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

The Navier-Stokes equation 2

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. 3

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 T Liquids are typically assumed to be incompressible and so p is a very weak function of  and T. Sound speed in gases is typically smaller than in liquids (air ~ 343 m/s, water ~ 1484 m/s, iron 5120 m/s). 4

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 = m; smallest scale of turbulent eddy that feels viscosity in atmosphere ~10 -3 m. 5

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

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

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

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

Notes on shear stress (i) Any shear stress, however small, produces relative motion. (ii) If  =0, du/dy=0, but  ≠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. y U 0 10

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

Units Dynamic Viscosity e.g. SI: 12

e.g. SI: Kinematic Viscosity 13

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