Question

HW 8 Poiseuille flow: Fully developed laminar pipe flow (in cylindrical coordinate) - The simplified z-momentum equation - The boundary conditions = No slip at r=R The Navier-Stokes equation for 2D (x,y) incompressible flow DV P -Op+uv2V + pg dt - Assumptions: 1. 2. 3. 4. 5. 6. Finite velocity at r=0 - Final velocity solution of Poiseuille flow - The rz component of the NS equation (in cylindrical coordinate) - Volume flow rate (Q = ſ vedA)

Answer 1

Answert The Naviey - Stokes equation you 2D (X,Y) incompdessible flow - DV Р dt -up + UD2W+p Assumptions 1) Flow is laminas. 2) flow is incompressible. 3) flow is steady 4) flow is unidirectional 5) Flow is axisymmetrical 6) flow is fully developed. 7) Gravity is zero. • The é,2 component of the nis equation & (in cylindrical coordinate ) 3'&'- momentum & Р auy ət tue But aur tuz de Ə ap de due อื่น ut Curso t + Pge 2 E dz?

'Z'- momentum - Pl dua ət + Uų but the way of 3 (a bolest on the Park • The Simplified 2- momentum equation - suz < (e ap dz the la OE to to 3 de (z do tema) - tu equation (1) • The boundary Conditions & no slip at & =R (Boundary condition 2) finite velocity at a=0 (Boundary condition 2) 9 Uring equation (1) = Code) -(0) ta * Intregate both side = -)-() ta 2² zu a uz 3 du zu a Uz - 3 dre ap dz C G equation (2) 2u

* Intregate it ; 3 U₂ = (ap) + C, loge & + (z eq.(3 2) 7 Applying the boundary condition @ in equatione & Do slip at E=R llz=0 0 ( ap dz ita zu C=0 . Applying the boundary condition in equation 3 : » no slip at t = R ☆ U2=0 'ap dz co P² 4 +(2 3 eq = -( ap R2 4д & Putting ly and Ca in equation (3) uu + C loge & +62 U2 'ap e² - lla Charles ap 22 g² 4д R² 4K

2 >> Uz - Gul & final relocity solution of Poiseuille y low : L'ele ( (2) R?) eq.(4) • Volume flow date (Q = (v2 dA) : 7 Q = 6 ZTEUR de = Q = JR 2T & Leu =R²) de 2 Qar- :) RS ap 2 dz ㅠ gu JP : 22