Understanding Fluid Pressure and Its Properties
Pressure
Definition
Pressure is defined as force per unit area.
Fluid Pressure
If we have a liquid in a container, the fluid pressure exerted on the bottom of the container is defined as follows: To define the pressure at another point (internal fluid), we isolate the top of the fluid.
We take an elemental surface (AS) around the point and consider the force (AF) exerted on that surface.
P = (AF / AS)
The pressure at the point is obtained when AS approaches 0:
P = Lim (AF / AS) (AS -> 0) = (dF / dS)
In this way, we define the pressure inside the fluid.
Properties of Fluid Pressure
Pascal’s Principle
1. Pressure at a Point
The pressure at a point in a fluid at rest is equal in all dimensions, i.e., it is a scalar.
Let’s consider a basic surface around the point and assume that the pressures in different directions are different. Each pressure gives rise to a force.
For the fluid to be at rest:
Pds = P’ds’cosθ
dS = dS’cosθ
P’dS = PdS
P = P’
The pressure does not depend on the direction. The pressure at a point at rest is equal in all directions.
2. Pressure in a Horizontal Plane
The pressure at all points in the same horizontal plane within a fluid at rest is the same.
Let’s pick two points on a horizontal plane and consider an elementary surface. We will assume that the pressures are different.
For the fluid to be in equilibrium:
PdS = P’dS
3. Contact Force
In a quiescent fluid, the contact force exerted by the fluid inside a portion of it on the next is normal to the interface.
In a quiescent fluid, force acts only due to pressure.
If we consider an expanse of water and separate two parts, assuming that the force is not normal, there will be movement produced by the tangential component.
Therefore, there can be only normal forces (forces of pressure).
4. Direction of Pressure Force
The force of pressure in a fluid at rest is always directed into the fluid (since the fluid cannot pull).
5. Free Surface
The free surface of a liquid at rest is horizontal.
If we imagine a surface that is not horizontal, the horizontal component of force will cause motion.
Note on Equilibrium
When we speak of balance, we mean absolute balance, for example, a liquid at rest inside a container at rest.
If the container moves with acceleration a, we speak of relative equilibrium. The same applies if the container rotates.
Pressure Units
[P] = F / L^2
In the International System: N/m^2 = Pa
In the CGS system: dyne/cm^2 = barye
1 bar = 10^5 Pa
Atmosphere
Atmosphere: Pressure that Hg exerts with a column height of 76cm.
1 atm = W/S = ρVg/S = ρShg/S = ρHg
1 atm = ρHg = 13.6 g/cm^3 * 76cm * 980cm/s^2 = 1.013 * 10^6 baryes = 1.013 bar