- #1

Adesh

- 735

- 188

Consider a small volume element ##\Delta \tau## in the fluid and let’s assume it to be a cuboid with dimensions ##\Delta x##, ##\Delta y## and ##\Delta z##. The seemingly backward face of ##\Delta \tau## have ##x## coordinate as ##x## and the forward face would have the ##x+\Delta x## as coordinate.

Now, forces acting in the ##x-##direction: backwards face = ##p(x) \Delta y \Delta z##

forwards face = ## -p(x+\Delta x) \Delta y \Delta z##

on the whole fluid =##F_x \Delta \tau##. For equilibrium we must have $$ \left(

p(x+\Delta x) - p(x) \right) \Delta y \Delta z = F_x \Delta \tau \\

\frac{\partial p}{\partial x} \Delta x \Delta y \Delta z = F_x \Delta \tau \\

F_x = \frac{\partial p}{\partial x}$$

Similarly, for other directions and therefore we have $$ \mathbf F = \nabla p$$ .

My problem is that Mr. Arnold Sommerfeld is saying that this pressure distribution is keeping the fluid from moving. He gives the reason that gravity has a potential and can be written like $$\mathbf F = - \nabla U$$ And for this he says

**Equilibrium is only possible if the external force has a potential**.

My problem is how is potential or pressure distribution is keeping the fluid from moving? I think it is the lower wall and side walls of the column that is preventing the fluid from moving. I’m sceptical because later on he proves that if the force were to be magnetic, then fluid will start to flow in a circular motion. What is the significance of ##\nabla P## and ##\nabla U##?

Please explain me what is he trying to emphasise.