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September 15th, 2014, 05:10 PM  #1 
Senior Member Joined: Jan 2014 Posts: 196 Thanks: 3  Understanding Conservation Law with no Sources
can someone elaborate on $\displaystyle u_t + \phi_x = 0$ , the fundamental conservation law. Is this describing a scenario where density is changing locally, but does not leave the boundary area, hence no sink or source term? Thank you 
September 16th, 2014, 01:33 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 1,882 Thanks: 609 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
The above equation is called the continuity equation and it basically states that given some fixed volume, the change in the amount of some conserved quantity (for example, mass) in the volume is equal to the amount passing through the boundary of that volume. Generally it is taken that a positive flux is for mass/whatever leaving the volume. For example, let's say I take some stuff out of a box. If the rate of change of mass ($\displaystyle u_t$) in a box is 1kg per second, then the law states that I have a flux out of the lid of the box ($\displaystyle \phi_x$) of 1 kg per second. You are right to say that the average density of mass inside the control volume would change, but you are not obligated to use the continuity equation to describe how it's internal components behave. You'd want to typically do this for diffuse systems (such as mass transport of gases/plasma or heat transport), where you would apply it everywhere. If you only care about the boundary of a container/geometry, you can apply the law at the boundary only if you wish. You are also correct to state that there is no sink/source term in the equation. Last edited by Benit13; September 16th, 2014 at 01:46 AM. 

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