CONVECTIVE FLUX, FLUID MASS CONSERVATION
The flux of any quantity in any direction is the rate per unit time per unit area
that the quantity crosses a face normal to that direction.
Quantities that can be fluxed in fluids include fluid mass, mass of a
dissolved contaminant such as salt, mass of a suspended contaminant such
as sediment, fluid momentum, fluid energy, heat, etc.
There are three fundamental mechanisms of flux of interest in fluid
mechanics:
Convective flux, by which the quantity is carried with the flow;
Diffusive flux, by which the quantity migrates from zones of high
concentration to zones of low concentration by random molecular motion,
and
Radiative flux, by which the quantity (e.g. heat) is carried by waves such as
electromagnetic waves (in the infrared spectrum in the case of heat).
Here we are concerned only with the first two kinds.
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
A flowing fluid carries any associated quantity with it convectively. We first
consider the convective flux of fluid volume and mass.
The tube shown below has rectangular cross-section with area A. The
fluid velocity u1 in the tube is taken to be constant on the cross-section.
A
u1
x1
u1t
u1tA
At time t = 0 we mark a parcel of fluid, the downstream end of which is
bounded by an orange face.
In time t the leading edge of the marked parcel moves downstream a
distance u1t, so that volume u1tA and mass u1tA has crossed the
face in time t.
Convective flux of fluid volume in x1 direction FC,vol,1 =
the volume that moves across a face per unit face area per unit
2
time = u1tA/(tA) = u1
CONVECTIVE FLUX, FLUID MASS CONSERVATION
The mass that crosses normal to the section in time t is density x volume
crossed = u1tA
The convective mass flux in the x1 direction across the section =
u1tA/(tA) = u1.
A
u1
x1
u1t
u1tA
The heat in a fluid is characterized by the heat capacity cp = heat needed
to raise one kg of the fluid 1 degree. In the SI system, [cp] = joules/kg/°K.
Where  denotes the temperature of the fluid in °K, then, the amount of heat
per unit volume in the fluid is cp ~ joules/m3.
The convective flux of heat in the x1 direction = FC,heat,1
heat that moves across a face per unit face area per unit time =
cpu1tA/(tA) = cpu1
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
The momentum in the x1 direction that crosses normal to the section in
time t = velocity x mass = u1 x u1tA = (u1)2 tA
The convective flux in the x1 direction of momentum in the x1 direction
across the section =
(u1)2tA/(tA)
= (u1)2
A
u1
x1
u1t
u1tA
Summary: the convective flux of a quantity in some direction = the
quantity per unit volume x the flow velocity in the direction it is being
fluxed.
Volume flux in x1 direction = volume/volume x u1 = FC,vol,1 u1
Mass flux in x1 direction = mass/volume x u1 = FC,mass,1 = u1
Heat flux in x1 direction = heat/volume x u1 = FC,heat,1 = cpu1
Flux in x1 direction of momentum in x1 direction = momentum/volume x u14=
u1 x u1 = FC,mom,11 = (u1)2
CONVECTIVE FLUX, FLUID MASS CONSERVATION
A scalar quantity such as volume, mass and heat can be fluxed threedimensional space (x1, x2, x3), so that the flux of a scalar quantity is a
vector. The vectorial convective flux of a scalar quantity = the quantity
per unit volume x the velocity vector.
Thus the flux vectors of volume, fluid mass and heat are:
FC,vol,i = ui, FC,mass,i = ui and FC,heat,i = cpui.
The flux of a scalar quantity across a face of specified direction ni is a scalar
quantity, given as the dot produce of the flux vector and the unit normal of
the face. Consider the illustrated face directed in the ni direction. The flux of
fluid mass across the face is given as uini, i.e
uini
i.e. the component of the vector uini that is
oriented normal to the face (and thus crosses it).
In the diagram, the pink vector denotes ni, the red
vector denotes uini, the blue vector denotes the
component of uini that is parallel to the face (and thus
does not cross it) and the green vector denotes the
component uini of ui that is perpendicular to the face
and thus is the mass flux across it.
ui
ni
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
Let Fi denote the flux vector of some quantity and ni denote the unit normal
vector of a face of area A. The discharge of the quantity across the face
Qquan is a scalar given as
Qquan   FinidA
A
The discharges of volume, mass and heat across the face are thus given as
Q   uinidA
A
, Qmass   uinidA
A
, Qheat   cpuinidA
A
Note that no subscript is used in the case of volume discharge
uini
If ui is oriented normal to the face and has the
value U which is constant across the face, the
(volume) discharge is given as
ui
Q  UA
ni
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
A vector quantity, such as momentum, has three components, each of
which can be fluxed in each of the three directions (x1, x2, x3), so that the
flux of a vector quantity is a matrix.
The flux of momentum ui in the xj direction is given as
FC,mom,ij = uiuj.
For example, the flux in the x2 direction of momentum in the x1 direction is
given as u1u2.
Now how can
momentum in the x1
direction be fluxed in
the x2 direction?
Consider a velocity
vector that crosses a
face diagonally.
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
More specifically, we assume that ui = momemtum per unit volume = (u1,
u2 ,0).
ni u
i
u2
u1
ui
u2
u1
x3
x1
x2
The animation illustrates how momentum in the x1 direction can be fluxed in
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the x2 direction. Note that in this case ni = (0, 1, 0).
CONVECTIVE FLUX, FLUID MASS CONSERVATION
So let’s compute the flux in the x2 direction of momentum in the x1 direction.
How much x1 momentum
crossed the face normal to
the x2 direction in time t?
u2t
u1t
Where A denotes the area
of the face, the volume of
fluid that crossed in time t =
Au2t.
u2
u1
The total amount of x1
momentum contained in this
volume = momentum/volume
* volume = u1Au2t.
x3
x1
x2
Flux of x1 momentum in x2
direction = u1Au2t/(At)
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=
u1u2
CONVECTIVE FLUX, FLUID MASS CONSERVATION
The flux uiuj of momentum in the i direction across a face in the nj direction
is a vector given as
uiu jn j
where ujnj denotes the component of the velocity that is actually normal to
the face.
Let nj denote a unit outward normal to a control volume
that is fixed in space (so that fluid can flow in and out).
The net inflow rate of the scalar
quantity fluid mass into the
control volume is
Qmass,inf low    uinidA
ni
dA
S
The net inflow rate Qmom, inflow,i of
the vector quantity fluid
momentum into the control
volume is
Qmom,inf low ,i    uiu jn jdA
S
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
Equation of fluid mass conservation:
/t(fluid mass in control volume) = net inflow rate of mass

