Question 8.4: Water at 60°C flows between two large flat plates. The lower...
Water at 60°C flows between two large flat plates. The lower plate moves to the left at a speed of 0.3 m/s. The plate spacing is 3 mm and the flow is laminar. Determine the pressure gradient required to produce zero net flow at a cross section. ( \mu=4.7 \times 10^{-4} \mathrm{Ns} / m^{2} at 60^{\circ} \mathrm{C} )
Governing equation: \mu \frac{\mathrm{d}^{2} u}{\mathrm{~d} y^{2}}=\frac{\mathrm{d} p}{\mathrm{~d} x}
u=\frac{1}{2 \mu} \frac{\mathrm{d} p}{\mathrm{~d} x} y^{2}+C_{1} y+C_{2}
At y=0, u=-U, \quad C_{2}=-U
At y=b, u=0 \text {, which yields }
0=\frac{1}{2 \mu} \frac{\mathrm{d} p}{\mathrm{~d} x} b^{2}+C_{1} b-UOr C_{1} =\frac{U}{b}-\frac{1}{2 \mu} \cdot \frac{\mathrm{d} p}{\mathrm{~d} x} b
u =\frac{1}{2 \mu} \frac{\mathrm{d} p}{\mathrm{~d} x}\left(y^{2}-b y\right)+U\left(\frac{y}{b}-1\right)
Now, Q=\int_{0}^{b} u \mathrm{~d} y=\int_{0}^{b}\left[\frac{1}{2 \mu} \frac{\mathrm{d} p}{\mathrm{~d} x}\left(y^{2}-b y\right)+U\left(\frac{y}{b}-1\right)\right] \mathrm{d} y
Or Q=-\frac{1}{12 \mu} \frac{\mathrm{d} p}{\mathrm{~d} x} b^{3}-\frac{U b}{2} a
For Q=0, \text { with } \mu=4.7 \times 10^{-4} \mathrm{Ns} / \mathrm{m}^{2}
\frac{\mathrm{d} p}{\mathrm{~d} x}=-\frac{6 U \mu}{b^{2}}=\frac{-6 \times 0.3 \times 4.7 \times 10^{-4}}{(0.003)^{2}}=-94.0 \mathrm{~N} / \mathrm{m}^{2} \cdot \mathrm{m}