A cylindrical drum of length \(l\) and radius \(r\) can rotate inside a fixed concentric cylindrical casing, the clearance space \(c\) between the drum and the casing being very small and filled with liquid of dynamic viscosity \(\mu\). To rotate the drum with angular velocity \(\omega\) requires the same power as to pump the liquid axially through the clearance space while the drum is stationary, and the pressure difference between the ends of the drum is \(p .\) The motion in both cases is laminar. Neglecting end effects, show that $$ p=\frac{2 \mu \operatorname{lr} \omega \sqrt{3}}{c^{2}} $$

Viscous flow refers to the type of fluid motion where the fluid's viscosity plays a significant role. Viscosity is a measure of a fluid's resistance to deformation and determines how 'thick' or 'sticky' a fluid is. In simpler terms, it's like the internal friction within the fluid.
In the context of the given exercise, the fluid in the clearance space between the drum and the casing experiences viscous flow due to the small clearance and the dynamic viscosity \( \mu \). This resistance is what makes the rotational power needed to rotate the drum significant. Understanding viscous flow is crucial for calculating the power needed in systems like pumps, engines, or any machinery involving fluid movement.

Rotational power is the amount of power required to keep a rotating object spinning at a constant angular velocity \( \omega \). It involves torque ÷ force causing object rotation and velocity.
For our problem, the power to rotate the cylindrical drum is derived using the formula:
\[ P_{rot} = T \omega = \frac{2 \pi \mu lr^2 \omega^2}{c} \]
This utilizes the concept that shear stress is applied on the drum's surface by the fluid's viscosity as it rotates. The key here is understanding how the viscosity and rotational speed contribute to the overall power usage for rotation.

The Hagen-Poiseuille equation describes the volumetric flow rate \( Q \) of a viscous fluid through a pipe or cylindrical annulus in laminar flow conditions. This principle helps us understand the pressure difference \( p \) required to pump the liquid through the clearance space:
\[ Q = \frac{\pi pr^4}{4 \mu l} \]
This is a pivotal equation, especially when analyzing systems where fluids move through narrow spaces. In our problem, it helps quantify how the pressure difference influences the flow rate and ties into the power needed to pump the liquid. The correct balance of these factors is crucial for efficient fluid dynamics.

Laminar flow refers to a smooth, orderly movement of fluid, where the fluid flows in parallel layers, with no disruption between them. It typically occurs at lower velocities and is characterized by a Reynold's number less than 2000. In our problem, it's given that the motion is laminar.
This type of flow is predictable and consistent, making calculations easier and more accurate. The Hagen-Poiseuille equation specifically applies to laminar flows, making it essential to ensure the flow type when applying such mathematical models.
Understanding laminar flow helps us predict how the fluid will behave in the clearance space between the rotating drum and casing, thereby enabling precise calculation of the rotational and pumping power required.