Curvilinear Coordinates - User page server for CoE

Curvilinear Coordinates

Outline:

1. Orthogonal curvilinear coordinate systems
2. Differential operators in orthogonal curvilinear coordinate systems
3. Derivatives of the unit vectors in orthogonal curvilinear coordinate systems
4. Incompressible N-S equations in orthogonal curvilinear coordinate systems
5. Example: Incompressible N-S equations in cylindrical polar systems

The governing equations were derived using the most basic coordinate system, i.e,
Cartesian coordinates:

x = xˆi + yˆj + zkˆ
grad f = ∇f = ∂f ˆi + ∂f ˆj + ∂f kˆ

∂x ∂y ∂z
div F = ∇ ⋅ F = ∂F1 + ∂F2 + ∂F3

∂x ∂y ∂z
ˆi ˆj kˆ

curlf = ∇ × F = ∂ ∂ ∂
∂x ∂y ∂z

F1 F2 F3

Laplacian = ∇2 f = ∂2 f + ∂2 f + ∂2 f
∂x2 ∂y2 ∂z2

Example: incompressible flow equations

∇⋅V =0

ρ DV = −∇ ( p + γ z) + µ∇2V

Dt

ρ ⎛ ∂V + V ⋅ ∇V ⎞ = −∇ ( p + γ z ) + µ∇2 V
⎝⎜ ∂t ⎟⎠

ρ ⎡ ∂V + 1 ∇ ( V ⋅ V ) − V × ω⎦⎥⎤ = −∇ ( p + γ z ) + µ ⎣⎡∇ (∇ ⋅ V ) − ∇ × ω⎦⎤
⎣⎢ ∂t 2

ω =∇×V

∇ (∇ ⋅ V) = 0 in the above equation, but retained to keep the complete vector identity

for ∇2V in equation.

However, once the equations are expressed in vector invariant form (as above) they can
be transformed into any convenient coordinate system through the use of appropriate definitions
for the ∇ , ∇ ⋅ , ∇ × , and ∇2 . Frequently, alternative coordinate systems are desirable which either

exploit certain features of the flow at hand or facilitate numerical procedures. The most general
coordinate system for fluid flow problems are nonorthogonal curvilinear coordinates. A special
case of these are orthogonal curvilinear coordinates. Here we shall derive the appropriate
relations for the latter using vector technique. It should be recognized that the derivation can
also be accomplished using tensor analysis

1. Orthogonal curvilinear coordinate systems
Suppose that the Cartesian coordinates ( x, y, z) are expressed in terms of the new

coordinates ( x1, x2, x3 ) by the equations
x = x ( x1, x2 , x3 )
y = y ( x1, x2 , x3 )
z = z ( x1, x2 , x3 )

where it is assumed that the correspondence is unique and that the inverse mapping exists.

For example, circular cylindrical coordinates

x = r cosθ
y = r sinθ
z=z

i.e., at any point P , x1 curve is a straight line, x2 curve is a circle, and the x3 curve is a
straight line.

The position vector of a point P in space is

R = xˆi + yˆj + zkˆ

R = (r cosθ ) ˆi + (r sinθ ) ˆj + ( z)kˆ for cylindrical coordinates

By definition a vector tangent to the x1 curve is given by:

R = xx1ˆi + yx1ˆj + zx1kˆ (Subscript denotes partial differentiation)
x1

So that the unit vectors tangent to the xi curve are

R R R x3
eˆ1 = x1 , eˆ2 = x2 , eˆ 3 = h3

h1 h2

Where hi = R are called the metric coefficients or scale factors
x1

hr = 1, hθ = r , hz = 1 for cylindrical coordinates

The arc length along a curve in any direction is given by

ds2 = dR ⋅ dR = h12dx12 + h22dx22 + h32dx32

Since dR = R xi dxi = hi dxieˆ i and R = hi eˆ i
xi

and since the xi are orthogonal: eˆi ⋅ eˆ j = ⎧0, i≠ j
⎩⎨1, i= j

An element of volume is given by the triple product

d∀ = (h1dx1eˆ1 × h2dx2eˆ2 ) ⋅ h3dx3eˆ3 = h1h2h3dx1dx2dx3

Where since the xi are orthogonal eˆ1 × eˆ2 = eˆ3

Finally, on the surface x1 = constant, the vector element of surface area is given by
ds1 = h2dx2eˆ2 × h3dx3eˆ3 = eˆ1h2h3dx2dx3

