The velocity profile for a laminar flow in a round pipe is given by the equation
ID: 1861501 • Letter: T
Question
The velocity profile for a laminar flow in a round pipe is given by the equation where Umax is the centerline velocity, R is the pipe radius and r is the radial distance from the pipe centerline what is the velocity gradient at the pipe wall? Note that the normal distance from the pipe wall is y=R-r, thus dy=-dr and du/dy=-du/dr. what is the shear stress at the pipe centerline? show that the velocity gradient, and hence the shear stress varies linearly from zero at the pipe centerline to a maximum at the pipe wall.Explanation / Answer
a)
u(r) = u_max [1 - (r/R)^2]
Differentiating it, du/dr = -2r*u_max/R^2
At pipe wall, r = R, so velocity gradient = du / dy = 2*u_max
b)
Shear stress, tau = mu*(du/dy).................where mu = absolute visosity
= mu*(-du/dr)
Thus, tau = mu*u_max *(2r / R^2)
At pipe centre line, r = 0.
Thus, shear stress = 0.
c)
tau = mu*u_max *(2r / R^2)
The above relationship shows that tau is linearly varying with r. When r is zero, tau is zero. When r is max., tau is max.
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