Lecture 17 Fluid Dynamics Handouts

Srinivasan 19 two laminar flow cases u h w w ee c245

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: P + ρg + η∇ 2 U + ∇(∇ ⋅ U ) Dt 3 DU = −∇P + ρg + η∇ 2 U Dt EE C245 ρ 17 U. Srinivasan © Cartesian Coordinates ρ DU = −∇P + ρg + η∇ 2 U Dt x direction ∂U x ∂U x ∂U x ∂U x + Ux +U y + Uz = ∂x ∂y ∂z ∂t EE C245 ρ U. Srinivasan © ∂ 2U x ∂ 2U x ∂ 2U x ∂Px − + ρg x + η 2 + 2 + ∂x ∂y ∂z 2 ∂x 18 9 Dimensional Analysis DU ∇P η∇ 2 U =− +g+ ρ ρ Dt • Each term has dimension L/t2 so ratio of any two gives dimensionless group EE C245 • inertia/viscous • pressure /inertia • flow v/sound v = = = P/ρU2 = = = Reynolds number Euler number Mach number • In geometrically similar systems, if dimensionless numbers are equal, systems are dynamically similar U. Srinivasan © 19 Two Laminar Flow Cases U h τw τw EE C245 y U. Srinivasan © τw τw High h P Low P y Ux U Ux Umax 20 10 Couette Flow • Couette flow is steady viscous flow between parallel plates, where top plate is moving parallel to bottom plate • No-slip boundary conditions at plates ∇ ⋅ U = 0 U = Uxix U = U x ( y )i x DU x η ∂ 2U x ∂ 2U x 1 ∂P =− + gx + + ∂y 2 Dt ρ ∂x ρ ∂x 2 U x = 0 at y = 0 ∂ 2U x = 0, and U x = U at y = h ∂y 2 EE C245 U h τw τw y Ux = Ux U U. Srinivasan © 21 Couette Flow • Shear stress acting on plate due to motion, , is dissipative • Couette flow is analogous to resistor with power dissipation corresponding to Joule heating τ w = −η U τw EE C245 h U. Srinivasan © τw RCouette = ∂U x ∂y = y =h τ w A ηA U = h PCouette = RU 2 22 11 Poiseuille Flow • Poiseuille flow is a pressure-driven flow between stationary parallel plates • No-slip boundary conditions at plates ∂ 2U x ∂ 2U x DU x 1 ∂P =− + gx +η + ∂x 2 ∂y 2 Dt ρ ∂x ∂P ∆P =− L ∂x ∂ 2U x ∆P =− , 2 ∂y Lη U x = 0 at y = 0, h Ux = EE C245 y τw τw Highh P Low P U max = Ux Umax U. Srinivasan © 23 Poiseuille Flow • Volumetric flow rate Q ~ [h3] Q = W ∫0 U x dy = h • Shear stress on plates, , is dissipative • Force balance • Net force on fluid and plates is zero since they are not accelerating • Fluid pressure force ∆PWh (+x) balanced by shear force at the walls of 2 WL (-x) EE C245 • Wall...
View Full Document

This note was uploaded on 04/01/2014 for the course CHBE 6200 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

Ask a homework question - tutors are online