|
| What is a fluid? At this stage let's stick to our natural interpretation of a fluid. Let's use our intuitive notion of the fluids as a first tool. Let's use this notion to guide our progress towards a mathematical description of what a fluid is. Later on, when certain mathematical background has been created, we will identify, classify, and analyze all the resulting mathematical tools, we will also make sure they, to some extend, coincide with our natural notions of what we commonly refer as to a fluid, and moreover, we will be able to extrapolate to regimes where our experience has not yet been. This, in principle is what a physical theory is. We start from daily notions, we create a mathematical background that describes it, and then we extrapolate this theory to make predictions of unseen, or untested scenarios. | ||
So, lets start with the question of how to describe a fluid. Historically there are two main descriptions. The one postulated by Euler, and the one by Lagrange. Both have the power to give you a full description of a fluid ,the difference is conceptual. I would like to make the analogy with a graph. By a graph I mean the collection of points $\{x,y\}$ in the set $R^2$, such that $y=f(x)$. This collection of points can be fully described by specifying the x-coordinate and the y-coordinate of each point. However there is another, perhaps less natural, way to do so. It is as follows, we will specify the slope and the intersect with the y-axis of every line which is tangent to f at the point of interest. The envelope of this family of tangent lines is our graph ''$f$''. Many other examples can be found in Physics and Math See for example: http://web.uvic.ca/~chem458/Handouts/Handout1_Lengendre_Transform.pdf From this we see that both ideas are equivalent and that through the appropriate transformation (Legendre Transformation) we can go from one to the second. In the case of fluids , the situation is very similar; The Eulerian description is equivalent to the Lagrangian description and the mathematical transformation between these two is called the material derivative. Lets explain these three concepts. | ||
| Let's imagine a fluid as a group of little "blobs", each moving with certain velocity, and each having certain mass. We will call them fluid particles. Now, if I tell you how the position and velocity of each fluid particle evolves in time, it will be a full description of the fluid. This way of describing a fluid is called the Lagrangian description. In other words, we put little ''receptors'' on each fluid particle, and these receptors send us the position and the velocity of the particles at any desired time. -----In the Lagrangian picture, we follow the fluid particle------ | ||
Now, another way to describe the fluid is as follows, we put these receptors fixed to the ''floor'', and at any desired time they will send the velocity of the nearest fluid particle. It seems at first that these description offers less information than the previous one. But it doesn't. It surely looks as if the Lagrangian description offers both position and velocities, and the new description, only offers velocity. However, the fact that we know where the transmitters are, save us the need to specify their location. We know beforehand where they are, and since they are fixed to the ''floor'', their positions won't change in time. These second description is called the Eulerian description of a fluid. ------In the Eulerian picture, we study a fixed geometrical point, letting the fluid particles "flow" above it---- | ||
| Both descriptions are equally valid, they both offer the same amount of information about the fluid, and they both have pros and cons. | ||
| As a first example, let's imagine for example a flow of fluid particles in some direction (don't worry about the details of the flow, at this point I'm only concerned with a qualitative exercise). Now lets imagine that the temperature of the fluid is a function of the position and the time i.e. the fluid particles at some instant (x,t) will have a temperature T(x,t). Now, since the fluid particles are moving/flowing, their temperatures are changing too. Now, if we want to specify the change of rate of the temperature in the Eulerian description, we write: | ||
\[ EulerianChange = \frac{\partial }{\partial t}\] | ||
| This term only contains the partial derivative of T. The reason, in my own words is as follows: We don't need to specify the change of T in space because we know, from our imaginary receptors, which are fixed, which point we are referring to. The change of T in space is zero in the Eulerian description. | ||
Now, from the Lagrangian description the change of T looks a bit more complex. I mean, if you are fixed with the fluid particle, and look at a the floor, you need to specify not only the change of T in time at your fluid particle, but also how your position changes in time. So, from the Lagrangian perspective, we write: | ||
\[ LagrangianChange= \frac{d}{dt}\] | ||
Material Derivative:
Let's compute the change in T a bit more formally. That is, taking the total derivative of T with respect to time, and let's see what we get. | ||
\[\frac{\Delta T(x_1,x_2,x_3,t)}{\Delta t}=\frac{T(x_1+\Delta x_1,x_2+\Delta x_2,x_3+\Delta x_3,t+\Delta t)-T(x_1,x_2,x_3,t)}{\Delta t}\] | ||
| Using the Taylor expansion, and considering $\Delta x_i$ small enough so higher derivatives are neglected. We obtain: | ||
\[\frac{\Delta T(x_1,x_2,x_3,t)}{\Delta t}=\frac{\partial T}{\partial t}\frac{\Delta t}{\Delta t}+\frac{\partial T}{\partial x_1}\frac{\Delta x_1}{\Delta t}+\frac{\partial T}{\partial x_2}\frac{\Delta x_2}{\Delta t}+\frac{\partial T}{\partial x_3}\frac{\Delta x_3}{\Delta t}\] | ||
