# Natural Convection: Uniform Wall Temperature

Hello, let us welcome you all in our eight

lecture of convective heat transfer course, in this lecture we will be discussing about

natural conviction, in earlier versions of this course what we have seen that there will

be a flow over the flat plate but in this case we will be seeing there is no flow over

a hot plate rather the flow is being created by virtue of the temperature gradient between

the plate and the free stream. So this type of convection is actually called

natural conviction and in this natural convection we will be actually concentrating in this

lecture on uniform wall temperature, so today’s lecture is based on natural convection uniform

wall temperature, okay. So let me tell you that what things will be covering in this

lecture at first I will be introducing you what is natural convection, how its velocity

and temperature profile looks like, okay. Whenever you are having a hot plate in vertical

orientation, remember for natural convection vertical orientation is utmost important in

the previous lectures you have found out the plate was kept horizontal and there was a

flow which is actually force conviction, in this case the plate will be vertical and natural

conviction will be occurring so I will be introducing you what is the velocity and temperature

profile. That too will be considering the plate is

being kept at constant temperature, okay. As I have told you that there is no concept

of forced velocity so what will be the velocity scale determination of that velocity scale

is very important, so I will be introducing you how velocity scale for natural convection

can be considered especially for constant temperature case, constant temperature of

the wall case, okay. We will be deriving the momentum and energy equation for flow around

a vertical hot plate which is kept at constant temperature, okay. And at the end you will be mentioning sets

of equation and Nusselt number correlations for both high and prandtl number fluid flow

around the vertical hot plate. So let me first introduce you with the situation

what happens in case of natural conviction as I have told you that in case of natural

convection there will be no predominate flow, flow will be rather generated by virtue of

convection so let me show you this is the plate whatever I am considering over here

is kept in the vertical orientation so g is actually perpendicularly g is actually coming

from top to bottom. So in the downward direction g is acting over

here gravity is acting over here and as we are interested in uniform wall temperature

case, so we are keeping that the wall is having at constant temperature TW, okay and let us

consider that the fluid which is there around is at in temperature T infinity and there

is no flow predominant flow in the fluid so a fluid is at rest away from the plate, okay. So free stream velocity is zero, okay. Now

as we are having high temperature TW over here in comparison T infinity which is a free

stream temperature will be finding out there will be some temperature profile because at

wall you are having very high temperature and away from the wall you are having low

temperature, so the temperature profile you will be reducing in such in this fashion.

So here I have shown you the temperature profile, and what we can consider wherever the temperature

has reached to ninety nine percent of the free stream temperature we can call that point

as thickness of the boundary layer, okay like this if we consider along the plate axial

direction that means x bar so we can find out locus of points okay which actually resembles

ninety nine percent of the free stream temperature. So if we join those we will be having the

thermal boundary layers so this actually is the thermal boundary layer, okay. On the other

hand to give this temperature profile we will be finding out there is some flow around the

plate, okay though away from the plate we are having zero velocity okay and at adjacent

to the plate due to your no slip boundary condition we will be having zero velocity. But around the plate you can find out that

to enhance the convection from wall to the freeze team you will be having some velocity

so we are considering that at wall we are having zero velocity and far away from the

wall once again we have regained the zero velocity and in-between we are having a velocity

profile like this, okay. We are not saying that this velocity profile will be symmetric

or it can be you know taking high value very near the wall and then slowly going as towards

the zero velocity, okay. So just like your concept of velocity bound

layer in case of forced convection here also velocity boundary can be constructed but in

this case as there is zero velocity in the free stream team so we have to reach something

around zero, ninety nine percent of the zero velocity so those points if we find out along

the axial direction axial location of the plate then and then join those points will

be finding out the velocity boundary layer. So this inner curve is actually our velocity

boundary layer and outer curve is our temperature or thermal boundary layer, okay. But here

