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CEE 262A H YDRODYNAMICS

CEE 262A H YDRODYNAMICS. Lecture 1* Introduction and properties of fluids *Adapted from notes by Prof. Stephen Monismith. The Navier-Stokes equation. What is a Fluid ? (Fluid vs. Solid) A substance which deforms continuously under the action of a shearing stress.

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CEE 262A H YDRODYNAMICS

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  1. CEE 262A HYDRODYNAMICS Lecture 1* Introduction and properties of fluids *Adapted from notes by Prof. Stephen Monismith

  2. The Navier-Stokes equation

  3. What is a Fluid ? (Fluid vs. Solid) • A substance which deforms continuously under the action of a shearing stress. • A perfectly elastic solid can resist a shear stress by static deformation; a fluid cannot. • An elastic solid can behave like a fluid beyond its yield point, at which point it behaves as a "plastic". • Viscoelastic fluids behave like fluids and solids (i.e. egg whites, which have a small tendency to return to their original shape). • Corollary: A fluid at rest must be in a state of zero shear stress.

  4. Liquid vs. Gas • Gases typically expand to fill the shape of container. • Liquids assume shape of only part of container. • Equation of state for pressure • Gases typically obey equations of state for the pressure e.g. the ideal gas law p = r R T • Liquids are typically assumed to be incompressible and so p is a very weak function of r and T. • Sound speed in gases is typically smaller than in liquids (air ~ 343 m/s, water ~ 1484 m/s, iron 5120 m/s).

  5. Continuum Hypothesis • Microscopic approach: Analyze molecular structure and associated collisions (e.g. pressure is due to the net exchange of momentum at a solid surface) • Macroscopic (continuum) approach: Analyze bulk behavior of fluid (e.g. pressure is force exerted by fluid per unit area of solid surface) • Continuum approach always assumes that scale of motion is much larger than mean free path • Almost always valid (e.g. can break down in upper atmosphere where density becomes very low); In air, mean free path = 10-8 m; smallest scale of turbulent eddy that feels viscosity in atmosphere ~10-3 m.

  6. Stress Force per area - defined by particular surface orientation Stress at a face is decomposed into a sum of the normal and tangential stresses.

  7. Normal stresses Fluid pressure ”p” Tangential Stresses Shear stress “t” Tangential force is a vector

  8. Shear strain angle will grow as f(t) stress strain rate Viscosity = “Resistance to shear” For fluids such as water, oil, air

  9. However, But As , , 0 For fluids:"Stress is proportional to strain rate". For solids:"Stress is proportional tostrain" (s=Ee) Where dynamic viscosity. This is a constitutive relation, which relates forces to material (fluid) properties.

  10. y 0 U Notes on shear stress (i) Any shear stress, however small, produces relative motion. (ii) If t=0, du/dy=0, but m≠0. (iii) Velocity profile cannot be tangent to a solid boundary - This requires an infinite shear stress. "No-slip" condition: u=0 at solid boundary.

  11. Types of fluids Bingham Plastic Real Plastic Shear-Thinning Fluid Newtonian Shear-Thickening Fluid 1 Shear-thinning: Ketchup, whipped creamShear-thickening: Corn starch in water Newtonian fluid: Stress is linearly proportional to strain rate.

  12. Units Dynamic Viscosity e.g. SI:

  13. e.g. SI: Kinematic Viscosity

  14. Dynamic vs. kinematic viscosity Force on plates F~ muA/H Air: 10 N (2 lb), Water: 1000 N (200 lb) Shear stress exerted on plates t=F/A~m u/H Air: 10-2 Pa, Water: 1 Pa Shear stress per unit fluid density f=F/rA~n u/H Air: 10-2 m2/s2, Water: 10-3 m2/s2 Water is dynamically more forceful, but kinematically less forceful, per unit density. Area A=1000 m2 (747 wing area) Flow speed u=1 m/s Air: r=1 kg/m3, m=10-5 kg/ms Water: r=103 kg/m3, m=10-3 kg/ms H=1 mm

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