Question 5 (10+10=20 Points): For an incompressible subsonic flow, the freestream airspeed is 20m/s Above a flat plate the flow speed decreases to 15m/s and below the flat plate increases to 25m/s The free-stream pressure is 101 kPa and rho =1.225kg/m^3 (a) Calculate the pressures on the top of the plate and on the bottom of the plate separately. (b) Calculate the lift created on the flat plate if the plate surface area is 0.5m^2

(a) To calculate the pressures on the top of the plate and on the bottom of the plate, we can use Bernoulli's equation, which states that the total pressure in a streamline is constant. The total pressure is the sum of the static pressure and the dynamic pressure.<br /><br />The dynamic pressure is given by the equation:<br /><br />$Q = \frac{1}{2} \rho v^2$<br /><br />where $Q$ is the dynamic pressure, $\rho$ is the density of the fluid, and $v$ is the velocity of the fluid.<br /><br />For the top of the plate, the velocity is $15m/s$, so the dynamic pressure is:<br /><br />$Q_{top} = \frac{1}{2} \times 1.225 \times (15)^2 = 86.625 Pa$<br /><br />The total pressure on the top of the plate is the sum of the free-stream pressure and the dynamic pressure:<br /><br />$P_{top} = P_{\infty} + Q_{top} = 101 \times 10^3 + 86.625 = 101086.625 Pa$<br /><br />For the bottom of the plate, the velocity is $25m/s$, so the dynamic pressure is:<br /><br />$Q_{bottom} = \frac{1}{2} \times 1.225 \times (25)^2 = 153.125 Pa$<br /><br />The total pressure on the bottom of the plate is the sum of the free-stream pressure and the dynamic pressure:<br /><br />$P_{bottom} = P_{\infty} + Q_{bottom} = 101 \times 10^3 + 153.125 = 101153.125 Pa$<br /><br />(b) To calculate the lift created on the flat plate, we can use the lift equation:<br /><br />$Lift = (P_{bottom} - P_{top}) \times A$<br /><br />where $Lift$ is the lift force, $P_{bottom}$ is the pressure on the bottom of the plate, $P_{top}$ is the pressure on the top of the plate, and $A$ is the surface area of the plate.<br /><br />Substituting the values we have:<br /><br />$Lift = (101153.125 - 101086.625) \times 0.5 = 33.375 N$<br /><br />Therefore, the lift created on the flat plate is $33.375 N$.