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If U = free stream velocity, u = velocity at y, and δ = boundary layer thickness, then in a boundary layer flow, the momentum thickness θ is given by
1. \(\theta = \mathop \smallint \limits_0^\delta \frac{u}{{{U_\infty }}}\left( {1 - \frac{u}{{{U_\infty }}}} \right)dy\)
2. \(\theta = \mathop \smallint \limits_0^\delta \frac{u}{{{U_\infty }}}\left( {1 - \frac{{{u^2}}}{{U_\infty ^2}}} \right)dy\)
3. \(\theta = \mathop \smallint \limits_0^\delta \frac{{{u^2}}}{{U_\infty ^2}}\left( {1 - \frac{u}{{{U_\infty }}}} \right)dy\)
4. \(\theta = \mathop \smallint \limits_0^\delta \left( {1 - \frac{u}{{{U_\infty }}}} \right)dy\)

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Correct Answer - Option 1 : \(\theta = \mathop \smallint \limits_0^\delta \frac{u}{{{U_\infty }}}\left( {1 - \frac{u}{{{U_\infty }}}} \right)dy\)

Displacement thickness

\({\delta ^*} = \mathop \smallint \limits_0^\delta \left( {1 - \frac{U}{{{U_\infty }}}} \right)dy\)

Momentum thickness (θ) \( = \mathop \smallint \limits_0^\delta \frac{U}{{{U_\infty }}}\left( {1 - \frac{U}{{{U_\infty }}}} \right)dy\)

Energy thickness \( = \mathop \smallint \limits_0^\delta \frac{U}{{{U_\infty }}}\left( {1 - \frac{{{U^2}}}{{U_\infty ^2}}} \right)dy\)

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