Lagrangian and Eulerian Perspectives
So welcome back to one more lecture on introduction to Soft Matter and last time, so in the lab we had a very nice session where we took a look at actual rheological characterization of some of these fluids and we took a look at how these fluids can be, so how some of the theoretical concepts that we discussed earlier on in the class.
How that looks like in a real experimental situation and some of the things that you probably noticed is that there was noise the curves are not as well behaved obviously as we are drawing it the theoretical setting. At the same time the idea of linearity which seemed most likely would see would have felt to you that must be valid in many cases seemed quite not so applicable in someone of the case at least one of the cases that we saw.
But those experiments you have to do multiple times it has to be repeated which we did not do for the interest of time you can also use different geometries to study to the same rheological characteristics. We did not discuss that because this is an introductory course in the introductory course, we are not delving into all the details.
So, if you take a more advanced course that those are some things that you would get to understand, but now that we have taken a look at the relaxation phenomena, we want to ask ourselves as to what are some of the simple models which can give us an elementary relaxation, more behaviour. And before we go so, we will look at a model known as Maxwell's model we will also look into the original paper which in which Maxwell offered this model to us.
(Refer Slide Time: 02:30)
And but before we do that, we want to take a quick look at a couple of important topics which are the Lagrangian and Eulerian perspectives. There is an important reason why I am discussing this and it is often not clear in many of the text books. When they derive the Maxwell’s model as to what is the basis for application. So, we want to make sure that our case is well understood.
So, let us discuss these two important perspectives. Now Lagrangian and Eulerian perspectives are two perspectives by which you can observe or explain deformation of a material or a flow. And these are two model, two perspectives which are based on the person who is observing the flow. And so, the two common with these two common ways to study a moving fluid or a deforming body.
So, the first is the Eulerian. In the Eulerian one, what we do is we look at a particular location and then we observe how all the particles are all the material points are flowing through that location and how that behaves. So here what we do is we look at a particular location and observe. So, this is just like, and observe how all the fluid passing through that location behaves. This is the Eulerian point of view.
(Refer Slide Time: 05:28)
So, what do we mean here? We mean that in this condition we will create, let’s say you have you will create a mesh and this makes meshes are static and you have all the different particles, say these are fluid particles. And these fluid particles all have a velocity and they are moving in different directions. Now instead of observing the individual fluid particles you say that I will confine my attention to this particular window.
So let us just say this particular window I am just taking an example and then you will say that, I will look at all the different fluid particles behaving how they and how they behave as they pass through this, but once they pass out I of this control window of this section I will not bother. Then it is to be studied in the next control window perhaps.
So, I only confined my attention to this small section and you can probably realize this is your control volume. And if you take this same control volume and then you say that the in the limit of the size as it shrinks down to 0, that will also become your differential control volume.
So this is Eulerian perspective where you do not track individual fluid particles but rather say but my the way I am going to look at it I am not going to bother about all the different fluid particles, rather I will keep my view fixated on certain meshes or grids. And these grids are your control volume and I will keep I will quantify the behaviour in these grids.
So, this was 1, so the second one is the Lagrangian perspective where you look at a particular piece of the fluid packet. So, bracketing and put a fluid packet and observe how it behaves as it moves from location to location. This is your called your Lagrangian point of view. So,
what do you do here? Let us say you have a fluid particle. And let us say it is possible for you
to identify that particular. So, how you identify that I am not discussing that.
But let us say somehow you are able to identify this particular particle and this particle
particular particle is now going to move through these different locations at different times
and this is your path of that. So, this is let us say at some time ot so this is some ott+ . So,
this is the initial position so your position now the position vector is a function of your time.
So, this is your Lagrangian point of view where you are looking at individual fluid particles
and you are following them. So, why are we discussing these two by way? Why is this
important? Well because when you apply Newton's laws, they are applicable for a fluid
particle when you are looking at it from a Lagrangian perspective.
So, Newton's laws are applicable to this mass as it moves through but it may not be it is not
applicable in the same way in the in its original form to this control volume. So, for
application to the control volume we have to do something different
(Refer Slide Time: 09:58)
So, the difference is in the control in the Lagrangian point of view is that, for example if you
have the position vector that is just simply a function of time here. The position vector value
itself is changing just as a function of time. Whereas, if you have if you are looking from the
Eulerian point of view, then and let us say you are tracking the velocity field in a differential
control volume, then your velocity vector is so you are u is now a function of your time as
well as X bar which where X bar is now the control volume location. u = u(t, x) .
So, when you apply the derivative if you want to apply that derivative you want to apply it
such that the derivative with respect to time is following a fluid particle. And if you are
looking at it from the Eulerian perspective, so if u is an Eulerian variable you cannot compute
this time derivative easily. So, for that what you have to do is now it is a multi there are two
variables here. So, you have to individually take derivatives and by the chain rule you will
know that you know the derivative will look something like this.
du u u dx u dy u dz
dt t x dt y dt z dt
= + + +
So, I am just applying the chain rule here, because here to evaluate this was not to evaluate
this, I must apply the chain rule and look at it from that perspective. Now you will notice that
in the case of the Eulerian system, these are nothing but your Eulerian velocities
(Refer Slide Time: 12:28)
So that is why you have the concept of when you take the Eulerian derivative, in the Eulerian
sense you have to take material derivative and that is usually a derivative of this form. So, I
am just so I am saying when you have it if this is an Eulerian variable, then when you take
the derivative following a particle you have to take it in this particular form. And this can be
any quantity, it can be a scalar it can be a vector and. So, with that small introduction we are
now ready to discuss constitutive relationships.