With similar results for x2 and x3 = constant
ds2 = eˆ2h3h1dx3dx1
ds3 = eˆ3h1h2dx1dx2

2. Differential operators in orthogonal curvilinear coordinate systems

With the above in hand, we now proceed to obtain the desired vector operators

2.1 Gradient ∇f = 1 ∂f eˆ1 + 1 ∂f eˆ 2 +1 ∂f eˆ 3
h1 ∂x1 h2 ∂x2 h3 ∂x3

By definition: df = ∇f ⋅ dR = fxi dxi
If we temporarily write ∇f = λ1eˆ1 + λ2eˆ2 + λ3eˆ3

Then by comparison

df = fxi dxi = λihidxi

λi = 1 ∂f
hi ∂xi

∇ = 1 ∂ eˆ1 + 1 ∂ eˆ 2 + 1 ∂ eˆ 3
h1 ∂x1 h2 ∂x2 h3 ∂x3

∇ = ∂ eˆ r +1 ∂ eˆθ + ∂ eˆ z for cylindrical coordinates
∂r r ∂θ ∂z

Note ∇xi = eˆ i
hi

λi = 1 ∂f
hi ∂xi

So that by definition ( curl(grad f ) = 0 )

∇ × ∇xi = ∇ × eˆi = 0
hi

Also eˆ1 = eˆ2 × eˆ3 = ∇x2 × ∇x3
h2h3 h2 h3

So that by definition ( ∇ ⋅(∇f × ∇g ) = 0 )

∇ ⋅ ⎛ eˆ1 ⎞ = ∇ ⋅ ⎛ eˆ 2 ⎞ = ∇ ⋅ ⎛ eˆ 3 ⎞ = 0
⎜ h2h3 ⎟ ⎜ h3h1 ⎟ ⎜ h1h2 ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠

2.2 Divergence ∇⋅F = 1 ⎡∂ (h2h3F1 ) + ∂ (h3h1F2 ) + ∂ ( h1h2 F3 ) ⎤
h1h2h3 ⎢ ∂x2 ∂x3 ⎥
⎣ ∂x1 ⎦

∇ ⋅ F = ∇ ⋅( F1eˆ1 ) + ∇ ⋅ ( F2eˆ2 ) + ∇ ⋅ ( F3eˆ3 )

∇ ⋅ ( F1eˆ1 ) = ∇ ⋅ ⎡ h2 h3 F1 ⎛ eˆ1 ⎞⎤ using ∇ ⋅(ϕu) = ϕ∇ ⋅u + u ⋅∇ϕ
⎢ ⎜ h2h3 ⎟⎥
⎣ ⎝ ⎠⎦

= eˆ1 ⋅ ∇ (h2h3F1 ) using ⎛ eˆ1 ⎞ = ⎛ eˆ 2 ⎞ = ⎛ eˆ 3 ⎞ = 0
h2h3 ∇⋅⎜ h2h3 ⎟ ∇⋅⎜ h3h1 ⎟ ∇⋅⎜ h1h2 ⎟
⎠ ⎠ ⎠
⎝ ⎝ ⎝