dV    uinidA

t V
S
Now the control volume is fixed in
space, so the equation can be
rewritten as:

V t dV  S uinidA  0
ni
dA
But
ui
dV
xi
V
 uinidA  
S
So
  ui 
V  t  xi dV  0
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
Now the condition
  ui 
V  t  xi dV  0
must be true for any volume V. The only way this can hold is if the
integrand vanishes everywhere:
 ui

0
t
x i
The above equation is the
equation of mass
conservation, or continuity
equation of a fluid.
For an incompressible fluid, i.e. one
for which  can be locally
approximated as constant, the relation
reduces to
ui
0
x i
ni
dA
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
But this result is predicated on the following theorem:, if A = A(xi) is a
continuous function of xi and
 AdV  0
V
simply-connected volume in a given domain, then
A0
everywhere within that domain.
The equivalent 1D theorem can be stated as follows. Let f(x) be a
continuous function such that for every set of values x1 and x2 within a given
domain

x2
x1
f ( x )dx  0
Then everywhere within that domain
f (x )  0
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
Proof: if

x2
x1
f ( x )dx  0
everywhere within some domain, then where x is a free variable within that
domain,

x
x1
f (x )dx  0
Now take the derivative with respect to x to obtain
d x
d0
f
(
x
)
dx

f
(
x
)

0

x
dx 1
dx
and the desired result is proved.
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
2D corner flow of an incompressible fluid
ui u1 u2


0
x i x1 x 2
ui = (0, - U)
Thus
u1
u
 2
x1
x 2
Now
L
u1 u1 x1L  u1 x10 U
~

x1
L
L
u2 u2 x 2 L  u2 x 2 0  U  0
U
~


x 2
L
L
L
ui = (U, 0)
x2
x1
L
Thus continuity is satisfied. The physical interpretation is as follows.
Since no mass can be stored in the control “volume” because the fluid is
incompressible, the inflow into the control volume must be precisely
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balanced by the outflow.