ya so welcome back to the part two of fluid
flow modeling and we will continue from where
we left over in the part one and in the part
one we have done the entire formulation along
with initial and boundary conditions up to
a point we have not looked at this specific
boundary conditions or ah special terms at
are applicable for the fusion welding and
the formation we gave as almost like for any
liquid that would under go fluid flow within
the domain so we will look at those special
terms that are applicable for welding now
the first thing that comes to our mind when
we looked at the ah the driving forces for
fluid flow that are happening in the ah weld
pool is the thermal buoyancy and that is basically
the buoyancy effect due to the temperature
differences within the domain so let us see
what it look like 
so basically the ah driving force because
of ah the thermal buoyancy is coming from
ah the density changes which are (( )) temperature
so in other words ah density is going down
as the temperature is going up and because
of density changes you would have ah combined
with the gravity you would have the resultant
flow and the flow is such that the ah liquid
that is having lower density will go up up
as in away in the gravity direction opposite
to the gravity and the one which is heavy
because of higher lower temperature that would
be settling down ok so this is the kind of
ah fluid flow that would be coming because
of thermal buoyancy and the way it is modeled
in the ah equations that we have written is
as follows you basically have a an assumption
that is made and i would just write the name
for you here the assumption is called boussinesq
assumption the reason why mean it to make
some assumption to combine the thermal buoyancy
with the fluid flow equation that we wrote
is because of an apparent clash between two
assumptions first assumption we made is inc
ah incompressible ok
incompressible we made an assumption ah that
one by rho ah by rho by dow t plus so the
continuity equation that we wrote we said
that this can be changed as rate of direction
is zero so in this process what we have said
is basically the ah material derivative of
rho is zero thats what we are saying basically
ok so this implies that you should not have
the density changing across the difference
control volumes taking in to account the addiction
term and that goes opposite in the phase of
the density changing because of the temperature
changes and therefore we must make an assumption
that ah the density changes are only to lead
to flow but not to lead to the ah in compressible
assumption been invalid so which means that
this assumption is valid ah whenever the changes
are less then five percent ok in such situations
you can five may be about ten percent its
fine so for small changes in the density because
of temperature variations it is possible for
you to ah make an assumption which goes by
the name of business assumption ah to say
that incompressible fluid flow equations are
still valid ok so under that constant we can
just then use the ah thermal buoyancy has
a volumetric term ok so it is going as a volumetric
term
volumetric are the body force term and which
means that for the by component ah assuming
that the gravity (( )) is along the y direction
for the y component you could then add a term
ah which add add a term which is going like
this 
ok with this is the term that would be added
to the equations so which means that ah be
have f y minus dow p by dow y so in that you
can add one more term like this this is basically
the reference temperature which would be usually
the melting point itself because the reference
temperature about which you can take the differences
and look at what would to be ah change in
the temperature and this is the compressibility
ok it is property of the material and this
is the acceleration duty gravity and this
is the average ah density ah over the temperature
ranges for which the melt flow is going to
be experiencing the flow
so by adding this term you basically have
take in to account how the density changes
are happening you can actually see that ah
beta into t into ah rho bar is nothing but
this is some what like ok something like that
so it is basically take into account how the
ah density changes are causing the ah fluid
flow to come up come about and thermo buoyancy
is essentially taking the density changes
(( )) may because of the temperature differences
in this manner ok so we could actually then
by analogy see also if a solutal buoyancy
can be looped at and solutal buoyancy is quit
state forward this term itself should be modified
in the following manner 
the density changes are because of solute
or concentration differences and not the temperature
differences is what distinguishes solutal
buoyancy from thermal buoyancy because other
then that there is no other difference ok
and what does it mean it means that this is
going to be this to be expressed as due to
concentration differences what is how ok so
if you express delta rho as temperature differences
ah then you get the beta term there is coming
up and the rho bar but if you want you can
express the change in the density ah due to
the concentration differences in the location
within the melt pool and concentration differences
are coming because concentration is function
of location within the melt pool that is actually
the situation when you have (( )) welding
or a weld pool in which you are adding a fear
war that is of different density ok in such
situations that is valid ok
so solutal buoyancy and thermal buoyancy are
both together as categories of buoyancy convection
which means that they are basically caused
by the density differences within the melt
pool at upon by the gravity to lead to the
