Modeling: Light irradiance and spatio-temporal decay

[herbeeb]

Hello everyone!

1. Homework Statement
I want to model the irradiance on a fluid element while it is flowing, from a fixed(above or below) light source that has an irradiance that shows first order decay with time and from x-naught to x-infinity .

here are the assumptions:
-Laminar flow at fixed rate (in x-direction)
-fluid element in fixed y-value from light source (above or below, doesn't matter.
-reflection/scatter effects are insignificant
-flow is in a tube (differential irradiance in y-direction)

Homework Equations

first order decay.

The Attempt at a Solution

I would like some direction regarding the best approaches to model such a situation, and am comfortable with solving differential equations. Thank you so much!

 

[KataKoniK]

Hello! This is a great question and it sounds like you have a good understanding of the assumptions and parameters for your model. To start, I would suggest looking into the Beer-Lambert Law, which describes the relationship between the concentration of a substance and the amount of light that can pass through it. In your case, the "substance" would be the fluid element and the "light" would be the irradiance from the fixed source.

Next, you can use the equation for first order decay to incorporate the time decay of the irradiance from the light source. This will give you a better understanding of how the irradiance changes as the fluid element flows through the tube.

To incorporate the laminar flow and fixed rate in the x-direction, you can use the Navier-Stokes equations, which describe the motion of a fluid. These equations can help you determine the velocity and flow rate of the fluid element, which will affect the irradiance on the element.

Finally, to incorporate the differential irradiance in the y-direction, you can use the equation for irradiance as a function of distance from the light source. This will allow you to see how the irradiance changes as the fluid element moves along the y-axis.

Overall, your approach should involve combining these equations and using them to create a model that can accurately predict the irradiance on the fluid element as it flows through the tube. I hope this helps and good luck with your modeling!