Part IV: Detailed Flow Structure Chap. 7: Microscopic Balances 1. Introduction • Fluid motion and forces on a fine scale • Conservation of mass & momentum (Navier-Stokes eq.) in small C.V.  Differential equation set with x,y,z and t (Cartesian coord.) 2. Conservation of mass Continuity equation

Mass flux in x-direction • Volume = x y z • Total mass = • Rate of change of mass: • Mass flow rate {in – out}, x-face: Mass flux in x-direction

• Mass flow rate {in – out}, y-face: • Mass flow rate {in – out}, z-face: • Dividing by x y z • x, y, z  0 Net efflux of mass flux

• Equation of continuity: Substantial derivative

• For any property (x,y,z,t), Rate of change with time as recorded by an observer moving with a fluid particle. • Note: Change of fish concentration with time in river. - Partial time derivative (on a bridge) - Substantial time derivative (in a canoe with local stream velocity) - Total time derivative (in a motorboat)

Vector notation Incompressible fluid • Constant density, no net efflux, , v is curl of any vector Ex.) - Check div v=0 at simple shear and extensional flows. - 참고: Trace(v) = div v

3. Conservation of linear momentum Momentum flow • Momentum = mv, momentum/volume= v, momentum flux=  or vv (1) x-component of linear momentum • x-mom. / vol. = , total x-mom. = • Rate of change of x-mom.: (=force) • Rate of {entering-leaving} x-mom.: Momentum change by bulk fluid motion

• Force acting on x-face: Stress • Force acting on x-face: Note: Normal stress includes isotropic pressure. Normal to surface (tension or compression) Parallel to the face (shear) • ij: i=face, j=coordinate direction in which stress is acting.  two directions  TENSOR !

• Force acting on y-face: • Force acting on z-face: Sign convention Fluid on the side of the face with the greater value of the coord. exerts positive stresses on the fluid with the smaller coord value.

• x-direction forces exerted by surrounding fluid on control volume: Body force • Body force in x-direction: Cauchy momentum equation

• Dividing by x y z • x, y, z  0 by EOC

• For y- and z-directions: Stress symmetry • Conservation of angular momentum

4. Newtonian fluid Intuitive development • Six unknown stresses • Constitutive equation: relation btw. stress & velocity field (rate of deformation) Newtonian, non-Newtonian (Generalized Newtonian or viscoelastic …) Newtonian relation considered here. • x-direction motion only: • Or, y-direction motion only:

Total stresses Isotropic pressure Extra stresses ij: i=j: normal stress ij: shear stress Symmetric !

“0” for incompressible fluid Constitutive equation • Properties of Newtonian fluids: - Symmetric stress - Instantaneous response of stress w.r.t. velocity gradients  Linear function between stress & velocity gradients - Existence of isotropic (same in all directions) stress (pressure) in no motion case (유도, 다음 슬라이드 참고만 하세요) • Newtonian constitutive eq. (incompressible case) Bulk viscosity “0” for incompressible fluid

“Mean normal stress = pressure” Momentum equation and Navier-Stokes equation • Differentiation of EOC w.r.t. x

• Similar forms in y- and z- directions Equivalent pressure

5. Curvilinear coordinates • Tables 7-1 ~ 7-10 (EOC, EOM, and CE using cylindrical and spherical coordinates) Cylindrical coordinates (r,,z) Spherical coordinates (r,,)

- The form of grad, div, curl in curvilinear coordinates (참고자료) Special cases: cylindrical, spherical coordinates (a) Cylindrical coordinates:

(b) Spherical coordinates:  P  r x z y

6. Boundary conditions • Differential equations set: EOC, EOM, CE • Boundary conditions for completely solving DEs. • Typical boundary conditions - Dirichlet (essential) B.C.: , (x,y) ;  is prescribed func on  - Neumann (natural) B.C.: , (x,y) ;  is prescribed func on  - Robin B.C. : , (x,y) ; , >0 on  • Examples) - No slip boundary condition: v=U at y=H (at solid-fluid boundary)

- Balance of normal stresses with surface tension - At fluid-fluid interface: - Continuous velocity and tangential stress - Balance of normal stresses with surface tension v1=v2 at interfacial boundary at liquid1-liquid2 boundary at liquid-air boundary (2H: curvature) at liquid-air boundary of right figure