C3_Shell_Momentum_Balance

C3_Shell_Momentum_Balance - Shell Momentum Balance for...

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Shell Momentum Balance for Steady Laminar Flow Definitions of momentum fluxes Molecular momentum transfer o Fluid at rest : static pressure p is due to momentum transfer among molecules in random motion. o Fluid in motion : viscous stress τ is due to molecular momentum transfer in the directions of velocity gradients in fluid flow ,, , ij p i j z y z π δτ =+ is the total molecular momentum flux xx xy xz yx yy yz zx zy zz p p p ττ τ ⎛⎞ + ⎜⎟ + ⎝⎠ π
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Shell Momentum Balance for Steady Laminar Flow is the flux of j-momentum transferred convectively across a unit area whose normal is parallel to i direction is the flux of momentum transferred convectively across a unit area whose normal is parallel to δ 1 Definitions of momentum flux Convective momentum transfer o Recall that mass flux is momentum density. Since momentum is conserved, momentum per unit volume (momentum density) can be transferred by bulk flow. This leads to a rather simple way to describe convective momentum transfer: ρ u 1 u u is momentum density at a point in fluid 11 21 23 3 1 uu u uu u u ρρ =+ + δδδ u j i u u u u 1 δ 1 1 δ
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Shell Momentum Balance for Steady Laminar Flow Quiz Call u the velocity vector at a given point in a flowing fluid, writing the total momentum flux by convection across each of the three mutually perpendicular planes (i.e. planes of constant x, y, z, respectively) x y z P (x,y,z) u
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Shell Momentum Balance for Steady Laminar Flow Definitions of momentum flux xx xy xz yx yy yz z y z z p p p ττ τ ⎛⎞ + ⎜⎟ =+ + ⎝⎠ π z z uu ρρ =
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Shell Momentum Balance for Steady Laminar Flow Momentum principle for a fixed elementary volume Momentum principle Net forces Momentum influx (inward arrows) Momentum outflux (outward arrows) where momentum flux ()( ) 33 11 j i j ij ii convective molecular Mu up ρ δτ == =+ + ∑∑ ±²³ ² ´± ² ²³ ² ²´ External forces are exerted by gravitational, magnetic, etc. Momentum influx Momentum outflux External forces Momentum accumulation For unsteady-state flow Momentum influx Momentum outflux External forces P(x,y,z) For steady-state flow
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Shell Momentum Balance for Steady Laminar Flow Example & Quiz net momentum transfer 0 m x in out x x m M F M −+ = ±²³²´ ()[] () zz xi i x i i x i x ix i x i i x MA u u A p uA up A A ρ δτ ρτ == =++ =+ + ∑∑ Net forces Momentum influx (inward arrows) Momentum outflux (outward arrows) The balance for x-momentum of a steady flow over a cubic control volume
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This note was uploaded on 09/01/2011 for the course PGE 312 taught by Professor Peters during the Fall '08 term at University of Texas.

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C3_Shell_Momentum_Balance - Shell Momentum Balance for...

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