1. AOE 5104 Class 3 9/2/08 Online presentations for today’s class:
Vector Algebra and Calculus 1 and 2
Vector Algebra and Calculus Crib
Homework 1 due 9/4
Study group assignments have been made and are online.
Recitations will be
5:30pm (with Nathan Alexander)
5pm (with Chris Rock)
Locations TBA
Which recitation you attend depends on which study group you belong to and is listed with the study group assignments

2. Unnumbered slides contain comments that I inserted and are not part of Professor’s Devenport’s original presentation.

3. Last Class… Vectors, inherent property of direction Algebra
Volumetric flow rate through an area
Taking components, eqn. of a streamline
Triple products, A.BxC, Ax(BxC)
Coordinate systems
Class 3. 8/29/06
Assignments: Online Presentation “Vector Algebra and Calculus 2”
Homework 1, due 9/5/06
Vector Algebra and calculus crib on course web site
Study groups

4. Cylindrical Coordinates
Coordinates r,  , z
Unit vectors er, e, ez (in directions of increasing coordinates)
Position vector
R = r er + z ez
Vector components
F = Fr er+F e+Fz ez
Components not constant, even if vector is constant z F ez e er R z  r y x

5. Spherical Coordinates
Errors on this slide in online presentation
Spherical Coordinates
Coordinates r,  , 
Unit vectors er, e, e (in directions of increasing coordinates)
Position vector
r = r er
Vector components
F = Fr er+F e+F e z er e F r e  r  y x

7. LOW REYNOLDS NUMBER AXISYMMETRIC JET
J. KURIMA, N. KASAGI and M. HIRATA (1983)
Turbulence and Heat Transfer Laboratory, University of Tokyo

8. Class Exercise Using cylindrical coordinates (r, , z)
Gravity exerts a force per unit mass of 9.8m/s2 on the flow which at (1,0,1) is in the radial direction. Write down the component representation of this force at
(1,0,1) b) (1,,1) c) (1,/2,0) d) (0,/2,0) z
(9.8,0,0)
(-9.8,0,0)
(0,-9.8,0) ez e er
9.8m/s2 R z  r y x

9. Vector Algebra in Components

10. Fluid particle: Differentially Small Piece of the Fluid Material
3. Vector Calculus
Fluid particle: Differentially Small Piece of the Fluid Material

11. Class 3. 8/29/06
Assignments: Online Presentation “Vector Algebra and Calculus 2”
Homework 1, due 9/5/06
Vector Algebra and calculus crib on course web site
Study groups

12. Concept of Differential Change In a Vector. The Vector Field.
Scalar field
=(r,t)
Vector field
V=V(r,t) V dV
V+dV
Differential change in vector
Change in direction
Change in magnitude

13. Change in Unit Vectors – Cylindrical System
de
er+der e ez
e+de
der P' er e P z er r d 

14. Change in Unit Vectors – Spherical System r
See “Formulae for Vector Algebra and Calculus” 

15. Example Fluid particle
Differentially small piece of the fluid material
The position of fluid particle moving in a flow varies with time. Working in different coordinate systems write down expressions for the position and, by differentiation, the velocity vectors.
V=V(t)
R=R(t)
Cartesian System O
Cylindrical System
... This is an example of the calculus of vectors with respect to time.

16. Vector Calculus w.r.t. Time
Since any vector may be decomposed into scalar components, calculus w.r.t. time, only involves scalar calculus of the components

17. High Speed Flow Past an Axisymmetric Object
Finned body of revolution fired from a gun into a supersonic wind tunnel flow for a net Mach number of 2. The plastic shell casing is seen separating. Vincenti, NASA
Shadowgraph picture is from “An Album of Fluid Motion” by Van Dyke

18. Line integrals

19. Integral Calculus With Respect to Space
D(r)
D(r) O n B r ds
D=D(r),  = (r) dS d
Surface S
Volume R A
Line Integrals
For closed loops, e.g. Circulation

20. For closed loops, e.g. Circulation
Mach approximately 2.0
Picture is from “An Album of Fluid Motion” by Van Dyke

21. Integral Calculus With Respect to Space
D(r)
D(r) O n B r ds
D=D(r),  = (r) dS d
Surface S
Volume R A
Surface Integrals
For closed surfaces
e.g. Volumetric Flow Rate through surface S
Volume Integrals

22. n dS Mach approximately 2.0
Picture is from “An Album of Fluid Motion” by Van Dyke

23. Differential Calculus w.r.t. Space Definitions of div, grad and curl
In 1-D
Elemental volume 
with surface S n dS
D=D(r), = (r)
In 3-D

24. Alternative to the Integral Definition of Grad
We want the generalization of
continued

25. Alternative to the Integral Definition of Grad Cylindrical coordinates
continued

26. Alternative to the Integral Definition of Grad
Spherical coordinates

27. Gradient  = high ndS (medium) ndS (large) n  = low Resulting ndS
(small)
Elemental volume 
with surface S
ndS
(medium)
= magnitude and direction of the slope in the scalar field at a point

28. Gradient
Component of gradient is the partial derivative in the direction of that component
Fourier´s Law of Heat Conduction

29. The integral definition given on a previous slide can also be used to obtain the formulas for the gradient.
On the next four slides, the form of GradF in Cartesian coordinates is worked out directly from the integral definition.

30. Differential form of the Gradient
Cartesian system
Evaluate integral by expanding the variation in  about a point P at the center of an elemental Cartesian volume. Consider the two x faces:
 = (x,y,z) k dz i P j
adding these gives
Face 2
Proceeding in the same way for y and z
we get
and
, so
Face 1 dx dy

31. . Gradient of a Differentiable Function, F
An element of volume with a local Cartesian coordinate system having its origin at the centroid of the corners, O x y z Δx Δz Δy M
Point M is at the centroid of the face perpendicular to the y-axis with coordinates (0, Δy/2, 0)
Other points in this face have the coordinates (x, Δy/2, z) . O
Gradient of a Differentiable Function, F
continued

34. Differential Forms of the Gradient
Cartesian
Cylindrical
Spherical
These differential forms define the vector operator 

35. Divergence Fluid particle, coincident with  at time t, after time
t has elapsed. n dS
Elemental volume 
with surface S
= proportionate rate of change of volume of a fluid particle

36. Differential Forms of the Divergence
Cartesian
Cylindrical
Spherical

37. Differential Forms of the Curl
Cartesian
Cylindrical
Spherical
Curl of the velocity vector V =
twice the circumferentially averaged angular velocity of
the flow around a point, or
a fluid particle
=Vorticity Ω
Pure rotation
No rotation
Rotation

38. Curl Elemental volume  with surface S e n dS Perimeter Ce Area 
dS=dsh
radius a
v avg. tangential velocity
= twice the avg. angular velocity
about e