CP 502 Advanced Fluid Mechanics Incompressible Flow of
CP 502 Advanced Fluid Mechanics Incompressible Flow of Viscous Fluids Set 01
Equations describing incompressible flow of viscous fluid: υ We will learn: l Physical meaning of each term l How to derive l How to solve Prof. R. Shanthini 09 June 2019 ρ
What do we mean by ‘Fluid’? l Physically: liquids or gases l Mathematically: l A vector field u (represents the fluid velocity) l A scalar field p (represents the fluid pressure) l fluid density (ρ) and fluid viscosity (μ) Prof. R. Shanthini 09 June 2019
Recalling vector operations l Del Operator: l Laplacian Operator: l Gradient: l Vector Gradient: l Divergence: l. Prof. Directional R. Shanthini 09 June 2019 Derivative:
Continuity equation for 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 Prof. R. Shanthini 09 June 2019
Continuity equation derivation Mass flux on left face Inflow at left face = Outflow at right face = Difference between inflow and outflow in the x direction per unit volume Prof. R. Shanthini 09 June 2019 Mass flux on right face
Continuity equation derivation Mass flux on left 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 Prof. R. Shanthini 09 June 2019 Mass flux on right face
Continuity equation Mass flux on left face derivation 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 Prof. R. Shanthini 09 June 2019 Continuity equation in general form
Continuity equation for incompressible flow Density is constant for incompressible flow: Divergence of u or Prof. R. Shanthini 09 June 2019
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ kinematic viscosity (constant) density (constant) ρ pressure external force (such as gravity) Prof. R. Shanthini 09 June 2019
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ υ Prof. R. Shanthini 09 June 2019 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 Prof. R. Shanthini 09 June 2019 ρ
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ Convection term Prof. R. Shanthini 09 June 2019 ρ
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ Viscous forces term Prof. R. Shanthini 09 June 2019 ρ
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 Prof. R. Shanthini 09 June 2019
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. ) Prof. R. Shanthini 09 June 2019
Navier-Stokes equation for incompressible flow of Newtonian (constant viscosity) fluid - derived from conservation of momentum υ Prof. R. Shanthini 09 June 2019 ρ
Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid υ Prof. R. Shanthini 09 June 2019 ρ
Continuity and Navier-Stokes equations for incompressible flow of Newtonian fluid in Cartesian coordinates Continuity: Navier-Stokes: x - component: y - component: z - component: Prof. R. Shanthini 09 June 2019
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 Prof. R. Shanthini 09 June 2019 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: Prof. R. Shanthini 09 June 2019 See the handout
Steady, incompressible flow of Newtonian fluid in a pipe - fully developed pipe Poisuille flow φ Fixed pipe r z Prof. R. Shanthini 09 June 2019 Fluid flow direction 2 a 2 a
Steady, incompressible flow of Newtonian fluid between a stationary outer cylinder and a rotating inner cylinder - fully developed pipe Couette flow φ a b Prof. R. Shanthini 09 June 2019 r aΩ
- Slides: 23