3 - Fluid has Mass As a fluid moves, it has interia from...

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1 Fluid has Mass As a fluid moves, it has interia from its mass, which cannot be neglected in more formal descriptions of stress. Total mass in volume V is: m = ρ dV where we must conserve mass, i.e.: Dm/Dt = D/Dt ( ρ dV) ∂ρ / tdV+ ρ v i v j dS = 0 ds v i v j Density(t) of material Movement of dS (surface) along a path The final material derivative: D ρ /Dt = ∂ρ / t + v i ∂ρ / x j Amount of “stuff” moving along a path the path follows the fluid current described by the fluid's velocity field v time position Continuity Equation for Mass the rate at which mass enters a system is equal to the rate at which mass leaves the system: If d ρ /dt is constant, the mass continuity equation simplifies to a volume continuity equation: •v = 0; i.e. the change in rate of local volume changes is zero. ***Depends on the biofluid as to whether this assumption holds or not D ρ /Dt + ρ∂ v i / x i = 0 ds v i v j Empirical Data on Water to Validate Assumptions Tait (1888): V(p) = V(p=1) – 0.31 V o log 10 (B+p/B+1) • V(p) is volume of water at pressure p • B = 2668+19t -0.3t 2 +0.0017t 3 ; t = temperature **For t = 40 o C: V = 1.0079 @ 1atm 0.988 @ 483 atm 0.97 @ 967 atm
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2 Compressibility of Blood(?) • Best way to measure biofluids (because of their heterogeneities) is using the speed of sound in the fluid: • Note: Blood pressures < 200 mm Hg = 1/3 atm • Thus: ρ = constant and dv i /dx i
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This note was uploaded on 09/02/2009 for the course BENG 112B taught by Professor Huang during the Spring '08 term at UCSD.

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3 - Fluid has Mass As a fluid moves, it has interia from...

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