Fluid Statics and Pressure Measurement

Let us determine the pressure at any point in a fluid a rest. For this, let us consider a wedge-shaped particle that is exposed on all sides to a fluid. This is presented as follows:

The cross section of the particle can be used to sketch a free-body diagram, as shown below:

The dimensions Δx, Δy, and Δz are small and approach zero as the particle shrinks to a point. Pressure and gravity are the only forces considered to be acting on the particle. Applying Newton’s second law in the x- and z-directions respectively, we obtain the following equations;

where:

• p_x, p_z, and p_s are average pressures acting on the three corresponding faces
• a_x and a_z are the accelerations
• ρ is the particle density

The net force equals zero in a static fluid. After simplification, with a_x = a_z = 0, these two equations become:

p_xΔz – p_sΔs(sinθ) = 0                              and

p_zΔx – p_sΔs(cosθ) – (ρg/2)ΔxΔz = 0

(ρg/2)ΔxΔz can be neglected as it is a higher-order term containing ΔxΔz, which is very small in comparison to the other terms. From the geometry of the wedge, we find that:

                                      Δz = Δs sinθ              and            Δx = Δs cosθ      

Since θ is chosen arbitrarily chosen, substitution into the pressure equations we obtain:

This shows that pressure at a point is the same in all directions and is applicable to both 2 dimension and 3-dimension cases.  

Let us consider an element of a fluid at rest, as illustrated below:

The element chosen has a volume, dx dy dz, and is sketched on a coordinate system where the positive z-direction is downward, coincident with the direction of the gravity force.

The following figure is a view of the element looking in the positive y-direction; the force acting on the right face is pdydz and that on the left face is (p + (∂p/∂x)dx)dydz, both normal to their respective surfaces. 

Summing forces in the x-direction, we have the following for a static fluid:

this means that pressure does not vary with respect to x. Similarly, for the y-direction we can assume the same as x-direction, hence:

These two equations show that there is no pressure variation in any lateral direction.

If we create a free body diagram for the z-direction and add up the forces, we obtain:

This shows that pressure does vary in a static fluid in the z-direction i.e., pressure increases with depth. This is mathematical shown by integrating both sides of the above equation:

where point 1 is a reference point such as the free surface of a liquid and point 2 is a point of interest. Since the density is a constant for incompressible fluids, we get: 

p₂ – p₁ = ρg(z₂ – z₁) = ρgΔz

where Δz is the depth below the liquid surface. This relationship is the basic equation of hydrostatics and is often written as:

Δp = ρgz