Hello everyone, in todays class we will discuss
about the basic equations of fluid mechanics
and heat transfer in basic fluid mechanics
course and basic heat transfer course you
might have derived these equations. So we
will not derive in this class these equations
only will write down the governing equations
of fluid flow and heat transfer. First will
write down the fluid flow equations for incompressible
and Newtonian fluid flow.
So incompressible flow equations 
in Cartesian coordinate, okay so if we consider
a Cartesian coordinate, let us say this is
your x, this is your y and this is your z,
so respectively will define the velocities
in x direction u, in y direction v and in
z direction w. So, if you consider a velocity
vector u then this is we can write u is the
velocity in the x direction then the unit
vector in that direction i, v, j, v is the
velocity in the y direction and w OK.
So will have some assumptions to write the
governing equations, so one assumptions already
we have told that it is incompressible flows
that means density is constant or you can
tell that Mach number is less than 0.3 Will
also consider Newtonian fluid flow and will
consider constant properties, so with those
assumptions if you write down the governing
equations. So the assumptions we have taken
one is obviously incompressible flow, second
is Newtonian fluid flow 
and we can write with constant properties.
So in fluid flow equations, so will first
write the continuity equation which you can
derive from the conservation of mass, right.
So will write the continuity equation first,
continuity equation okay, so you can derive
it from conservation of mass, so it will conserve
the mass, right, so it is conservation of
mass. So, as it is incompressible flow so
it will be we just in vector form divergence
of, so we can write this continuity equation
for incompressible flow in vector form as
divergence of u is equal to 0.
So we can write Del u by Del x plus Del v
by Del y plus Del w by Del z is equal to 0.
So, if you write this equation for 2 dimensional
then it will be just Del u by Del x plus Del
v by Del y is equal to 0. So if you write
this equation for 2 dimensional case then
it will be Del u by Del x plus Del v by Del
y is equal to 0. So, this is the continuity
equation.
Now, will write the momentum equations, so
in momentum equation we conserve the momentum.
So, momentum equations, conservation of momentum,
so as we are using incompressible flow and
for Stokesian fluid, we can write the Navier-Stokes
equations. So, this is also known as Navier-Stokes
equations, Navier-Stokes equations okay will
write in vector form, first will write in
general so Del by Del t rho u plus divergence
of rho u-u is equal to minus grad p plus divergence
of mu grad u plus rho b, okay.
So where rho is your density of the fluid
okay and mu is the viscosity of the fluid,
it is dynamic viscosity okay. So it is dynamic
viscosity. So you can write mu is equal to
dynamic viscosity of the fluid. And p, p is
your pressure. Okay so p is your pressure
and b is the body force, body force. So you
can have gravity as body force okay or if
you have a multi-physics problem, then electromagnetic
force or magnetic force you can have as source
term or body force term.
So, this equation now for the constant properties
if you write down this equation, it will be
Del u by Del t. So, rho you can take it outside
and divide both sides then divergence of u-u
is equal to minus 1 by rho grad p plus. So,
if mu is constant you can take it outside
and mu by rho will be nu. So, grad square
u, okay this equation what we are writing
constant properties and no body force or negligible
body force, okay negligible body force.
So, for a 2 dimensional situations you can
write down these equations u momentum equation
and v momentum equation or x momentum equation
and y momentum equation. So, for 2D, 2 dimensional
situation you can write x momentum equation
as Del u by Del t plus Del (uu) by Del x plus
Del u v by Del y is equal to minus 1 by rho,
so obviously in the x direction. So, pressure
gradient will be Del p by Del x plus nu Del
2 u by Del x square plus Del 2 u by Del y
square.
Okay similarly, y momentum equation you can
write as Del v by Del t plus Del (uv) by Del
x plus Del (vv) by Del y is equal to minus
1 by rho Del p by Del y is the pressure gradient
in the y direction plus nu Del 2 v by Del
x square plus Del 2 v by Del y square okay.
So, you can see the second term and third
term we have written the velocities inside
the derivative. So, these form are known as
conservative form of the governing equations,
conservative form, okay.
