fluids urgent job q2 and q3

I NEED IN MAX 6 HRS JUST Q2 AND Q3
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3. a) The uniform stream of velocity U? approaches a channel (of width b). After flowing a
distance le (“entrance length”) flow inside the channel will become “fully developed”. Show that
the entrance length le is proportional to the Reynolds number Re (based on the channel width b).
Fluid’s dynamic viscosity is ? and density is ?. (10 points)
U?
Boundary layer
Potential core
b
Boundary layer
Fully developed
flow
le
Entrance region
b) The uniform stream of velocity U? and temperature T? approaches stationary flat
plate whose temperature is TW (TW > T?). Fluid’s kinematic viscosity is ?, thermal diffusivity is
?, and Prandtl number Pr = 10.
Estimate distances xv and xT (measured from the leading edge) where velocity boundary
layer reaches thickness ?v ~ L and thermal boundary layer reaches thickness ?T ~ L, respectively.
What is the ratio xT / xv ? (15 points)
?????
U?
U?
T?
T?
TW
x
Page 2 of 4
4. A fluid is flowing from A to B through the network of tubes due to the known pressure
difference pA – pB (see schematic). Obtain an expression for the flow rate Q as a function of fluid
viscosity (?), pressure difference (pA-pB), tube radius (R) and tube length (L).
Neglect elbow effects and disturbance in various tube junctions. Consider simple laminar flow
through straight circular pipe throughout the system. Neglect body forces.
To save you some integration time: mean velocity in a circular pipe is one half of the maximum
velocity.
Page 3 of 4
Navier-Stokes Equation in Cylindrical Coordinates
? 1
?
?
? v?
? vz
?r r ??
?z
2
1 ? ? ? ? 1 ?
?2
2
? ?
?
?r ??
r ?r ? ?r ? r 2 ?? 2 ?z 2
V ?? ? vr
r ? momentum
?vr
v
1
1 ?p
2 ?v ?
?
? ?V ?? ? vr ? v? 2 ? ?
? g r ? ? ? ? 2vr ? r2 ? 2 ? ?
?t
r
? ?r
r
r ?? ?
?
? ? momentum
?v?
vv
1 ?p
2 ?v v ?
?
? ?V ?? ? v? ? r ? ? ?
? g? ? ? ? ? 2 v? ? 2 r ? ?2 ?
?t
r
? r ??
r ?? r ?
?
z ? momentum
?vz
1 ?p
? ?V ?? ? vz ? ?
? g z ? ?? 2 v z
?t
? ?z
Page 4 of 4

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