Modeling a waterfall

Question

I'm modeling a waterfall, the water streams out of a rectangular box with a width of $\text{W}$, a length of $\text{L}$ and a hight of $\text{H}$, so the volume of the water that is streaming from a higher point to the lower point equals $\text{V}_\text{water}=\text{W}\times\text{L}\times\text{H}$. To calculate the acceleration of the water I used:

$$\text{a}(t)=\frac{\text{d}\text{v}(t)}{\text{d}t}=\frac{\text{d}^2\text{x}(t)}{\text{d}t^2}=\frac{\gamma}{\text{m}}\times \text{v}(t)^2-\text{g}$$

Where $\gamma$ is the drag coefficient (in kg/m) and $\text{m}$ is the mass of the water and $\text{g}$ is the acceleration due to gravity.

Question: Am I right about this equation?

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For example, when I want to find $\text{x}(t)$ (all the initial conditions I assmumed to be zero). I have 6kg of water and the drag coefficient of water I set $7.0\cdot10^{-6}$ and $\text{g}=9.81$. Then I find for $\text{x}(t)$ (when I plot it): But the distance is way to high for such a short time. So where am I going wrong? (the horizontal is in seconds and the vertical in meters)

Answer 1

First of all, I think $x(0) \neq 0$ even though you claim that all the initial conditions are assumed to be zero. I think you forget to put in the value of $x(0)$ because the value of $x(t) - x(0)$ seems reasonable. In addition, $v(0) = 0$ and $v(\infty)$ seems to approach a constant, which seem reasonable for your model. However, next time, please provide a solution with variable, not numerical, so other people can understand what you try to ask better.

To comment on your model, it seems to suggest that the waterfall behave like a ball free falling but with air resistance. For a very rough estimate, this may be fine. I suggest you think about fluid dynamics, like Bernoulli's equation or Navier-Stoke's equation. Nevertheless, I think your question on modeling waterfall is interesting and quite complicated. I don't know how to model it if waterfall is not a laminar flow. Waterfall splash into small drops of water and fall down like a rain? What would the radius of those drops be? I wonder if there are experimental results to show the speed of waterfall at different height.