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05 Basic Laws - 5 Basic Laws 5.1 Continuity"conservation of...

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5 Basic Laws 5.1 Continuity (“conservation of mass”) 5.1.1 Integral via Reynold’s Transport Theorem Assuming continuum assumption is valid, Amount of matter in a material region is constant if the time rate of change of mass in a material region is zero The mass in a material region is 0 MR MR MR MR M dV DM d dV Dt dt ρ ρ = = = ��� ��� Apply Reynold’s transport theorem with ( 29 ( 29 , , i i F x t x t ρ = of change of mass in net flux into/out of arbitrary a arbitrary region region through the arbitrary surface 0 AR S AR dV V d S t ρ ρ = + ��� �� ur ur 1 42 43 1 442 4 43 where V ur is the velocity of the flow with respect to the normal velocity of the arbitrary surface. To continue the derivation of the derivative form of the continuity equation, a fixed region will be assumed. of FR change of mass in net flux into/out of free a fixed region region through the free surface 0 FR S dV V d S t ρ ρ = + ��� �� ur ur 142 43 1 442 4 43 5.1.2 Mass flux and volumetric flow rate Mass flux : " " S m V d S ρ = �� ur ur & Volumetric Flow Rate : " " S Q V d S = �� ur ur 5.1.3 Differential in conservation and substantial derivative forms How do we go from integral equations to differential equations? Use Gauss to convert the surface to a volume integral [ ] [ ] dS dV = �� ���
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Gibbs: ( 29 S V V d S V dV ρ ρ �� �� ��� ur ur ur Index: ( 29 i i i i S V u n dS u dV x ρ ρ �� ��� Combine all terms into a single [ ] 0 dV = ��� So continuity becomes ( 29 0 V V dV t ρ ρ + = �� ��� ur Because dV is an arbitrary infinitesimal volume element, the integral must equal 0!!! [ ] 0! = Conservative form of continuity is Gibbs: ( 29 0 V t ρ ρ + = �� ur Index: ( 29 0 i i u t x ρ ρ + = Physical interpretation of Conservative form t ρ = rate of accumulation or depletion of mass per unit volume at a point P ( 29 V ρ = �� ur net flow rate (flux) of mass @ P per unit volume The second term can be further decomposed via the chain rule Gibbs: 0 V V t ρ ρ ρ + + = �� �� ur ur Index: 0 i i i i u u t x x ρ ρ ρ + + = Material derivative form : rewritten using the material derivative of continuity 1 1 or 1 rate of expansion MR MR i i D DV V Dt V Dt D u Dt x ρ ρ ρ ρ = - = - �� = - = ur Physical interpretation of Material or Substantial derivative form
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1 D Dt ρ ρ = fractional rate of change of density of a fluid particle V - = �� ur fractional rate of change of volume of fluid particle 5.1.4 Special Cases: Steady flow: ( 29 0 V ρ = �� ur Incompressible flow (NOTE: “flow” NOT “fluid” 0 0 D V Dt ρ = = ��� ur , incompressibility results in kinematic constraint 5.2 Linear Momentum Equation Newton’s Second Law for a continuum using a Eulerian Method of Description “the time rate of change of the linear momentum of material region is equal to the sum of the forces on the material region” "linear momentum" MR MR d p D p F dt Dt p mV = = = ur ur ur ur ur 5.2.1 Integral via RTT Rate of change of momentum of a material region MR i MR MR d d u dV VdV F dt dt ρ ρ = = ��� ��� ur ur Two types of forces: MR F ur
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