# Pressure in Fluids: Conceptual Q & Maths Explained

• crazy student
In summary, the equations for pressure in fluids are derived by considering the mass above a particular point in the fluid. For a fluid at rest, the pressure is equal from above and below, as well as on the sides. This can be shown mathematically and is unique to fluids. Pressure is a scalar and has no direction, so the statement that pressure is equal in all directions is true. The isotropy condition, where pressure does not depend on the orientation of the local surface element, is what ensures balance of forces on either side of a surface.
crazy student
I have a conceptual question about fluid...
we derive the equations about pressure in fluid by considering the mass above a particular point in fluid.
and for fluid at rest, the pressure above equals pressure from below.
how about on the perpendicular plane? is the pressure on the sides equal to pressure from above and below also?
my textbook says that 'At any point in a fluid at rest, the pressure is the same in all directions at a given depth',
is it an experimental statement or can it be shown mathematically?

If the pressure was not the same on either side of your fluid "element", there would be an unbalanced (net) lateral force on it, and it would move. The fact that we're considering hydrostatics means that must not happen.

crazy student said:
is the pressure on the sides equal to pressure from above and below also?

To tackle this specific point, the whole point of the analysis was to show that pressure varies only with height. Since "above", "below", and "on the sides", generally describes three different heights, I don't see why the pressure would be the same at any of these points. For instance, you already know that the pressure below is higher than the pressure above, so how could the pressure on the sides be equal to the pressure "above and below"?!

cepheid said:
If the pressure was not the same on either side of your fluid "element", there would be an unbalanced (net) lateral force on it, and it would move. The fact that we're considering hydrostatics means that must not happen.

But this is not sufficient to demonstrate that the horizontal pressure is the same as the vertical pressure. It only requires that horizontal pressure be the same in two sets of opposite directions and that vertical pressure be the same in both directions, not that they all be equal. The equality in all directions is unique to fluids. It is certainly not a requirement for solids.

I think you would have to look microscopically at the nature of fluids and get into the statistical mechanics of ensembles of particles to justify the statement of equal pressure in all directions. The pressure is the same in all directions because the kinetic energy distribution of the particles in a fluid is isotropic. The interaction of an infinitesimal volume of fluid with its surroundings will involve collisions of energetic particles with isotropic momentum distribution, etc etc etc

OlderDan said:
But this is not sufficient to demonstrate that the horizontal pressure is the same as the vertical pressure.
Pressure is a scalar. It has no direction, so there's no such things as horizontal and vertical pressure.

OldeDan is correct, what we have to show is that the scalar pressure, $$p=p(x,y,z,t)$$
at most, and not $$p(x,y,z,t,\vec{n})$$ where $$\vec{n}$$ is the normal to the surface element we're looking at, at the point (x,y,z) at time t
This is not completely trivial, since forces would balance on either side of a surface, as long as p were an even function of $$\vec{n}$$
Since, then:
$$p(\vec{x},t,\vec{n})\vec{n}dA+p(\vec{x},t,-\vec{n})(-\vec{n})dA=\vec{0}$$
The condition that the pressure does not depend on the orientation of the local surface element is called the isotropy condition.

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## 1. What is pressure in fluids?

Pressure in fluids is the force exerted by a fluid per unit area. It is a measure of how much force is being applied over a given area. In simpler terms, it is the amount of "push" or "squeeze" that a fluid is exerting on its surroundings.

## 2. How is pressure in fluids calculated?

The formula for calculating pressure in fluids is pressure = force ÷ area. This means that pressure is directly proportional to the force applied and inversely proportional to the area over which the force is applied.

## 3. What is the difference between gauge pressure and absolute pressure?

Gauge pressure is the pressure measured relative to atmospheric pressure, while absolute pressure is the total pressure at a given point, including atmospheric pressure. This means that gauge pressure can be either positive (above atmospheric pressure) or negative (below atmospheric pressure), while absolute pressure is always positive.

## 4. How does pressure change with depth in a fluid?

According to Pascal's law, pressure in a fluid increases with depth. This is because the weight of the fluid above exerts a greater force on the fluid below, resulting in an increase in pressure. Therefore, the deeper you go in a fluid, the higher the pressure will be.

## 5. What is the relationship between pressure and density in fluids?

Pressure and density in fluids have an inverse relationship. This means that as density increases, pressure decreases and vice versa. This is because an increase in density means there are more particles in a given volume, resulting in a greater force being applied to the surrounding area, thus decreasing pressure.

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