Is $b$ in the drag force formula $F=-bv$ constant for a certain medium and object?

Question

I heard that $F=-bv$, where $F$ is the drag force, $b$ is the damping coefficient, and $v$ is the velocity of an object, can be used to calculate the drag force exerted on an object moving at a moderate velocity.

1. What range is this moderate velocity referring to?

2. Does $b$ in the drag force formula $F=-bv$ have to be constant for a certain medium and object?

3. If the answer to question 2 is a "yes" my question is: should $b$ not vary in the damped oscillation formula $x(t)=Ae^{-b/2m}cos(ωt)$? It sounds counterintuitive that $b$ has to be constant in that formula. I've done an experiment where I changed the value of spring constant in spring-mass systems damped in water, and my results (which seem to be precise) show different values of $b$ for different spring constants.

Answer 1

1. It is not referring to some absolute range of velocities, rather it means the flow of fluid around the object is laminar flow.

We can establish whether or not the flow is laminar by computing the so-called dimensionless number $\mathbf{Re}$, i.e. Reynolds number:

$$\mathbf{Re}=\frac{vD}{\nu}$$

where:

• $v$ is the velocity
• $D$ a characteristic dimension of the object (like its diameter)
• $\nu$ the kinematic viscosity of the fluid

Laminar flow occurs for $\mathbf{Re}<2300$ and turbulent flow for $\mathbf{Re}>2900$ (in between these numbers is the so-called 'transitional regime'). In the laminar regime, viscous drag forces are said to dominate $F$ and in the turbulent regime inertial forces dominate it.

In the case or turbulent flow the drag force is of the form:

$$F=-cv^2$$

so the velocity dependence is on the square of the velocity.

2. In the either laminar or turbulent regime $b$ and $c$ resp. are considered constant and invariant to $v$.
3. I've not checked your formula ($x(t)=Ae^{-b/2m}\cos(ωt)$) but why does it "sounds counterintuitive that b has to be constant in that formula"? As stated above: in the relatively narrow velocity interval (laminar flow!) $b$ should be constant.