flow ok and how would the ah average magnitude
look like it would be for typical ah weld
pool shapes about in the order of mili meters
per second which means thats a very weak convection
you are talking about so there is a reason
why many of the ah models are usually ignore
the (( )) because very weak ok we will see
what is the strong term shortly ah but mili
meter per second also is the kind of magnitude
you will get ah how did we arrive at that
kind of a magnitude thats scaling analysis
we will do ah look at it in a ah small exercise
as a part of the ah ah tutorials ah later
on in this course ok
and the other driving force that would be
playing a role ah in the case of welding ah
for the melt pool conversion is a thermal
marangoni so i will just call the marangoni
convection marangoni convection is basically
ah caused by the surface tension differences
ah as a function of the distance and that
can be understood by the small following steps
ok so step by step we can understand how it
comes above so first thing at we must appreciate
is that the surface tension ok forwardly this
is the surface tension from the ah liquid
to the vapor ok so liquid vapor you can say
the surface tension is a function of a temperature
which very ah well know because ah at boiling
temperature the surface tension is not acting
all the atoms are escaping into the vapor
ah readily so this no surface tension acting
because surface tension is nothing but forces
that are keeping the ah atoms in the liquid
together bound to the rest of the bulk o
so therefore you can imagine that the surface
tension must be a function of temperature
so how does it vary usually it looks like
that you normally start the plot of a surface
tension in the liquid domain ah with x axis
going from melting point because below the
melting point it is not liquid so it start
from the melting point and the way surface
tension is going to change is from a high
value it keeps going down ok so in other words
the temperature coefficient of them the ah
materials surface tension temperature coefficient
ah is negative but pure metals but pure metals
ah so that at temperatures close to the boiling
point surface tension is negligible it has
gone to zero and it melting point is very
high because thats when the bounds are ah
most active in keep in the liquid atoms bound
to the rest of the bulk so this is the variation
that we are in know ok how does this variation
cause the fluid flow to take place the weld
pool is a question and that will be illustrated
now by drawing is small schematic on showing
which part of the ah volume element of the
surface is going to going which direction
so let us consider a schematic of the ah liquid
pool and look at elements and see how the
surface tension variation would be ah effected
so let us look at this pool as follows this
is the liquid pool and rest of it is all solid
ah and together this is the work piece and
in this let us look at a plot and we would
then ah look at how the different plots are
to be approximated so at the function of the
distance ok let us first plot the temperature
ok so how would the temperature look like
when you go from the center of the pool away
is something that we can plot and that means
i am actually now looking at the top most
layer in this region ok so how does a temperature
ah distribution look like it looks like this
here is a melting point ok so i am putting
points here because thats what is defining
the fusion zone end and that the center of
the pool you would have the temperature very
high so you would have a peak temperature
like this so you would have a temperature
profile like that ok so that that is the width
of the fusion zone as i can be seen also on
the top ok now this is how the temperature
profile is going to look like hot of the center
and as you go away the temperature is falling
and width reaches the melting point that is
are melt pool width and then further away
it goes to even the ambient temperature which
is of the temperature of the solid far away
from the ah center of the melt pool
and if this is how the temperature where is
need then how would the surface tension variation
look like is something we can plot ok you
can see that as a function of distance you
can see that ah at the highest temperature
that is at high temperatures you would have
low sigma and that low temperatures close
to melting point we have high sigma right
so from that what we can do is that at high
temperatures you have a small value 
and at low temperatures you have a large value
so which means that surface tension is going
to look something that kind ok so goes ah
some a small value it goes to high values
away from here so which means that on the
liquid melt pool how does this kind of a surface
tension difference is going to be play a role
in the flow is something that we can ah illustrate
and for that i would just draw the pool much
bigger and then show you 
consider the elements like this ok at the
center and element here and an elements here
ok here you see that because a temperature
is high the liq surface tension is low so
here it is low sigma this is high sigma then
here as a high sigma ok and the reason why
the liquid in the melt pool is going to convert
because of surface tension gradients is essentially
to minimize the surface energy contribution
from the whole surface and the way it can
minimize is by having more area which is having
low surface tension or high area ok more area
high high area with low surface tension or
smaller area with surface tension both ways
the multiplication of surface tension with
area can be minimized which means that the
way this should move is that if this can expand
this layer can expand and if this can contract
so then you achieve the surface tension contribution
