For steady continuity a non-zero gradient du/dx in the flow will have to be accompanied by a non-zero dv/dy, hence v cannot be zero everywhere. I think you have made too many assumptions and the apparent conflict in your results is indicative of this. Once you reach fully developed flow, du/dx is zero and the pressure gradient is balanced by the viscous stress. Then v is zero. But in the development region you cannot make this assumption.
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What you are trying to do is the study of the developing boundary layer along a flat plate (top and bottom walls), so you have to use the Prandtl equations.
Just consider that the flow evolves until the thickness of the BL is equal to H/2 (H the distance between the two walls). You can use the law delta(x) to compute this lenght that depends on the viscosity of the fluid (or Re number). A FD solver can be used to check the theory. |
You've made way too many assumptions. Your solution is u=constant everywhere, which doesn't even make sense.
du/dy=0 is only at the inlet, not everywhere. This should be obvious in a pipe flow because u=something at the inlet but u=0 on walls because of the no-slip condition. From where do you get dp/dx=0 ? Even for a flat plate dp/dx is not zero and for a pipe it must eventually be a non-zero constant when the flow in the pipe is fully developed. From where do you get v = 0? https://www.cfd-online.com/Forums/vb...4ffa0f34-1.gif This is the advection-diffusion equation for a parallel flow (flows very small gradients in the y direction). If you solve this equation, you will indeed get a result that looks like a flow because it is an advection equation. But it has limitations. For learning finite differences this is actually a really good equation to try and solve because it contains non-linear advection on top of diffusion. Usually diffusion problems are 1st problem introduced. |
I think you should not make any assumption since the velocity is developing.
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