Hi, need to submit a 1500 words essay on the topic Project 1: Transport of a pollutant . course(Mathematical.Download file to see previous pages... Also, we’ll only do this in one dimension. The con

Hi, need to submit a 1500 words essay on the topic Project 1: Transport of a pollutant . course(Mathematical.

Also, we’ll only do this in one dimension. The conservation of mass in one dimension can be written quite simply as: which can be stated in words as “the time rate of change of material in a fluid parcel is equal to the spatial rate of change of the (molecular) diffusive flux plus any in situ production”. Note that we’ve used different derivative forms on either side of the equation. The first term (the one on the left hand side) is usually called the Lagrangian derivative or the “complete derivative”, since it quantifies the time rate of change following the fluid parcel. We must express our conservation law in this form since it is only from the perspective of the fluid parcel that we can guarantee the conservation of mass. The second term is simply the molecular diffusive flux divergence (k is the molecular diffusivity). That is, mass will only accumulate (or deplete) in a fluid parcel if the amount of material diffusing into it from one side exceeds (or is less) than the amount diffusing out. In order for this to occur, the diffusive flux must change with distance. With the assumption that the molecular diffusivity is constant in space (a reasonable assumption in most systems) the above equation can also be expressed in the possibly more familiar Fick’s Law form as but we’ll stick the other form for now. ...

Of course the corresponding meaning or value of k would be different, but the relationship would be the same. Now it is more common to view things in a space-fixed frame of reference, rather than a framework which follows fluid parcels. This space-fixed frame is referred to as an Eulerian frame of reference, and we can translate to this frame of reference by taking partial derivatives: which simply states that the rate of change in stuff in a fluid parcel is the sum of the time variation in the overall distribution (the first term) and the downstream change of stuff associated with its displacement. A useful way to think of the difference between Eulerian and Lagrangian derivativesis to consider a weather analogy. Standing in your garden one evening you notice that the temperature is dropping. Is it due to local radiative cooling, or is a cold front coming in? Your friend in a hot air balloon passing overhead, since he or she moves with the air mass, feels du/dt (here we’re using u to mean the temperature), while you feel ?u/?t at your fixed point in space. Your friend might actually feel the temperature going up (maybe the sun hasn’t set just yet), but because the cold air front is moving in, you feel a decrease. Another way of saying this is to recognize that the change in stuff at a location is the sum of the time rate of change of the distribution summed with the divergence of the advective flux. The advective flux is just the concentration times the velocity. We can then rewrite our conservation equation as: The above equation is perfectly general and both mathematically and physically correct in all respects.