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Math Help - Fluid mechanics problem

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    Fluid mechanics problem

    I have a fluid mechanics problem where a fluid passes through a pipe of sinusoidal profile. I solved the problem by numerical integration and found a definitive and physically meaningful answer. However, when I solved the problem analytically and obtained the closed form flow equation, the expression which involves trigonometric and inverse trigonometric functions makes no sense, because when I substitute the two limits (i.e. -L/2 and L/2 where L is the length of the tube which expands over a complete wavelength, that is full sinusoidal cycle), some terms turned out to be undefined as they involve tan(pi/2) while others are zero as they involve sin(pi). I checked the analytical expression again and again and concluded it is technically sound on pure mathematical ground. Is this an artefact of trigonometric functions? How can this be reconciled with the physical reality and numerical results?
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  2. #2
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    Quote Originally Posted by Contributor View Post
    I have a fluid mechanics problem where a fluid passes through a pipe of sinusoidal profile. I solved the problem by numerical integration and found a definitive and physically meaningful answer. However, when I solved the problem analytically and obtained the closed form flow equation, the expression which involves trigonometric and inverse trigonometric functions makes no sense, because when I substitute the two limits (i.e. -L/2 and L/2 where L is the length of the tube which expands over a complete wavelength, that is full sinusoidal cycle), some terms turned out to be undefined as they involve tan(pi/2) while others are zero as they involve sin(pi). I checked the analytical expression again and again and concluded it is technically sound on pure mathematical ground. Is this an artefact of trigonometric functions? How can this be reconciled with the physical reality and numerical results?
    Why not show us some of your mathematics so we know exactly what your talking about.
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    Quote Originally Posted by Danny View Post
    Why not show us some of your mathematics so we know exactly what your talking about.
    It is very lengthy!
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    MHF Contributor ebaines's Avatar
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    Without seing any details all I can suggest is that the domains for an inverse function may not be the exact same as the range of the function itself. Hence arctan(tan(x)) may not equal x. For example if  x = \pi, then \arctan(\tan(x)) = \arctan(\tan(\pi)) = \arctan(0) = 0 \ne x.

    Another thought - sometimes mathematical techniques fail due to subtlities at singularities, such as when we unwittingly try to multiply 0 times infinity. Perhaps your mathematical technique has such an issue? A numerical technique may skip right by such a singularity without even noticing.
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    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by Contributor View Post
    I have a fluid mechanics problem where a fluid passes through a pipe of sinusoidal profile. I solved the problem by numerical integration and found a definitive and physically meaningful answer. However, when I solved the problem analytically and obtained the closed form flow equation, the expression which involves trigonometric and inverse trigonometric functions makes no sense, because when I substitute the two limits (i.e. -L/2 and L/2 where L is the length of the tube which expands over a complete wavelength, that is full sinusoidal cycle), some terms turned out to be undefined as they involve tan(pi/2) while others are zero as they involve sin(pi). I checked the analytical expression again and again and concluded it is technically sound on pure mathematical ground. Is this an artefact of trigonometric functions? How can this be reconciled with the physical reality and numerical results?
    Check your results by finding the normal depth via the chezzy manning equation, the critical depth, and by using the momentum equation on the wave. Continuity and bernoulli will also need to be used here.

    See what you get.
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