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I'm making laminar fluid flow devices and want to be able to calculate the velocity as a function of distance from the channel edges. As the channels are relatively wide compared to their height I'm treating the effect of the parabolic velocity profile as negligible in the horizontal plane (i.e. uniform velocity horizontally). I can measure the average velocity across a plane perpendicular to the direction of flow (from the flow rate), but can't seem to derive the expression that relates mean velocity and maximum velocity (which I believe should be u[mean]=0.5*u[max]). I'm sure I'm probably making a really basic mistake, but here's my working, which ends up giving me u[mean]=(2/3)*u[max]

Where u is the velocity as a function of position relative to the channel centre; r is the distance from the channel centre and R is maximum distance from the channel centre (i.e. the channel is 2*R) wide.

u=u[max]*(1-(r/R)^2)

integrate to give:

u[net] = u[max]*(r-(r^3)/(3*(R^2)))

evaluate between R and -R to give:

u[net] = u[max]*(4/3)*R

divide by the channel width to give the average velocity (u[mean])

u[mean] = 2/3*u[max]

If anyone could tell me where I'm going wrong I'd be really greatful, as I can't see why I don't end up with u[mean]=1/2*u[max], which is what I keep getting for the relationship when I look it up online.

Thanks in advance

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# Mean velocity for parabolic velocity profile

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