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Math Help - Fluid Mechanic Integration

  1. #1
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    Fluid Mechanic Integration

    I have a fluid mechanics question, i don't know which equation i should integrate and how?

    For turbulent flow in a smooth, circular tube with a radius R, the velocity profile varies according to the following expression at a Reynolds number of about 10^5.

    Vx= Vxmax * [(R-r)/R)]^(1/7)

    where r is the radial distance from the center and Vmax the maximum velocity at the center. Derive equation relating the average velocity ( bulk velocity ) Vav to Vmax for an incompressible fluid.
    ( Hint: The integration can be simplified by substitution z for R-r )
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  2. #2
    MHF Contributor chisigma's Avatar
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    Is...

    \displaystyle v_{m} = \frac{v_{0}}{R} \ \int _{0} ^{R} (1-\frac{r}{R})^{\frac{1}{7}}\ dr (1)

    ... that setting x=\frac{r}{R} becomes...

    \displaystyle v_{m} = v_{0} \ \int_{0}^{1} (1-x)^{\frac{1}{7}}\ dx = \frac{7}{8}\ v_{0} \ |-(1-x)^{\frac{8}{7}}|_{0}^{1} = \frac{7}{8}\ v_{0} (2)

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by chisigma View Post
    Is...

    \displaystyle v_{m} = \frac{v_{0}}{R} \ \int _{0} ^{R} (1-\frac{r}{R})^{\frac{1}{7}}\ dr (1)

    ... that setting x=\frac{r}{R} becomes...

    \displaystyle v_{m} = v_{0} \ \int_{0}^{1} (1-x)^{\frac{1}{7}}\ dx = \frac{7}{8}\ v_{0} \ |-(1-x)^{\frac{8}{7}}|_{0}^{1} = \frac{7}{8}\ v_{0} (2)

    Kind regards

    \chi \sigma
    but the result is (49/60)* Vmax, we should use hint, i think
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  4. #4
    MHF Contributor chisigma's Avatar
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    Effectively I tried to symplify the problem integrating in the only r variable ... integrating in the r and \theta variables it results...

    \displaystyle  v_{m} = \frac{v_{0}}{\pi \ R^{2}}\ \int_{0}^{2 \pi}\ \int_{0}^{R} (1-\frac{r}{R})^{\frac{1}{7}}\ r \ dr \ d\theta (1)

    ... that setting again x= \frac{r}{R} becomes...

    \displaystyle  v_{m} = \frac{v_{0}}{\pi}\ \int_{0}^{2 \pi}\ \int_{0}^{1} x\ (1-x)^{\frac{1}{7}} \ dx \ d\theta=

    \displaystyle = 2\ v_{0}\ \int_{0}^{1} x\ (1-x)^{\frac{1}{7}}\ dx =

    \displaystyle = 2\ v_{0} \ \frac{7}{8}\ |x\ (1-x)^{\frac{8}{7}} - \frac{7}{15} \ (1-x)^{\frac{15}{7}}|_{0}^{1}= \frac{49}{60} (3)

    Kind regards

    \chi \sigma
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