SpecialProblemsME106_s11 - O. Sava s ME 106 Fluid Mechanics...

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¨ O. Sava¸ s Jan 11, 2011 ME – 106 Fluid Mechanics Spring 2011 SCP – Special/Computational Problems 1. Consider a rigid sphere of radius a submerged in a liquid of density ρ . The total force on the sphere due to the hydrostatic pressure distribution is F = Z S p n dA where n = ˆ e r is the unit normal vector. Write out the Cartesian components of the integral using the spherical polar coordinate variables ( r, θ, φ ) and integrate over the surface of the sphere to deduce that F =( F x ,F y ,F z )= Z 2 π φ =0 Z π θ =0 p ( θ ) n ( θ, φ ) dA ( θ, φ )=(0 , 0 ,ρgV ) where V is the volume of the sphere. 2. Consider a reference frame x which is translating at a constant velocity of U with respect to an inertial reference frame X . The fluid velocity vector ±eld measured in the x reference frame is u . (a) Show that the integral mass conservation equation written in the translating reference frame remains unchanged; d dt Z V ρdV = Z V ∂ρ ∂t dV + Z S ρ u · d A =0 (b) Using the above result, show that also the momentum conservation equation written in the
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SpecialProblemsME106_s11 - O. Sava s ME 106 Fluid Mechanics...

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