Applying Newton’s second law in that moving fluid particle in a steady flow we get,
\(\sum F_S=ma_S\)
Now considering the assumptions mentioned above, the significant forces that will be acting in the s-direction are the pressure and the component of the weight of the particle in the s-direction.
Therefore, we can write the above equation as:
P dA − (P + dP) dA − W sin = mV (dV/ds );
where is the angle between the normal of the streamline and the vertical z-axis at that point, m = V = dA ds is the mass, W = mg = g dA ds is the weight of the fluid particle, and sin = dz/ds. Substituting we get,
−dP dA − ρg dA ds (dz/ds) = ρ dA ds V (dV/ds)
After simplification, we get
−dP − ρg dz = ρV dV
Now, V dV = 1/2 d(V2) and dividing each term by gives
dP/ρ + 1/2 d(V2) + g dz = 0
Integrating we get,
P/ρ +V2/2 + gz = constant
Dividing by g we get,
(P/ρg)+(V2/2g)+z=Constant
This is the famous Bernoulli equation, widely used in fluid mechanics for steady, incompressible flow along a streamline in inviscid regions of the flow.