So, now let us write these equations for steady
2 dimensional flow in non-conservative form.
So, if you do so, then if it is steady then
this term and this term you can drop down
and these you can see how you can write.
So, if you consider to 2D steady flow okay
and in non-conservative form we are writing,
in non-conservative form we want to write,
conservative form. So, if you write in this
form, so, you will actually invoke the continuity
equation and you know this for 2 dimensional
situation Del u by Del x plus Del v by Del
y is equal to 0 is continuity equation and
you will get Del u by Del x (okay) plus u
Del u by Del x then you will get u Del u by
Del y plus, 1 minute.
So u Del sorry Del v by Del y plus v Del u
by Del y is equal to minus 1 by rho Del p
by Del x plus nu Del 2 u by Del x square plus
Del 2 u by Del y square okay. So, if you do
that, so, you can see this 2 terms okay if
you take u common then it will be Del u by
Del x plus Del v by Del y. So, now you can
see that it is a continuity equation. So,
for incompressible flows obviously, this will
become 0, okay. So, this will become 0.
So, these 2 terms will become 0 combined,
because it is actually continuity equation.
So, you can write the equation as u Del u
by Del x plus v Del u by Del y is equal to
minus 1 by rho Del p by Del x plus nu Del
2 u by Del x square plus Del 2 u by Del y
square. So, this is the x momentum equation
in non-conservative form for 2 dimensional
steady flow.
Similarly, you can write the y momentum equation
Similarly, you can write y momentum equation
as u Del v by Del x plus v Del v by Del y
is equal to minus 1 by rho Del p by Del y
plus nu Del 2 v by Del x square plus Del 2
v by Del y square okay. So, these equations
are written in non-conservative form. Now
let us write the 1 dimensional okay 1D Navier-Stoke
equation without pressure term. Okay because
this is a very simple equation and we consider
it as model equation in our CFD course. So
that we can discretize this equation using
different schemes.
So, it is a model equation 1 dimensional unsteady
Navier-Stokes equation without pressure term.
So, obviously you can write it as, so it is
actually known as Burgers equation. So, it
is a fundamental partial differential equation
from fluid mechanics and it relates to the
1 dimensional Navier-Stokes equation for incompressible
flow with the pressure term removed.
So, for a given u which is function of one
space coordinate x and time and kinematic
viscosity, kinematic viscosity nu you can
write this equation viscous Burgers equation,
viscous Burgers equation as Del 
u by Del t plus u Del u by Del x is equal
to nu Del 2 u by Del x square okay. So, you
can see here you have the temporal term Del
u by Del t okay you have the convective term
as well as you have the diffusion term okay.
So, it represents a model equation, where
you have all the terms involved and it is
simple equation where pressure term is removed.
And similarly, you can write in Burgers equation
just dropping the convective term sorry you
can write the inviscid Burgers equation dropping
the diffusion term. So, this is your convective
term and this is your diffusive term.
Now, you can write inviscid Burgers equation
dropping a diffusive term, so, you can write
simple Del u by Del t plus u Del u by Del
x is equal to zero okay. So, this is kind
of a wave equation 1 dimensional wave equation
with a varying speed u. So, this u is actually
wave speed which is varying, so obviously
this is known as nonlinear model equation,
nonlinear model equation.
Now, if you consider u as a constant speed
some way speed c then the equation you can
write as Del u by Del t plus c Del u by Del
x is equal to zero, if c is constant okay,
wave speed then this equation is known as
first order wave equation okay and you can
see this is linear because c is constant,
so it is a linear equation.
And now, we can write Del u by Del t is equal
to minus c Del u by Del x. If you take the
time derivative, in both sides, then you can
write Del 2 u by Del t square is equal to
minus c, c is constant. So you can take it
outside, you change the variable so you can
write Del of Del x of Del u by Del t because
we are taking the time derivative.
Now, you can substitute Del u by Del t with
this equation, so you can write c Del of Del
x minus c Del u by Del x. So, you can see
now it will be minus-minus plus and c square
Del 2 u by Del x square. So, you got this
equation Del 2 u by Del t square is equal
to c is the speed c square Del 2 u by Del
x square and this is known as second order
wave equation, second order wave equation.