| Talking the limit $\Delta t \rightarrow 0$ and realizing that $\frac{\Delta x_i}{\Delta t}\rightarrow u_i$ where $u_i$ is the $i^{th}$ component of the velocity. We can write | ||
\[\frac{dT}{dt}=\frac{\partial T}{\partial t}+\vec{u}\cdot \nabla T\] The last equation can be seen as both, the total derivative of T with respect of time, or as the transformation needed to go from the Lagrangian description to the Eulerian one, and vice-versa. This is the link between the two descriptions, the bridge between the two point of views. From the equation follows: \[LagrangianChange = EulerianChange + (\vec{u}\cdot \nabla)\] From now on we will adopt the following notation: \[\frac{D}{Dt}:=\frac{\partial }{\partial t}+(\vec{u}\cdot \nabla )\] And we will call the operator $\frac{D}{Dt}$ the material derivative. Some authors might call it the convective or Lagrangian derivative too. This operator can act upon any scalar or vectorial quantity (In the case of a vectorial quantity you have to be careful when applying $\nabla$, since the result is a tensor! ). And it tells you how such quantity changes in time as a function of the velocity, and the variation of such quantity in a given fluid particle. Now we need to finish our mathematical background. The next big mathematical tool we will use is the so called Reynolds transport theorem. In order to demonstrate this theorem we will introduce two very important concepts, namely the concept of a control volume, and a fluid volume. This two new concepts are deeply related to the Eulerian and Lagrangian descriptions. In fact, I would propose to use the names Control-volume/Eulerian-volume, and Fluid-volume/Lagrangian-volume interchangeably. In the Eulerian description, where we attached receptors all over the floor, and get readings about the velocities of the nearest fluid particles, we can take the idea a bit further. We can place some holographic projectors on the floor, and project right in the middle of the fluid, a box, or a sphere, or any 3D object. Now this is our Control-Volume or Vc. The Vc has some particular characteristics, for example, it doesn't interact with the fluid regardless how many particles come in one side and how many go out the other, or how fast they do. It is simply an hologram and does not interact with the fluid. We can also make it expand or contract at any desired rate. We can make it go left or right, along the flow, or transverse to it....its motion is completely up to us. Now back to the Lagrangian description, where we track the fluid particles. In this model we will define the fluid volume as follows: We will freeze time. Then we will place the holographic projectors on the floor. Project any 3D body right in the middle of the fluid. We will then "paint" of a different color all the fluid particles inside this 3D body. Next we will turn off the holographic projector. And finally un-freeze the picture. The fluid volume is the volume that all the newly-painted fluid particles create. This Fluid-Volume, or Vf will flow along the rest of the fluid particles, it will deform, tore apart, put back together and so on. We don't care...we only care about keeping an eye on the fluid particles that were once part of the holographic image. Since we were very very smart and we painted them with a different color!, it's "easy" to keep track of them. The cool thing is this: at the instant when we froze time, lets say t=0 Vf was exactly the same as Vc. However, the Vf motion depends on the flow of the fluid while the Vc motion depends only on us. At the same time we see that, by definition, the number of fluid particles in Vf won't change, but the number of fluid particles inside Vc might!; They might compress so each will occupy less volume, and more fluid particles will fit inside Vc, or the opposite, they will expand and then less of them will fit inside Vc. If we make Vc bigger, more particles will fit into it...and so on. This being said, I will proceed to formulate Reynolds transport theorem: Reynold's Transport theorem (RTT): Let's $\phi$ be a scalar quantity, defined in some region $\Omega$. Then: \[\frac{d}{dt}\int_{V_f(t)}\phi(x,t)\,d\Omega=\int_{V_f(t)}\frac{\partial\phi(x,t)}{\partial t}\,d\Omega+\oint_{\partial V_f(t)}\phi(x,t) \left[\vec{u}_f\cdot\vec{n}\right]\,d\sigma\] Where $\partial V_f$ is the surface of Vf, and $\vec{u}_f$ is the velocity at which the boundaries of Vf is moving. $\vec{n}$ is the normal vector to the surface of Vf. The demonstration of this theorem is not very difficult, but since for a good understanding of it, a great deal of drawing is involved, I will omit it, and refer you to: Just keep in mind that they use the following notation: \[\phi\rightarrow b\rho\] \[\int_{V_f(t)} b \rho\,d\Omega\rightarrow B_{sys} \] \[\frac{d}{dt}\int_{V_c(t)}\phi\,d\Omega=\int_{V_c(t)}\frac{\partial\phi}{\partial t}\,d\Omega+\oint_{\partial V_c(t)}\phi [\vec{u}_c\cdot\vec{n}]\,d\sigma\] Finally we subtract both equations. We must also realize that \[\int_{V_c(t)}\frac{\partial\phi}{\partial t}\,d\Omega=\int_{V_f(t)}\frac{\partial\phi}{\partial t}\,d\Omega\] The reason for this is simple. Since the RTT holds at any time, we can just pick t=0 where our two volumes were exactly the same. Therefore these terms cancel out, and we are left with: \[\frac{d}{dt}\int_{V_f(t)}\phi\,d\Omega=\frac{d}{dt}\int_{V_c(t)}\phi\,d\Omega+\oint_{\partial V_c(t)}\phi\left[(\vec{u}_f-\vec{u}_c)\cdot\vec{n}\right]\,d\sigma\] We must realize that $\vec{u}_f$ is simply the velocity of the fluid $\vec{u}$ and so we drop the subindex. This result is super important!. I will say, it's the cuspid of our mathematical background. Now that we have this, and the material derivative we can start doing some physics!. |
No comments:
Post a Comment