How does a pressure distribution keep the fluid from moving?

In summary, the author is saying that a lack of acceleration is necessary for equilibrium, and that a pressure gradient must be zero for equilibrium to be maintained in the absence of a gravitational force.
  • #1
Adesh
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Let’s say we have a unit volume of some fluid in a column on the Earth surface. Let ##\mathbf F## be the gravitational force that acts on the unit volume of the fluid.

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.
 
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  • #2
Adesh said:
My problem is how is potential or pressure distribution is keeping the fluid from moving?
The pressure distribution is preventing the fluid from accelerating. If the fluid starts from rest, a lack of acceleration is sufficient to keep it at rest. A lack of acceleration is also required to keep the fluid at rest.

The author is expecting you to consider the interior of the fluid volume. He does not seem concerned here with the conditions at the boundaries of the volume.
 
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  • #3
jbriggs444 said:
The pressure distribution is preventing the fluid from accelerating.
Wasn’t there any pressure in fluid when there was no gravitational force? I mean will there be a pressure when there is no external conservative force?
 
  • #4
Take an ideal fluid. Then Euler's equation reads
$$\rho \mathrm{D}_t \vec{v}=\rho (\partial_t \vec{v} + (\vec{v} \cdot \vec{\nabla} \vec{v})=-\vec{\nabla} P+ \rho \vec{g},$$
were I assumed only the gravitational force close to Earth as an external force. In static situations the left-hand side vanishes and then
$$\vec{\nabla} P=\rho \vec{g}.$$
 
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  • #5
Adesh said:
Wasn’t there any pressure in fluid when there was no gravitational force? I mean will there be a pressure when there is no external conservative force?
The pressure gradient in such a case must be zero. As @vanhees71 showed above.

The pressure can take on any value. For instance, it depends on how much air you put in the tire.
 
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  • #6
Will there be no pressure (or any value of pressure) when the fluid is subjected to a non-conservative force?
 
  • #7
Adesh said:
Will there be no pressure (or any value of pressure) when the fluid is subjected to a non-conservative force?
You cannot have an equilibrium in such a case. There can be pressure, certainly. But the fluid will also be accelerating.
 
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  • #8
Thank you so much.
 

Related to How does a pressure distribution keep the fluid from moving?

1. How does pressure distribution affect the movement of fluid?

Pressure distribution refers to the way that pressure is distributed throughout a fluid. This pressure distribution affects the movement of fluid by creating a force that opposes the movement of the fluid. This force is known as drag and it is caused by the differences in pressure between different areas of the fluid.

2. What is the relationship between pressure distribution and fluid movement?

The relationship between pressure distribution and fluid movement is that pressure distribution determines the direction and speed of fluid movement. When there is a difference in pressure between two points in a fluid, the fluid will move from the high pressure area to the low pressure area. This movement is what keeps the fluid from being stationary.

3. How does pressure distribution keep the fluid from flowing in a single direction?

Pressure distribution keeps the fluid from flowing in a single direction by creating a balance of forces within the fluid. When there is a difference in pressure between two points in a fluid, the fluid will move from the high pressure area to the low pressure area. However, as the fluid moves, it creates areas of high pressure behind it, which then balances out the pressure distribution and stops the fluid from moving in a single direction.

4. Can pressure distribution be manipulated to control the movement of fluid?

Yes, pressure distribution can be manipulated to control the movement of fluid. By changing the shape or size of an object in a fluid, the pressure distribution can be altered, which in turn changes the direction and speed of the fluid. This principle is used in many applications, from designing airplane wings to regulating the flow of water in pipes.

5. How does pressure distribution affect the flow rate of a fluid?

The flow rate of a fluid is directly affected by the pressure distribution within the fluid. When there is a large difference in pressure between two points in a fluid, the flow rate will be faster as the fluid moves from the high pressure area to the low pressure area. On the other hand, when the pressure distribution is more uniform, the flow rate will be slower as the fluid does not have as strong of a driving force to move it.

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