Navier Stokes Equation and Plane Couette Flow — Lesson 1

This lesson covers the analytical solution of Navier Stokes equations for simplified problems and simple geometry. It explains the continuity equation for Cartesian coordinate and the momentum equation for laminar incompressible flow with constant fluid properties. The lesson also discusses the components of viscous stress tensor for incompressible nutrient fluid and the vorticity component. It further elaborates on the exact solution of Navier Stokes equations, the concept of fully developed flow, and the calculation of shear stress and vorticity. The lesson concludes with the explanation of plane couette flow and two-layer couette flow.

Video Highlights

00:58 - Discussion on the continuity equation for Cartesian coordinate and the X, Y, Z momentum equations.
05:22 - Discussion on the formation of boundary layer due to viscous effect in the flow inside two infinite parallel plates.
10:00 - Derivation of the continuity equation and Navier stoke equation for steady laminar incompressible fully developed flow.
20:53 - Explanation of the concept of plane couette flow or shear driven flow between two infinite parallel plates.
32:55 - Discussion on the special cases of plane couette flow.
36:58 - Explanation of the concept of two-layer couette flow where two different fluids are present inside the two parallel plates.

Key Takeaways

- The Navier Stokes equations are derived from the Reynolds transport theorem.
- The continuity equation for Cartesian coordinate and the momentum equation are essential for understanding laminar incompressible flow with constant fluid properties.
- The components of viscous stress tensor for incompressible nutrient fluid and the vorticity component are crucial in fluid dynamics.
- The exact solution of Navier Stokes equations is limited to laminar, one and two-dimensional flows with constant properties and simple geometry.
- The concept of fully developed flow is essential in understanding the behavior of fluid flow in different conditions.
- The calculation of shear stress and vorticity is crucial in understanding the forces acting on a fluid element.
- The plane couette flow and two-layer couette flow are special cases of fluid flow.