ME 57006700 Fluid Dynamical Phenomena An Intermediate Graduate Level Fluid Mechanics Class

1
ME 5700/6700Fluid Dynamical PhenomenaAn
Intermediate Graduate Level Fluid MechanicsClass

• Mathematics Review

2
Differential Calculus Review
1. Gradient Operator
y
z
x
Gradient of a Scalar (i.e., density,
temperature,etc)
2. Divergence
3. Curl
3
Differential Calculus Review
4. Laplacian -
5. Total Differential
4
Integral Calculus Review
1. Gauss Theorem Divergence theorem
y
z
x
Flux of A across S in the n direction
Divergence of A in The volume V
S3
S2
S1
5
Integral Review
2. Stokes Theorem - Circulation
Flux of the curl of through the surface S
The circulation around a closed contour is
defined as the line integral of the tangential
component of the velocity
6
Integral Review
3. Leibnizs Theorem time derivative of a
volume integral
d/dt because only a function of time remains
after integration VS is the velocity of the
integration boundary
7
Integral Review
3. Leibnizs Theorem One Dimensional Example
8
Integral Review
3. Leibnizs Theorem One Dimensional Example
9
Integral Review
3. Leibnizs Theorem time derivative of volume
integrals
For a fixed Volume (limit of integration is NOT a
function of time)
But, for a Material Volume which consists of the
same fluid particles and whose bounding surface
moves with the mass of the fluid
Reynolds Transport Theorem