CP502 Advanced Fluid Mechanics

1. CP502 Advanced Fluid Mechanics Incompressible Flow of Viscous Fluids Set 01

2. υ ρ Equations describing incompressible flow of viscous fluid: We will learn: • Physical meaning of each term • How to derive • How to solve

3. 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 (ρ) and fluid viscosity (μ)

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

5. Continuity equationfor incompressible (constant density) flow - derived from conservation of mass where u is the velocity vector ux, uy & uz are velocity components in x, y & z directions

6. Continuity equationderivation Mass flux on left face Mass flux on right face Inflow at left face = Outflow at right face = Difference between inflow and outflow in the x direction per unit volume

7. Continuity equationderivation Mass flux on left face Mass flux on right face Difference between inflow and outflow in the y direction per unit volume Difference between inflow and outflow in the z direction per unit volume Thus net rate of inflow/outflow per unit volume

8. Continuity equationderivation Mass flux on left face Mass flux on right face Net rate of inflow/outflow per unit volume = rate of increase in mass per unit volume = rate of change of density Continuity equation in general form

9. Continuity equationfor incompressible flow Density is constant for incompressible flow: Divergence of u or

10. 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)

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

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

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

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

15. 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

16. 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.)

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

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

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

20. 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

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

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

23. 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