MFGT 242: Flow Analysis
Chapter 4: Governing Equations for Fluid
Flow
Professor Joe Greene
CSU, CHICO
1
Governing Equaitons
•
•
•
•
•
•
Introduction to Vector Analysis
Mathematical Preliminaries
Conservation of Mass
Conservation of Momentum
Conservation of Energy
Boundary Conditions
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Introduction to Vector Analysis
• Various quantities used in fluid mechanics
– Scalars: needs only single number to represent it
• Temperature, volume,density
– Vectors: needs magnitude and direction
• Velocity, force, gravitational force, and momentum
– Tensors: needs two vectors to describe it (later)
• Vectors
– Determined by its magnitude (length) and direction
– Usually represented by boldface and shown with
• unit vectors, ex, ey, and ez with length of unity and point in each of the
respective coordinate directions
• Coordinate system: RCCS (Rectangular Cartesian Coordiante System
• Vector, v, can be expressed as linear combination of unit vectors
v = vxex + vyey,+ vz ez
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Vector Operations
• Vector addition and subtraction
• If a vector u is added to another vector v, then the result is a third
vector w whose components equals the sums of the corresponding
components of u and v. wx = ux + vx
• Similar results holds for subtraction of one vector from another
• Vector products (Fig 5.2)
– Multiplication of scalar, s, with vector, v, is a vector sv.
– Dot product
u  v  u v cos 
• Dot product of two vectors yields a scalar
– Which is the area of the rectangle with sides u and vcos 
– Dot product can be expressed with the Kronecker delta symbol, ij = 0
for i  j and ij = 1 for i = j
u  v  u x vx  u y v y  u z vz
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Vector Operations
• Cross Product
• Cross product of two vectors, u and v, is a vector, whose
– magnitude is the area of the parallelogram with adjacent sides u and v,
namely, the product of the individual magnitudes and the sine of the
angle  between them.
– Direction is along the unit vector, n, normal to the plane of u and v and
follows the right-hand rule. u  v  u v n sin 
– Representative nonzero cross products of the unit vectors are:
e x  e y  e z , e y  e z  e x , e z  e x  e y , e x  e z  e y ,
e x  e x  0, e y  e y  0, e z  e z  0
– Cross product may be conveniently reformulated in terms of the
individual components and also as a determinate
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Vector Operations
• Vector differentiation



  ex
 ey
 ez
x
y
z
– Introduction of the (del or nabla) operator 
• In RCCS, if a scalar s=s(x,y,z) is a function of position, then a
constant value of s, such as s=s1, defines a surface
• The gradient of a scalar s at a point P, designated grad s (equal to s)
is a directional derivative.
– Defined as a vector in the direction in which s increases most rapidly
with distance, whose magnitude equals the rate of increase
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Vector Operations
• Gradient is shown with unit vector r
r
dx
dy
dz
ex  e y  ez
dr
dr
dr
– Whose magnitude can readily be shown to be unity,
s
s
s
s  ex
 ey
 ez
– Vector s
x
y
z
– Component of in the direction of r is the dot product
s dx s dy s dz
s  r 


x dr y dr z dr
– Note: a differential change is s, and hence its rate of change in the r
direction are given by
s
s
s
ds  dx  dy  dz
x
y
z
ds s dx s dy s dz



dr x dr y dr z dr
– Then,
ds
s  r 
dr
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Physical Properties
• Density
– Liquids are dependent upon the temperature and pressure
• Density of a fluid is defined as
mass
M

