Outline Time Derivatives & Vector Notation Differential Equations of Continuity Momentum Transfer Equations

Lagrangian Perspective z Lagrangian coordinate system Motion of a particle (fluid element) The position of the particle is relative to the position of an observer pathline 2 1 y x

Lagrangian Perspective z Local time derivative pathline 2 1 Local spatial derivative y x

Lagrangian Perspective Total differential/change for any property  Total time derivative

Lagrangian Perspective Fluid velocity If the observer follows the fluid motion Substantial time derivative

Eulerian Perspective flow Motion of a fluid as a continuum z Motion of a fluid as a continuum flow Fixed spatial position is being observed rather than the position of a moving fluid particle (x,y,z). y x

Equation of Continuity differential control volume:

Differential Equation of Continuity

Differential Equation of Continuity In cylindrical coordinates: If fluid is incompressible:

Equations of Motion Fluid is flowing in 3 directions For 1D fluid flow, momentum transport occurs in 3 directions Momentum transport is fully defined by 3 equations of motion

Differential Equation of Motion

Differential Equation of Motion

Navier-Stokes Equations Assumptions Newtonian fluid Obeys Stokes’ hypothesis Continuum Isotropic viscosity Constant density

Navier-Stokes Equations

Navier-Stokes Equations

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Steady state flow

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Unidirectional flow

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: No viscous dissipation (INVISCID FLOW) Euler’s equation

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: No external forces acting on the system Inviscid flow:

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: No external forces acting on the system Viscous flow:

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Semi-infinite system

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Laminar flow (no convective transport)

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Laminar flow (no convective transport)

Quiz 9 – 2014.01.17 Derive the equation giving the velocity distribution at steady state for laminar, downward flow in a circular pipe of length L and diameter D. Neglect entrance and exit effects. TIME IS UP!!!