You want to determine how fast a rock will settle in mud, which behaves like a Bingham plastic. The first step is to perform a dimensional analysis of the system.

(a) List the important variables that have an influence on this problem, with their dimensions (give careful attention to the factors that cause the rock to fall when listing these variables), and determine the appropriate dimensionless groups.

(b) Design an experiment in which you measure the velocity of a solid sphere falling in a Bingham plastic in the lab, and use the dimensionless variables to scale the answer to find the velocity of a $2 \mathrm{in}$. diameter rock, with a density of $3.5 \mathrm{~g} / \mathrm{cm}^{3}$, falling in a mud with a yield stress of $300 \mathrm{dyn} / \mathrm{cm}^{2}$, a limiting viscosity of $80 \mathrm{cP}$, and a density of $1.6 \mathrm{~g} / \mathrm{cm}^{3}$. Should you use this same mud in the lab, or can you use a different material that is also a Bingham plastic but with a different yield stress and limiting viscosity?

(c) If you use a suspension in the lab with a yield stress of $150 \mathrm{dyn} / \mathrm{cm}^{2}$, a limiting viscosity of $30 \mathrm{cP}$, and a density of $1.3 \mathrm{~g} / \mathrm{cm}^{3}$ and a solid sphere, how big should the sphere be and how much should it weigh?

(d) If the sphere in the lab falls at a rate of $4 \mathrm{~cm} / \mathrm{s}$, how fast will the $2 \mathrm{in}$. diameter rock fall in the other mud?