= 1 ∂ (h2h3F1 )
h1h2h3 ∂x1

Treating the other terms in a similar manner results in

∇ ⋅F = 1 ⎡∂ (h2h3F1 ) + ∂ ( h3h1 F2 ) + ∂ ( h1h2F3 )⎤⎥
h1h2h3 ⎢ ∂x2 ∂x3
⎣ ∂x1 ⎦

∇ ⋅F = 1 ⎡∂ ( rF1 ) + ∂ ( F2 ) + ∂ ( rF3 )⎤⎥⎦
r ⎢⎣ ∂r ∂θ ∂z

= 1 ∂ (rF1 ) + 1 ∂ ( F2 ) + ∂ ( F3 ) for cylindrical coordinates
r ∂r r ∂θ ∂z

h1eˆ1 h2eˆ2 h3eˆ3

2.3 Curl ∇ × F = 1 ∂ ∂ ∂

h1h2h3 ∂x1 ∂x2 ∂x3

h1F1 h2 F2 h3F3

∇ × F = ∇ × ( F1eˆ1 ) + ∇ × ( F2eˆ2 ) + ∇ × ( F3eˆ3 )

∇ × ( F1eˆ1 ) = ∇ × ⎡⎢( h1 F1 ) ⎛ eˆ1 ⎞ ⎤
⎜ h1 ⎟ ⎥
⎣ ⎝ ⎠ ⎦

= − eˆ1 × ∇ ( h1 F1 ) using ∇ × (ϕu) = ϕ∇ × u + ∇ϕ × u and ∇ × eˆi =0
h1 hi

= − eˆ1 × ⎡ 1 ∂ ( h1F1 ) eˆ1 + 1 ∂ ( h1F1 ) eˆ 2 + 1 ∂ ( h1F1 ) eˆ 3 ⎤
h1 ⎢ h1 h2 h3 ⎥
⎣ ∂x1 ∂x2 ∂x3 ⎦

= − eˆ 3 ∂ (h1F1 ) + eˆ 2 ∂ (h1F1 )
h1h2 ∂x2 h3h1 ∂x3

= 1 ⎢⎡ h2eˆ 2 ∂ − h3eˆ 3 ∂ ⎤ ( h1 F1 )
h1h2h3 ⎣ ∂x3 ∂x2 ⎥
⎦

h1eˆ1 h2eˆ2 h3eˆ3

∇×F = 1 ∂ ∂ ∂
h1h2h3 ∂x1 ∂x2 ∂x3

h1F1 h2 F2 h3F3
eˆr reˆθ eˆ z

∇ × F = 1 ∂ ∂ ∂ for cylindrical coordinates

r ∂r ∂θ ∂z

F1 rF2 F3

2.4 Laplacian acting on a scalar ∇2 f = 1 ⎡ ∂ ⎛ h2h3 ∂ ⎞ + ∂ ⎛ h3h1 ∂ ⎞ + ∂ ⎛ h1h2 ∂ ⎞⎤
h1h2h3 ⎢ ∂x1 ⎜ h1 ∂x1 ⎟ ∂x2 ⎜ h2 ∂x2 ⎟ ∂x3 ⎜ h3 ⎟⎥
⎣ ⎝ ⎠ ⎝ ⎠ ⎝ ∂x3 ⎠⎦

∇2 = ∇⋅∇

= 1 ⎡∂ ⎛ h2h3 ∂ ⎞ ∂ ⎛ h3h1 ∂ ⎞ ∂ ⎛ h1h2 ∂ ⎞⎤
h1h2h3 ⎢ ⎜ h1 ∂x1 ⎟+ ∂x2 ⎜ h2 ∂x2 ⎟+ ∂x3 ⎜ h3 ∂x3 ⎟⎥
⎣ ∂x1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎦

∇2 = 1 ⎡∂ ⎛ r ∂ ⎞ + ∂ ⎛ 1 ∂ ⎞ + ∂ ⎛ r ∂ ⎞⎤
r ⎢⎣ ∂r ⎜⎝ ∂r ⎟⎠ ∂θ ⎜⎝ r ∂θ ⎟⎠ ∂z ⎜⎝ ∂z ⎟⎠⎥⎦