I have shown you only the representative case quickly I will be showing you what happens

in case of you know two different types of extend of fluid high prandtl number and loop

prandtl number of fluids, where you will be finding out there is contest between the velocity

boundary layer and the thermal boundary layer. Sometimes velocity boundary layer it is becoming

dominant and sometime thermal boundary layer becomes dominant okay, so this is one representative

where I have shown velocity boundary layer thickness delta is smaller than thermal boundary

layer thickness delta t but reverse cases also possible, okay. Quickly we will be coming

to those situations by the way let us then quickly try to understand that what are the

governing equations involved with this one. So first we will be introducing from fluid

mechanics what is the continuity equation, okay. We are considering that we are having

incompressible fluid over her around this vertical plate so we are actually as neglecting

the density changes so if you consider that one then continuity equation simplifies to

del u del x plus del u del y is equals to zero okay. Then if you see the momentum equation

let us consider two dimensional plates over here in x and y coordinates x-bar and y-bar

coordinator rather. So we will be finding out that our x momentum

equation it goes like this so this is your inertia term pressure gradient term, viscous

term and subsequently this term will be there due to bouncy okay now as the plate is actually

aligned with the x- axis so the whole term of the buoyancy will be coming in the x momentum

equation there will be no representation of the buoyancy in the y momentum equation. Y momentum equation is very simple in terms

of v bar so this is the inertia pressure gradient in terms of subsequently the viscous terms,

okay. So after this continuity and momentum equation as we are in convective heat transfer

so energy equation will be important so let us see the energy equation, so energy equation

you see in the left hand side we are having as usual convection and right hand side we

are having as usual conduction term. They are linked up with the alpha which is

nothing but your diffusivity, okay thermal diffusivity then as we have done earlier let

us try to non dimensionalize this equations by taking different scales, so first for the

locations x-bar and y-bar will be non dimensionalizing with L which is the characteristic length

scale or you can length of the plate okay and we actually construct x and y as non dimensionalize

parameters, non dimensionalize lengths scale, okay. In plate direction and cross plate direction,

okay. For temperature let us consider theta, now we have mention the temperature of the

plate is TW and free stream temperature is T infinity so with the help of this two we

are non dimensionalize and temperature T so T – T infinity by TW – T infinity this

we are considering as theta, okay. So as a result near the wall theta will become one

and away from the wall theta will become theta will tend towards zero, okay. For velocity we are considering some characteristic

velocity scale U not okay this will be also U not so some characteristic velocity scale

U0 we are considering so U and V these are U- bar/ U not and V equals to V bar by U not

okay, pressure we are characterizing by P-bar by rho u square so u square is having actually

a unit operation so p-bar by u square is actually capital P which is non dimensionalize pressure,

so if you take all this non dimensionalization schemes and try to modify this continuity

momentum and energy equations. Our system of equations will be changing,

so let us try to see from there what this equations will be taking the form but before

that let us see what is this U not because U not is some fictitious quantity what we

have taken okay so to take this U not what we are considering that what is the order

of your inertia term so if you see the x momentum equation the inertia term will be becoming

U not square by L okay. So this is U not into U not and here it is

x so that means L, so U not square by L is the dominant term over there, so what we are

considering U not square by L which is the inertia term order is equals to the last term

that means the buoyancy term over here g beta T – T infinity, okay. So if you consider

that so g beta (T – T infinity) is equals to U not square by L so here actually we have

equated the inertia order along with the buoyancy order if you do so, so you can get the idea

what will be the characteristic velocity U not. So U not will be root g beta L Tw minus T

infinity okay, next let us try to find out that what will be the what will be the coefficient

in our viscous term so if you see the viscous term in the previous equation I will show

you in the viscous term we will be having gamma and then which is kinematic viscosity

from there here we will be getting U not and here you will be getting L square and from

this side we will be getting U not square by L so if you divide the whole equation by

U zero square by L the coefficient in front of the vicious term will become gamma U not

square by L and U not square by L okay. So after reduction you will be getting this

is nothing but gamma by U not L okay, so as we have already found out what is U not we

can straight away put this U not value over here to get what is the coefficient of the

viscous term, so if you see the coefficient after cancellation and putting this value

of U zero it is becoming gamma not square divided by g beta L cube) TW – T infinity.