So, now, if you consider the full Navier-Stokes
equations and if it is very low Reynolds number
flow, Reynolds number, what is the Reynolds
number? Reynolds number is the ratio of inertia
force by viscous force. If the Reynolds number
is very low, then your inertia force is very
low, so viscous force is dominant.
In that case, you can drop the convective
term from the Navier-Stokes equation and flow
is known as Creeping flow. So, this equation
is known as Stokes equations, Stokes equation.
So Stokes equation, so here Reynolds number,
okay Reynolds number is very small, okay Re
is equal to rho some characteristic velocity
u, characteristic length L divided by viscosity
mu and rho is the density.
So if Reynolds number is less than 1 then
inertia term you can drop because viscous
force will dominate. Okay so inertia, inertia
terms may be dropped from Navier-Stokes equation,
from Navier-Stokes equation. So, this is a
Creeping flow approximation, Reynolds number
less than 1, so this is your Creeping flow
approximation. So, in that case you can write
the governing equations as obviously the continuity
equation will be as it is and momentum equation
will be just Del rho u by Del t.
So, inertia term will be not be there, you
just write the pressure gradient term plus
the viscous term and obviously you can have
the body force term. So, now if you have a
high Reynolds number flow, okay so in high
Reynolds number flow inertia force will be
dominating. So, in that case you can drop
the viscous term and that equation is known
as Euler equation.
So, will write now Euler equation 
where you have a high Reynolds number flow,
so Reynolds number is very high okay. So,
in that case the viscous terms can be or so,
Euler equation we can write for high Reynolds
number flow, very high Reynolds number flow,
Reynolds number flow. So, in that case you
can drop the viscous term, so the viscous
term may be dropped from the Navier-Stokes
equation. So we can write the continuity equation
will be as it is, only the momentum equation
you can drop the viscous term.
So, you can write as Del rho u by Del t plus
divergence of rho u-u is equal to minus grad
p. So, there will be no viscous term just
rho b. Now, if you consider a fluid flow,
over a flat plate then you can have the boundary
layer flow. So, already you have studied the
boundary layer equations, so will write down
the boundary layer equation.
So, obviously boundary layer flow you can
see that due to the no slip condition because
velocities of those fluid particles sitting
on the solid wall will be zero. So, from zero
to the free stream velocity there will be
velocity gradient and you can distinct two
different regions, one is viscous region okay
where velocity variation will be there and
you can have the inviscid region outside these
boundary layer, edge of the boundary layer
that is known as inviscid region.
So, with these certain assumptions, you can
write the Navier-Stokes equation in a droppings
you can write the Navier-Stokes equation dropping
some term. So, that is known as boundary layer
equations and the assumptions for writing
this boundary layer equation is that there
is no flow separation and the boundary layer
thickness is much-much smaller than the characteristic
length and if it is a high Reynolds number
flow.
So, for that you can write the boundary layer
equation and you can write for 2D steady incompressible
flow. So, if you consider a flat plate you
have free stream velocity at u infinity. So,
when it will flow over this flat plate there
will be one region which is known. So, there
will be a distinct region where you will have
viscous region and another is inviscid region.
So, in inviscid region your velocity will
be at free stream velocity. So, whatever free
stream velocity you have that will be your
free stream velocity and inside this viscous
region there will be change in the velocity
from zero to free stream velocity okay.
So, this is known as the edge of boundary
layer and the distant at any location x, if
it is x the distance of this edge of boundary
layer is known as Delta which is your boundary
layer thickness, boundary layer thickness.
So with Delta and L is the length of the plate
then Delta by L is much-much smaller than
1 and it is valid when you have a Reynolds
number greater than 100 okay.
So for that you can write the equation, continuity
equation Del u by Del x plus Del v by Del
y is equal to zero and you can have the boundary
layer equation Del 2 u by Del x plus v Del
u by Del y is equal to free stream velocity
u infinity du infinity by dx that you can
write the pressure gradient in terms of the
free stream velocity plus nu Del 2 u by Del
y square, so here from the viscous term Del
2 u by Del x square is dot because that is
much-much smaller than the Del 2 u by Del
y square.
