Understanding Terminal Velocity in Viscous Fluids

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration. For a sphere settling in a viscous fluid, this occurs when the downward force of gravity is balanced by the upward buoyant force and the drag force exerted by the fluid.

Forces Acting on a Settling Sphere

Three main forces act on a sphere settling in a viscous fluid:

• Gravitational Force (\(F_g\)): Acts downwards. It is equal to the mass of the sphere multiplied by the acceleration due to gravity (\(g\)). Mass is volume times density of the sphere (\(\rho_s\)). So, \(F_g = V \rho_s g\), where \(V = \frac{4}{3}\pi r^3\) is the volume of the sphere with radius \(r\).
• Buoyant Force (\(F_b\)): Acts upwards. According to Archimedes' principle, this force is equal to the weight of the fluid displaced by the sphere. It is given by \(F_b = V \rho_f g\), where \(\rho_f\) is the density of the fluid.
• Drag Force (\(F_d\)): Acts upwards, opposing the motion. This force depends on the velocity of the sphere, the properties of the fluid (viscosity), and the size and shape of the sphere. For slow speeds (low Reynolds number, typically \(Re < 1\)), the drag force on a sphere is given by Stokes' Law: \(F_d = 6\pi \eta r v\), where \(\eta\) is the dynamic viscosity of the fluid and \(v\) is the velocity of the sphere.

Reaching Terminal Velocity

Terminal velocity (\(v_t\)) is reached when the net force on the sphere is zero, meaning the sum of upward forces equals the downward force:

\[F_g = F_b + F_d\]

Substituting the formulas for the forces (using Stokes' Law for drag):

\[\frac{4}{3}\pi r^3 \rho_s g = \frac{4}{3}\pi r^3 \rho_f g + 6\pi \eta r v_t\]

Rearranging the terms to solve for \(v_t\):

\[6\pi \eta r v_t = \frac{4}{3}\pi r^3 \rho_s g - \frac{4}{3}\pi r^3 \rho_f g\]

\[6\pi \eta r v_t = \frac{4}{3}\pi r^3 (\rho_s - \rho_f) g\]

\[v_t = \frac{\frac{4}{3}\pi r^3 (\rho_s - \rho_f) g}{6\pi \eta r}\]

Simplifying the expression:

\[v_t = \frac{4 \pi r^3 (\rho_s - \rho_f) g}{18 \pi \eta r}\]

\[v_t = \frac{2 r^2 (\rho_s - \rho_f) g}{9 \eta}\]

Since the diameter of the sphere is \(D = 2r\), the radius is \(r = D/2\). Substituting this into the equation for \(v_t\):

\[v_t = \frac{2 (D/2)^2 (\rho_s - \rho_f) g}{9 \eta}\]

\[v_t = \frac{2 (D^2/4) (\rho_s - \rho_f) g}{9 \eta}\]

\[v_t = \frac{(D^2/2) (\rho_s - \rho_f) g}{9 \eta}\]

\[v_t = \frac{D^2 (\rho_s - \rho_f) g}{18 \eta}\]

Relationship with Diameter

The formula for terminal velocity is \(v_t = \frac{D^2 (\rho_s - \rho_f) g}{18 \eta}\). In this equation, \((\rho_s - \rho_f) g / (18 \eta)\) is a constant for a given sphere-fluid system (assuming \(\rho_s > \rho_f\)).

Therefore, the terminal velocity \(v_t\) is directly proportional to the square of the diameter \(D\).

\[v_t \propto D^2\]

Analyzing the Options

Let's look at how terminal velocity relates to the given options based on the derived formula \(v_t = \frac{D^2 (\rho_s - \rho_f) g}{18 \eta}\):

• The Reynolds number: Reynolds number (\(Re\)) is \(Re = \frac{\rho_f v D}{\eta}\). The terminal velocity depends *on* the properties that determine the Reynolds number (\(v\), \(D\), \(\rho_f\), \(\eta\)), but \(v_t\) itself is a specific velocity achieved under force balance. The derivation above assumes a low Reynolds number where Stokes' Law applies. So, \(v_t\) varies as the square of the diameter, and this relationship is valid in the low \(Re\) regime. The question asks what \(v_t\) varies *as*, and \(D^2\) is the direct proportionality factor from the formula, assuming the flow regime allows for Stokes' Law.
• The square of its diameter: As shown in the derivation, \(v_t \propto D^2\). This directly matches our finding.
• Its diameter: \(v_t\) is proportional to \(D^2\), not just \(D\).
• Viscosity of the fluid: The formula shows \(v_t \propto \frac{1}{\eta}\). Terminal velocity is inversely proportional to the fluid viscosity, not directly proportional.

Thus, the terminal velocity of a sphere settling in a viscous fluid varies as the square of its diameter, assuming the conditions for Stokes' Law are met (low Reynolds number). For higher Reynolds numbers, the drag force formula becomes more complex, and the direct \(D^2\) proportionality no longer holds precisely, but in many standard problems regarding terminal velocity of small spheres, the Stokes' Law regime is implied or assumed.

Revision Table: Terminal Velocity Concepts

Additional Information: Factors Affecting Settling

While the derivation focused on a spherical particle in a Newtonian fluid under low Reynolds number conditions, other factors can influence settling velocity in more complex scenarios:

• Particle Shape: Non-spherical particles experience different drag forces.
• Particle Concentration: At high concentrations, particles interact, affecting settling (hindered settling).
• Fluid Properties: Non-Newtonian fluids have viscosity that depends on shear rate.
• Flow Regime: At higher Reynolds numbers, the drag force dependence changes (e.g., proportional to \(v^2\) in turbulent flow), and the \(D^2\) proportionality for terminal velocity from Stokes' Law is no longer accurate.

However, for a single sphere settling in a viscous fluid under typical conditions where terminal velocity is discussed at an introductory level, the low Reynolds number regime and Stokes' Law are usually assumed, leading to the \(v_t \propto D^2\) relationship.