Can You Solve for Po to Determine a Shape That Satisfies Pe(t) = Pa(t)?

[Apophenia]

Note: I think I get too specific about the problem when what I am asking about a specific part of the whole problem.

The problem: Find a shape with volume V(t) and surface area Ab(t) such that Pe(t) = Pa(t).
In the interest of time I will not refer to what all variables signify; I will keep it mathematic unless you request something explained.

Im fairly certain I know how to go about solving most of what I am about to present but I would like some advice where noted.

To simplify I am going to define: dPo/dt = Po, dV/dt = V, dr/dt = r
Basically a bolded variable to signify any derivative with respect to time. I may not explicitly state that for other introduced variables so please note that.

Also constants will be presented as a lowercase in []; i.e., [a], [x]... Variables will be regular text and assumed variable with time unless otherwise noted: so Ab(t) = Ab. Also note that if two variables are side by side I will separate them by an operator to avoid Po(t) = Po being misconstrued as P*o.

Basically what I am trying to do is equate the two eventual sides of an equation making up Pe(t) = Pa(t).

For Pe(t) and the lefthandside(LHS) the governing equations are as follows:

1) V = rAb
2) r = [a]Po^[n]
3) Po+*(1/V)*Po - [c]*([d]-[e]Po)*(Ab/V)*Po^[n] = 0

We know the value of r(0) and 2) is just an empirical relation between r and Po.

Now ignore the right-hand-side of the equality and say that we already know this; Pa(t) is a known function.

For the LHS first we get Po and then Pe.

Pe is related to Po by: Pe = Po/[m]

My question is specific to solving for Po by finding a shape; namely we need to find two unknowns (Ab(t) and V(t)) that constitute one shape (out of an infinite number of possibilities) that can satisfy the equality.

Now when I do this I use Eulers method to "discretize" Po. We know Po(0)

The problem is incomplete with what's currently given because the variables with which we ultimately need are Ab and V? This whole problem is basically finding a shape (a shape per every interval of time that is; so it is a shape changing with time) that satisfies Pe = Pa. THIS IS WHERE IM LOOKING FOR SOME ADVICE. How can you resolve this incompletion in the most general sense? For example I know you can simply assume a cylinder and use:
4) Ab = 2*pi*r'*h
5) V = pi*(r'^2)*h

(the derivative of r wrt to time in the edits is not the bolded r just dr/dt)

We can get r' via dr/dt and knowing r'(0)... the r' here is radius while the r in dr/dt is referring to the distance at which the shape surface diminishes. Think of dr/dt as the rate at which the surface of the shape diminishes perpendicular to Ab so in the case of a cylinder (neglecting the ends of the cylinder) the radius r' at some time = the current radius - the current dr/dt*(time step).

These two satisfy 1)., taking the derivative wrt time of 5) and plugging into 4) gives 1). The initial values, Ab(0) and V(0) must satisfy 3).

I think I am walking around the answer to my question in what I just wrote however. I think 1) is the general sense and any equations such as 4) and 5) satisfying 1) are compatible.

Basically, I wanted to keep the solution general.

 

[jvicens]

Thank you for your detailed explanation of the problem you are trying to solve. It seems like you have a good understanding of the mathematical equations involved and are on the right track in finding a solution.

In terms of finding a shape that satisfies the equation Pe(t) = Pa(t), there are a few things to consider. First, as you mentioned, there are an infinite number of possible shapes that could satisfy this equation. Therefore, it may be helpful to narrow down the possibilities by considering any constraints or limitations on the shape. For example, is there a specific material or manufacturing process that the shape must adhere to? This could help guide your search for a suitable shape.

Secondly, it may be helpful to think about the physical properties of the shape, such as its volume, surface area, and curvature. These properties can also be expressed in terms of the variables you have defined, such as V(t) and Ab(t). By considering these physical properties, you may be able to come up with a more specific equation or set of equations that the shape must satisfy.

Lastly, it may be useful to consider using numerical methods, such as Euler's method, to approximate the shape. As you mentioned, this can help "discretize" the problem and give you a more concrete solution. However, be aware that numerical methods may not always give an exact solution and may require some additional iterations or adjustments to find a more accurate shape.

Overall, my advice would be to continue exploring different equations and properties of the shape and use numerical methods to help narrow down the possibilities. I hope this helps and I wish you luck in finding a suitable solution to your problem.A scientist