CP502 Advanced Fluid Mechanics Flow of Viscous Fluids Set 01

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) R. Shanthini 15 Mar 2012

Recalling vector operations Del Operator: Laplacian Operator: Gradient: Vector Gradient: Divergence: Directional Derivative: R. Shanthini 15 Mar 2012

Continuity equation for 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 R. Shanthini 15 Mar 2012

Continuity equation derivation Mass flux out of differential volume Rate of change of mass in differential volume Mass flux into differential volume R. Shanthini 15 Mar 2012

Continuity equation derivation Mass flux into differential volume = Mass flux out of differential volume + Rate of change of mass in differential volume = + R. Shanthini 15 Mar 2012

known as one dimensional Continuity equation Continuity equation in 1-dimension known as one dimensional Continuity equation R. Shanthini 15 Mar 2012

Continuity equation in 3-dimension where u, v, w are velocities in x, y, and z directions divergence R. Shanthini 15 Mar 2012

Continuity equation for incompressible flow Density is constant for incompressible flow: or R. Shanthini 15 Mar 2012

kinematic viscosity (constant) Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ kinematic viscosity (constant) density (constant) pressure external force (such as gravity) R. Shanthini 15 Mar 2012

Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ ρ υ R. Shanthini 15 Mar 2012

Acceleration term: change of velocity with time Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ Acceleration term: change of velocity with time R. Shanthini 15 Mar 2012

Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ Advection term: force exerted on a particle of fluid by the other particles of fluid surrounding it R. Shanthini 15 Mar 2012

Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ 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) R. Shanthini 15 Mar 2012

Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ Pressure term: Fluid flows in the direction of largest change in pressure R. Shanthini 15 Mar 2012

Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ Body force term: external forces that act on the fluid (such as gravity, electromagnetic, etc.) R. Shanthini 15 Mar 2012

Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum ρ υ change in velocity with time body force = advection + diffusion + pressure + R. Shanthini 15 Mar 2012

Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid ρ υ R. Shanthini 15 Mar 2012

Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid in Cartesian coordinates Continuity: Navier-Stokes: x - component: y - component: z - component: R. Shanthini 15 Mar 2012

Steady, incompressible flow of Newtonian fluid in an infinite channel with stationery plates - fully developed plane Poiseuille flow Fixed plate Fluid flow direction h x y Steady, incompressible flow of Newtonian fluid in an infinite channel with one plate moving at uniform velocity - fully developed plane Couette flow Fixed plate Moving plate h x y Fluid flow direction R. Shanthini 15 Mar 2012

Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid in cylindrical coordinates Continuity: Navier-Stokes: Radial component: Tangential component: Axial component: R. Shanthini 15 Mar 2012

Steady, incompressible flow of Newtonian fluid in a pipe - fully developed pipe Poisuille flow Fixed pipe 2a φ r z Fluid flow direction 2a R. Shanthini 15 Mar 2012

Steady, incompressible flow of Newtonian fluid between a stationary outer cylinder and a rotating inner cylinder - fully developed pipe Couette flow φ aΩ a b r R. Shanthini 15 Mar 2012