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When problems overwhelm our ability to model them, dimensional analysis allows us to cut through the mental block and establish some quantitative insight without thinking.

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As shown in the above figure, a liquid with density \( \rho \) and viscosity \( \mu \) flows through a pipe with diameter \( d. \) In a section of the pipe with length \( L, \) the liquid flows with a speed of \( v. \) Using the Buckingham \(\pi\) theorem, determine the pressure difference \( \Delta P = P_1 - P_2 \) in terms of the fluid properties \( d, L, \rho, \mu \) and \( v .\)

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The sand timer in the above figure has a due time of \( T. \) The radius of the hole is \( r, \) the initial height of the sand is \( H, \) and the density of the sand is \( \rho. \)
Using the Buckingham \(\pi\) theorem, determine \( T \) in terms of the properties \( r, H, \rho, \) and the gravitational constant \( G .\)

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As shown in the above figure, a liquid with density \( \rho \) flows through a pipe with diameter \( d. \) In a section of the pipe with length \( L, \) the liquid flows with a speed of \( v_1 \) at the center and \( v_2 \) at the edge of the flow. The pressures are \( P_1 \) at the center and \( P_2 \) at the edge. Using the Buckingham \(\pi\) theorem, express the viscosity \( \mu \) of the fluid in terms of the liquid properties \( d, L, \rho, \Delta v = v_1 - v_2 \) and \( \Delta P = P_1- P_2 .\)

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