How does a motion sensor work?
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Answer:
Explanation:
First we should recall how Newton's laws relates shear stress to a fluid's velocity profile:
where tau is the shear stress, mu is viscosity, v is the fluid's velocity and y is the direction perpendicular to flow.
Now, in this case we have a parabolic velocity profile, and also we know that the fluid's velocity is zero at the boundary (no-slip condition) and that the vertex (maximum) is at and the velocity at that point is
We can put that in mathematical terms as:
From the no-slip condition, we can deduce that and so we are left with just two terms:
We know that the vertex is at and so we can rewrite the last equation as:
where k and h are constants to be determined. First we check that :
So we found that h was the maximum velocity for the fluid, now we have to determine k, for that we need to make use of the no-slip condition.
And thus we find that the final expression for the fluid's velocity is:
where v is in mm/s and y is in mm.
In SI units it would be:
To calculate the shear stress, we need to take the derivative of this expression and multiply by the fluid's viscosity:
for we have:
Which is our final result