Question 8.4: Water enters a rectangular duct at a rate of 10 m³/s as show...
Water enters a rectangular duct at a rate of 10 m ^{3}/s as shown below. Two of the faces of the duct are porous. On the upper face, water is added at a rate shown by the parabolic curve, while on the front face water leaves at a rate determined linearly by the distance from the end. The maximum values of both rates are as given in Fig. 8.16. What is the average velocity leaving the duct if it is 1m long and has a cross-section of 0.1 m ^{2}?
Consider the top surface. The water enters the top surface in a parabolic manner. Let us first find this parabolic curve.
Let w=a y^{2}+b y+c, where w is the flow rate per unit length at top face. Following are the boundary conditions:
at y = 0, w = 0
at y = 1, w = 3
\text { at } y=0, \frac{ d w}{ d y}=0
\text { So, } \quad c=b=0, a=3
\text { Thus, } w=3 y^{2}
Similarly, consider the front surface.
let u = my + d
when y = 1, u = 0 we get: d = 5
y = 0, u = 5 m = -5
So, u = – 5y + 5
For steady incompressible flow, the continuity equation gives
\int_{c s} \vec{V} \cdot d \vec{A}=0
Choose the interior of the duct as a control volume.
Thus, -10-\int_{0}^{1} 3 y^{2} d y+\int_{0}^{1}(5-5 y ) d y+V_{2}(0.1)=0
\text { or }-10-1+\left(5-\frac{5}{2}\right)=-0.1 V_{2}
Finally, V_{2} = 85 m/s