Now this by g beta L cube (TW – T infinity) by gamma not square is actually special quantity

or special non dimensional number we call that one as actually Grashof number, okay. So this is kinematic viscosity so this is

Grashof number so we are getting this coefficient of the viscous term is becoming one by root

over of Grashof number okay, so Grashof number gives the physical significance that what

the effect of buoyancy and what is the effect of viscosity, okay. So that is actually non

dimensional number, so once if we see that what our equations are taking the form after

non dimensionalization so you will be getting inertia term is having simplified U del u

del x plus V del u by del y because we have divided with the coefficient whatever we had

over there throughout so we have got one by root over of Gr in front of the viscous term

and as we have equated the buoyancy and inertia force so there is no coefficient in front

of the buoyancy force. Because we have divided with the coefficient

of the buoyancy term so coefficient of the inertia term so you are finding out that here

it is throughout divided so it is becoming one okay, why momentum equation similarly

you too will be getting U del v del x plus V del v del y is equals to minus del p del

y plus one by root over of Gr del square v by del x square okay, here theta term will

not be there as plate is vertically oriented, okay. Similarly for energy equation you will

be finding out along with one by root Gr we will be having Prandtl number also over here. Because in this case in front of energy equation

in place of gamma we had alpha so to bring the effect of the Grashof number what we can

do, we can write down alpha by gamma into gamma okay so that gamma will be giving the

effect of the Grashof term but an alpha by gamma will become the Prandtl number, so here

you can see root over of Gr into Prandtl number is coming over here, okay. Next let us try to do some sort of scale analysis

from all these three equations whatever we have derived, so let us start from continuity

equation now we know that what we can do, for the along the direction of the plate we

can assume that x which is along the direction of the plate is of the order of one and for

the cross direction or the vertical or the horizontal direction we can call y is of the

order of the del not, del not is actually the some thickness. Whether it can be velocity boundary layer

thickness or thermal boundary layer thickness that comes later on whenever you are considering

different prandtl number, now from continuity we can straight away write down that du, del

u del x need to be in order of one so if x is order of one then obviously u will be order

of one and in case of Y if it is order del not , del v del y needs to be in order of

one. So V becomes order of del not, okay. Now let us see the viscous term if you see

the viscous term what is the order of the viscous term let me show you the viscous term

once again, so if you see the viscous term once again over here one by root over of Gr

then we are having del square u del x square plus del square u del y square okay, so in

this case you can find out that the order will be coming will be finding out that this

term is becoming dominant, okay. Because Y is of order del not. So this term will becoming dominant but x

is of order one so this term is becoming dominant and order will become one by root over of

Gr into one by del square, okay so this is the order of viscous term one by root over

of Gr int one by del not square okay. In a similar fashion if you try to see from the

inertia term so in case of inertia you will be finding out the order as we are having

V and Y over here, so this will be cancelling out okay, so delta not and delta not will

be cancelling out. U is of order one so this becomes order one

similarly all this quantity U and X these are of order one so the total inertia term

is becoming of order one, so inertia term is having order one, okay. Whereas the viscous

term is having this order right. So if we equate the inertia and viscous term to get

the momentum equation back, so we get one by root over of Gr del square is of order

one and from here we get del not is actually Gr to the power minus one by four okay. So we have got the thickness of the boundary

layer is of the order of Grashof number to the power minus one by four. So let us take

some stretching terms formation, so from small y as we have already known that what is the

order in the y direction we can take y into Grashof number to the power one-fourth is

actually capital Y and similarly to satisfy the continuity we need to take for small v

also, so small v into Grashof of number to the power one by four is actually capital

V, okay. So if we use this stretching transformation

our continuity equation will not be changing because in the first term there is no capital

Y or capital V, in the second term we are having capital V and capital Y but this Grashof

number to the power one by four will be cancelling from denominator and numerator okay, so continuity

equation will not change, in x momentum equation we can find out in inertia term there is no

change. As it is of order one already we have told,

okay. In case of your viscous term you can find out that your cross wise viscosity that

means del square u del y square is actually releasing one Gr to the power half so that

Gr to the power half and its coefficient are earlier coefficient one by root over of Gr

is cancelling out and there is no coefficient in front of the del square u del y square

term but del square u del x square will be keeping the coefficient one by root over of

Gr, okay buoyancy remains as usual as theta and pressure gradient remains as minus del

p del x, okay. As we have not done any transformation for x so this term will be remaining same

so this is my x momentum equation. In a similar fashion Y momentum equation can

be constructed so this is Y momentum equation so as we are having over here V so what we

have done the coefficient of V actually has taken this one Gr to the power minus one by

four similarly over here we are having two V’s and one Y so ultimately Gr to the power

minus one by four will be coming out, so that we have taken common over here, in the right

hand side for the pressure gradient term we will be having Gr to the power one by four

okay because Y is over here. For the viscous term so for the first term

del square v del x square there will be only V over here so Gr to the power minus one by

four will be coming out and earlier we are having Gr to the power minus half so that

will be giving rise Gr to the power minus three by four and for the last term del square

v del y square so we will be having one V at the bottom sides so that along with one

by root over of Gr will be giving rise to the Gr to the power minus one by four okay.

