Oil Spill - Differential Equations

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SUMMARY

The discussion focuses on solving differential equations (DE) related to oil spill modeling. Key points include the flow dynamics where oil enters and exits cells, represented mathematically as $s_i(t)$ for the amount of oil in cell i. The rate of change of oil in each cell is defined by the equation $\frac{ds}{dt}= s$ with an initial condition of s(0)= S. Participants emphasize the importance of understanding the flow rates and their derivatives to accurately model the system.

PREREQUISITES
  • Understanding of differential equations (DE)
  • Familiarity with flow rate concepts in fluid dynamics
  • Basic calculus, particularly derivatives and limits
  • Knowledge of initial value problems in mathematical modeling
NEXT STEPS
  • Study the application of differential equations in fluid dynamics
  • Learn about numerical methods for solving differential equations
  • Explore initial value problems and their significance in modeling
  • Investigate real-world applications of oil spill modeling using DE
USEFUL FOR

Mathematicians, environmental scientists, and engineers involved in modeling fluid dynamics and oil spill scenarios will benefit from this discussion.

echianne
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Hello everyone. I hope anyone can help me with this problem. I will greatly appreciate it. Willing to compensate anybody to answer this problem correctly for me.
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echianne said:
Hello everyone. I hope anyone can help me with this problem. I will greatly appreciate it. Willing to compensate anybody to answer this problem correctly for me. View attachment 11832View attachment 11833
This is a free forum. We volunteer our time... payment is not necessary. However in order to help you best we need to have an idea of just where you are running into problems. What are you able to do with this?

-Dan
 
topsquark said:
This is a free forum. We volunteer our time... payment is not necessary. However in order to help you best we need to have an idea of just where you are running into problems. What are you able to do with this?

-Dan
Thank you so much. I have come up with different systems of DE but nothing seemed to work. I think there is something wrong with how I understood the problem.
 
echianne said:
Thank you so much. I have come up with different systems of DE but nothing seemed to work. I think there is something wrong with how I understood the problem.
What have you tried?

-Dan
 
The crucial information is in the last two sentences: "So at time t, the ith cell contains $s_i$ wL $ft^3$ of oil. Oil flows out of cell i at a rate equal to $s_i(t)$ wv $ft^3/s$ and flows into cell i at a rate equal to $s_{i-1}(t)$ wv $ft^3/s$. It flows into the first cell at rate S $ft^3/s$".

That is just saying that the rate oil flows into one cell is the rate at which it flows out of the previous cell.
The DERIVATIVE of the amount of oil ($s_i(t)$) at a given x (at a given cell) is the rate at which oil is flowing in minus the rate at which it is flowing out: $\Delta s_i(t)= (s_{i-1}(t)- s_i(t))\Delta t$.
$\frac{\Delta s_i(t)}{\Delata t}= (s_{i-1}(t)- s_i(t))= -(s_i(t)- s_{i-1})$.

Taking the limit as the length of each of each cell goes to 0, $\frac{ds}{dt}= s$ with initial value s(0)= S.
 

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