 3
volume L
– mass per unit volume, and
– indicates the inertia or resistance to an accelerating force.
• Examples,
–
–
–
–
Water: density = 1 g/cc = 62.3 lb/ft3
Steel: density = 7.85 g/cc; Aluminum: density: 2.7 g/cc
PP: density= 0.91 g/cc; HDPE: density= 0.95 g/cc
Most plastics: density = 0.9 to 1.5 g/cc
• Specific Gravity
– Density of material divided by density of water (Unit-less)
• Examples,
– Water: specific gravity = 1.0
– Most plastics: density = 0.9 to 1.5
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Velocity
• Velocity is the rate of change of the position of a fluid particle with
time
– Having magnitude and direction.
• In macroscopic treatment of fluids, you can ignore the change in
velocity with position.
• In microscopic treatment of fluids, it is essential to consider the
variations with position.
• Three fluxes that are based upon velocity and area, A
– Volumetric flow rate, Q = u A
– Mass flow rate, m = Q =  u A
– Momentum, (velocity times mass flow rate) M = m u =  u2 A
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Mathematical Preliminaries
• Assumptions
– Fluid is a continuous flow in a surrounding environment
– The values of velocity, pressure, and temperature change smoothly
and are differentiable
• Material Derivative
– Some fluid properties change with position and time
• velocity, pressure, temperature, density
   x, y, z, t 
• Use chain rule for differentiation
• Then,
– Material Derivative
– Accounts for
» motion of fluid
» changing position with time
Dt
t
 v  v 
Dv
v
Dt
t
 v    ; Or for veloci ty
D

dt
t
 v    ; then,
d

dt x
y
z
t
 vx  v y  vz 
d 



dt x t y t z t t
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



d  x  y  z 
Compressible and Incompressible Fluids
• Principle of mass conservation
t
 (  v)

– where  is the fluid density and v is the velocity
• For injection molding, the density is constant
(incompressible fluid density is constant)
• Flux
v  0
• The flux v of an extensive quantity, X, for example, is a vector that
denotes the direction and rate at which X is being transported (by
flow, diffusion, conduction, etc.) (per unit area)
• Examples
– Mass, momentum, energy, volume
– (Volume transported per unit time pre unit area) or m/s
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Basic Laws of Fluid Mechanics
• Apply to conservation of Mass, Momentum, and Energy
• In - Out = accumulation in a boundary or space
Xin - Xout = X system
• Applies to only a very selective properties of X
– Energy
– Momentum
– Mass
• Does not apply to some extensive properties
– Volume
– Temperature
– Velocity
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Basic Laws of Fluid Mechanics
• Conservation of Mass
– If V(t) is a material volume of fluid flowing
continuously, then, the mass contained in V(t) does not
V (t )
change.
m    x, t dV
– Mass is given by
t
   v   0

– Conservation says rate of change is zero.
dt V(t )
 x, t dV  0
d
dt
0
dm
– For incompressible fluid, the density is constant and
– For Material Derivative t     v   v     0

v  0
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Basic Laws of Fluid Mechanics
• Conservation of Momentum
– Momentum is mass times velocity
– Time rate of change of fluid particle momentum in a
material, V(t), is equal to the sum of the external forces
dt V(t )
vdV   Fext
dDt 
    P      g
 Dv 
Force
=
Pressure
Force
Viscous
Force
Gravity
Force
•Or rewritten by expanding the material derivative
t
v   P       g    vv

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Basic Laws of Fluid Mechanics
• Energy
– Total energy of the fluid in a material volume V(t) is given by the
sum of its kinetic and internal energies.
 Dt 
C p      q   p  v    v
 DT 
Energy
volume
= Conduction
Energy
Compression Viscous
Energy
Dissipation
• Or expanded out as
 t

C p   v  T     q   p  v    v
 T

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Boundary Conditions
• Apply conservation of mass, momentum and energy to
injection molding causes the appliction of the equations to
specific problem
– Example of injection molding surface
– Pressure BC
Mold Wall
Mold
Wall
Flow
Mold
Wall
Mold Wall
• Pressure gradient in normal direction (90° from flow) is zero
– The mold walls are solid and impermeable
• Melt flow rate, Q, or pressure, P, is specified at the inlet.
• The pressure is zero at surface or flow front. (Fountain effect)
– Temperature BC
• Temperature profile through cavity is described as uniform at the
injection point,
• Temperature at mold walls is initially constant and varies as the melt
hit the mold wall and heats up.
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• Mold walls are cooled by heat transfer fluid