The dimensional parameters used to describe the operation of a ship or airplane propeller (sometimes called a screw propeller) are rotational speed, \(\omega,\) diameter, \(D,\) fluid density, \(\rho\) speed of the propeller relative to the fluid, \(V\), and thrust developed, \(T .\) The common dimensionless groups are called the thrust coefficient and the advance ratio. Propose appropriate definitions for these groups.

The first step is to understand what each of the parameters mean and how they impact the operation of a propeller. The rotational speed (\(\omega\)) refers to how quickly the propeller is spinning, the diameter (\(D\)) refers to the size of the propeller, fluid density (\(\rho\)) refers to the mass per unit of volume of the fluid in which the propeller is operating, speed of the propeller relative to the fluid (\(V\)) refers to how quickly the propeller is moving through the fluid and thrust developed (\(T\)) refers to the force being exerted on the propeller in the opposite direction of its motion. Knowing all these, we can now proceed to define the dimensionless groups: the thrust coefficient and the advance ratio.

The thrust coefficient (\(C_T\)) provides a measure of the effectiveness of a propeller in creating thrust. It is a dimensionless quantity obtained by dividing the thrust (\(T\)) by the product of fluid density (\(\rho\)), the square of the propeller's rotational speed (\(\omega^2\)), and the square of the propeller's diameter (\(D^2\)). The formula is defined as: \(C_T = T / (\rho * \omega^2 * D^2)\).

The advance ratio (\(J\)) relates the forward speed of the propeller to the speed at which it is revolving. It is a dimensionless quantity obtained by dividing the speed of the propeller relative to the fluid (\(V\)) by the product of the rotational speed (\(\omega\)) and the propeller's diameter (\(D\)). The formula is defined as: \(J = V / (\omega * D)\).