So we will get started with the 23rd lecture
of the course and what we saw in the last
two lectures is some problems on stability
of a system okay. The first problem that we
discussed on stability was that of a well-mixed
system so there was no spatial gradients and
we only had time dependency okay. So time
dependency has to be retained because you
are talking about how things behave as you
progress in time.
So the governing equations of the system were
couple of ordinary differential equations,
which are actually linked with each other.
So they are coupled ordinary differential
equations. Then we did the problem on the
reaction diffusion system and the reaction
diffusion system it was the partial differential
equation. So that was the level of complexity
we added from an ODE we went to a partial
differential equation.
But then we simplified things a little bit
by saying that we will consider only one variable
and there is only concentration okay just
to illustrate the ideas. So today now what
we will do is we will actually look at a fluid
flow problem and in the fluid flow problem,
it is going to have more than one variable
the different velocity components okay and
the pressure. There is also going to be temperature,
which is going to come from the energy balance.
But then we will again keep life a little
bit simple by considering only a single phase.
So we were looking only at one phase and then
after we finish this problem then we will
get to doing actually multiphase flow problems
where we have to worry about the tracking
the interface okay. So that is just to tell
you the gradual evaluation in the complexity
of the problems that we are trying to solve.
So today what we are going to look at is this
problem of natural convection 
and this is also called the Rayleigh-Benard
problem after the scientist who actually analyzed
this particular system and we are going to
follow that procedure and try to get some
insight into this problem of natural convection
okay and we are looking at single phase as
far as the liquid is concerned okay.
Single phase but now the system will be governed
by coupled partial differential equations
okay. So because there is only one liquid
we do not worry about things like and of course
this is going to be bounded between solids.
We do not worry about things like the kinematic
boundary condition, the normal stress boundary
condition of the interface. We do not have
to worry about interface deformation.
After this we will resolve problems where
we have to worry about those, also will include
those effects in the model okay. So now what
is this problem of natural convection?
We will keep things simple like we always
do. Look at 2 flat plates, this is the y direction
and this is the x direction and this is the
z direction okay. Now we have in this coordinate
system 2 flat plates, one is at y=0 and the
other is at y=H. These flat plates are extending
to infinity in the x direction and in the
z direction okay. So we have rectangular plates
extending to infinity in the x and z directions.
The spacing 
between the plates is H. Now we want to talk
about this problem of natural convection okay.
So as opposed to so convection means we are
going to have movement and natural convection
as you all know is going to cause by density
differences okay. So if you have a layer of
liquid or a fluid at the bottom, which is
having a lower density then the layer at the
top, then it will have a tendency to rise
up.
Because the density is lower because of the
buoyancy it has a tendency to go up and when
it goes up the fluid which is at the top will
have a tendency to come down, which is heavier
and so you can have motion, it has circulation
set in okay. Normally, the natural convection
that we talk about is caused by density differences,
which are going to be induced by temperature
gradients.
So if there is a layer of fluid when there
is a temperature gradient, the hot fluid at
the bottom which is at a higher temperature
will have a lower density and this guy has
a tendency to rise up okay. So what we are
going to do is we are going to solve this
problem subject to a temperature gradient
okay and I am going to call the temperature
here T0 because corresponding to y=0.
And I am going to call the temperature here
TH, so basically what I am saying is that
there are 2 plates. The lower plate is at
a temperature T0, the top plate is at the
temperature TH okay. So T0 and TH are the
temperatures of the 2 plates as the first
thing and natural convection arises because
of density gradients okay. These density gradients
can be induced by temperature gradients.
So you all know that density is the function
of temperature okay and therefore we need
to basically include the effect of this density
dependent C on temperature and to be able
to proceed okay.
Now if you have a configuration of this kind,
let us consider first the case where case
1 T0<TH, which means the lower plate is colder
than the upper plate okay. So what does this
mean? You have less dense fluid at the top,
more dense fluid at the bottom okay. So that
is the configuration where you will have stability
always in the sense that there is nothing
which is going to cause this liquid to go
up okay. It is a stable configuration.
So here the less dense fluid is on top of
a more dense layer okay and this is a stable
configuration and we do not expect to see
any convection. What about the reverse case?
The reverse case is when T0 is>TH that is
the lower plate is hotter than the upper late
okay. When T0 is>TH the less dense fluid is
below the 
more dense fluid. Buoyancy forces this fluid
to rise up okay.