So, you can have this Del 2 u by Del y square
and for flow over flat plate as u infinity
is constant. So, this will be zero for flow
over flat plate. Okay so, you can have the
equation as u Del u by Del x plus v Del u
by Del y is equal to nu Del 2 u by Del y square
and you have Del p by Del y is zero. These
are the equations of fluid flow. So, now,
another transport equation will derive from
the Navier-Stokes equation.
So, if you have 2 dimensional unsteady Navier-Stokes
equation, then we can write the, derive the
vorticity transport equation, vorticity transport
equation okay. So, in vorticity transport
equation, how can you derive? So, you have
the momentum equations, so you have momentum
equations Del u by Del t, so 2 dimensional
flow okay.
So, we are writing Del u by Del t, u Del u
by Del x conservative form we are writing
u Del u by Del x plus v Del u by Del y is
equal to minus 1 by rho Del p by Del x plus
nu Del 2 u by Del x square plus Del 2 u by
Del y. So, this is the x momentum equation.
Now, if you write the y momentum equation
you will get Del v by Del t plus u Del v by
Del x plus v Del v by Del y is equal to minus
1 by rho Del p by Del y plus nu Del 2 v by
Del x square plus Del 2 v by Del y square.
So you can see that this is your x momentum
equation and this is your y momentum equation
okay. So, now, if you do this mathematical
algebra Del of Del y of x momentum equation
minus Del of Del x of y momentum equation
then you can write down this equation as Del
rho sorry. So, you can write this equation
with the vorticity if you define as Del v
by Del x minus Del u by Del y. So, vorticity
we are defining omega sometime in book you
will get this as a angular velocity, but in
this case we are defining as the vorticity.
Okay so, vorticity is equal to 2 into angular
velocity. So, vorticity is Del v by Del x
minus Del u by Del y, if you do this mathematical
algebra, you can write this equation as Del
omega by Del t Del omega by Del t plus u Del
omega by Del x plus v Del omega by Del y and
what you are doing actually, you are actually
canceling this pressure term okay. So, doing
this mathematical algebra you can see Del
of Del y so it will be Del 2 p by Del x Del
y and this term will be De l2 p by Del x Del
y, so it will be canceled.
So, you are actually canceling this pressure
term and you will have mu Del 2 omega by Del
x square plus Del 2 omega by Del y square
okay and this equation is known as Vorticity
transport equation. So, this equation along
with the stream function equation if you solve
then you can have the, you can solve the fluid
flow equation.
So stream function equation now you define
the stream function. So, u is defined as Del
psi by Del y and v as minus Del psi by Del
x. So, you have the omega which is your Del
v by Del x minus Del u by Del y. So, you can
see that if you do the Del v by Del x it will
be minus Del 2 psi by Del x square and minus
Del u by Del y it will be minus Del 2 psi
by Del y square, then you can solve Del 2
psi by Del x square plus Del 2 psi by Del
y square is equal to minus omega. So, this
is your stream function equation.
So, if you solve this equation you can get
the value of psi and if you solve the psi
you can find the velocity u and v. Once you
solve the u for the u v, you can solve this
vorticity transport equation because u v are
known. So, in that way you can also solve
the fluid flow equations.
Now, let us consider the energy equation,
will now have the conservation of energy and
will write down the energy equation in terms
of temperature.
So, now we are writing energy equation, conservation
of energy 
and we are going to write this equation in
terms of temperature. So, it will be Del of
Del t rho CpT plus divergence of rho Cp u
T is equal to divergence of K grad T plus
q triple prime plus mu phi okay.
So, here k is the thermal conductivity of
the fluid thermal conductivity 
and q is the heat generation per unit volume,
heat generation per unit volume and this is
the viscous dissipation term, viscous dissipation
term okay due to the share there will be convection
of mechanical energy to intermolecular energy
which in turn will rise the temperature and
phi is the viscous dissipation function, so,
that I am not going to write.
