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Unformatted text preview: ME 309 Fall 2008 Section 1 (Merkle) Solution: Homework # 7 Due Fri 12 September 2008 Problem 7.1 A constant linear acceleration is applied to a container half filled with water (density = 1000 kg/m 3 ). The container dimensions are L = 100 mm long by W = 10 mm wide by H 20 mm high (water height in the cart at rest = 10 mm ). The acceleration is in the horizontal direction and is parallel to the length, L . Assuming there is no sloshing, find: a) A relation for the variation of pressure in the x, y, and zdirections (take x as the direction of acceleration, y in the direction perpendicular to the paper, and z in the vertical direction) and a relation for the equilibrium shape of the free surface as a function of the acceleration rate; b) The maximum height of the surface as a function of acceleration rate and the magnitude of acceleration that will just bring the water to the top of the container to allow spillage; c) The pressure in the bottom left corner and the bottom right corner when the free surface just reaches the top of the container. d) Explain how you have used the conservation of mass and the conservation to solve this problem. Solution: Known : L = 0.10, W = 0.01, H = 0.02, Undisturbed water level = 0.01 m . Fluid = water (density = 1000) Find: a) Relation for p ( x, y, z) and equilibrium surface shape. b) Maximum surface height and acceleration required to bring the water to the top of the container; c) The pressure in the two bottom corners when surface reaches top of container. Sketch : above (for unaccelerated case) Assumptions : Free surface shape determined by momentum equations; conservation of mass used to compute surface location; Incompressible fluid, steady state, equilibrium surface—leading to rigid body problem. Spillage occurs just as the free surface reaches the top of the container. F, a L H Analysis : a) A fluid in a container under constant acceleration is a rigid body system. Conservation of momentum reduces to the following scalar equation in the direction parallel to the acceleration: a F V sys sys M dt dM = = The mass of the system is constant. The acceleration is constant and includes gravitational acceleration as well as the unknown linear acceleration. The volume of the system is a constant (incompressibility assumption with conservation of mass). The only forces acting are the pressure and gravity. The pressure gradient is given by: ρ a x =dp/dx,  ρ g z = dp/dz and dp/dy=0. Integrating these gives the pressure distribution as: z g x a p p z x ref ρ ρ + + = p = (independent of y ). For simplicity, take x = 0 at midpoint of length, and z = 0 at the surface height at x = 0 where the pressure is atmospheric. The integration constant can then be evaluated to give the equation for the pressure as a function of x, y and z (no dependence on y ): z g x a p p z x atm ρ ρ + = Note z will be negative below the surface indicating that the pressure increases with depth (as expected)....
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 Spring '08
 MERKLE
 Fluid Dynamics, Acceleration, pistons, Hydraulic fluid, free surface

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