= 1 ∂ ⎛ r ∂ ⎞ + 1 ∂ ⎛ 1 ∂ ⎞ + 1 ∂ ⎛ r ∂ ⎞
r ∂r ⎜⎝ ∂r ⎠⎟ r ∂θ ⎜⎝ r ∂θ ⎠⎟ r ∂z ⎝⎜ ∂z ⎠⎟

2.5 Laplacian acting on a vector ∇2F = ∇ (∇ ⋅ F) − ∇ ×(∇ × F)

Using ∇f = 1 ∂f eˆ1 + 1 ∂f eˆ 2 + 1 ∂f eˆ 3
h1 ∂x1 h2 ∂x2 h3 ∂x3

and ∇⋅F = 1 ⎡∂ (h2h3F1 ) + ∂ (h3h1F2 ) + ∂ ( h1h2 F3 ) ⎤
h1h2h3 ⎢ ∂x2 ∂x3 ⎥
⎣ ∂x1 ⎦

∇(∇⋅F) =

1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ ( h3h1F2 )+ ∂ ( h1h2 F3 ) ⎤ ⎤ eˆ1
h1 ∂x1 ⎢ ⎢ ∂x2 ∂x3 ⎥ ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦

+ 1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ (h3h1F2 ) + ∂ ( h1h2 F3 )⎥⎤ ⎤ eˆ
h2 ∂x2 ⎢ ⎢ ∂x2 ∂x3 ⎥
⎣ h1h2 h3 ⎣ ∂x1 ⎦⎦ 2

+ 1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ (h3h1F2 ) + ∂ ( h1h2 F3 ) ⎤ ⎤ eˆ 3
h3 ∂x3 ⎢ ⎢ ∂x2 ∂x3 ⎥ ⎥
⎣ h1h2 h3 ⎣ ∂x1 ⎦⎦

h1eˆ1 h2eˆ2 h3eˆ3
Using ∇ × F = 1 ∂ ∂ ∂

h1h2h3 ∂x1 ∂x2 ∂x3

h1F1 h2 F2 h3F3

∇×(∇×F) =

= 1 ⎡∂ ⎛ h3 ⎡∂ ( h2 F2 ) − ∂ ( h1 F1 ) ⎤ ⎞ − ∂ ⎛ h2 ⎡∂ (h1F1 ) − ∂ ( h3 F3 )⎥⎤ ⎞⎤ eˆ1
h2h3 ⎢ ⎜⎝⎜ h1h2 ⎢ ∂x2 ⎥ ⎟⎠⎟ ∂x3 ⎝⎜⎜ h1h3 ⎢ ∂x1 ⎠⎟⎟⎥⎦⎥
⎣⎢ ∂x2 ⎣ ∂x1 ⎦ ⎣ ∂x3 ⎦

+ 1 ⎡∂ ⎛ h1 ⎡∂ (h3F3 ) − ∂ ( h2 F2 )⎥⎤ ⎞ − ∂ ⎛ h3 ⎡∂ (h2F2 ) − ∂ ( h1F1 ) ⎤ ⎞⎤ eˆ 2
h1h3 ⎢ ⎝⎜⎜ h2h3 ⎢ ∂x3 ⎠⎟⎟ ∂x1 ⎜⎜⎝ h1h2 ⎢ ∂x2 ⎥ ⎟⎟⎠⎥⎦⎥
⎢⎣ ∂x3 ⎣ ∂x2 ⎦ ⎣ ∂x1 ⎦

+ 1 ⎡∂ ⎛ h2 ⎡∂ (h1F1 ) − ∂ ( h3 F3 ) ⎤ ⎞ − ∂ ⎛ h1 ⎡∂ (h3F3 ) − ∂ ( h2 F2 )⎥⎤ ⎞ ⎤ eˆ 3
h1h2 ⎢ ⎝⎜⎜ h1h3 ⎢ ∂x1 ⎥ ⎠⎟⎟ ∂x2 ⎝⎜⎜ h2h3 ⎢ ∂x3 ⎠⎟⎟ ⎥
⎣⎢ ∂x1 ⎣ ∂x3 ⎦ ⎣ ∂x2 ⎦ ⎦⎥

