CP502 Advanced Fluid Mechanics

1. CP502 Advanced Fluid Mechanics Flow of Viscous Fluids and Boundary Layer Flow [ 10 Lectures + 3 Tutorials ] Computational Fluid dynamics (CFD) project Midsemester (open book) examination

2. What do we mean by ‘Fluid’? • Physically: liquids or gases • Mathematically: • A vector field u (represents the fluid velocity) • A scalar field p (represents the fluid pressure) • fluid density (d) and fluid viscosity (v)

3. Recalling vector operations • Del Operator: • Laplacian Operator: • Gradient: • Vector Gradient: • Divergence: • Directional Derivative:

4. Continuity equationfor incompressible (constant density) flow - derived from conservation of mass where u is the velocity vector u, v, w are velocities in x, y, and z directions

5. υ ρ Navier-Stokes equationfor incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum kinematic viscosity (constant) density (constant) pressure external force (such as gravity)

6. υ υ ρ ρ Navier-Stokes equationfor incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum

7. υ ρ Navier-Stokes equationfor incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum Acceleration term: change of velocity with time

8. υ ρ Navier-Stokes equationfor incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum Advection term: force exerted on a particle of fluid by the other particles of fluid surrounding it

9. υ ρ Navier-Stokes equationfor incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum viscosity (constant) controlled velocity diffusion term: (this term describes how fluid motion is damped) • Highly viscous fluids stick together (honey) • Low-viscosity fluids flow freely (air)

10. υ ρ Navier-Stokes equationfor incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum Pressure term: Fluid flows in the direction of largest change in pressure

11. υ ρ Navier-Stokes equationfor incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum Body force term: external forces that act on the fluid (such as gravity, electromagnetic, etc.)

12. υ ρ Navier-Stokes equationfor incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum change in velocity with time body force advection = + diffusion + pressure +

13. υ ρ Continuity and Navier-Stokes equationsfor incompressible flow of Newtonian fluid

14. Continuity and Navier-Stokes equationsfor incompressible flow of Newtonian fluid in Cartesian coordinates Continuity: Navier-Stokes: x - component: y - component: z - component:

15. Moving plate Fixed plate h h Fluid flow direction Fluid flow direction y y x x Fixed plate Fixed plate Steady, incompressible flow of Newtonian fluid in an infinite channel with stationery plates- fully developed plane Poiseuille flow Steady, incompressible flow of Newtonian fluid in an infinite channel with one plate moving at uniform velocity - fully developed plane Couette flow

16. Continuity and Navier-Stokes equationsfor incompressible flow of Newtonian fluid in cylindrical coordinates Continuity: Navier-Stokes: Radial component: Tangential component: Axial component:

17. φ 2a Steady, incompressible flow of Newtonian fluid in a pipe- fully developed pipe Poisuille flow Fixed pipe r z Fluid flow direction 2a

18. φ a r b aΩ Steady, incompressible flow of Newtonian fluid between a stationary outer cylinder and a rotating inner cylinder- fully developed pipe Couette flow