Question 8.5: Water at 20 ºC is flowing between a two dimensional channel ...
Water at 20 ºC is flowing between a two dimensional channel in which the top and bottom walls are 1.5 mm apart. If the average velocity is 2 m/s, find out (a) the maximum velocity, (b) the pressure drop, (c) the wall shearing stress, and (d) the friction coefficient [μ = 0.00101 kg/m . s].
(a) The maximum velocity is given by
U_{\max }=\frac{3}{2} U_{ av }=\frac{3}{2}[2]=3 m/s
(b) The pressure drop in a two dimensional straight channel is given by
\frac{ d p}{ d x}=\frac{-2 \mu U_{ max }}{b^{2}} \text {, where } b=\text { half the channel height }
=\frac{-2(0.00101) 3}{[1.5 /(2 \times 1000)]^{2}}
= – 10773.33 N/m ^{3}
or -10773.33 N/m ^{2} per metre
(c) The wall shearing stress in a channel flow is given by
\tau_{y x}=-\mu \frac{\partial u}{\partial y}=-b \frac{ d p}{ d x}=\frac{-1.5}{2 \times 1000}
or \tau_{y x}=8.080 N / m ^{2}
(d) Re (based on channel height and average velocity)
=\frac{\rho U_{ av }(2 b)}{\mu}=\frac{1000 \times 2 \times(1.5 / 1000)}{0.00101} \approx 2970
which is more than 2300 but the flow is not turbulent as well. Laminar approximation is
quite logical. However, C_{f}=\frac{12}{\operatorname{Re}} = 0.004.