answer the questions on paper An approximation for the boundary-layer shape in F

Question

answer the questions on paper

An approximation for the boundary-layer shape in Figs. 1.6b and P1.51 is the formula U(y) U sin (pi y/2delta), 0 y delta where U is the stream velocity far from the wall and delta is the boundary layer thickness, as in Fig. P. 151. If the fluid is helium at 20 degree C and 1 atm, and if U = 10.8 m/s and delta = 3 cm, use the formula to (a) estimate the wall shear stress tau w. in Pa, and (b) find the position in the boundary layer where tau is one-half of tau w. P1.51

Explanation / Answer

Shear stress tau = mu*(du/dy)

du/dy = U*pi/(2*delta) Cos(pi*y/(2*delta))

Thus, tau = mu*U*pi/(2*delta) Cos(pi*y/(2*delta))

Putting y = 0 we get, Wall shear stress, tau_w = mu*U*pi/(2*delta)

Viscosity mu = 199*10^-7 kg/(m-s)

a)

tau_w = 199*10^-7 *10.8 *3.14/(2*0.03)

tau_w = 0.01124 Pa

b)

tau = Half of tau_w = 0.5*0.01124 = 0.00562 Pa

0.00562 = mu*U*pi/(2*delta) Cos(pi*y/(2*delta))

0.00562 = (199*10^-7)*10.8*3.14/(2*0.03) Cos(3.14*y/(2*0.03))

y = 0.02 m = 2 cm

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