As the ball falls through a liquid, its velocity V is a...

As the ball falls through a liquid, its velocity $V$ is a function of the diameter $D$ of the ball, its density $\rho_{b},$ the density $\rho$ and viscosity $\mu$ of the liquid, and the acceleration due to gravity $g .$ Determine the relation between $V$ and these parameters.

Key Concepts

Dimensionless Groups

Dimensionless groups are combinations of variables that cancel out all the physical units and provide insight into the scaling and similarity properties of a system. In the analysis of a falling ball in a liquid, such groups can include the ratio of densities (?_b/?) and a group involving the viscosity (such as ?/(??(gD)D)), which encapsulate the effects of buoyancy, inertia, and viscous drag. These groups determine the functional dependence of the terminal velocity on the involved parameters.

Dimensional Analysis

Dimensional analysis is the technique used to study the relations between physical quantities by comparing their fundamental dimensions such as mass, length, and time. In the context of this problem, it is applied to relate the ball’s terminal velocity to other measurable parameters (diameter, densities, viscosity, and gravity) by ensuring that both sides of the equation have consistent units.

Buckingham Pi Theorem

The Buckingham Pi Theorem is a key tool in dimensional analysis that allows one to reduce the number of variables by forming dimensionless groups. By selecting a set of repeating variables (typically those that represent the basic dimensions) and combining them with the remaining variables, one can express the relationship between the quantities using fewer, dimensionless parameters. In this problem, it is used to derive a relation of the form V = ?(gD) f(?_b/?, ?/(??(gD)D)).

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