So, if you can neglect this viscous dissipation
term and write this equation for a constant
properties, then you can write constant properties,
constant properties and neglecting viscous
dissipation term. We can write this equation
as Del T by Del t constant properties, so
rho Cp you can take it outside. So, Del T
by Del t plus divergence u T is equal to alpha
is, alpha is the thermal diffusivity grad
square T plus q triple prime by rho Cp okay.
So, you see alpha is nothing but k by rho
Cp which is known as thermal diffusivity.
So it is constant properties we have considered,
so rho Cp you have taken outside and we have
divided. So, alpha is equal to k by rho Cp
thermal diffusivity. So, in 2 dimensional
situation you can write this equation as Del
T by Del t plus u Del T by Del x plus v Del
T by Del y is equal to alpha Del 2 T by Del
x square plus Del 2 T by Del y square plus
q triple prime rho Cp.
Okay so this is the energy equation in 2 dimensions
neglecting the viscous dissipation term. So,
now if you consider that velocity is zero,
so if you consider this heat conduction in
solid, then you can make this velocity as
zero and you can write this energy equation
as heat conduction equation in a solid.
So now will write this equation as now will
write heat conduction equation, heat conduction
equation in solid okay, so we are considering
solid, so obviously velocity you can make
zero, okay you can make zero. So your convective
term will become zero and you will have the
equation as Del of Del t rho Cp T. So, this
your convective is zero because u is zero
is equal to divergence of k grad T plus q
triple prime okay.
So now, if you write for constant properties
then you can write Del T by Del t is equal
to alpha grad square T plus q triple prime
by rho Cp. So, alpha is your thermal diffusivity
and this equation is known as Fourier Biot
number, a Biot equation, Fourier Biot equation
okay. So, q triple prime is the heat generation
per unit volume in the solid okay.
Now, if you neglect this heat generation term
then you can write this equation as, so neglecting
heat generation okay you can write this equation
as Del T by Del t is equal to alpha grad square
T and this equation is known as diffusion
equation, heat diffusion equation okay. Now
if you consider steady heat conduction equation
then you can drop the temporal term, then
you can write this equation steady with heat
generation, steady with heat generation.
So, this equation you consider, so if it is
a steady then the first term you can make
zero then you will get alpha grad square T
plus q triple prime by rho Cp is equal to
zero or you can write grad square T plus q
triple prime by k is equal to zero and this
equation is known as Poisson equation.
Now, if you have steady and negligible heat
generation, so steady and without heat generation,
so this is a simple one because you have no
heat regeneration and you have steady state
so Del T by Del t is equal to zero. So, you
will have grad square T is equal to zero which
is known as Laplace equation okay.
And in 2 dimension if you want to write these
equations, then you can write Del T by Del
t is equal to alpha Del 2 T by Del x square
plus Del 2 T by Del y square neglecting the
heat generation term and obviously in 1D you
can write Del T by Del t is equal to alpha
Del 2 T by Del x square. Similarly, if you
have species transport in the with the fluid
flow then you can write the species transport
equation okay and we can write in terms of
the mass fraction of the species.
So, will write species transport equation,
species transport equation, so now will write
in terms of mass fraction which will denote
with Yi, i is the n species i, so that is
your mass fraction of species i and rho is
the density of the mixture, density of the
mixture. And will consider Wi which is your
reaction rate or formation rate of species
i okay or formation rate. So, this is known
from the chemical kinetics, known from chemical
kinetics.
So, if you write the governing equation then
you can write it as Del of rho Yi divided
by Del t is equal to divergence of rho u,
Yi is equal to the diffusion term divergence
of rho Di, n is the diffusion coefficient
Di to n, so i to n species and grad Yi plus
omega i. So, this is your diffusivity, okay.
So, you we can see that all these equations
Navier-Stokes equations, the energy equation,
the vorticity transport equation and the species
transport equation you can write this equation
in a general form, okay convective diffusive
equation.
So will write the general transport equation,
general transport equation okay, so will write
for any variable phi, phi is any variable
okay it may be u, it may be v or it may be
T okay. So, all these transport equations
can be written in the following form for any
general variable phi. So, this you can write
as Del rho phi by Del t which you have the
temporal term plus you have convective term,
divergence of rho u phi.
