how to know the fluid resistance is depend on [tex] v [/tex] or [tex] v^2 [/tex] ??
Are you asking why fluid resistance is velocity dependent? Or are you asking how to determine whether the dependence goes linear or whether it goes quadratic with velocity?
Re: fluid resistance
Well, you could look it up. Or you could run some experiments. A simple experiment might use a tub or water and a submerged block of wood, with a spring or rubber band providing a measurable force to move the block, and a stopwatch.
Experiments in the tub are always fun.
I'm going to guess it depends on [tex]v^1[/tex]... let us know when you find out.
It is almost certainly a power series expansion in powers of velocity, with the linear term being the most important contributor. In other words:
[itex] F(v) = a v + bv^2 + cv^4 + ...[/itex]
It would be a fun experiment to determine a, b, c,...
The [tex]v^2[/tex] in the formula for turbulent current is an energy thing: the kinetic energy of the fluid whirls goes with [tex]v^2[/tex], and energy equals force times deplacement.
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