So if you just look at the buoyancy effect,
the less dense fluid has a tendency to rise
up. So what is it that is going to prevent
this motion? What is it that is going to prevent
this less dense fluid from going up? Basically,
the viscous force. Viscosity is like a friction
is going to prevent this liquid from going
up okay. So basically what I am saying is
viscosity acts as a friction and opposes this
tendency for the liquid to go up.
I am just trying to tell you that there are
2 forces that you have to look at one is the
buoyancy force which is trying to push this
guy up and the viscous force which is trying
to prevent it from moving up. So what does
that mean? It means that when the temperature
difference here T0-TH is sufficiently small
okay, the buoyancy force is going to be less
okay in comparison to the viscous force.
Viscous force of course is going to be decided
by the viscosity times the velocity gradient
okay. So that is going to be dominating the
buoyancy force. The viscous force will dominate
the buoyancy force when T0-TH is sufficiently
low, but what is going to happen if we keep
increasing the temperature of the bottom plate,
there is going to be a time, which comes or
there is going to be a value of this lower
plate temperature, which comes when the buoyancy
force is going to dominate over the viscous
force.
And then liquid is going to start moving okay.
So again we have a situation where there is
a critical parameter and this critical parameter
experimentally you can think of as the temperature
of the lower plate for a value of this parameter,
the lower plate temperature>a sudden value
I expect that to be natural convection.
If the temperature is lower than that critical
value there is going to be no natural convection
because viscosity is basically going to prevent
the motion. All I am trying to tell you is
that just because you have a small temperature
gradient you do not have to expect a natural
convection to take place okay. It is not that
any small delta T is going to give you convection.
You need to have significant amount of delta
T.
And what we want to do is we want to see if
we can determine what this critical value
of delta T is by posing this problem as a
stability problem okay and that is basically
what our strategy is. Our objective is to
identify this delta T and get his. Yeah “Professor
- student conversation starts.” Delta T
would depend upon on whole bunch of things
and that is what the analysis will tell us.
The analysis will tell us, it will depend
upon the properties of the fluid, it will
depend upon the gap between the plates and
what are these different things on which this
is going to depend upon the analysis is going
to tell us yes but it will depend upon the
fluid, it will depend upon how strong the
density variation is with temperature okay.
It will depend upon the thermal conductivity
of the fluid; it will depend upon many things
okay. “Professor - student conversation
ends.”
So here what I am saying is if T0-TH is sufficiently
low 
then F viscous is more than F buoyancy and
the liquid is static. If T0-TH is>a critical
value, if buoyancy will be>F viscous and we
expect to see convection okay. So the question
is how do you go about determining this critical
temperature or temperature difference? And
like he says it is going to depend upon the
fluid properties, it is going to depend upon
space etc.
So let me just call this delta T critical,
which is T0=TH or delta T is T0-TH okay has
a critical value above which we have convection
okay. So what we want to do is find out what
this critical value is. So we want to find
delta T critical and this is done by posing
the problem as a stability problem. So we
want to ask the question in the context of
the stability framework that we were introduced
earlier okay.
The another way to look at this whole thing
is supposing there is very small delta T,
then what it means is the mechanism of heat
transfer that we are going to have is going
to be that of only conduction, that is conduction
alone is enough for you to transfer the heat
from the lower plate to the upper plate. If
the delta T becomes high then conduction alone
is not enough for you to do the heat transfer.
And so in order to facilitate the heat transfer
in addition to conduction you have convection,
which is necessary for you to transfer the
heat okay. There is one way to look at it
also okay, so that is what I am saying is
for low delta T, conduction alone can transfer
the heat. For high delta T, conduction and
convection are required for the heat transfer
from the lower plate to the upper plate okay.
Now clearly what 
we need to do is we need to write down the
how do we go about solving this problem of
stability? We need to write out the governing
equations. So what are the governing equations
that are required? One is the continuity equation
and the momentum equations in the x and y
direction or rather yeah in the x and y direction.
We are going to assume its infinity in the
z direction.
And we also need the energy balance equation
because we need to worry about how the temperature
is changing okay. We need to include the energy
balance equation also. So the governing equations
are and why do I need both x and y direction?
Because when the hot liquid here has a tendency
to go up okay, this guy the cold liquid from
here is going to have tendency to come down
so it is going to get something like a circular
vortex okay and although I am extending this
to infinity what I expect to see is I am expecting
to see a periodically repeating pattern of
these kind of cylindrical rolls, so basically
this means that I have this kind of a situation
all of the same size okay.
