Disclaimer: I am a programmer, not a mathematician. Apologies if anything is unclear, I'm happy to provide additional information or update as needed.

Consider a 2D region of space defined as $$ D = \left\{\left(y, z\right)~|~ y~\epsilon~Y, z~\epsilon~Z \right\} $$ where $$ Y = \left\{y~|~0 \leq y \leq B\right\} $$ $$ Z = \left\{z~|~ z_0(y) \leq z \leq H \right\} $$ and $z_0(y)$ is a single-valued continuous function describing the lower boundary of the region. Examples of this region could be a rectangle, a trapezoid, or some other arbitrary shape. In this example, the region represents water in a river channel.

Assume each location $\left(y, z\right)~\epsilon~D$ has a concentration $C(y,z) \geq 0$. At the bottom boundary of the region, we know that $$ C\left(y, z_0(y)\right) = C_b(y) $$ while at the upper boundary of the region $$ C(y, H) = 0 $$ The concentration at each location $\left(y, z\right)~\epsilon~D$ is defined as $$ C(y, z) = C_b(y)~C^*\left(\zeta\right) $$ where $\zeta = \frac{z}{H - z_0(y)}$ and the function $C^*$ is a normalized concentration function that describes the shape of the concentration profile in $z$. $C^*$ is monotonic and $$ 0 \leq C^* \left(\zeta\right) $$ Note that by definition $$ C(y, H) = C^*\left(1\right) = 0 $$ and $$ \frac{C\left(y, z_0(y)\right)}{C_b(y)} = C^*\left(\zeta_0(y)\right) = 1 $$ where $\zeta_0(y) = \frac{z_0(y)}{H - z_0(y)}$. Finally, assume that the quantity $$ m = A\bar{C} = \int_0^B \int_{z_0(y)}^H C(y, z)~dz~dy $$
is known. We can potentially assume additional constraints, e.g. that $C_b(y)$ is continuous.

My question: Is this system sufficiently constrained to compute the value of $C_b$ for all $y~\epsilon~Y$, i.e. is there a unique function $C_b(y)$ that satisfies the system? If so, how should I approach solving this system numerically? If there is no unique solution, what additional information (other than $C_b(y)$ itself) would be needed to sufficiently constrain the system?


1 Answer 1


My intuition is that additional constraints are needed in order get a unique value for the system. I think I have a solution based on the additional assumption that the value of $C_b(y)$ is linearly proportional to the normalized concentration at $y$, i.e. $$ C_b(y) = C_b^*C^*\left(\zeta_0(y)\right) $$ where $C_b^*$ is some constant. Then $$ m = \int_0^B \int_{\zeta_0(y)}^1 C_b^* C^*(\zeta_0(y)) C^*(\zeta)~d\zeta~dy = C_b^* \int_0^B C^*\left(\zeta_0(y)\right) \int_{\zeta_0(y)}^1 C^*(\zeta)~d\zeta~dy $$ We can the solve for $C_b^*$ as $$ C_b^* = \frac{m}{\int_0^B C^*\left(\zeta_0(y)\right) \int_{\zeta_0(y)}^1 C^*(\zeta)~d\zeta~dy} $$ Once you know $C_b^*$, you can compute $C_b(y)$ and therefore $C(y)$.


I think that there may be an incongruity with the definition of $C^*$ that makes this setup impossible. I try to require that $C^*(\zeta_0) = 1$, but since $\zeta_0$ varies in $y$ it means $C^*$ cannot logically be a single-valued continuous function of $\zeta$ only. The shape of $C^*$ is fundamentally different for each $y$ since the bottom boundary $\zeta_0$ is a function of $y$. What I need is some way to transform the interval $[z_0, H]$ to $[0,1]$ and define $C^*$ such that $C^*(0) = 1$ and $C^*(1) = 0$.


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