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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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written 6.3 years ago by | • modified 6.3 years ago |
Navier-Stokes equation in x-direction is given by
u∂u∂x+v∂u∂y+w∂u∂z+∂u∂t=Fx−1ρ.dpdx+v(∂2u)∂x2+(∂2u∂y2+∂2u∂z2)
u∂u∂x+v∂u∂y+w∂u∂z=Inertia force,∂u∂t=local accelration, Fx=all body forces,
∂p∂x=pressure gradient, ∂2u∂x2+∂2u∂y2+∂2u∂z2=shear tensor(viscous force),
v=μρ,kinematic viscosity.
Assumptions:
The flow is only x-direction therefore v=w=0
Flow is steady flow ∂u∂t=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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