Why does the drag force increase with velocity and viscosity?

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The drag force increases with velocity and viscosity due to the relationship between turbulence and the drag coefficient, which varies with the Reynolds number. The exponent n, indicating the power to which velocity is raised in drag calculations, increases as turbulence intensifies, complicating the mathematical modeling of drag. While n is often considered variable and somewhat arbitrary, it serves to simplify problem-solving in fluid dynamics. For practical applications, drag coefficients can remain constant in certain velocity ranges, typically following F ~ v^2 or F ~ v relationships. Understanding these dynamics is crucial for accurate predictions in fluid mechanics.
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I know that the drag force is proportional to the velocity to some power, n, and that n increases with velocity and with the viscosity of the medium.

I want to know why n increases, and how (mathematically) it increases. Some general speed ranges for v and v^2 in STP would be appreciated as well.
 
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n increases because turbulence increases.
The drag is so shape dependent, that I don't if there are formulas for n.
The fact that n depends on v means that n is really meaningless,
and is just used to make problems tractable
 
This sort of thing is usually discussed by specifying how the drag coefficient varies with Reynolds number (which depends on velocity). You should find some nice plots if you search for those terms. Although these can be complicated, there are large regions where the drag coefficient remains approximately constant (F ~ v^2) or goes like 1/R (so F ~ v).
 
Alright, I looked up some things on Reynolds number as you suggested.

Thanks for the info.
 

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