a$.$ A sphere translating through a viscous fluid under the action of gravity will

experience a drag force $F=6aU$ if the Reynolds number is small. Obtain the

steady state fall velocity $($also called the terminal velocity$)$ of the sphere from a

force balance on the sphere. You may denote the density difference between

the sphere and the fluid as $.$

b$.$ Suppose now we have two particles of radius a which are separated $($center to

center$)$ by a distance d$,$ where and Calculate a first correction

$($i$.$e$.$ to order $(\frac{a}{d}))$ to the terminal velocity of fall of the spheres which takes into

account the fact that each will cause the other to fall with a different velocity than

they would fall by themselves. Consider two cases: line of centers horizontal,

and line of centers vertical. Will the two spheres move closer together or further

apart? Hint: Use the Oseen tensor defined above.

c$.$ Now consider three particles falling in a line, separated initially by an equal

distance d $.$ Calculate the velocities of fall to $O(\frac{a}{d})$ for each particle. Discuss

any relative change in position with time