Okay so phi is convected by the velocity u
is equal to divergence of gamma grad phi where
gamma is your diffusion coefficient, so this
is your diffusion terms, so gamma is your
diffusion coefficient, diffusion coefficient
and plus some source term S phi. Okay it may
be pressure gradient term for the Navier-Stokes
equations okay or heat generation term for
the energy equations.
So, these all these equations whatever we
have written in todays class okay we can write
in this form. So here you can see you have
a temporal term, convective term and diffusive
term and with some source term is phi okay.
So, you can see that if you write phi is equal
to 1 okay and gamma and this is your S phi,
then what your equation you are going to get?
Okay so you see, so if you put phi is equal
to 1, it is incompressible force over density
obviously, is constant. So, this term will
become zero.
And if you put these terms, so phi is equal
to 1 and density is constant, so gamma dot
rho u is equal to now, if you gamma if you
put zero then this diffusion term will become
zero and S phi if you put zero then you are
going to get the equation divergence of u
is equal to zero, so that means it is your
continuity equation okay.
So, from this general transport equation you
can see if you put phi is equal to 1, gamma
is equal to zero and S phi is equal to zero
you are going to get continuity education.
Similarly, if you put phi is equal to u and
gamma is equal to mu and Del phi S phi as
Del p by Del x, S phi as minus Del p by Del
x, then you are going to get x momentum equation
okay without the body force term or if you
write body force term then you can add it
here.
Or for velocity v if you put gamma as mu,
S phi as minus Del p by Del y, you are going
to get y momentum equation. Similarly, if
you write for w and gamma as mu and S phi
as minus Del p by Del z then you are going
to get z momentum equation and similarly for
vorticity transport equation you can see.
So, if you write omega phi is equal to omega
and gamma is equal to nu, okay gamma is equal
to Nu sorry gamma is equal to mu because rho
is there in the left hand side and if it is
source stem is zero then you will get vorticity
transport equation, vorticity transport equation.
And for energy equation if you put phi is
equal to t and gamma as k by Cp and S phi
as q triple prime by Cp then you will get
energy equation.
And if you put mass fraction phi is equal
to mass fraction Yi and gamma as rho Di and
this is as omega i then you will get mass
fraction equation or species transport equation,
species transport equation. So, why I have
written this equation? Because it is a general
transport equation which represent any of
these equations okay putting the values of
phi or diffusion coefficient gamma or the
source term S phi you can write any equations
from this general transport equation.
So, when we discretize these equations, okay
first will discretize in general transport
equation, so that you have a convective term,
temporal term and the diffusion term and if
you can discretize this equation using some
discretization method may be finite difference
method or finite volume method then you can
innovate discretize the other equations like
Navier-Stokes equations or the energy transport
equation. Only thing is that separately you
have to discretize the pressure term.
So, for that reason we have written this equation
and when we will use finite volume method
specifically that time will consider this
general transport equation and will discretize.
So, in todays class we have started with writing
the basic fluid flow equations, conserving
the mass, conserving the momentum, we have
written the continuity equation and the Navier-Stokes
equations and from this general Navier-Stokes
equation we have written the Burgers equation
which is your 1 dimensional Navier-Stokes
equation without the pressure term.
So, we have written for viscous Burgers equation
and inviscid Burger equation dropping the
viscous term and from there we have written
the first order wave equation and second order
wave equation for a constant speed c, constant
wave speed c. Then, we have also written the
vorticity transport equation and from there
we have written stream function equation,
so that these vorticity transport equation
and the stream function equation combined
if you solve you will be able to solve a fluid
flow problem.
Then we consider the energy equation, so we
from this energy equation in general we have
written for a, with the fluid flow, then putting
the velocity as zero we have written the heat
conduction equation in a solid and after that
we have written the species transport equation
in terms of the mass fraction Yi.
And at last we have written all these equations,
transport equations in a general transport
equation for any variable, general variable
phi and where you have the temporal term,
convective term then diffusive term as well
as you have a source term S phi. Thank you.
