Download Lecture 14c - TTU Physics

Sect. 14.5: Fluid Dynamics
Section 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!
Types of Fluid Flow
• 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
Paths of the particles look
qualitatively like this!
We’ll assume this type of flow
Types of Fluid 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
Paths of the particles can
Look qualitatively like this!
We’ll not discuss this type.
Viscosity
• 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.
Ideal Fluid Flow
• 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
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
Equation of Continuity
• 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!
m2 = mass of fluid
in this volume
m1 = 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) gives
r1A1v1 = r2A2v2
(2)
• Fluid is incompressible so, r = constant
 (2) gives:
A1v1 = A2v2
ρAv  “mass flow rate”
Units: mass per time interval
or kg/s
(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
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
Or:

A1v1 = A2v2
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 Continuity
A1v1 = 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 Flow
rcap = 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 Duct
Speed 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