I need to solve in the explicit form the PDE of the depth of the current of fluid moving in a non uniform inclined channel under gravity effect. See the attached file.
You certainly do not need to solve this equation since it is self-contradictory. Working with a local fluid depth h(x,t) means that the ground is flat in y-direction. If the ground, however, is curved along y-direction as for 'a circular cross‐section boundary' the flow will develop (as walking at a rainy day will show you wherever you look) a depth which also depends on y. For this dependence your equation gives no local law. So, you certainly have to rework your model equation before you can solve it.
The model is based on shallow water approximation and the height h(x,t) is invariant in the spanwise direction, see Fig. 1 in the attached file. The model itself is correct and has been experimentally verified in numerous cases. The model I presented in my question is a modification of the model adopted in the attached paper in Section 2.1.2, with an additional forcing term. Without the forcing term, I can find out a solution with the method of characteristic, which is verified by the experiments. They are both reported in the paper. Thank you for requesting the clarification.
Dear Sandro.
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I did not look with the precaution that merits the passage of the equation (9) to the equations (15) and (28), but I worried with the dimensionality of its terms. Approximate solutions for numerical calculation sometimes contain terms out of the same dimension, but in the case of an analytical solution that is not possible.
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Maybe not I have not done the analysis of the coefficients properly, I will try to look more carefully.
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Dear Rogerio,
the equation is intended expressed in non-dimensional form after choosing the proper length and time scales. An example of non-dimensional formulation of a similar equation is reported in the manuscript attached to my answer. Thank you.
Hello Sandro: the main difficulty for your problem is that you have the variable "h" in the right hand term of the equation, and "h" is your unknown variable.
Without "h" in righhand side of the equation ( but with a f(x,t) function) , you could solve this equation with the method of characteristics.
I really do not know how to have an explicit form of the solution.
Have you tryied to write it in an non dimensional form? may be the analytical solution will be easier ? good luck!
Considering your problem, you may ask Hubert Chanson from University of Queensland to help you. May be you know him. I know him well, we collaborate together. I have his mail. He works on this kind of problem ( but not with a cylindrical bottom...)
best regards
hubert
Dear Hubert, thank you for the suggestion. The solution is possible in implicit form, I was looking for an eventual (if any) solution in explicit form. The equation is already in non dimensional form. Thank again, I already have the email of Prof. Chanson. Sandro
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