to the energy of the surface ah coming down
and this means that the direction of the flow
is such that it goes like that radially outward
ok radially outward and the surface tension
variation is only at the very surface ah of
the liquid pool and it does not have an effect
on the entire pool so which means that this
ah recirculation to close the loop ah to satisfy
the contently equation is going to happen
ah inside and it could be like this ok
and it could penetrate to the entire depth
or only of the surface depending upon the
following situation ok
so this con convection penetrates more depth
of pool if p viscosity is less a for more
depth ok and you know already the viscosity
is actually function of temperature which
means that if that temperature variation of
viscosity is not strong ok it weak function
of temperature that what happens is that at
high temperature is viscosity is less and
which means that you could have the pool ah
entire convection in the [vocalized-noise]
depth of the pool and in the case the viscosity
is the strong function of temperature it means
that at high temperature it is less viscus
at low temperature is more viscus which means
viscosity gradient up and that means at this
liquid the bottom is very discuss it cannot
be moved easily so then these ka surface convection
will be limited to the top layer either way
it is basically radially outward and what
is it mean by saying radially outward it means
that it would actually make the pool shallow
and wide ok so that is the reason why that
is how it will be effected
so marangoni convection essentially leads
to a melt pool which is shallow and wide and
the magnitude of the ah surface tension driven
flow ah we can estimate it using a small scaling
analysis which i would discuss in a tutorial
ah session later on and that would come out
to be a boat a meter per second ok so magnitude
ok so in other words ah surface tension driven
flow ah will be the strongest ah component
of the velocity in a liquid pool and it will
be on the order of the minutes per second
which should be actually slightly hard then
the ah traversate of the torch because traversate
of a torch is generally about ah couple of
hundred mm ah per minute or you know ah of
that order and that means that is a very very
strong convection the reason why it such a
strong convection is because of the ah strong
ah change in the surface tension as a function
of temperature and temperature is also changing
very dramatically with respect to the distance
so couple to that we will have this as the
ah strong driving force
now ah there is a small modification in the
way the fluid flow will take place when there
is a ah solutal effect on this and thats where
you actually bring the ah idea of solutal
marangoni ok what we changes i will try to
re draw on this itself to show you are difference
ok so what we mean by solutal marangoni is
effect of ah solutal elements on the marangoni
convection so that the surface tension is
not a function of only temperature its a function
of composition also and composition also are
a function of distances in a particular manner
so let us see what should be the effect solute
effects and particularly we let us take the
effect of some surfactant elements such as
oxygen or sulfur in steels 
so what these elements do is basically the
following these elements ah tend to go to
the surface and segregate their so that they
can reduce a surface tension ok so such elements
which migrate to the surfaces radially ah
are basically the surfactant elements ah some((
)) elements like for example ah nickel in
ah liquid iron for example they dont behave
like that they will be in the substitutional
positions wherever iron is their and they
will not segregate selective to one boundary
however let us say boron sulfur oxygen etcetera
they will tend to migrate to the boundary
and then ah segregate their ah change the
composition locally so that the surface (( )) is
lowered ok so such elements if their present
they are going to change the surface tension
behavior of that particular liquid metal
and how does it change in what way so that
is given by this relationship i will show
here ah this for an (( )) for example what
happens is very close to the melting point
the effect of the surface surfactant elements
is very high which means that there will reduce
the surface tension significantly but the
reduction of surface tension is not effective
as you increase the temperature so as you
go further and further at higher temperatures
amount of reduction is not significant and
after some temperature it is not at all present
so in other words the plot would look like
that goes up and down ok so in other words
whenever there are surfactant elements the
surface tension at low temperatures close
to the melting point goes up and then at a
peak value after that it just keeps coming
down so we could then say that the plot is
going to look like that 
some some t naught beyond which it goes ah
so which means that surface tension coefficient
of temperature is not negative always ok it
is actually positive at low temperatures here
ok i say and in this region negative at very
high temperatures so in other words the sine
of the surface tension coefficient is going
to change and if that changes then our argument
that we have discussing here is going to also
change and there is because the temperature
is actually changing ok so which means that
this plot is going to be modified and you
can see that at high temperatures it is still
low and at low temperature is again its low
but is going to be 
so you have got at t naught ah the the distance
correspond to the t naught is where the surface
tension is maximum and so on and so this kind
of a variation would immediately change the
