ENE 503 – Computational Fluid Dynamics
WEEK 2: MATHEMATICAL TERMS
MATHEMATICAL TERMS:
Derivatives:
Full
dx du
- The derivative of u w.r.t. x, where u = u (x)
Partial
x u
- The derivative of u w.r.t. x, where u = u (x, y, z, t)
Substantial u N
x N dt
DN
.
At this point it is possible to define an extensive property, N in terms of its corresponding intensive property, as below:
ms
dm η N
where m
sis the mass of the system.
Since dm d ρ , then
s
d
ρ
η
N
with being the volume of the system.
s Vector differential operator
Del
k z j y i x
The gradient: Consider ξ x , y , z , t a scalar field
The gradient of ξ is defined as:
z k j ξ y i ξ x ξ ξ
which gives a vector field.
The divergence: Consider V x , y , z , t U i Vj W k
a vector field The divergence of V
is therefore:
z W y
V x V U
.
The curl: The curl of V is
y k U x j V x W z
i U z V y V W
x
Definitions - Rectangular coordinates:
z k j f y i f x f f
z A y
A x
A
x y z
.A
Identities:
The curl: The curl of V is
y k A x j A x A z i A z A y
A
z y x z y x
A
2 2 2 2 2 2 2
z f y
f x
f f
k A j
A i
A
x 2 y 2 z2
2
A
A A)
(
A
2
z k A B y A B x A B
z j A B y A B x A B
z i A B y A B x A B
z z z y z x
y z y y y x
x z x y x x
)B (A
τ volume bounds
which surface
the is S where
τ d
f τ
d f
τ S
τ S
