MrsDrPoe: Conservation of Mass, Part I

Welcome to Thesis Tuesday on the blog! Today we'll be introducing one of the physical conservation laws- conservation of mass.

For those of you unfamiliar with this law, when you hear the word "conservation," what does it make you think of? An activist yelling "save the trees" or a family trying to use as little electricity as possible to save money? These ideas are similar but not exactly what is conveyed in the law of conservation of mass. This law states that mass can not be created or destroyed; it can only be transformed (unless you're the One True and Living God who can do whatever He wants). If we think about our life experiences, this concept makes sense...

Let's look at an example: suppose I wanted to make some brownies. The box says I'll need to combine water, oil, eggs, and the brownie mix in a bowl. When I put each of these ingredients in and stir, each ingredient is still present; the mass has "become" something else, but every bit that I put in the bowl remains. Although many of us would like it, there isn't some black hole in the bowl that takes the fat/oil out while I stir.

If we examine this concept mathematically, the law of conservation of mass tells us that the total mass in a control system is always the same, or there is no change in the mass in a control system. But as fluid folks we don't want to know what happens in a control system...we'd much rather know about the control volume (CV). To see what this law means for CVs, we employ the handy-dandy Reynolds Transport Theorem (RTT) to get:

Dm/Dt = d/dt(int(rho)dV) + int(rho*(V.n))dA

Here m is the mass; rho is the density; V is the velocity vector; and n is the unit outward normal vector. The left side of our equation is the change of mass in a control system. The first term on the right side of the equation represents how the mass in the control volume changes with respect to time; the second term on the right side represents the net rate of mass flowing across the control surface (into/out of the control volume). Since we just said that the change of mass in a control system is zero because of the conservation of mass, the equation becomes:

0 = d/dt(int(rho)dV) + int(rho*(V.n))dA

This is known as the integral from of the conservation of mass or the continuity equation (int(...)dV means we are integrating over a volume; int(...)dA means we are integrating over a surface or area). In differential form, the equation is:

0 = d(rho)/dt + d(rho*u)/dx + d(rho*v)/dy + d(rho*w)/dz

Again rho is the density of the fluid, and u, v, and w are the scalar x-,y- and z-velocities.

To recap:
Mass is conserved ALWAYS- it cannot be created nor destroyed.
We can use this fact and the RTT to come up with the continuity equation.
There are two main forms of the continuity equation.

Let these concepts brew for a week, and we'll come back to them next Thesis Tuesday. Have a great day!