Consider a small sphere made of material density $\rho_p$ that is moving in a viscous fluid which has a fluid flow velocity $\vec{u}$. Later in this class we will see that the movement of the sphere in the fluid can be described using an equation of motion given as follows:
$m \frac{dV}{dt} = 6\pi R \mu_f (u - V) + mg$ (1)
where $m$ is the sphere mass; $V$ is the speed of the sphere; $u$ is the flow speed; $\mu_f$ is the fluid viscosity; and $R$ is the sphere radius; and $g$ represents the acceleration due to gravity.
Part a: Divide throughout by the sphere mass, and use the information that we are dealing with a sphere gemoetry, to simplify the above motion equation.
Part b: Non-dimensionalize the above simplified equation of motion, using a characteristic velocity $U_0$ and a characteristic length scale $L_0$. Hint: Note that both velocities can be written as a combination of their non-dimensional counterparts and the characteristic value; eg $u = U_0 \bar{u}$, $V = U_0 \bar{V}$; and we have not given a characteristic time variable explicitly.
Part c: If you have done the non-dimensionalization in part b above properly, you should be able to identify two non-dimensional terms/numbers that define the physics of the sphere motion. Describe what each of these two numbers physically indicate. Hint: A common way to describe these would be to say they represent the ratio between two kinds of forces etc..
Part d: Estimate the sphere velocity at the instant where the sphere stops accelerating, using the two non-dimensional numbers.
