Flux - definition: (same format for all types of transport, momentum, energy, mass)

Transcription

1 Fundamentals of Transport Flu - definition: (same format for all types of transport, momentum, energy, mass) flu in a given direction Quantity of property being transferred ( time)( area) More can be transported if the contact area is larger More can be transported if the time allowed is longer Three transport laws: momentum flu: τ Momentum flu = shear stress µ d = ρ d viscosity ( ρv ) heat flu: q A d = α ( ρc T ) d thermal diffusivity p molar diffusivity d c d * ( ) molar flu: A J flu A = D AB transport coefficient gradient 1

2 Momentum Transport (fluid mechanics) Momentum transport is based on the principle of conservation of momentum (Newton s second law of motion) The property we follow is the flu of momentum (quantity of momentum transferred per unit time per area, τ y ) We use a transport law to relate the flu to the gradient of momentum (Newton s Law of Viscosity) We combine momentum conservation with the transport law to calculate velocity fields ( v(,y,,t) ) Effect of angle of attack on flow around an airfoil Source: Illustrated eperiments in fluid mechanics: the NCFMF book of film notes, MIT Press, 1972 angle of attack 2

3 Effect of flow rate on flow around a cylinder Source: Young, et al., A Brief Introduction to Fluid Mechanics, Wiley 1997 Entrance flow field in pipe flow Entrance region Boundary layer Fully developed flow 3

4 Flow around comple obstacles standard with fairing with fairing and gap seal Flow around a wing Source: Illustrated eperiments in fluid mechanics: the NCFMF book of film notes, MIT Press,

5 Momentum Flu Momentum = mass * velocity p = mv vectors v = V top plate has momentum, and it transfers this momentum to the top layer of fluid V y H v = 0 v (y) momentum flu How is F related to V? F A V = + µ H Stress on a y-surface in the -direction v = µ v = µ y ( y = 0) v ( y = H ) H 0 (Note choice of coordinate system) τ y = µ dv dy Newton s Law of Viscosity 5

6 τ y = 9 stresses at a point in space force kg m / s = area area 2 = τ y ( kg)( m / s) ()( s area) Momentum Flu f ê y f = A( τ eˆ + τ eˆ yy y y + τ eˆ ) y stress on a y-surface in the -direction A surface whose unit normal is in the y-direction in the y-direction flu of -momentum τ = y µ dv dy Newton s Law of Viscosity viscosity m / s Pa = m g = = cm s [ = ] µ µ [ = ] Pa s [ ] kg m s [ ] [ ]Poise µ water = 1cP = 10 2 Poise 6

7 Q: dv τ = y µ dy What do we want to do with? A: We want to calculate velocity profiles, v (y). Q: How do we do that? A: mass balance momentum balance momentum transport law combined v (y) Control Volume Instead of following a single body in motion, we choose a small volume in space and write our balances (mass, momentum, energy) on that. d d dθ r dr θ dv = d dy d dv = dr rdθ d 7

8 To solve for velocity and stress fields, we choose a differential-sied control volume within the fluid domain. d d Apply: Mass balance Momentum balance Newton s 2nd Law: The time rate of change of momentum of a system is equal to the summation of all forces acting on the system and takes place in the direction of net force d p d( mv) F = = = ma dt dt flowing system (open system): rate of sum of forces + net momentum = accumulation acting on system flowing in of momentum in out 8

9 vector review: same vector, different coordinate systems v v = v v y y v1 = v2 v vr = v v θ rθ v = v = vector magnitude () v v = vˆ = unit vector θ r P y We will choose coordinate systems for convenience. Net? Learn how to calculate velocity profiles. sketch system choose coordinate system choose a control volume perform a mass balance perform a momentum balance (will contain stresses) substitute Newton s Law of Viscosity, solve differential equation that results apply boundary conditions dv τ = y µ dy 9