Today’s Lecture Objectives: Review (Learn) fluid dynamics Conservation equation Mass Momentum Energy

Terminology Fluid particle Scalar vs. Vector Boundary conditions Steady State vs. Unsteady State Stream lines Gradient Lines Incompressible vs. Compressible fluids …… Examples

Stream Lines and Gradient lines Wind Gradient Lines Temperature Jet airspeed

Steady-state vs. Unsteady-stat https://www.youtube.com/watch?v=IDeGDFZSYo8

Conservation equations Fluid Dynamics: Conservation equations

Fluid particle vs Control Volume Pay attention to the orientation of the coordinate system

Important operations Total derivative for fluid particle which is moving: V z any scalar y Vector and scalar operators: x scalar vector

Total Derivative Shows how variable  change during time while traveling from A to B Change in  are introduced by changing boundary conditions over time and by moving from part A to B B  V z A mathematical definition y Total derivative allows all arguments vary x Physical interpretation …

Gradient of a scalar and Divergence of a vector Where is the gradient of temperature largest? - nabla operator Divergence of a vector Vector filed You can also write that

Notation Density () in our problem change is so small that we can assume constant (most of the time) Book (handouts) vs. Class notes We are going to use these interchangeably Vx ≡ u Vy ≡ v Vz ≡ w

Continuity equation -conservation of mass Mass flow in and out of fluid element Infinitely small volume Volume V = δxδyδz Volume sides: Ax = δyδz Ay = δxδz Az = δxδy Change of density in volume = = Σ(Mass in) - Σ(Mass out) ………………. same See equation 2.2 in notes = ….

Shear and Normal stress τyx

Momentum equation –Newton’s second law dimensions of fluid particle Stress components in x direction forces per unit of volume in direction x ……………….. ……………… ……………. total derivative

Momentum equation Sum of all forces in x direction Internal source y direction z direction

Newtonian fluids Viscous stress are proportional to the rate of deformation (e) Elongation: Shearing deformation: For incompressible flow Viscous stress: viscosity

Momentum equations for Newtonian fluids After substitution: x direction: y direction: z direction:

Momentum equations for Newtonian fluids Same like previous slide with some rearrangements: x direction: y direction: z direction:

Momentum equations for Newtonian fluids Finally: x direction: y direction: z direction: