C3_Shell_Momentum_Balance

# C3_Shell_Momentum_Balance - Shell Momentum Balance for...

This preview shows pages 1–7. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 =
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 23

C3_Shell_Momentum_Balance - Shell Momentum Balance for...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online