Subsequent changes of side will be giving you del p del y is equals to something in

the right hand side which is having everywhere Grashof number, okay. So this is the final

form of the momentum equation we can write down, okay. Similarly energy equation so you see left

hand side will be having no change because this is your convection side having order

one so U and X those are of order one, V and Y they are having stretching’s but Grashof

number is releasing in denominator and numerator those will be cancelling out, in the right

hand side conduction term you see for del square theta del y square if we convert that

into capital Y, capital del y square. So one Grashof number to the power half will

be releasing so that Grashof number to the power half and one by root over of Gr will

be cancelling out and leaving out this one by Pr into del square theta del y square but

same thing will not happen for del square theta del x square so you will be finding

out one by root over of Gr Pr okay del square theta del x square right, so these are the

equations form equations of different forms after this stretching term formation. Now

let us take the limit of Grashof number tends to infinity. So this Grashof number tends to infinity comes

out whenever you are having significant amount of temperature drop between the wall vertical

wall and the free stream compared to the you know viscous effects, okay. So here we take

this Gr number tends to infinity if you take Grashof number tends to infinity in all the

equations wherever you are having Grashof number to the power minus power. So those terms can be cancelled, so you see

from x momentum equation last term can be cancelled okay and from Y momentum equation

whole right hand side can be cancelled okay everywhere we are having minus power okay

of the gross of number and from the energy equation this delta square theta del x square

can be cancelled, so ultimately the equations comes out to it continuity in this fashion

x momentum in this fashion this is your Y momentum equation, simply del p by del y is

equals to zero and finally this is my energy equation which is important for this course

okay. So delta square theta del x square is absent over here, right. So let us take further simplicity let us consider

that from here we can get del p del y equals to zero so that means P is a function of x. And let us take that P which is function of

x is actually a constant okay, so if we take a constant then obviously in x moment equation

minus del p del x is equals can be dropped down okay, so here you see from x moment and

equation del p del x we have dropped down to make our equations simpler in look okay.

So this is my x momentum equation, y moment term equation we have not shown over here

because this derivation now we have actually taken from y momentum equation. This is my energy equation as usual whatever

we have derived in the last slide, let me tell you the boundary conditions also so at

Y equals to zero which is nothing but adjacent to the wall okay, adjacent to the wall obviously

no sleep and no penetration will be giving me u equals to zero, V equals to zero. And

from the non-dimensionalization of the theta we will be finding out that at the wall temperature

is Tw so this is becoming Tw minus t infinity by Tw minus T infinity which is equal to equals

to one, okay. And away from the wall so Y tends to infinity,

capital Y tends to infinity, we will be finding out there is no velocity so u equals to zero

okay, and theta will be becoming zero because t infinity minus t infinity by t w minus t

infinity, is the theta which is nothing but zero okay, so these are the equations and

boundary condition subsequently. Let us try to modify this things using our stream function

concept, all of us know that u is nothing but del sai del y and V equals to minus del

sai del x where sai is the stream function, okay from fluid mechanics we know this.

Let us try to get that what will be the equation forms involving this stream function, so obviously

continuity equation will not be giving us something so I am not writing continuity equation

over here, basically this things have been derived from the continuity equation only

so let us see the momentum equation x momentum equation so if you plug in this u over here

and V over here then we will be writing down del sai del y into del square sai del x del

y. So this is nothing but your del u del x minus

del sai by del x this is nothing but your V and del square sai by del y square this

is nothing but your del u by del y. On the right hand side we are having del square u

del y square is nothing but del cube sai by del y cube plus your by in theta miss giving

theta. And in energy equation first term is actually your u del sai by del y is actually

your u and del theta by del x is remaining like this for V we are writing minus del sai

del y and this is your del theta by del y which is coming from the second term of the

convection. In the right hand side we are having one by

Pr del square theta del y square okay as usual from the energy equation. Boundary condition

