What is the context of the problem? In other words, how did this problem arise - how was it formulated?
Hi Subhotosh Khan, hi everyone,
thank you all for responding and asking the right questions. Sorry to keep you waiting.
I realize it was a long shot hoping to receive adequate input while submitting this problem without context, but I was restricted in available time.
Further on, I will go into more detail on the formulation using more and other names/symbols for certain parameters, so please
disregard the equations shown in my previous posts as I simplified/generalized them.
Anyway, I will try to concisely explain the physical mechanism this problem arises from:
It's all about soil mechanics and hydraulics and their interaction. See first image below. There's a small sketch showing a longitudinal (in the direction of the filling progress) profile along a layer of soil gradually placed from a pumped soil-water mixture from left to right. The soil particles consist of 2 grain sizes: sand (coarse) and silt (fine). The sand will settle rather immediately from the mixture and the process is controlled such that the height of that sand layer equals 'H'. On the other hand, the silt takes more time to settle on the bottom and flows further and is here assumed to settle equally (same thickness) along a distance = D
s. As the process evolves, the sand will be placed on top of the increasing layer of settled silt, burying it. this leads to a buried thickness (height) h
b, which is smaller than h, the height of the silt settled just in front of the progressing sand layer. This is due to a change in the volume of pores between the particles, expressed by a so-called bulking factor B
F.
Now, what I'd like to know is an expression that represents the unburied height of the silt layer (at the sand front) for a certain progress of the filling process. In short: 'h' as a function of 'x'. In the picture below you can find my formulations of the corresponding math. I'd like to think this is a solid derivation of the described process. Apart from h
b, h and x all symbols represent constants (e.g. bulking factors and fine to coarse ratio).
The formulation of the differential equation is based on the fact that, apart from factors B
u and F, both green hatched areas (incremental volumes of sand vs silt) have the same size. For every 'dx', a 'dh' is added to 'h'.
HOWEVER, above formulation is only valid as long as x < D
s. This is due to the fact that, from that point onwards, 'h' is not the sum any more of all previously settled dh. This is what the red text aims to point out. Although too simplified, the problem i want to convey is demonstrated in the small sketch below:
Hence my attempt to again subtract the 'dh' from previous steps in the differential equation. It cannot be as simple as subtracting h(x - D
s) from h(x).
Based on numerical calculations, the graph of the function should be something like this:
I expect a periodic function much like a damped oscillator, eventually converging to a certain value...
So please, enlighten me
Thanks!