way the fluid flow will happening for the
same logic what would do is just follows it
would change in the following manner
so you would see that at the peak temperature
at very high temperatures its ah low and and
it very low temperature again its low and
here it is high so you have a situation where
ah you want to expand but this wants to ah
this this also wants expand and here wants
to contract so in other words you now see
that you have two look ok and if you want
to rotate it around then you would have so
you have radially have a and this is radially
inward ok so you would see that ah at the
center it is away but in the ends it is inward
ok so this inward flow at the ends this way
inward flow if the ends is going to basically
make the ah ah ah pool not very wide but set
in arrow some which means that the flow is
going to be like this flow near edges of pool
is inward which implies that the pool is deep
and narrow
so in other words if i what the super pause
that kind of a change over this pool shape
when you would look more like this 
because of the heat going inwards of the pool
must be narrow over and it is also experimentally
observed that in steels that contain a surfactant
elements sulfur and oxygen the same g t a
w with everything ah all the parameters same
for the weld pool the weld pool is known to
be its like the deeper and narrow compare
to when it is not having those kind of an
elements ok so this is how a surface tension
is going to changed the weld pool shape and
we can rationalize that from this surface
tension variation ok and solutal effects are
not playing a role only from the surfactant
elements it is playing role also from the
normal aligning elements because the aligning
elements also can for example change the surface
tension in the case of decibel meters so one
has to always be attention to how the surface
tension is changing as a function of composition
if composition changing as a function of distance
so that ultimately you have to inspect whether
surface tension is changing as a function
of distance the movement its change as a function
of distance that immediately it starts seeing
the flow coming because of that variation
so this is how the solutal marangoni machine
is going to play role ok
and let me just erase it so that we can come
to the end of this lecture some more ah body
force surface so the other body force that
is going to play role in the convection is
the electromagnetic force lorentz force so
electromagnetic force is going to be given
as so the current density vector j and the
magnetic field induced by the current itself
ok and this can be calculated and you can
see that this is basically a force which is
going into the an (( )) equation in the body
force terms so simply you can ah add this
in to the body force term and you need calculate
the current density vector ah in entire domain
separately find out what it the force that
is added upon the liquid and add that force
to the respective body force terms and that
is also basically play role in the convection
on the pool and you can see that convection
of pool is depending upon the um current density
vector and current can be going inward or
outward depending upon the polar rate so change
in the polarity also means that we are going
to change the way the convection is going
to cha ah happening in the melt pool because
of the lorentz force that is acting ok so
this is a body force term ok
so let us then ah look at a map of all the
terms that we have done till now and for which
process which ah ah which of the ah special
terms are going to be ah effective ah let
us this do that as a map by making a tabler
fashion which we did in the beginning of this
ah fashion so the term is the buoyanciter
so let us take the thermal buoyancy thermal
buoyancy is going to be given as a ah body
force term which means that it is going to
come into the equation itself as a additional
term so which means that if we what look at
a formulation of the fluid flow ah modeling
in any paper then you can look at the equation
that they are solving and you should be able
to see the body force them their ok and if
it is their that means their taking into account
the thermal buoyancy and what processes is
relevant actually any process that is happening
on earth where the acceleration due to gravity
is present then this term should be valid
it must be present everywhere
but we must play attention that most of the
times the welding is then in a flat geometry
so that the ah highest temperature is on the
top and the gravity is actually downwards
ok so in other words when you change the geometry
from flat geometry to over head the buoyancy
convection is going to playing role ok so
it is not acting in the same direction so
pay attention to the geometry but otherwise
it is every where effective but everywhere
effective but negligible in the sense compare
to negligible compared to the marangoni convection
but it is present everywhere and it is strongly
effected by the geometry
ok solutal solutal buoyancy is also a body
force term but it is only to be taking to
account when there is a solute change that
is present which means that concentration
changes are are to be their which means that
in the case of let us autogenous or homogeneous
filler welds not present so that is if you
do not have any filler and if are going to
weld the material using autogenous welding
then there is no concentration differences
that are happening and therefore there is
no solutal buoyancy and if you are also doing
to do a welding with filler but the filler
is actually homogeneous filler which means
the same material as the base material then
again there is no concentration changes and
therefore there is no solutal buoyancy in
the dissimilar welding it is very strongly
