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Concrete is pumped from a cement mixer to the place it is being laid, instead of being carried in wheelbarrows. The flow rate is 200.0 L/min through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the pump is $8.00 \times 10^6 \textrm{ Pa}$ . Verify that the flow of concrete is laminar taking concrete's viscosity to be $48.0 \textrm{ N/m}^2\cdot\textrm{s}$, and given its density is $2300 \textrm{ kg/m}^3$.
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Final Answer

$N_R = 2.54 < 2000$. The flow is laminar.

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OpenStax College Physics Solution, Chapter 12, Problem 55 (Problems & Exercises) (1:32)

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Video Transcript

This is College Physics Answers with Shaun Dychko. The Reynolds number for the concrete is two times its density times its speed through the hose times the radius of the hose divided by the viscosity of concrete. Now the speed is something we need to calculate from the volume flow rate. So the volume flow rate is the cross sectional area of the hose times its speed and the cross sectional area is <i>pi</i> times its radius squared. And we’ll divide both sides by <i>pi r</i> squared to solve for <i>v</i>, and we get <i>v</i> is <i>Q</i> over <i>pi r</i> squared. And this is what we substitute in for <i>v</i> in the Reynolds number formula. So Reynolds number is two times density times <i>Q</i> over <i>pi r</i> squared, substituting for <i>v</i>, times <i>r</i> over <i>nu</i>, and this works out to <i>2 rho Q</i> over <i>pi r nu</i>. So that’s two times 2300 kilograms per cubic meter, density of concrete, times 200 liters per minute converted into cubic meters per second, because we always want <i>mks</i> units in our formulas, meters, kilograms and seconds. So this 200 liters per minute multiply by one minute for every 60 seconds to put seconds in the denominator, and then multiply by one cubic meter for every 1000 liters to put cubic meters in the numerator. And then divided by <i>pi</i> times the radius which is half the diameter converted into meters, and then multiply by the viscosity 48.0 Pascal seconds. This works out to 2.54 which is significantly less than 2000, so this flow is definitely laminar.