CEE 262A

HYDRODYNAMICS

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 = 10-8 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”

22

1

LT

ML

normalF

AtangentialF

Tangential force is a vector

Tangential Stresses Shear stress “”A

Fp normal

A

0lim

T

A

FT

A

tangential

0lim

7

Shear strain angle will grow as f(t)

y

u tt

u

x

For fluids such as water, oil, air

t

stress strain rate

Viscosity = “Resistance to shear”

8

However, y

tu

tan

As , , 0

dy

du

dt

d

t y

dt

d t

But

dy

du

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 tostrain" (=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

U0 10

dydu /

dy

du

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 creamShear-thickening: Corn starch in water

11

Units

Dynamic ViscositydydU /

]/[][][ dydU

222][

LT

M

LT

ML

Area

Force

dy

dU

TLT

L 11

T

ML

LT

MT

LT

M

2][

s PaNs/m2 e.g. SI:

12

]/[][][ v

T

L

M

L

LT

Mv

23

][

LT

M][

3][L

M

e.g. SI: Stokes 1/10 24 sm

Kinematic Viscosity

v

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: 10-2 Pa, Water: 1 Pa

Shear stress per unit fluid density f=F/~ u/H

Air: 10-2 m2/s2, Water: 10-3 m2/s2

Water is dynamically more forceful, but kinematically less forceful, per unit density.

Flow speed u=1 m/s

Air: =1 kg/m3, =10-5 kg/ms

Water: =103 kg/m3, =10-3 kg/ms

Area A=1000 m2

(747 wing area)

H=1 mm

14