present because it is strong component change
across the melt pool and in the situations
with (( )) added which is different from the
base material again it is present so it is
not present in autogenous and homogeneous
and it is present in rest of the situations
and in the dissimilar welding every say dissimilar
welding it is important ok
now what about marangoni marangoni term as
you can see is because of the surface tension
change and that means there must be a surface
ah thats available and surface is available
normally and this is actually a surface boundary
condition unlike the remaining terms it is
not appearing the equation they appearing
as a boundary condition and it is present
in every situation where is a free surface
available are the situations where there is
no free surface for example let us sake you
know ah arch welding the when there is a flux
so the surface of the liquid metal is covered
by flux it is not exposed to the vapor directly
so there fore the surfaces tension that is
relevant is the surface tension between liquid
metal and liquid flux and that surface tension
is not a strong function of temperature so
which means that whenever the liquid melt
pool is covered by a flux completely and there
is no free surface of the liquid metal exposed
then you can say that the marangoni convection
is very much very ah surprised and not very
ah strongly present and which means that if
you have flux implies not strong
and it is very strong for example in in ah
materials which are being welded where the
temperature changes are drastically ah changing
from the center to the edge of the pool for
example laser welding an electron may welding
the peak temperature is very high and as you
go in a small distance in a small melt pool
the temperature drastically comes to the ah
fusion zone temperature which is the melting
point temperature so which means that you
basically have these effects very strong in
l b w or e b w simply because a gradient are
strong ok and what about the ah electromagnetic
force and that is also body force term and
naturally the electromagnetic force term should
not be present when there is no electromagnetic
ah force in due to the ah process itself that
is if it is an arch weld you will have that
force but it is a laser weight is not their
so you can say that in laser being welding
is absent and in every arch welding it is
present ok and it is depends from polarity
also ok
so as you can see that the ah different driving
forces are to be kept in mind and depending
upon the welding process you must choose ah
which have these are active and you may see
that some or boundary conditions some or body
force terms which come in the equation and
then some can be neglect neglected in some
conditions and some cannot be etcetera so
that way you can actually see that not or
all ah used in every ah situation and one
must be attend to the process conditions to
pick the right kind of a term for the fluid
flow ok so with this we basically ah summarize
by also telling what we did not look at so
let us next ah cris scribble those words what
we did not look at
so these are only for the research in interest
mainly because some of you may want to further
ah go into the research of these subjects
so what is this that we did not look at rippling
ok so surface ripples are known in welds ok
normally a good weld ah with a whats called
the good weld appeal ah is going to look like
that and each each of these are basically
arch shaped ah surface undulations which are
ah suppose to be very uniformed spaced for
a good weld and this surface rippling is basically
ah due to ah the free surface deformation
and that is not considered till now and there
are ways to model it you will come to it if
there is a adequate interest among the participants
and ah there are other things that we have
not consider for example vaporization
ah vaporization we would look at it briefly
when when we discuss the conduction mode to
the ah key whole mode and otherwise we are
not looking at so called knudson layer vaporization
and the related effects we have also not looked
at the effect due to the nozzles on the flow
that is the gas flow coming from the nozzles
to to protect the layer is going to effect
the liquid pool flow we did not consider it
can be dumpt into the surface boundary if
necessary but otherwise you have not considered
ok so like this there are some effects that
we need to keep in eye on to know whether
ah they are important in the process otherwise
we nee need to consider ok so with that i
would ah summarize by closing the formulation
and the formulation can be closed as saying
that in (( )) to that equation plus we can
say like this the formulation is close like
this ah (( )) equation plus continuity plus
special terms which are like this coming ah
plus the boundary conditions which are basically
in the case of marangoni you have got a special
boundary condition otherwise no flow condition
or rigid body condition or no slip condition
etcetera they can be the boundary condition
and then you have the initial conditions which
are simple because initially everything is
at zero ok and you have got special terms
also to take in to account single domain approach
approach the terms that are require to make
it is single domain
so this entire thing now is enough for us
to go head a solve the problem for a fluid
flow in a melt pool ok so will briefly discuss
later on how to go wards solving or what kind
of software are able to solve this for you
etcetera and ah we need to always keep in
eye on what all goes in and how much can then
be expected to come out ok so with this we
close the ah fluid flow modeling ah lesson
and we would discuss further on the solutions
in a later lesson
thank you