Combining those two terms gives

∇2F = ∇(∇⋅F) − ∇×(∇×F) =

= 1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ ( h3h1F2 )+ ∂ ( h1h2 F3 ) ⎤ ⎤ eˆ1
h1 ∂x1 ⎢ ⎢ ∂x2 ∂x3 ⎥ ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦

− 1 ⎡∂ ⎛ h3 ⎡∂ ( h2 F2 ) − ∂ ( h1F1 )⎥⎤ ⎞ − ∂ ⎛ h2 ⎡∂ (h1F1 ) − ∂ ( h3 F3 )⎤⎥ ⎞ ⎤ eˆ1
h2h3 ⎢ ⎝⎜⎜ h1h2 ⎢ ∂x2 ⎟⎠⎟ ∂x3 ⎝⎜⎜ h1h3 ⎢ ∂x1 ⎠⎟⎟ ⎥
⎣⎢ ∂x2 ⎣ ∂x1 ⎦ ⎣ ∂x3 ⎦ ⎦⎥

+ 1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ (h3h1F2 ) + ∂ ( h1h2 F3 )⎥⎤ ⎤ eˆ
h2 ∂x2 ⎢ ⎢ ∂x2 ∂x3 ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦ 2

− 1 ⎡∂ ⎛ h1 ⎡∂ ( h3 F3 ) − ∂ ( h2 F2 )⎥⎤ ⎞ − ∂ ⎛ h3 ⎡∂ ( h2 F2 ) − ∂ (h1F1 )⎥⎤ ⎞⎤ eˆ 2
h1h3 ⎢ ⎝⎜⎜ h2h3 ⎢ ∂x3 ⎠⎟⎟ ∂x1 ⎜⎝⎜ h1h2 ⎢ ∂x2 ⎟⎟⎠⎥⎥⎦
⎢⎣ ∂x3 ⎣ ∂x2 ⎦ ⎣ ∂x1 ⎦

+ 1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ (h3h1F2 ) + ∂ ( h1h2 F3 ) ⎤ ⎤ eˆ 3
h3 ∂x3 ⎢ ⎢ ∂x2 ∂x3 ⎥ ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦

− 1 ⎡∂ ⎛ h2 ⎡∂ (h1F1 ) − ∂ ( h3 F3 )⎥⎤ ⎞ − ∂ ⎛ h1 ⎡∂ ( h3 F3 ) − ∂ ( h2 F2 )⎥⎤ ⎞⎤ eˆ 3
h1h2 ⎢ ⎝⎜⎜ h1h3 ⎢ ∂x1 ⎠⎟⎟ ∂x2 ⎝⎜⎜ h2h3 ⎢ ∂x3 ⎠⎟⎟⎦⎥⎥
⎣⎢ ∂x1 ⎣ ∂x3 ⎦ ⎣ ∂x2 ⎦

For cylindrical coordinates (r,θ , z) , h1 = hr = 1, h2 = hθ = r, h3 = hz = 1 , and use the definition

of Laplacian operator acting on a scalar ∇2 f

∇2 f = 1 ⎡ ∂ ⎛ r ∂ ⎞ + ∂ ⎛1 ∂ ⎞ + ∂ ⎛ r ∂ ⎞⎤
r ⎢⎣ ∂r ⎜⎝ ∂r ⎟⎠ ∂θ ⎜⎝ r ∂θ ⎟⎠ ∂z ⎜⎝ ∂z ⎟⎠⎥⎦

