We have steady, laminar, incompressible and fully-developed flow of a fluid thro

Question

We have steady, laminar, incompressible and fully-developed flow of a fluid through
the pipe shown in the figure.

a. Using Navier-Stokes equations in cylindrical coordinates show that the velocity profile vz can be written as vz/Vzmax=1-(r/R)2. Where Vzmax is the centerline velocity and given as Vzmax= [-R2/(4)][ d/dz(p+gz)].

b. Show that the average velocity V in the pipe V= Vzmax/2.

c. Using conservation of momentum and energy between points 1 and 2 show that the
losses hf=[2w/(g)]L/R.

d. In a fully-developed laminar pipe flow wall shear stress w is a function of density ,
kinematic viscosity v, average velocity V and pipe diameter D. Using Buckhingham II
theorem show that w/( V2)=f(Re) where Re=(VD)/v .

e. Define function f(friction factor) as f=8w/( V2) and using results obtained in parts
a, b and c show that in a fully developed pipe flow hf=f(L/D)V2/(2g) and for a fullydeveloped
laminar pipe flow f=64/Re.

We have steady, laminar, incompressible and fully-developed flow of a fluid through the pipe shown in the figure. a. Using Navier-Stokes equations in cylindrical coordinates show that the velocity profile vz can be written as vz/Vzmax=1-(r/R)2. Where Vzmax is the centerline velocity and given as Vzmax= [-R2/(4?)][ d/dz(p+?gz)]. b. Show that the average velocity V in the pipe V= Vzmax/2. c. Using conservation of momentum and energy between points 1 and 2 show that the losses hf=[2?w/(?g)]L/R. d. In a fully-developed laminar pipe flow wall shear stress ?w is a function of density ?, kinematic viscosity v, average velocity V and pipe diameter D. Using Buckhingham II theorem show that ?w/( ?V2)=f(Re) where Re=(VD)/v . e. Define function f(friction factor) as f=8?w/( ?V2) and using results obtained in parts a, b and c show that in a fully developed pipe flow hf=f(L/D)V2/(2g) and for a fullydeveloped laminar pipe flow f=64/Re.

Explanation / Answer

plz see this link http://people.rit.edu/~pnveme/ExplictSolutions2/1Dim/LaminarFlowPipe/LaminarFlowPipe.html

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