will not be changing much, so at Y equals to zero, U equals to zero means del sai del

y equals to zero, V equals to zero means from here we can get if we integrate once we will

be finding out sai equals to zero and energy equation temperature boundary condition is

not changing theta equals to one. And for away from the wall Y tends to infinity we

are having del sai del y tends to zero and theta tends to zero, okay. So this is your

u tends to zero and theta tends to zero right. Let us do now little bit further try to have

some stretching transformation from this stream function equations, let us define that X star

is equals to e to the power alpha one small okay, so we are taking this exponential relationship

where Alpha1 is one constant, okay. Let us do similar things for Y star, sai star and

theta star okay, so Y star is equals to e to the power alpha two capital, sai star is

equals to e to the power alpha three sai and Theta star is equals to e to the power alpha

one theta, this we are doing for finding out the similarity variable in terms of x and

Y, okay. So here we are having four unknown alpha one,

alpha two, alpha three and alpha four we will putting back this stretching variables in

your vowelling equations to find out the similarity variable. So if we put all these stretching

transformation in our equation and convert all those without star terms to star terms

we will be having one co-efficient of e to the power alpha one plus two alpha two minus

two alpha three from our inertia term in x momentum equation. And in viscous term we

will be having e to the power alpha three a two minus alpha three okay, as we are having

Y cube so three alpha two and as we are having sai it is becoming alpha three okay, alpha

three and alpha two like this, okay. And for the theta term we are having e to

the power minus alpha four okay, similar thing can be done for the convection side so our

equation is becoming of e to the power alpha one plus two alpha two minus two alpha three

minus alpha four okay, so sai, theta and Y and x so all alpha one, two alpha two, two

alpha three and alpha four will be coming into picture, and in the conduction side we

are having e to the power two alpha two minus alpha four okay, so theta is giving rise alpha

four and Y square is giving rise to two alpha two, right. And from the boundary condition

so Y equals to zero means, Y star equals to zero okay, and Psi star equals to zero means

Psi equals to zero but here the second uequals to zero, V-zero boundary condition gives rise

del sai sorry, u equals to zero boundary condition gives del sai del y e to the power alpha two

minus alpha three equals to zero, okay. So and your temperature boundary condition

gives e to the power minus alpha four theta equals to one okay, in a similar fashion away

from the wall Y star tends to Infinity we can find out del sai star del y star e to

the power alpha two minus two into alpha three tends to zero okay, so this is nothing but

your u tends to zero and e to the power minus alpha four theta tends to zero this is nothing

but your theta tends to zero, okay. So now from here all these things if we equate the

order from both the sides we would like to find out what is the relationship between

alpha one, two alpha two, two alpha three and alpha four okay. So from the boundary conditions, first boundary

conditions clearly we can see that alpha four leads to be zero otherwise theta star cannot

be equals to one okay, because theta equal to one or the boundary conditions otherwise

we will be making it complicated. So definitely needs to be equals to zero okay, as we need

to have this one equals to one, okay. And then from here we can see that this three

alpha two minus three alpha three definitely needs to be minus three four okay, as we have

already prove that alpha four equals to zero so three alpha two minus alpha three also

needs to be zero, so three alpha two minus alpha three needs to be zero, okay. So from here we are getting alpha three equals

to three alpha two okay, in a similar fashion if we equate the inertia and viscous terms

co-efficient then we can find out relation between alpha one and alpha two in this fashion,

so alpha one plus two alpha two minus three alpha three needs to be equals to zero, because

inertia and buoyancy terms these two needs to be of say molder okay, so here you can

find out we are getting Alpha1 in terms of alpha two as alpha one equals to four alpha

two okay. So here we have obtained X eight equals to e to the power four alpha two into

x, Y star equals to e to the power alpha two into y, sai star equals to e to the power

three alpha two into sai and theta star equals to theta as alpha four equals to zero okay. Now let us try to consider that e to the power

alpha two equals to c, so ultimately we will be getting x star equals to c to the power

four, y star equals to c y, sai star equals to c cube and theta star equals to theta.