= 1 ∂ ⎛ r ∂ ⎞ + 1 ∂2 + ∂2
r ∂r ⎜⎝ ∂r ⎟⎠ r2 ∂θ 2 ∂z 2

= 1 ∂ + ∂2 + 1 ∂2 + ∂2
r ∂r ∂r 2 r2 ∂θ 2 ∂z2

( )∇2F ⎛ 1 2 ∂F2 ⎞ ⎛ F2 2 ∂F1 ⎞
= aeˆ r + beˆθ + ceˆ z = ⎝⎜ ∇ 2 F1 − r2 F1 − r2 ∂θ ⎠⎟ eˆ r + ⎜⎝ ∇2 F2 − r2 + r2 ∂θ ⎟⎠ eˆθ + ∇2 F3 eˆ z

a=

= 1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ ( h3h1F2 ) + ∂ ( h1h2 F3 ) ⎤ ⎤
h1 ∂x1 ⎢ ⎢ ∂x2 ∂x3 ⎥ ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦

− 1 ⎡∂ ⎛ h3 ⎡∂ ( h2 F2 )− ∂ ( h1F1 )⎤⎥ ⎞ − ∂ ⎛ h2 ⎡∂ (h1F1 ) − ∂ ( h3 F3 )⎥⎤ ⎞⎤
h2h3 ⎢ ⎜⎝⎜ h1h2 ⎢ ∂x2 ⎟⎠⎟ ∂x3 ⎜⎝⎜ h1h3 ⎢ ∂x1 ⎟⎠⎟⎥⎥⎦
⎢⎣ ∂x2 ⎣ ∂x1 ⎦ ⎣ ∂x3 ⎦

= ∂ ⎡1 ⎡∂ ( rF1 ) + ∂ ( F2 ) + ∂ ( rF3 )⎥⎤⎦ ⎤⎥⎦
∂r ⎣⎢ r ⎢⎣ ∂r ∂θ ∂z

− 1 ⎡∂ ⎛ 1 ⎡∂ ( rF2 ) − ∂ ( F1 )⎦⎥⎤ ⎞ − ∂ ⎛ r ⎡∂ ( F1 ) − ∂ ( F3 )⎦⎥⎤ ⎞⎤
r ⎢ ⎝⎜ r ⎢⎣ ∂r ∂θ ⎟⎠ ∂z ⎝⎜ ⎢⎣ ∂z ∂r ⎠⎟⎦⎥
⎣ ∂θ

= ∂ ⎡1 ⎡ F1 + r ∂F1 + ∂F2 + r ∂F3 ⎤⎤
∂r ⎢⎣ r ⎢⎣ ∂r ∂θ ∂z ⎥⎦ ⎦⎥

− 1 ⎡∂ ⎛ 1 ⎡ F2 + r ∂F2 − ∂F1 ⎤⎞ − ∂ ⎛ r ∂F1 − r ∂F3 ⎞⎤
r ⎢ ⎝⎜ r ⎢⎣ ∂r ∂θ ⎥⎦ ⎠⎟ ∂z ⎝⎜ ∂z ∂r ⎟⎠⎦⎥
⎣ ∂θ

= ∂ ⎡1 F1 + ∂F1 + 1 ∂F2 + ∂F3 ⎤
∂r ⎢⎣ r ∂r r ∂θ ∂z ⎦⎥

− 1 ⎡∂ ⎛ 1 F2 + ∂F2 − 1 ∂F1 ⎞ − r ∂2 F1 + r ∂2 F3 ⎤
r ⎢ ⎜⎝ r ∂r r ∂θ ⎟⎠ ∂z 2 ∂z∂r ⎥
⎣ ∂θ ⎦

= ⎛ −1 F1 ⎞ + 1 ∂F1 + ∂2 F1 + ⎛ −1 ∂F2 ⎞ + 1 ∂2 F2 + ∂2 F3
⎜⎝ r2 ⎠⎟ r ∂r ∂r 2 ⎜⎝ r2 ∂θ ⎟⎠ r ∂r∂θ ∂r∂z

