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If you were to measure the viscosity of a fluid you would do the following experiment https://www.wikihow.com/Measure-Viscosity

The formula for dynamic viscosity is the following one:

$n=\frac{2gr^{2}(p_s-p_l)}{9v}=\frac{ms^{-2}m^{2}(kgm^{-3})}{ms^{-1}}=kgm^{-1}s^{-1}= Pa·s$

where n is dynamic viscosity of the liquid in Pa s, g is gravity ($9.81{ms^-}^2$), r the radius of the ball in meters, $p_s$ density of sphere, $p_l$ density of liquid and v the velocity at which the ball travels through the fluid.

This source and many more state that the dynamic viscosity of water is $8.90*{10^-}^4$ Pa s https://www.engineersedge.com/physics/water__density_viscosity_specific_weight_13146.htm

If we plug this value for viscosity and try to find a value for the velocity at witch the sphere travels through the water (in this case). Then we come up with the following.

$8.90*{10^-}^4=\frac{2(9.81)r^{2}(p_s-1000)}{9v}$ (Density of water is 1000kg per meter cube)

Let's say that we drop an aluminum ball of 0.01m radius. Aluminum has a density of 2700kg/$m^3$

$8.90*{10^-}^4=\frac{2(9.81)(0.01)^{2}(2700-1000)}{9v}$

The final value for $v$ is 416.4 ${ms^-}^1$

Is the sphere really traveling through water at that high speed. Is the formula wrong? Or is it my math or what? I'm so confused.

Matthew
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1 Answers1

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Your equation comes from an application of Stokes' law whilst equating the viscous drag to the apparent weight of the object when the object is travelling at a terminal speed.

However Stokes' law is not applicable in your example the test for this being to evaluate a parameter called Reynolds' number $R_{\rm e} = \dfrac {\rho_{\rm fluid}R_{\rm shere}v_{\rm terminal}}{\eta_{\rm fluid}}$ which in your example is approximately four million far in excess of the upper limit for Stokes' law to be applicable which is about ten.

In your regime the fluid friction is no longer proportional to the speed but more likely to the speed squared so you will need to do a different analysis.
There is an answer in PSE which might help you and the analysis is done in many textbooks and Internet sites; here is one as an example.

Farcher
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