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### 05

1. 1. Basic Fluid Dynamics
2. 2. Momentum <ul><li>P = mv </li></ul>
3. 3. Viscosity <ul><li>Resistance to flow; momentum diffusion </li></ul><ul><li>Low viscosity: Air </li></ul><ul><li>High viscosity: Honey </li></ul><ul><li>Kinematic viscosity </li></ul>
4. 4. Reynolds Number <ul><li>The Reynolds Number ( Re ) is a non-dimensional number that reflects the balance between viscous and inertial forces and hence relates to flow instability (i.e., the onset of turbulence) </li></ul><ul><li>Re = v L/  </li></ul><ul><li>L is a characteristic length in the system </li></ul><ul><li>Dominance of viscous force leads to laminar flow (low velocity, high viscosity, confined fluid) </li></ul><ul><li>Dominance of inertial force leads to turbulent flow (high velocity, low viscosity, unconfined fluid) </li></ul>
5. 5. Poiseuille Flow <ul><li>In a slit or pipe, the velocities at the walls are 0 (no-slip boundaries) and the velocity reaches its maximum in the middle </li></ul><ul><li>The velocity profile in a slit is parabolic and given by: </li></ul>x = 0 x = a u(x) <ul><li>G can be gravitational acceleration or (linear) pressure gradient (P in – P out )/L </li></ul>
6. 6. Poiseuille Flow S.GOKALTUN Florida International University
7. 7. Entry Length Effects Tritton, D.J. Physical Fluid Dynamics, 2 nd Ed. Oxford University Press, Oxford. 519 pp.
8. 8. Re << 1 (Stokes Flow) Tritton, D.J. Physical Fluid Dynamics, 2 nd Ed. Oxford University Press, Oxford. 519 pp.
9. 9. Eddies and Cylinder Wakes Re = 41 Tritton, D.J. Physical Fluid Dynamics, 2 nd Ed. Oxford University Press, Oxford. 519 pp. Re = 30 Re = 40 Re = 47 Re = 55 Re = 67 Re = 100
10. 10. Eddies and Cylinder Wakes S.Gokaltun Florida International University Streamlines for flow around a circular cylinder at 9 ≤ Re ≤ 10.(g=0.00001, L=300 lu, D=100 lu)
11. 11. Eddies and Cylinder Wakes Streamlines for flow around a circular cylinder at 40 ≤ Re ≤ 5 0.(g=0.0001, L=300 lu, D=100 lu) ( Photograph by Sadatoshi Taneda. Taneda 1956a, J. Phys. Soc. Jpn., 11, 302-307. ) S.Gokaltun Florida International University
12. 12. Separation Tritton, D.J. Physical Fluid Dynamics, 2 nd Ed. Oxford University Press, Oxford. 519 pp.
13. 13. Laplace Law <ul><li>There is a pressure difference between the inside and outside of bubbles and drops </li></ul><ul><li>The pressure is always higher on the inside of a bubble or drop (concave side) – just as in a balloon </li></ul><ul><li>The pressure difference depends on the radius of curvature and the surface tension for the fluid pair of interest:  P =  /r </li></ul>
14. 14. Laplace Law P in P out r  P =  /r ->  =  P/r, which is linear in 1/r (a.k.a. curvature)
15. 15. Young-Laplace Law <ul><li>With solid surfaces, in addition to the fluid1/fluid2 interface – where the interaction is given by the interfacial tension    -- we have interfaces between each fluid and the surface  S1  and  S2 </li></ul><ul><li>Often one of the fluids preferentially ‘wets’ the surface </li></ul><ul><li>This phenomenon is captured by the contact angle </li></ul><ul><li>cos  = (  S2 -  S1   </li></ul>
16. 16. Young-Laplace Law <ul><li>Zero contact angle means perfect wetting; </li></ul><ul><li> P =  cos  /r </li></ul>