3 1 sin 1 2 a Ur r ψ θ See handout Streamlines are close to the body

# 3 1 sin 1 2 a ur r ψ θ see handout streamlines are

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3 1 sin 1 2 a Ur r ψ θ = See handout  Streamlines are close to the body! Recirculation is absent sphere drags the entire surrounding fluid with it circulation streamlines sphere pushes fluid out of the way   Other three-dimensional body shapes Happel & Brenner, Low Reynolds number hydrodynamics (1965) Disk normal to the freestream: F=(32/3) Ua µ Disk parallel to the freestream: F=16 Ua µ See White (1991) Viscous Fluid Flow EXACT SOLUTIONS OF NAVIER-STOKES EQS A) Parallel Flows: i.e. velocity is unidirectional B) Similarity solutions C) Generalized Beltrami Flows- difficulties - non-linear PDEs - no general solution is known 0 , v=w=0 u N 0 0 3D Flow V V ω ω = ∇× × = JG JG JG JG & Reference C.Y. Wang (1991) “Exact solutions of the steady-state N-S eqs.” Ann. Rev. of Fluid Mech . Vol.23, pp. 159-177. R.Berker (1963) Almost all of the particular solutions are for the case of incompressible Newtonien flow with constant transport properties, 2 2 . 0 p V DV p V Dt DT c k T Dt ρ µ ρ = = −∇ + = + Φ JG JG JG P: total hydrostatic pressure. i.e. it includes the gravity term for convenience. or P=p+ gz P p g ρ ρ = ∇ JG y x z g JG g gk = − JG G ( ) ( ) 1) Convective accel. V. . 2) Non-linear solutions V. does not vanish. vanishes ∇ → ∇ → JG JG GROUP A: PARALLEL FLOWS 0 , v=w=0 u parallel flow: only one velocity component is different from zero. i.e. all fluid particles moving in one direction. N N 0 0 0 0 u=u(y,z,t) ; v 0 , w 0 u v w u x y z x + + = = y-comp. of momentum eq. 2 2 2 2 2 2 v v v v p v v v u v w t x y z y x y z ρ µ + + + = − + + + 0 p y = z- comp. of mom. eq. ( ) 0 p=p x,t p z = x- comp. of mom. eq. 2 2 2 2 2 2 2 2 2 2 (A) u u u u p u u u u v w t x y z x u p u u t x x y z y z ρ µ ρ µ + + + = − + + + = − + + Linear dif. eq. for u(y,z,t) ( ) . u 0 0 (In general) t t steady = = 1) Parallel flow through a straight channel Couette Flows a) both plates stationary x y 2 2 dp d u dx dy µ = BC's y=0 u=0 y=a u=0 2 1 1 2 2 1 2 1 2 2 dp 0 . dx dp 1 dp dx 2 dx 1 dp 1 dp 0, 0= c 2 dp 2 dx dx 2 dx velocity profile p const y c u c u y y c y c c a a a y y u a a a µ µ µ µ µ µ = = = + = + + = + = − ⎞ ⎛ = ⎟ ⎜ ⎠ ⎝ Shear stress distribution 1 1 2 2 yx dp dp dp y y a a dx dx dx a τ ⎞⎡ = = ⎠⎣ Volume flow rate 3 0 2 . ( ) 12 1 / 12 a A l dp Q V d A u ldy a dx dp V Q A a dx µ µ = = = − = = − JG JG average vel.  #### You've reached the end of your free preview.

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