Chapter 1, Problem 23P

As described in Prob. 1.22, in addition to the downward force of gravity (weight) and drag, an object falling through a fluid is also subject to a buoyancy force that is proportional to the displaced volume. For example, for a sphere with diameter $d\left(\text{m}\right)$, the sphere's volume is $V=\pi d^3\text{/}6$ and its projected area is $A=\pi d^2\text{/}4$. The buoyancy force can then be computed as $F_b=–\rho Vg$. We neglected buoyancy in our derivation of Eq. (1.9) because it is relatively small for an object like a parachutist moving through air. However, for a more dense fluid like water, it becomes more prominent.

(a) Derive a differential equation in the same fashion as Eq. (1.9), but include the buoyancy force and represent the drag force as described in Prob. 1.21.

(b) Rewrite the differential equation from (a) for the special case of a sphere.

(c) Use the equation developed in (b) to compute the terminal velocity (i.e., for the steady-state case). Use the following parameter values for a sphere falling through water: sphere diameter $=1$ cm, sphere density $=2700{\text{ kg/m}}^{\text{3}}$, water density $=1000{\text{ kg/m}}^{\text{3}}$, and $C_d=0.47$.

(d) Use Euler's method with a step size of $\Delta t=0.03125$s to numerically solve for the velocity from $t=0$ to 0.25 s with an initial velocity of zero.