# How does inertia forces depend on speed?

I was learning about Reynolds number in fluid mechanics and it is given as

$$Re= \frac{\rho V L}{u}$$

Physically, compares the inertia and viscous forces acting in a fluid. I get that density will affect inertia force, but how the velocity affects inertia force?

If I take two bodies having same mass, one moving with a higher speed and the other with lower, and if I subject them to same acceleration won't the inertia force be same in both the cases?

The inertial force in Reynolds number is to do with how much mass of the fluid must be 'pushed along' when the body is going at velocity $$v$$ - not to do with the inertia of the body.

Reynolds number is the ratio of inertial force to viscous force, per unit area - the inertial force is $$v^2\rho$$

Inertial force is a confusing name but it's the force needed to change the momentum of all the fluid that impacts on the body when moving at velocity $$v$$.

The body of area $$A$$ comes across a volume $$Av$$ of fluid per second, the mass of that fluid is $$Av\rho$$.

If we presume it must bring it up to the same speed $$v$$, the change in momentum per unit area is $$v^2\rho$$, so that's the force per unit area - and is called 'inertial force'.

The viscous drag force, for a sphere, is from Stokes Law

$$F=6\pi r\eta v$$

so force per unit area, dividing by $$\pi r^2$$ and ignoring small numerical constants is $$\frac{\eta v}{r}$$.

Reynolds number, the ratio of inertial force to viscous drag is $$Re = \frac{v^2\rho}{\eta v/r} = \frac{\rho v r}{\eta}$$