1Fluid has MassAs 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 = ρdV where we must conserve mass, i.e.:Dm/Dt = D/Dt (ρdV)∂ρ/∂tdV+ ρvivjdS = 0dsvivjDensity(t) of materialMovement of dS(surface) along a pathThe final material derivative:Dρ/Dt = ∂ρ/∂t + vi∂ρ/∂xjAmount of “stuff” moving along a paththe path follows the fluid current described by the fluid's velocity field vtimepositionContinuity Equation for Massthe rate at which mass enters a system is equal to the rate at which mass leaves the system:If dρ/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 notDρ/Dt + ρ ∂vi/∂xi= 0 dsvivjEmpirical Data on Water to Validate AssumptionsTait (1888):V(p) = V(p=1) – 0.31 Volog10(B+p/B+1)•V(p) is volume of water at pressure p•B = 2668+19t -0.3t2+0.0017t3; t = temperature**For t = 40oC: V = 1.0079 @ 1atm0.988 @ 483 atm0.97 @ 967 atm
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