Now from here let us try to construct the similarity variable, so we can easily write

down from this two equation y star by x star to the power one by four so y star by x star

to the power one by four will become small y divided by capital y to the power one by

four okay, so here you see c and c to the power four by one by four is actually cancelling

out it side, so this becomes the similarity variable, okay. On the other hand if you see the third equation

and the first equation here from also we can write down sai star by x star to the power

three by four is equals to sai by x to the power three by four okay. So here also we

can get that what will be the functional dependence between the sai and X and as usual theta star

equals to theta will be remaining over there. Now as we have got the idea that what can

be my similarity variable, so let us write down this similarity variable as y by x to

the power one by four okay, so I am writing over here eta which is a similarity variable

y by x to the power one by four and as I do not know what is the constant term will be

coming in front of that let me keep that one as A. So I am defining eta equals to A into y divided

by x to the power one fourth, in the same fashion let us define sai as a function of

eta so what we are writing sai is actually x to the power three by four from here we

can get that one, x to the power three by four into F okay and as we do not know what

will be the constant we are keeping the constant as beta okay. And theta always from this one

we can write down theta can be expressed as a function of eta, so theta is actually theta

of eta, right. So now our task simplifies to finding out the values of A and B okay,

to explicitly find out the similarity variable and stream function dependence over eta. So let us do that so in order to do that first

we need to find out the derivatives of different terms and put in the equations, so first let

us find out what is the eta’s derivatives so similarity variables derivative, so del

del x of eta is minus eta by four x and del del y of eta is A by x to the power one by

four okay, and let us side by side do del theta by del x also, so del theta by del x

will be important for your energy equation, so this becomes del theta by del eta into

del eta by del x okay, so del eta by del x I can pick up from here and del theta by del

x will become theta dash, okay. So minus eta by four x theta dash in a similar

fashion del theta by del y we can write down del del eta of theta into del del y of eta

okay, so these becomes theta dash and this del theta by del x is nothing but A by x to

the power one by four so this is the del theta by del y coming out to be. So let us now put

everything in, at first everything let us put in no, before that we need the value of

del sai by del x also so this is del sai by del x okay, so for finding out del sai by

del x what we are doing the previous equation whatever we have over here we are making the

derivative with respect to x, so this is also function of x and F is also a function of

x so we are having derivative of two multipliers. So we are using that multiplier rule, so here

you see first we have kept f constant and we have made the derivative of x to the power

minus three by four okay, x to the power three by four and here we have kept x to the power

three by four constant and here we have made the derivative of F, as F is the function

of eta so d eta of dx also comes into picture using the value of eta so d eta of dx we can

simply this and it will be coming as B by four x rest to the power one fourth into three

F minus eta f dash okay. In a similar fashion del sai by del y can be calculated this is

simple, so B into x rest to the power three fourth as x is not a function of Y so this

will not be making derivative this time. So only F dash is having and A by x to the

power one fourth is coming due to B eta by dy, okay, so simplified form is AB x to the

power half into F dash. second derivative of sai with respect to Y you will giving AB

x to the power half F double dash into A by x to the power one by four okay, so ultimately

it is becoming A square B x to the power one fourth F double dash. Third derivative of

sai which is also coming in our viscous term so this is, this will be becoming A cube B

F triple dash okay, and cross derivative of sai del square sai del x del y will be, this

you can achieve by making derivative with respect to x of this term or derivative of

this term with respect to Y, okay. So both will be giving you same result and

final simplified answer you will be getting as AB by four x into two F dash minus eta

f double dash so all this derivatives we have obtained for sai okay, and theta also we have

obtained only term left is that del square cube so all this terms you can get in this

fashion. So let us put everything in our momentum equation

so this we have put in the momentum equation, so you can find out this is your u okay so

this is your u del u del x okay, and this one is your V so this one is your V okay,

and this part is u actually your del u del y okay, and in this side we are having over

here del square u del y square, del square u del y square and theta term will remains

like this okay, in the momentum equation. Further simplification of this one u will

be giving you in this fashion the equation will come out in this fashion okay, where

this two term can be cancelled okay, so if you do the simplification this is our equation

involving A and B. Remember here also still A and B prevails. From your energy equation

if you do the same things so this is your u, this is del theta del x, this is my V and

this is my del theta del y and in this side we are having del square theta del y square

into one by Prandtl number. Simplification of this one if you do for the

simplification this will come further one steps simplification if you do then it will

be coming like this and at the end you can find out this term and this term can be cancelled,

so theta double dash plus three Prandtl number B by four A F theta dash is equals to zero

becomes an energy equation. So this is the momentum equation this is the energy equation,

but still here A and B terms remains so let us try to find out A and B. So first let us see that what is happening

over here, here in both this cases AB by four is the co-efficient and here we are having

