1. 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

2. 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

3. • 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

4. • Equation of continuity:
Substantial derivative

5. • 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)

6. 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

7. 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

8. • 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 !

9. • 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.

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

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

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

13. 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:

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

15. “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

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

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

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

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

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

23. 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)

24. - 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