Navier-Stokes equation in x-direction is given by

$u \frac{∂u}{∂x}+v \frac{∂u}{∂y}+w \frac{∂u}{∂z}+\frac{∂u}{∂t}=F_x-\frac{1}{ρ}.\frac{dp}{dx}+v(\frac{∂^2u)}{∂x^2 }+(\frac{∂^2u}{∂y^2 }+\frac{∂^2 u}{∂z^2})$

$u \frac{∂u}{∂x}+v \frac{∂u}{∂y}+w \frac{∂u}{∂z}$=Inertia force,$\frac{∂u}{∂t}$=local accelration, $F_x$=all body forces,

$\frac{∂p}{∂x}$=pressure gradient, $\frac{∂^2 u}{∂x^2}+\frac{∂^2 u}{∂y^2}+\frac{∂^2 u}{∂z^2}$=shear tensor(viscous force),

v=$\frac{μ}{ρ}$,kinematic viscosity.

Assumptions:

The flow is only x-direction therefore v=w=0

Flow is steady flow $\frac{∂u}{∂t}$=0

The fluid is inviscid i.e. $μ$=0 Body forces acting are only due to gravity and is given by $F_x=\frac{F_b}{v}=\frac{body \ forces}{volume}$ ![enter image description here][1] $\frac{u^2}{2}+ρgz+\frac{p}{ρ}$=constant Divide by ‘g’ $\frac{u^2}{2g}+ρz+\frac{p}{ρg}$=constant

The above equation is Bernoulli’s equation for fluid motion