Adesh]

We divided in order to get rid of the volume element and obtain a differential equation for p. $F_x$ is the x-component of $\vec F$ . Equation 3b is based on equilibrium of the fluid.

I thought that the pressure at the $x$-faces of rectangular cell was caused due to the external force only. I mean to say that $p(x)$ at the negative $x$ surface was caused because we had a $x$ component of the force. So, for me invoking $F_x$ in the summation of pressures at opposite faces seems a quite hard thing to conceive. According to me, pressures at opposite faces should cancel each other by themselves.

Is there any way by which I can be helped in changing my perspective regarding the above situation? I mean can you help me?

 

[Adesh]

How pressure can be there in the absence of any external force on fluid?

 

[hutchphd]

I thought that the pressure at the xxx-faces of rectangular cell was caused due to the external force only

External force here means any force external to the cell in question...in particular from the cell adjacent. In limit of infinitesimal cells this gives the smooth pressure function.

 

[Adesh]

External force here means any force external to the cell in question...in particular from the cell adjacent. In limit of infinitesimal cells this gives the smooth pressure function.

What is the source of $p(x) \Delta y \Delta z$ and of $F_x$?

 

[hutchphd]

I believe the Fx represents any external field (like say gravity) and p(x) the contact pressure from the adjacent fluid (in the infinitesimal limit ultimately).

 

[Adesh]

I believe the Fx represents any external field (like say gravity) and p(x) the contact pressure from the adjacent fluid (in the infinitesimal limit ultimately).

Thank you.

 

[Orodruin]

According to me, pressures at opposite faces should cancel each other by themselves.

This is only true if the pressure is the same on both sides. This is not necessarily the case and for small $\Delta x$, the pressures will differ by approximately $\Delta x\, \partial_x p$.