CONVECTIVE FLUX, FLUID MASS CONSERVATION

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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.
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 time u1?t?A/(?t?A)
u1
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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.
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
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 u1 ?u1 x u1
FC,mom,11 ?(u1)2
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
A scalar quantity such as volume, mass and heat
can be fluxed three-dimensional 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).
ui
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.
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
The discharges of volume, mass and heat across
the face are thus given as
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
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).
The animation illustrates how momentum in the x1
direction can be fluxed in the x2 direction.
Note that in this case ni (0, 1, 0).
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
So lets 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.
u1
The total amount of x1 momentum contained in this
volume momentum/volume volume
?u1?Au2?t. Flux of x1 momentum in x2 direction
?u1?Au2?t/(?A?t)
u2
x3
x1
x2
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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
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
The net inflow rate Qmom, inflow,i of the vector
quantity fluid momentum into the control volume is
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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
Now the control volume is fixed in space, so the
equation can be rewritten as
But
So
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
Now the condition
must be true for any volume V. The only way this
can hold is if the integrand vanishes everywhere
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
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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
simply-connected volume in a given domain, then
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
Then everywhere within that domain
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
Proof if
everywhere within some domain, then where x is a
free variable within that domain,
Now take the derivative with respect to x to
obtain
and the desired result is proved.
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CONVECTIVE FLUX, FLUID MASS CONSERVATION
2D corner flow of an incompressible fluid
ui (0, - U)
Thus
Now
ui (U, 0)
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 balanced by the
outflow.