And this is extending to infinity in the z
direction. The point I am trying to make here
is the system is extending to infinity in
both x and z direction. To keep my life simple
what I am going to do is I am going to exploit
the fact that the thing is extended to infinity
in the z direction and look for solutions,
which are independent of z okay. So we are
just saying the things are independent of
z just to keep it mathematically tractable.
In the x direction also it is extending to
infinity but I am not going to use the argument
that it is going to be spatially uniform in
the x direction, I am going to look for a
solution just periodic in the x direction
okay and the reason why I am doing this is
because of the temperature gradient I am going
to say that what do I expect physically? I
am expecting that this guy goes up, this hot
fluid comes down.
And this is going to occur at some kind of
a regular periodic or spatially periodic interval
okay and that is one of things which we want
to find out. How does a system behave when
your delta T critical is exceeded okay? When
you say convection is going to take place
but how exactly is the liquid going to move,
just like we saw yesterday in the reaction
diffusion problem the velocity was 0.
But when it became unstable you have a solution
which is like a parabolic thing with a maximum
at the center okay. So now beyond a delta
T critical what exactly is going to be the
pattern? So I have already given you the answer
that one possible pattern is this kind of
a periodic cylindrical roll okay. So this
is called a cylindrical roll clearly because
this is circular, x as infinity to the cylinder
and that was the cylindrical roll okay.
So this is one possible pattern 
and one of things we really want to find out
is things like what is the spacing etc, etc.
Yeah, it is fine. “Professor - student conversation
starts.” No, the actual case in the sense
that the actual case is when you are doing
experiment. When you are doing an experiment
you would have walls at these 2 ends okay
and then you need to actually how to worry
about the boundary conditions.
So supposing you have a very long length in
the x direction okay. If you forget the end
effects where the boundary condition is going
to prevail and if you focus somewhere in the
center then this is one possible pattern that
you can get okay. Now as we go along I will
talk about that other pattern is also possible.
This is just for easy visualization, you can
have other patterns like hexagons etc possible
when you consider 3-dimensional thing.
When you have variations in the x, y and z
direction but then just to keep math simple
right now we are just looking at it this way,
but then experimentally and then as we go
along I will explain to you when what decides
what pattern and all that okay. So different
patterns are possible. “Professor - student
conversation ends.”
Now I am beginning to read the Subham’s
mind as dangerous okay. So let us write the
governing equations.
Equations are the continuity equation, which
is divergence of u=0 okay and says I am neglecting
things in the z direction, I do not write
the momentum equation in the z direction.
I am just going to write the equation in the
x and y direction okay. Momentum equation
in the x and y directions what is that? This
is in the x direction right. So this is x
direction and the gravity is not in the x
direction and then I have this.
Gravity is there I think downwards, so it
is not in the y direction and so just give
me a minute. Yeah “Professor - student conversation
starts.” Yeah, so the question is what I
have written is wrong and this equation is
valid only for an incompressible equation
when you say that the density is constant,
it does not change with x, y time okay. So
his objection is I should use the full-fledged
form of the continuity equation, which is
there for a compressible fluid okay.
I think that is a very valid objection. In
fact, I was expecting that objection.
So the density variation has to be included
clearly okay. So the equation of continuity
is that is the general equation okay and so
now the question is, ideally I have to include
the density variation like this, use this
form of the equation of continuity. In fact,
I need to go back to the momentum equation
also and make some changes because I have
actually pulled out the density from my derivative
term and I need to modify that term as well
okay. “Professor - student conversation
ends.”
So what is it that I am doing here? So now
the important thing whenever you are trying
to do an analysis is to develop a model, which
is as simple as possible okay but which can
capture the essential physics of the process
okay. So what you are saying is correct, I
need to use this particular form of the equation
of continuity.
I have actually used the fact that density
is constant and I have actually simplified
when I wrote the momentum equations and stuff
like that. So that has been an approximation
which has been made okay. So idea whenever
you are solving any problem is to keep the
model as simple as possible so that you can
basically solve it mathematically and try
to get some physical insight okay.
That is what we are trying to do here. If
you want to get the most accurate solution
and to the 8th decimal place of the 9th decimal
place, then you need to sit here and you know
put the density inside your continuity equation
and solve the full-fledged model without making
any assumptions or any approximations. So
the question now is what is the simplest thing
we can do which will capture the physics which
will retain the physics and give us insight
into the problem that we are studying okay?
So clearly density is a function of temperature,
temperature is changing with x and y because
of the density the temperature gradient. So
what we want to do is we want to keep the
model simple and so that I can possibly solve
it analytically and get physical insight okay
and to answer his question how does this critical
delta T depend upon thermal diffusivity, viscosity
etc, etc. Otherwise what are you going to
do?
