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

This preview shows page 133 - 144 out of 229 pages.

3 1 sin 1 2 a Ur r ψ θ = See handout
Image of page 133
Image of page 134
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
Image of page 135
Image of page 136
Image of page 137
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
Image of page 138
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
Image of page 139
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 =
Image of page 140
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
Image of page 141
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 ⎞⎡ = = ⎠⎣
Image of page 142
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.
Image of page 143
Image of page 144

You've reached the end of your free preview.

Want to read all 229 pages?

  • Spring '11
  • M

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes