3.044 MATERIALS PROCESSING
LECTURE 13
T
σ, P
c
heat transfer solid mech., fluid flow diffusion, phase trans.
Mission for the next part of the course:
Focus on fluid flow, How does fluid flow relate to solid mechanics?
How do we connect all three categories?
Fluid Flow Terminology
· free flow - against low resistance medium (air)
· confined flow - against a solid boundary
VS.
· incompressible flow - e.g. water, most melted/liquid matter
· compressible flow - e.g. gases, polymer melts (mildly)
· laminar flow - very uniform motion, low velocity
· turbulent flow - chaotic motion, high velocity
Fluid Flow is the diffusion of momentum
Date: April 2nd, 2012.
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Does pressure
change density?
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2
LECTURE 13
1-D Flow:
Two flat plates:
In steady-state → constant v-profile:
Must be applying a constant Force:
resulting velocity profile
driving force
F
A
N
=Pa
m2
V0
μ
L
N·s 1
( )=Pa·s
m2 s
=
3.044 MATERIALS PROCESSING
3
Differentially small element:
∂Vx
∂y
→ only applies to simple fluids, Newtonian fluids
→ We’ll return to non-Newtonian fluids later
Newton’s Law of Viscosity: τyx = −μ
Compare:
1.
x
τyx = −μ ∂V
∂y
What’s Different?
q̄ = −k ∇T
vector
2.
V S.
grad scalar
q = −k ∂T
∂x
¯ = −μ∇ v̄
σ̄
2nd rank tensor
Stress is equivalent to a Flux of Momentum:
Stress
F
A
N
kg
Units: 2 =
m
m · s2
Momentum Flux
m·v
Units:
Momentum Balance (1-D):
3.
kg ms
kg
=
2
m ·s
m · s2
grad vector
∂c
j = −D ∂x
4
LECTURE 13
momentum accumulated
momentum in
momentum out
momentum generated
A τyx |y − A τyx |y+δy +
N
·m2
m2
Fx · V
∂ (ρvx )
V
∂t =
N
·m3
m3
N
·m2
m2
m3 ·
kg m
·
m3 s2
Divide through by V = Aδy
τyx |y − τyx |y+δy
δy
+ Fx =
∂ (ρvx )
∂t
Plug in Newton’s Law of Viscosity
∂vx ∂vx ∂ (ρvx )
μ
+ Fx =
−
Δy ∂y y+Δy
∂y y
∂t
μ
∂vx
∂ (ρvx )
Δ
+ Fx =
Δy ∂y
∂t
1-D Fluid Flow Equation
μ
∂ 2 vx
∂ (ρvx )
+ Fx =
2
∂t
∂y
Assume ρ is constant
∂ 2 vx
∂vx
μ 2 + Fx = ρ
∂y
∂t
2
μ ∂ vx F x
∂vx
=
+
∂t
ρ ∂y 2
ρ
Compare:
1.
∂vx
=
∂t
2
μ ∂ vx F x
+
ρ ∂y 2
ρ
diffusivity of momentum
2.
∂ 2c
∂c
= D
∂x2
∂t
diffusivity
3.
∂ 2T
∂T
= α
∂x2
∂t
thermal diffusivity
3.044 MATERIALS PROCESSING
Diffusivity of Momentum is also known as Kinematic Viscosity
ν=
Units:
Pa · s
kg
m3
Momentum diffuses through a flowing fluid
At steady state:
∂vx
μ ∂ 2 vx
=
=0
∂t
ρ ∂y 2
∂ 2 vx
=0
∂y 2
Boundary Conditions:
@ y = 0, v = v0
@ y = L, v = 0
Solve:
vx = Ay + B
v0
vx = v 0 − y
L
=
μ
ρ
N · s · m3
m2
=
m2 · kg
s
5
6
LECTURE 13
In order to solve simply:
⇒ Use diffusion or conduction solutions and compare vx to c or T
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3.044 Materials Processing
Spring 2013
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