You are going to have bunch of equations,
you will go to the computer, write a finite
difference code and keep running simulations
and say oh now it is not convecting, now it
is convecting and you will have no clue as
what is going on okay. So we want to basically
get out of that situation where we are just
going blindly to the computer and doing some
calculations.
So we have made an approximation here like
you have just pointed out and this approximation
is called the Boussinesq approximation okay.
So let me just write down a few things.
We want a mathematical model, which is simple
but can capture the essential and important
physics. This gives us physical insight into
what is going on okay. Otherwise if we do
not simplify 
then I will have a bunch of computational
results, which we can make head or tail out
of okay.
We will not be able to interpret computational
results. We will have a whole bunch of results
coming out of that calculations and then you
say if I change my density I got this and
when I change this I got that but then at
the end of the day it will be lost okay but
then also you should be careful that even
simplify things too much then nothing is happening
okay. So I mean that is the important thing.
But do not simplify too much that you go not
get any convection, no matter how much you
are heating it okay. That is although you
should be careful about okay, but do not simplify
too much and that I think is the key thing
do not simplify too much to lose essential
physics okay. So that is the game you have
to do and what I have done now is actually
and the way I have written these equations
is we have done what is called the Boussinesq
approximation.
So what is this Boussinesq approximation?
The Boussinesq approximation is the thing
where we are making this simplification okay.
So now we have to retain the density dependency
on temperature correct because if you do not
have the density dependency on temperature
there is no way we are going to have any convection.
So this has to be included, do you have what
to include the density dependency on temperature
wherever density is occurring which means
I have to include it there, I have to include
it may be here and maybe modify this equation
suitably okay or is it possible for me to
include the density dependency only on one
term, which is going to be crucial and treat
in all the other terms as if density is being
constant okay.
Because if it is a liquid you really do not
expect a very, very significant change in
the density. If it is a gas, yes there will
be significant changes in density as you change
temperature. So the one term in which you
want to actually retain the density dependency
on temperature you guys want to take half
a minute and identify which term you want
to retain this thing in. The way I have written
it density is occurring in here, here, there
and in the equation of continuity right.
So which term do you think we need to retain
the density dependency on temperature? In
the gravitational term because that is the
one which is going to give you the buoyancy
force okay. The rho g term is the one so if
I have the density dependence on temperature
retain in the rho g term and for all practical
purposes everywhere else I am going to assume
density is constant okay.
And basically that is the approximation this
Boussinesq approximation that I am talking
about which is basically telling you that
everywhere else I am going to treat this as
if is the constant rho 0, but in rho g because
that has to be retained for me to get my because
eventually it is the density gradient in the
direction of the gravity, which is actually
causing the motion okay that has to be retained.
So basically what this means is here we keep
rho as a function of temperature only in the
rho g term in the y equation okay. This is
necessary to get natural convection at all
other places 
this we treat density as being equal to a
constant, which is equal to rho 0 okay. So
what I am going to do and that is the reason
my equation of continuity is written that
way as simplest I did okay.
So we use the divergence of u=0 for 
an incompressible fluid and I am going to
quickly pull a fast one here put a rho 0 here
and a rho 0 there. So just put rho 0 here
because at these places I do not want to include
my density dependency or temperature but that
term over there rho g term I keep rho as a
function of temperature okay. So only in the
rho g term, I retain the temperature dependency.
And we are going to keep life simple which
is assumed a very linear relationship for
the temperature, density dependency or temperature
rho0 times 1+beta times T=T0 okay. So rho
0 is the density at T=T0 and everywhere else
it is varying linearly. So you just keep this
linear dependency and what does this mean?
I need to have beta as positive or negative.
Density has to decrease with temperature so
beta is negative.
As temperature increases density has to decrease
okay. So now what I have done therefore is
a simplified model okay, that simplification
is called the Boussinesq approximation and
again that is the whole motivation for any
modeling any exercise you do is see the idea
is if you have to even solve the full-fledged
problem at the end of the day this critical
delta T that you are getting with this Boussinesq
approximation maybe let us say 80 degree Celsius
okay.
With all this complications may be 81 degree
Celsius and that you would not be able to
do with your computations, you would not be
able to get at that value okay. So I mean
as an engineer for a 1 degree you are willing
to compromise if you can you know do a simplified
model and get some insight. So that is the
motivation okay. Of course, if somebody is
teaching a computational fluid dynamics course,
he may possibly argue the other way that is
different.
