A large plate moves with speed v 0 over a stationary plate on a layer of oil (see Fig.). If the velocity profile is that of a parabola, with the oil at the plates having the same velocity as the plates, what is the shear stress on the moving plate from the oil? If a linear profile is assumed, what is the shear stress on the upper plate?
Text Solution
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Sol. For a parabolic profile, v 2 = ay. When y = d, v = v 0 . Hence, 
= ad, a = 
/d. Therefore,
v 2 = ( 
/d) (y) = ( 
)(y/d) v = v 0 
dv/dy = [(v 0 ) (1/ 
)(½)(y –1/2 )]
τ = µ(dv/dy) = µ[(v 0 ) (1/ 
)(½)(y –1/2 )]
For y = d, τ = µ[(v 0 ) (1/ 
)(½)(d –1/2 )] = µv 0 /(2d). For a linear profile, dv/dy = v 0 /d, τ = µ(v 0 /d).
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