Question

3. (3 points) A tank of diameter D is filled with water up to a height h above the bottom of the tank (Figure 3). At the bottom of the tank is a hole of diameter d. Assume that the water flows out of the hole with a laminar flow and that the difference in atmospheric pressure between the top and the bottom of the tank is negligible Figure 3: A lank draining a) What speed will the water have just as it exits the bottom of the tank? Assume that the hole is small enough so that the water level in the tank is constant (b) As the stream of water gets further from the bottom of the tank, i will narrow toa smaller diameter than d. You can observe the same process for water flowing out of a tap. What w be the diameter of the (laminar) water stream a distance r below the bottom of the tank? (c) If you no longer assume that the level in the tank is constant, but rather that the top of the water has a speed downwards as the tank drains, show that the speed of the water exiting the bottom of the tank is given by: 2gh at an instant when the water level in the tank is b

Answer 1

Bernoulli's principle: P_1 + 1/2 rho v^2_1 + g h_1 = p_2 + 1/2 rho v^2_2 + rho g h_2 Equation of continuity A_1 V_1 = A_2 V_2 Applying the above formulas for point 1 at the top and point 2 just at opening of hole: pi d^2 /4 v_1 = pi D^2 /4 V_2 = > d^2 v_1 = D^2 V_2 P_1 = P_2 = P atm, V_2 = 0 as level is not changing -h_1 + h_2 = h P atm + 1/2 rho v^2_1 = 0 = P atm + 0 + rho g (h) v_1 = squareroot 2 g h P' = P_1 = P atm, h_1 - h' = x pi d^2 /4 v_1 = pi d'^2 /4 V' P' + 1/2 rho v'^2 + rho g h' = P_1 + 1/2 rho v^2_1 + rho g h_1 v'^2 = v^2_1 + 2 g x v^2_1 d^4 /d'^4 = v^2_1 + 2 g x = > d'