Lecture14c

Sect. 14.5: Fluid Dynamics

• We’ve done fluid statics. Now, Fluid Dynamics (fluid flow), which is much more interesting! COURSE THEME: NEWTON’S LAWS OF MOTION!

NOW• Sects. 14.5 - 14.7: Methods to analyze the dynamics of

fluids in motion.

• First, we need to discuss FLUID LANGUAGE. We’ve introduced a lot of this language while talking about fluid statics. But, there is some other terminology we need to discuss before we discuss

Newton’s Laws (Especially Newton’s 2nd Law!) in Fluid Language!

Section 14.5: Fluid Dynamics

• Two main types of fluid flow:

1. Laminar Flow (or Streamline Flow)– Steady flow

– Each particle of the fluid follows a smooth path

– The paths of the different particles never cross each other

– Every fluid particle arriving at a given point has the same velocity

– The path taken by the particles is called a streamline

Types of Fluid Flow

Paths of the particles lookqualitatively like this!

We’ll assume this type of flow

• Two main types of fluid flow:

2. Turbulent Flow– Irregular flow which has small whirlpool-like regions– It’s turbulent flow when the particles go above some critical speed

Streamlines can cross each other

Types of Fluid Flow

Paths of the particles can Look qualitatively like this!

We’ll not discuss this type.

• Viscosity is a measure of the amount of internal friction in the fluid.

• This internal friction or VISCOUS FORCE, comes from the resistance that two adjacent layers of fluid have to moving relative to each other.

• Viscosity causes part of the kinetic energy of a fluid to be converted to internal energy.

Viscosity

• We make four simplifying assumptions in our treatment of fluid flow to make the analysis easier:

1. The fluid is nonviscous

Internal friction is neglected

2. The flow is steady The velocity of each point remains constant

3. The fluid is incompressible

The density remains constant

4. The flow is irrotational

The fluid has no angular momentum about any point

Ideal Fluid Flow

Streamlines• The path a particle takes in steady flow is

a streamline

• The velocity of each particle is tangent to a streamline

• A set of streamlines

is called a

TUBE OF FLOW

• Consider a fluid moving through a pipe of nonuniform diameter. The particles move along the streamlines in steady flow.

• The mass m1 in the small portion of pipe of length Δx1, crossing area A1 in some time Δt, must be exactly the same as the mass m2 in length Δx2, crossing area A2 in the same time Δt.

• Why? Because no fluid particles “leak” out of the pipe!

The fluid has

Conservation of Mass!

Equation of Continuity

m1 = mass of fluid

in this volume

m2 = mass of fluid

in this volume

Conservation of Mass: m1 = m2 (1)

For point 1 & point 2, the definition of density ρ in terms of mass m & volume V gives: m = ρV.

For points1 & 2, use V = Ax (1) givesA1v1 = A2v2 (2)

• Fluid is incompressible so, = constant

(2) gives:A1v1 = A2v2 (3)

– (3) is called the EQUATION OF CONTINUITY FOR FLUIDS

– The product of the area and the fluid speed at all points along a pipe is constant for an incompressible fluid

ρAv “mass flow rate”Units: mass per time interval

or kg/s

Av “volume flow rate”Units: volume per time interval

or m3/s

• Mass flow rate (mass of fluid passing a point per second) is constant: ρ1A1v1 = ρ2A2v2

Equation of Continuity

PHYSICS: Conservation of Mass!!• For an incompressible fluid (ρ1 = ρ2 = ρ)

Then A1v1 = A2v2

Or: Av = constant Where cross sectional area A is large, velocity v is

small, where A is small, v is large.

• Volume flow rate: (V/t) = A(x/t) = Av

Implications of Equation of ContinuityA1v1 = A2v2

The fluid speed v is low where the pipe is wide (large A)

The fluid speed v is high where the pipe is constricted (small A)

• The product, Av, is called the volume flow rate or flux. Av = constant says that the volume that enters one end of the pipe in a given time equals the volume leaving the other end in the same time (If no leaks are present!)

• PHYSICS: Conservation of Mass!!

A1v1 = A2v2 Or Av = constant

• Small pipe cross section larger v

• Large pipe cross section smaller v

Example: Estimate Blood Flowrcap = 4 10-4 cm, raorta = 1.2 cm

v1 = 40 cm/s, v2 = 5 10-4 cm/s

Number of capillaries N = ?

A2 = N(rcap)2, A1 = (raorta)2

A1v1 = A2v2

N = (v1/v2)[(raorta)2/(rcap)2]

N 7 109

Example: Heating DuctSpeed in duct:

v1 = 3 m/s

Room volume:

V2 = 300 m3

Fills room every

t =15 min = 900 s

A1 = ?

A1v1 = Volume flow rate = (V/t) = V2/t

A1 = 0.11 m2