(Refer Slide Time: 13:40)
So, what we want to do now is we want to look at constitutive equations for 1D response of his viscoelastic materials. And to do that, now we had discussed if some time ago just scrolling through the notes to find where we had discussed that. Okay. So, we had discussed some time ago that 1D response can be interpreted as a combination of elastic and viscous responses.
This is something that we had said long ago, when we had discussed what a viscoelastic phenomenon is. So, we are saying that in the case of 1D response we can obviously go back to that idea where the viscoelasticity is represented as a combination of elastic response and viscous response.
So, let us say take the case of elastic body. We had just said that there was 1 analog that we can use and that analog allows us to interpret our model elastic response and that analog was your spring. So, your spring represents energy storage and your dashpot this represents energy dissipation my system to the surroundings.
(Refer Slide Time: 17:15)
So that will help us create this first in the simplest of the viscoelastic models. Where what we will do is we will put them in series, but before we go here, as we have been doing frequently in this course whenever we introduce an interesting and important topic we try to see if we there is if we can go back and look at the original works where this was represented.
So, this is the manuscript by James Clerk Maxwell which is titled on the dynamical theory of gasses. It was published in 1866 and this is the manuscript where he proposes the Maxwell's model. The history of why he was trying to look into the dynamical behaviour of gases is an interesting something you can look up later on by itself but we are not going to concern ourselves with that.
So, in the beginning he talks about the theories of the Constitutions of bodies and if you know at that time, they were not they had still not seen an atom. This is 1866 this is still before we
have been able to image atom and we have been conclusively able to prove that materials exist as atoms and molecules.
So, this is pre-dating their time but there are different philosophical or there are people who had sort of guessed that material is going to be made a lot of small particles. So, in the beginning he talks about that and that is quite interesting to read because he says that the theories of the constitution of bodies suppose them either to be continuous and homogeneous or to be composed of a finite number of distinct particles or molecules. So, he is talking about the two ways in which we believe that the material exists.
The first is the continuum model where the continuum case where the material is just believed to be homogeneous at any scale even if you go down smaller and smaller and in the other case it is always supposed to be composed of a finite number of particles of molecules. And then he goes over the different ideas that are prevalent at that time says that if we adopt a statical theory and suppose the molecules of a body kept at rest in their positions of equilibrium actions of forces in the directions of the lines joining their centres, we may determine the mechanical properties of a body so constructed if distorted so that the displacement of each molecule is a function of its coordinates when in equilibrium.
So, he is saying that if you knew the internal forces, then you would be able to figure out how the body is going to react to that. So, this this is something that you can read at your leisure it is downloadable it is also available on archive I believe one of the archive platforms has all of Maxwell's papers.
Even a section where he goes into the idea’s where, so some of this I had referred to in one of the earlier classes that the opinion of the observed properties of visible bodies apparently at rest are due to the action of invisible molecules in rapid motion is to be found in Lucretius. So, he is referring to the Roman and the Greek literature on the idea that matter is composed of small indivisible parts.
And we know that there was a similar school of belief in India at that time, which is called the vaisesika system of which Kanade is probably the most well-known propounded which also believed that the materials are composed of small atoms and molecules. So, he goes through all that and then he introduces this particular equation where thus there is a solid body.
So, there is a solid body that has so
dF dS F
dt dt T
=−. Where the there is a body which has
both elasticity and viscosity at the same time. So, this T is a relaxation time scale, so we will
go down, we will what we will do is we will try to derive this from our first principles and we
will see whether we end up using this we end up with this equation or not.
So, let us go back to the Maxwell's model. So here we said what we are going to do is, we are
going to put into series, let’s say this is represented by E and the modulus here and this
represents a viscosity of . So, the viscosity and the E are constants and you are applying a
force and as a result both the spring and the dashpot will experience some displacement.
So, let us say that the displacement observed here is s x , the subscript s standing for spring
and the displacement by the dashpot is xd , d being a subscript for dashpot. And these
themselves are massless, so we will also make a note that these are massless. So now the total
displacement in the system is, so let us say that this spring dashpot combination is
representing a Lagrangian material point.
So now if that is the case then what we want to do is, we want to apply force balance and we
want to see where that leads us to, but before we do that then let us quickly write that from
geometry of the current system, from geometry, the total displacement in the this body is
s d x = x + x . We want to apply force balance so consider these separate entities.
So, I am just going to create a separate plane here and draw that. So, if the force applied is F,
then my force balance on the system requires me and since these are massless that the forces
on both sides will be F(t) for the spring and, similarly for the dashpot. So, if I were to write
the so from force balance, we have F(t) of the force F(t) is equal to Fs(t), where Fs(t) is the
force being experienced by the spring and Fd(t) is the force that is being expressed by the
And our question that we want to answer is we have to find a relationship. So, the question is
find a relationship between x and F(t). So this is where we are trying to get. So now what
do we know about Fs(t), and Fd (t) so Fs(t) we know is the force that is in the spring.
( ) s s F t = Ex
( ) d d F t x
(Refer Slide Time: 26:33)
So, if that is the force in the spring then from our analogy the relationship between the force and the displacement in the spring should be this because this is a linear spring and the force in the dashpot should be. So, what we will do is we will stop here today and the next class we will take out from here I will try to derive the formula.