Welcome back. With this segment we are going to start a new unit, and this will take us away from elliptic problems. We are going to start looking at parabolic problems. Right? And we're going to, going to stick with linear parabolic PDEs in three dimensions, but for a scalar variable. The kinds of problems we are looking at, therefore, are very similar to ones we've looked at, we've already considered. They are the, the unsteady heat conduction problem in three dimensions, or the unsteady mass diffusion problem also in three dimensions. You will recall that previously we studied the steady state versions of these two physical problems, and, because we were looking at the steady state versions, those, particular PDEs, are what we call elliptic PDEs. When we bring back the the time dependence, and say they're unsteady problems, we have parabolic PDEs. Okay. So with that somewhat verbal introduction let's get on with it, right?. Linear. Parabolic PDE in a scalar variable, in three dimensions as well. [SOUND] Okay? And, like I said, the physical problems we are considering here are unsteady. Heat conduction. Unsteady heat conduction and mass diffusion. In 3D. Okay? And just remember that unsteady here means that we're talking of time dependent. All right. So, what is the situation we have here? I don't have with me today my my basis vectors, but we don't really need them. We have our body, right? We have basis vectors here, three dimensional. Everything that we talked about, the, the steady state heat conduction problem holds. Okay, so we have surfaces on which we are going to specify Dirichlet and Neumann conditions for the temperature if you're doing heat conduction or the concentration, if we are doing mass diffusion. That's fixed, okay, that remains the same. We have a source dom. We have the notion of the conductivity tensor, or the diffusivity tensor. All the same. The additional component is that we are saying now that at every point in the domain. Either the temperature if it's the heat conduction problem. Or the concentration if it's the As diffusion problem, is changing with time. Because of flexes, or because of the source term. Okay? So at every point, we will have an addition, a time dependent term. Its going to be a first order time dependence, because that is the nature of parabolic problems, right? And that is indeed the nature of the heat con, of the time dependent on unsteady heat conduction and master fusion problem, right? Their first order in time. Okay, so with that setting, let's let's wr-, essentially write out the strong form. Okay, and as we've been doing, let's begin by drawing a picture. The figure, these are our basis vectors E1, E2, E3. Right, our domain, that, right? Three dimensional, of course. We have three basis vectors here. Omega. Right? Somewhat mercifully perhaps, we are back to a scalar problem. Right? So, we don't need to worry about the three different, decompositions of the boundary, right? So we have here, partial of omega u, right? Because u is now a scalar once again, and here we have partial, I believe omega j is how we denoted it, right? All right. Now point here has position vector X which we will use, right?. And at this point the picture here, the usual pill box argument that's given is the following, right? We look at a little elemental volume. Okay. What we see is that we have a flux is coming into it. Okay? And since we've already introduced the notion of a flux before, we can use it now, okay? So, The flux is coming into it, and, and for certain it could be exiting some part also, right? So next one. So this is our flux factor J, okay? Now, inside that little volume element, we have some source stone. And that source stone, you recall, if we're de, dealing with a heat conduction problem, would represent local heating, right, through some external source. Or for the mass diffusion problem, it would represent a local supply of mass, okay? So let me write an F if I can there. Okay? That's F. Now, what we are seeing in this unsteady description of the problem, the description of the unsteady problem is that the result of the fluxes, the net flux into that volume element and the effect of the source dom they're combined effect is to change either the temperature for unit time. Right? With respect to time either the concentration changes, or the temperature changes, with respect to time. Okay, so let's also add in here essentially a, just to portray this, let me say that there is a d u, with respect to d t dom also coming up, right? T of course is time, okay? So, the strong form is the following. As always, given the data, given, I think we were still calling it Ug back then, we had Jn, which is our influx condition. Our source f, right? And the constitutive relation that we are now very familiar with, All right, using coordinate notation kappa i j being the conductivity tensor, right? Given all of this, now, we have one extra piece of not quite detail, but it really is a coefficient that is relevant to the problem. And I want to put it down here to have relevance to make connection with the physical problems that we are trying to keep at the back of our mind. Heat conduction and as diffusion problem. That quantity is going to be denoted as rho. Okay. I'll tell you once we set up the problem what rho is. Okay. So given all of these, what we're trying to do, is the following. Find u, okay, such that the following holds, right. Rho, partial of u, with respect to time, equals minus ji comma i plus f in. Now, here is this, this part is important. When you are doing the steady problem you specify the, all our steady, all our previous problems were time independent. Right? They were all steady state problems. We specify the PDEs therefore only over a spacial dimension, over a spacial domain omega, right? Subset of R 3, in general, in the 3D case. But now we have time dependence as well. So we say that this PDE holds in a combination of the spatial dimension, or the spatial domain and the time interval of interest. And that is indicated by a cross 0 comma capital T, okay? So the closed 0 to capital T is our time interval of interest. 'Kay? And when we write omega across that time interval we are just saying that our PDE holds over a certain spacial domain omega. And over a time interval 0 to T. Okay? All right. As before we have boundary conditions. We have u equals ug on the Dirichlet boundary. We have our Neumann condition, minus ji, ni equals j sub n on partial omega j. Is our problem complete with specifying boundary conditions? No. We need initial conditions as well. Alright? Because it's a first order problem in time, we have a single initial condition. Okay? And the way we do that is to say that U, which can be a function of position. It is indeed in general a function of position and this is what we saw in our steady state problems. Right. So, we have U as a function of, as, as, as parametrized we have position and at time t equals 0, okay? Equal sum U. Not function of position only, okay? All right? And perhaps this is best clarified by also saying here that U is a function of position. And time. Okay? All right. That indeed does complete the specification of our problem. PDE boundary conditions and initial conditions. What I'm going to do here is just make one or two remarks. Okay? The first remark is that, we need to say something about this new coefficient we've introduced, rho. Okay? For a heat conduction problem And for heat conduction problems, can you tell me what rho is? Yeah. For heat conduction problem, rho would be the, rho is the specific heat. [SOUND] Okay? And, and in the case of heat conduction, do you also know where our p d comes from? What, what physical principle leads to our PDE? It's actually the first law of thermodynamics. Okay? So in that setting rho is the specific heat, okay, and the way we've written it, rho would be the specific heat per unit volume. Okay? So it does as we went you know row with specific heat per unit volume. The specific heat also can be determined as a can also be defined per unit mass. Okay but in our in our setting for the way we've set up the problem. Loads the specific heat per unit volume. Right? It turns out that if however we were looking at the the mass diffusion problem. Rho is equal to 1. Okay? We don't need a notion of spe, of specific of anything like a specific heat in the context of mass diffusion. Mass diffusion just rises from a, physical principal, which is, the conservation principal. Okay, so that is the setting for, for, for, for this problem, sorry, that is the, sort of, setting of context for For, for the physical problems. Okay, so let me see. Is there anything else we need to really talk about here? Actually I believe not. So, we have laid down our strong form of the linear parabolic PDE in scalar varia, in a scalar variable in 3D. It connects up with our heat conduction, mass diffusion. Physical problems and [COUGH] we'll end the segment here. When we return we will do the usual take the usual steps that we've taken before right? The weak form and then talk about the the finite element formulation.
