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FLUID MECHANICS I Assignment 4 : Pressure distributions in moving fluids and friction Note: 1 problem on friction force is also included. For more problems, see class examples, tutorial and past exams.

Question 1: Problem P2.139 Question 2: Problem P2.156 Question 3 Figure 8.14b page 546 in the textbook shows the streamlines for the flow of air at 20[◦ C] in a corner. The velocity field for this flow is approximately: u(x, y) = v(x, y) =

V0 x L V0 − y L

where V0 = 10[m/s] is a velocity scale, L = 2[m] is a length scale, x is positive to the right, and y is positive upwards. Gravity acts in the z direction. Friction plays a negligible role in establishing the pressure field of this flow. 1. Show that the pressure field for this approximately: p(x, y) = p0 −

¢ ρV02 ¡ 2 x + y2 2L2

where p0 is the pressure at (0[m], 0[m]). 2. If the pressure at (2[m], 0[m]) is atmospheric pressure (105 [P a]), what is p0 . 3. Estimate the net force due to pressure acting on the wall 0[m] < x < 2[m]. The wall has a unit depth in the z direction and atmospheric pressure acts on the lower face of the wall.

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Question 4: Problem P2.141a,b The tank contains water and is 50[cm] wide. What is the net force due to pressure acting on the rear wall of the tank (the wall running from point A to the free surface of the water). You may assume that water has stopped sloshing.

Question 5 The sketch shows the laminar boundary layer region very close to a flat plate with flow of air over the plate. T he laminar velocity field is approximately: ¶ µ y y2 u(x, y) = u0 2 − 2 δ δ µ 2 ¶ δ y y3 v(x, y) = u0 − x 2δ 2 3δ 3 where δ is the thickness of the region above the plate in which the flow is influenced by the plate and u0 is the fluid speed far above the plate (also referred to as the freestream speed). For laminar flow, the thickness is a function of position along the plate: r x δ(x) = 5 ν u0 where ν is the kinematic viscosity of the air. 1. Derive an expression for the drag force that acts on the top side of the plate. The plate has a length L and a width (into the page) of b. 2. Estimate the drag force when air at 20[◦ C] flows over a plate 0.5[m] long and 2.0[m] wide. The freestream speed is 3[m/s]. 3. When the plate is flipped so it is perpindicular to the flow, the drag force is due to pressure. In this case, a reasonable estimate of the drag force is D = 0.6ρu20 Lb where ρ is the air density. Compare the size of the drag force in this case to that which exists when the flow is parallel to the plate.

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Figure 1: Schematic of laminar boundary layer on a flat plate.

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