outline time derivatives & vector notation
DESCRIPTION
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 xTRANSCRIPT
Outline
1. Time Derivatives & Vector Notation
2. Differential Equations of Continuity
3. Momentum Transfer Equations
Lagrangian Perspective
x
y
z
pathline
• Lagrangian coordinate system
• Motion of a particle (fluid element)
• The position of the particle is relative to the position of an observer 1
2
1 p1 o1
1 p1 o1
1 p1 o1
x x x
y y y
z z z
2 p2 o2
2 p2 o2
2 p2 o2
x x x
y y y
z z z
o1 o1 o1, ,x y z
Lagrangian Perspective
x
y
z
pathline
1 1
,( , , )p p p
t t
r x y z trt t
1
22 p2 o2
2 p2 o2
2 p2 o2
x x x
y y y
z z z
Local time derivative
1 p1 o1
1 p1 o1
1 p1 o1
x x x
y y y
z z z
p1 p1
p p, p
p p
( , , )
x x
r x y z trx x
Local spatial derivative
Lagrangian Perspective
t x y zt x y z
Total differential/change for any property
Total time derivative
d x y zdt t x t y t z t
Lagrangian Perspective
x y zv v v v i j kFluid velocity
If the observer follows the fluid motion
x y zD v v vDt t x y z
x y zx y zv v vt t t
Substantial time derivative
Eulerian Perspective
flow
x
y
z
Motion of a fluid as a continuum
Fixed spatial position is being observed rather than the position of a moving fluid particle (x,y,z).
Equation of Continuity
differential control volume:
Differential Equation of Continuity
yx zvv v
t x y z
yx zx y z
vv vv v vt x y z x y z
yx zvv vD
Dt x y zv
Differential Equation of Continuity
In cylindrical coordinates:
1 1 0
r zrv v vd
dt r r r z
2 2 1where , tan yr x yx
If fluid is incompressible:
1 0
r r zvv v vr r r z
Equations of Motion
For 1D fluid flow, momentum transport occurs in 3 directions
Fluid is flowing in 3 directions
Momentum transport is fully defined by 3 equations of motion
Differential Equation of Motion
yxxx zxx x x xx y z x
v v v v pv v v gt x y z x y z x
xy yy zyy y y yx y z y
v v v v pv v v gt x y z x y z y
yzxz zzz z z zx y z z
v v v v pv v v gt x y z x y z z
Differential Equation of Motion
yxxx zxxx
xy yy zyyy
yzxz zzzz
Dv p gDt x y z x
Dv p gDt x y z y
Dv p gDt x y z z
D pDt
v g
Navier-Stokes Equations
Assumptions
1. Newtonian fluid
2. Obeys Stokes’ hypothesis
3. Continuum
4. Isotropic viscosity
5. Constant density
Navier-Stokes Equations
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
x x x xx
y y y yy
z z z zz
Dv v v vp gDt x x y z
Dv v v vp gDt y x y z
Dv v v vp gDt z x y z
2D pDt
v g v
Navier-Stokes Equations
2
2 2
2 2 2 2
2
1 1 2
1
r r r rr z
r r r rr
r r r rr z
v vv v v vv vt r r r z
rv v v vp gr r r r r r z
v vv v v vv vt r r r z
pr
2 2
2 2 2 2
2
2 2
2 2 2
1 1 2
1 1
r r r rr z
z z zz
rv v v vg
r r r r r z
v vv v v vv vt r r r z
v v vp g rz r r r r z
Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
1. Steady state flow2 2 2
2 2 2
2 2 2
2 2 2
x x x x x x xx y z x
y y y y y y yx y z y
z z z zx y z
v v v v v v vpv v v gt x y z x x y z
v v v v v v vpv v v gt x y z y x y z
v v v v pv v vt x y z
2 2 2
2 2 2z z z
zv v vg
z x y z
Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
2. Unidirectional flow2 2 2
2 2 2
2 2 2
2 2 2
2 2
2
x x x x x xx y z x
y y y y y yx y z y
z z z z zx y z z
v v v v v vpv v v gx y z x x y z
v v v v v vpv v v gx y z y x y z
v v v v vpv v v gx y z z x y
2
2 2zvz
Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
3. No viscous dissipation (INVISCID FLOW)2 2 2
2 2 2
0 0
0 0
x x x xx x
y
z
v v v vpv gx x x y zg
g
xx xv pv gx x
Euler’s equation
Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
4. No external forces acting on the system
0xxvvx
Inviscid flow:
0 constantxx
v vx
Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
4. No external forces acting on the system
2 2 2
2 2 2x x x x
xv v v v
vx x y z
Viscous flow:
Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
5. Semi-infinite system
2 2 2
2 2 2x x x x
xv v v vvx x y z
, y x z y
x
z
Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
6. Laminar flow (no convective transport)2
2x x
xv v
vx y
2
2 0xvy
1 2 xv c y c
Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
6. Laminar flow (no convective transport)
0 02 0 1
Boundary conditions:
1 at 0 lower plate , ,
2 at upper plate , 0x
x
y v v vc v cy v
00 0 or 1x x
v yv y v v v
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!!!