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Starting from the Navier Stokes equation for an incompressible Newtonian fluid derive Bernoullis equation stating the assumptions
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Navier-Stokes equation in x-direction is given by

uux+vuy+wuz+ut=Fx1ρ.dpdx+v(2u)x2+(2uy2+2uz2)

uux+vuy+wuz=Inertia force,ut=local accelration, Fx=all body forces,

px=pressure gradient, 2ux2+2uy2+2uz2=shear tensor(viscous force),

v=μρ,kinematic viscosity.

Assumptions:

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

Flow is steady flow ut=0

The fluid is inviscid i.e. μ=0 Body forces acting are only due to gravity and is given by Fx=Fbv=body forcesvolume ![enter image description here][1] u22+ρgz+pρ=constant Divide by ‘g’ u22g+ρz+pρg=constant

The above equation is Bernoulli’s equation for fluid motion

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