Steady Flow Between Rotating Cylinders — Lesson 3

This lesson covers the exact solutions of the Navier Stokes equation in a cylindrical coordinate system. It delves into the study of steady axisymmetric flows between rotating cylinders, specifically focusing on the flow in the annulus between two rotating cylinders. The lesson explains the concept of laminar steady incompressible axisymmetric torsional flow with constant fluid properties, also known as circular Couette flow. It further discusses the continuity equation, the velocity in the Theta direction, and the assumptions made for the calculations. The lesson also provides a detailed explanation of the R momentum equation, the Z momentum equation, and the Theta momentum equation. It concludes with a discussion on special cases where both cylinders rotate with the same angular velocity.

Video Highlights

00:54 - Discussion on the characteristics of the inner and outer cylinders.
03:00 - Explanation of the concept of constant speed rotation of the cylinders.
06:29 - Explanation of the Theta momentum equation and its implications.
16:54 - Discussion on the concept of torque at the outer cylinder.
21:39 - Explanation of the special case where the gap between the two cylinders is very small.
26:26 - Discussion on the special case where both the cylinders rotate with the same angular velocity.

Key Takeaways

- The Navier Stokes equation in a cylindrical coordinate system can be used to study steady axisymmetric flows between rotating cylinders.
- The flow in the annulus between two rotating cylinders is known as circular Couette flow.
- The continuity equation and the velocity in the Theta direction are crucial in understanding this flow.
- The R momentum equation, the Z momentum equation, and the Theta momentum equation provide further insights into the flow dynamics.
- Special cases where both cylinders rotate with the same angular velocity provide unique insights into the flow dynamics.