So what we want to do now is we want to written
down the modeling equations, we want to find
the steady state okay and what is the steady
state that you have? The steady state you
have is going to be one which is stationary
where is the liquid is not moving okay.
So let me go slight sneak into this corner
and write this thing as rho 0*1+beta times
T-T0 that is my rho. That is the only place
where I am keeping my temperature dependency.
So I told you that we are going to look at
this as the stability problem.
And in order to find the stability problem
we need a steady state to find the stability
right and what is the steady state? What is
the one possible steady state? The one possible
steady state is the one where the liquid is
not moving, the u is 0, v is 0 okay. So a
steady state is the one where the liquid is
stationary that means u=0, v=0. There is no
motion right. I mean on that view clearly
we expect that that is going to true for low
delta T.
And in fact you will see that that is going
to be true for normally what the delta T is.
So u=0, v=0 is a steady state for all values
of the parameter but then it is stable for
low delta T, it is unstable for last delta
T, which is a reason you see the convection
okay. So when we do this you will understand
it better but what I want to do now is find
out the corresponding variation in the pressure
and the temperature.
Liquid is not moving fine. So how do I find
the variation of the pressure and temperature?
Go to the momentum equation. Momentum equation
in the x direction tells you dp/dx=0, pressure
is independent of x. This was base state;
this is my steady state okay. So this I should
write as if my steady state uss, vss because
this is a steady state whose stability I am
interested in.
And the idea is when this guy become unstable,
I have my natural convection just like when
I have the u=0 becoming unstable I had the
concentration varying in my reaction diffusion
problem. In my y direction what is the story?
In my y direction, you just put uss=0, vss=0,
you get dp/dy=-rho0 g times 1+beta times Tss-T0
okay, but I do not know what the steady state
profile is for temperature and how do I find
that?
I find that by I never wrote the temperature
equation is it? Oh I need to write the temperature
equation. So in order to find the temperature
profile I need to write the temperature equation
which is the energy equation.
Let me come here 
and here again I have a rho I keep that as
rho 0 okay. So that is your energy equation
in the simplified form, accumulation, convection,
conduction and I keep density constant here
okay but that is not important and steady
state that goes off. So there is no convection
for that particular thing, so there is no
motion and again now it is infinite in the
x direction okay.
So we have a steady state solution, which
is infinite in the x direction and there is
no variation of the boundary. So you expect
that this will also be 0 and we only have
d square T/dy square=0 okay and this is 0
since infinite in x direction and so the steady
state is going to be given by d squared Tss/dy
squared=0 and you have Tss=T0. I mean I need
to use the boundary conditions okay.
Boundary conditions are at y=0; T is T0, at
y=H; I have T=TH. So you will get a linear
profile. Tss is going to be of the form Ay+B
and we can actually calculate what the temperature
is going to be. Temperature is going to be
linear. So clearly if you have a solid slab
where nothing is moving, your temperature
gradient is linear, but only conduction is
taking place and that is the situation we
have here.
We have only conduction taking place and so
I have a linear temperature profile that is
my base steady state. Once I calculate what
the steady state is I will substitute it here
and I can calculate how my pressure is varying
in the y direction okay. So that is the idea
so what we have done today is just found this
steady state. Now clearly in fact if I have
a little bit more guts, I will actually solve
this problem.
At y=0 I need to get T0, so this B must be
T0 and at y=H I must get TH right. So y=0
I get T0 and y=H I get TH that is my profile.
So that is my linear profile for my steady
state temperature okay and what I do is I
substitute this here.
And I can find Tss, dpss/dy can be found as
–rho0 g times 1+beta times Tss-T0 is TH-T0
times y/H something like this okay. I just
substituted the Tss here. The point I am trying
to make here is that no matter what H is,
no matter what TH is, no matter what T0 is
this is always a steady state okay. So this
steady state where the liquid is not moving
is going to be valid always okay.
But then as we just argued earlier when the
delta T becomes more than a critical value,
this is not going to be something which you
are going to experimentally observe. You will
experimentally observe this only when delta
T is lower than a critical value okay. So
the fact that the guy starts moving in actual
experiment means that this guy has become
unstable so we want to find this delta T critical
by solving the stability of the steady state.
We are going to find out when is this guy
becoming unstable okay and just like we have
got some relationship for diffusion coefficient
the other day, we are going to find relationship
to find out when this guy starts moving and
for that we start with the governing equations,
have the steady state, do the linearization
and solve that okay.
