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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: R. Shanthini Directional 15 Mar 2012 Derivative: 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 IF v I F y J v y J G G H y KH y Kxz Mass flux out of differential volume Rate of change of mass in y x z differential volume t y x R. Shanthini 15 Mar 2012 z Mass flux into differential volume vxz Continuity equation derivation Mass flux into differential volume = Mass flux out of differential volume + Rate of change of mass in differential volume vxz R. Shanthini 15 Mar 2012 = IF v I F y J v y J G G H y KH y Kxz + y x z t Continuity equation in 1-dimension v v y y t v 0 y t known as one dimensional Continuity equation R. Shanthini 15 Mar 2012 Continuity equation in 3-dimension af af a f u v w 0 t x y z where u, v, w are velocities in x, y, and z directions V 0 t divergence R. Shanthini 15 Mar 2012 Continuity equation for incompressible flow Density is constant for incompressible flow: u v w 0 x y z or V 0 R. Shanthini 15 Mar 2012 Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ kinematic viscosity (constant) ρ 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 υ υ R. Shanthini 15 Mar 2012 ρ ρ Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ Acceleration term: change of velocity with time R. Shanthini 15 Mar 2012 ρ Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ Advection term: force exerted on a particle of fluid by the other particles of fluid surrounding it R. Shanthini 15 Mar 2012 ρ Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ ρ viscosity (constant) controlled velocity diffusion term: (this term describes how fluid motion is damped) R. Shanthini 15 Mar 2012 Highly viscous fluids stick together (honey) Low-viscosity fluids flow freely (air) Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ ρ 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 υ ρ 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 υ ρ change body in = advection + diffusion + pressure + force velocity with time R. Shanthini 15 Mar 2012 Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid υ R. Shanthini 15 Mar 2012 ρ 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 y x h Fixed plate Steady, incompressible flow of Newtonian fluid in an infinite channel with one plate moving at uniform velocity - fully developed plane Couette flow Moving plate Fluid flow direction y R. Shanthini 15 Mar 2012 x Fixed plate h 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 φ Fixed pipe r z R. Shanthini 15 Mar 2012 Fluid flow direction 2a 2a Steady, incompressible flow of Newtonian fluid between a stationary outer cylinder and a rotating inner cylinder - fully developed pipe Couette flow a r b aΩ R. Shanthini 15 Mar 2012