2.25 Advanced Fluid Mechanics October 15th 2008 Couette & Poiseuille Flows . 1 Some of the fundamental solutions for fully developed viscous ﬂow are shown next. The ﬂow can be pressure or viscosity driven, or a combination of both. We consider a ﬂuid, with viscosity µ and density ρ. (Note: W is the depth into the page.) • a) PLANE Wall-Driven Flow (Couette Flow) Parallel ﬂow: u(y) = u(y)x̂, ﬂow between parallel plates at y = 0 and y = H, wall-driven, and resisted by ﬂuid viscosity. Umax = max(Utop , Ubottom ) Umin = min(Utop , Ubottom ) Uavg = 0.5 ∗ (Utop + Ubottom ) u(y) H ŷ Q′ = Q/W Q′ = H 2 (Utop − Ubottom ) (signed) τw,top = −µ (Utop −Ubottom ) H x̂ u(y) = (Utop − Ubottom )(y/H) + Ubottom • b) PLANE Pressure-Driven Flow (Poiseuille Flow) (Stationary walls) Parallel ﬂow: u(y) = u(y)x̂, ﬂow between parallel plates at y = 0 and y = H, pressure driven. Umax = − dP dx H2 8µ Uavg = 23 Umax Q′ = − dP dx τw = >0 u(y ) H y H3 12µ H dP 2 (− dx ) (τw on either wall in the x direction) 1 −dP dx 2.25 Advanced Fluid Mechanics Problem x u(y) = U H 2 ) dP y 2µ (− dx )( H (1 − y H )) • c) TUBE Pressure Driven Flow (Poiseuille Flow) (Stationary walls) Parallel ﬂow: u(r) = u(r)x̂, ﬂow along a tube. −dP dx Umax = Uavg = >0 U R2 ) dP 4µ (− dx ) U R2 )U 8µ Q = − dP dx r̂ dP dx ) − πR4 8µ u(r) 2R = D x̂ τw (2πRL) = πR2 L(− dP dx ) (τw in the x direction) f= ΔP L 1 ρU 2 D 2 = 16 ReD u(r) = U R2 ) 4µ r 2 (− dP dx )(1 − ( R) )