A square by Pr. So first let us try to see from B what we

can say it is that B by four A equals to one, so B by four A you see here in the energy

equation we are having B by four A if we can make this one this equation look like a simplified

one, so first let us try to make B by four A equals to one and also try to make that

A cube B is actually equals to one, A cube B is here so A cube B is equals to one, so

if we make like this then from this two equations we can get the value of A and B by replacing

A from here to here we can get the value of A and B and that comes out to be one by quadruple

root of and B it will becomes out to be four to the power three by four. Once we know the value of A and B getting

the similarity variable is not difficult so similarity variable becomes Y by four x to

the power one by four and sai becomes four x to the power three four into F okay. so

once again if you give back all this A and B values in your equations whatever we have

shown over here this two equations so my final form of equation comes out to be F triple

dash plus three F F double dash minus two F dash square plus theta is equals to zero

this is the momentum equation and energy equation comes out to be theta double dash plus three

Pr F theta dash equals to zero okay, so this and this we have made actually one okay and

boundary conditions that eta equals to zero, F equals to zero, F dash equals to zero, theta

equals to zero okay, and at eta tends to Zero F dash is zero and theta tends to zero okay,

so these are the equations and boundary conditions for natural convection from uniform wall temperature,

okay. Next let us show you that what happens in

case of large Prandtl number case and small Prandtl number case okay, in case of large

prandtl number case the previous equations will be changing in this form, F triple dash

plus theta equals to zero plus theta triple dash plus three F theta dash equals to zero

okay, and for small prandtl number these will be becoming three F F double dash minus two

F dash square plus theta equals to zero and theta double dash plus three F theta dash

equals to zero corresponding boundary conditions I have written over here which are more or

less same okay, at eta tends to zero, eta equals to zero F and F dash both are zero,

theta equals to one, eta tends to infinity F dash and theta equals to zero and for prandtl

number tends to zero. At eta equals to zero F is zero and theta

equals to one and as eta tends to infinity, F dash and theta both tends to zero corresponding

Nusselt number also I have also written if we solve this two equations numerically then

we can find out Nusselt number becomes point five zero three Rayleigh number to power one

by four okay, so this is rally number is very important this is nothing but Grashof into

prandtl number, okay. And here in this case prandtl number tends to zero we give Nusselt

number equals to point six Rayleigh nuumber to the power one by four Prandtl number to

the power one by four, okay. In summary let us see what we have learnt

we have seen what is the choice of velocity scale for non- dimensionalization, we have

seen u not is nothing but root over of g beta L delta t okay, so delta Tw minus T infinity

as wall temperature was known. Governing equation for natural convection around vertical hot

plate at constant temperature so this was the generalized governing equation, this is

from the momentum equation, this is from the energy equation and corresponding boundary

conditions this we have derived. Different Nusselt number correlations also

we have mentioned for low prandtl number and high prandtl number case, in low prandtl number

case point six was the co-efficient and for large prandtl number case it was point five

zero three. And in case of large prandtl number Rayleigh number came into picture, okay here

also Rayleigh number came with Pr as having power of one by four, okay. Now let us test

your understanding what you have understood in this lecture. Help me in identifying the velocity and temperature

profiles for natural convection in different prandtl number limits, okay. I have shown

here two cases so red lines are for temperature profile and black line is for the velocity

profiles, okay. So here temperature profile very promptly goes to zero okay, but velocity

profile prevails and it is having a very high velocity profile so delta is high compared

to delta t. And in this case we are having delta t higher

compared to delta okay, so you need to tell me which one is for large prandtl number,

which one is for small prandtl number so I think all of you can guess the correct answer

is first this one is for large prandtl number okay, so that is why velocity boundary layer

is having higher thickness compared to thermal boundary layer just opposite happens over

here in case of low prandtl number, okay. So with this I will end this lecture, thank

you in my next lecture we will be having natural convection from a plate which is having uniform

hit flux in between if you are having any query please keep on posting in the discussion

forum, thank you.

## One Reply to “Natural Convection: Uniform Wall Temperature”

Aata Majhi Satakli