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10. The Continuity Equation

10. The Continuity Equation. CH EN 374: Fluid Mechanics. Review: Macroscopic/Integral Balances. Total change of property. Change within the control volume. Flow in and out across the control surface. Control Volume. Control Surface. What if we are interested in flow on a microscopic level?.

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10. The Continuity Equation

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  1. 10. The Continuity Equation CH EN 374: Fluid Mechanics

  2. Review: Macroscopic/Integral Balances Total change of property Change within the control volume. Flow in and out across the control surface. Control Volume Control Surface

  3. What if we are interested in flow on a microscopic level? • We do a differential balance.

  4. Differential Mass Balance Accumulation = Input - Output

  5. Differential Mass Balance

  6. Differential Mass Balance

  7. The Continuity Equation Or, more compactly…

  8. How is this similar to the material derivative? • How is this similar to the Reynold’s Transport Theory?

  9. Incompressible Flow • What simplification can we make?

  10. Problem • Can the unsteady, two-dimensional velocity field below be approximated as incompressible?

  11. Problem • The component of velocity in a steady, two-dimensional incompressible flow field is , where and are constants. The y-component is unknown. Generate an expression for as a function of and .

  12. Streamlines • A streamline is a way to visualize flow: https://www.youtube.com/watch?v=DOUfyDHxkYQ

  13. The Stream Function • Describes streamlines mathematically. • Continuity equation for 2d incompressible flow: • Define stream function so that: • Substituting into continuity:

  14. The Stream Function

  15. The Stream Function • What’s the point? • is a smooth function of x and y. • So for constant you can get x(y) or y(x) • Constant values represent streamlines

  16. Problem Consider the steady, incompressible, two-dimensional velocity field: What would we do if we wanted to plot some streamlines for x > 0?

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