− 1 ⎡1 ∂F2 + ∂2 F2 − 1 ∂2 F1 − r ∂2 F1 + r ∂2 F3 ⎤
r ⎢ ∂θ ∂θ∂r r ∂θ 2 ∂z 2 ∂z∂r ⎥
⎣ r ⎦

= −1 +1 ∂F1 + ∂2 F1 − 1 ∂F2 + 1 ∂2 F2 + ∂2 F3
r 2 F1 r ∂r ∂r 2 r2 ∂θ r ∂r∂θ ∂r∂z

⎡ 1 ∂F2 − 1 ∂2F2 + 1 ∂2F1 + ∂2F1 − ∂2 F3 ⎤
+ ⎢− ⎥
⎣⎢ r2 ∂θ r ∂θ∂r r2 ∂θ 2 ∂z2 ∂z∂r ⎥⎦

= −1 F1 + 1 ∂F1 + ∂ 2 F1 − 2 ∂F2 + ⎡1 ∂2 F1 + ∂2 F1 ⎤
r2 r ∂r ∂r 2 r2 ∂θ ⎢ ∂θ 2 ∂z 2 ⎥
⎣ r 2 ⎦

= ⎛1 ∂F1 + ∂2 F1 + 1 ∂2 F1 + ∂2 F1 ⎞ − 1 F1 − 2 ∂F2
⎜ ∂r ∂r 2 r2 ∂θ 2 ∂z 2 ⎟ r2 r2 ∂θ
⎝ r ⎠

= ⎛1 ∂ ⎛ ∂F1 ⎞ + 1 ∂2 F1 + ∂2 F1 ⎞ − 1 F1 − 2 ∂F2
⎜ ∂r ⎜⎝ ∂r ⎠⎟ r2 ∂θ 2 ∂z 2 ⎟ r2 r2 ∂θ
⎝ r ⎠

= ∇2 F1 − 1 F1 − 2 ∂F2
r2 r2 ∂θ

b=

= 1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ (h3h1F2 ) + ∂ ( h1h2 F3 ) ⎤ ⎤
h2 ∂x2 ⎢ ⎢ ∂x2 ∂x3 ⎥ ⎥
⎣ h1h2 h3 ⎣ ∂x1 ⎦⎦

− 1 ⎡∂ ⎛ h1 ⎡∂ ( h3 F3 ) − ∂ ( h2 F2 )⎥⎤ ⎞ − ∂ ⎛ h3 ⎡∂ ( h2 F2 ) − ∂ ( h1F1 )⎤⎥ ⎞⎤
h1h3 ⎢ ⎜⎝⎜ h2h3 ⎢ ∂x3 ⎠⎟⎟ ∂x1 ⎜⎝⎜ h1h2 ⎢ ∂x2 ⎟⎟⎠⎥⎦⎥
⎢⎣ ∂x3 ⎣ ∂x2 ⎦ ⎣ ∂x1 ⎦

= 1 ∂ ⎡1 ⎡∂ (rF1 ) + ∂ ( F2 ) + ∂ ( rF3 )⎦⎥⎤ ⎤
r ∂θ ⎢⎣ r ⎢⎣ ∂r ∂θ ∂z ⎦⎥

− ⎡∂ ⎛ 1 ⎡∂ ( F3 ) − ∂ ( rF2 )⎦⎥⎤ ⎞ − ∂ ⎛ 1 ⎡∂ ( rF2 ) − ∂ ( F1 )⎥⎤⎦ ⎞⎤
⎢ ⎜⎝ r ⎣⎢ ∂θ ∂z ⎠⎟ ∂r ⎜⎝ r ⎢⎣ ∂r ∂θ ⎟⎠⎦⎥
⎣ ∂z

= 1 ∂ ⎡1 ⎡ F1 +r ∂F1 + ∂F2 + r ∂F3 ⎤⎤
r ∂θ ⎢⎣ r ⎢⎣ ∂r ∂θ ∂z ⎥⎦ ⎥⎦

− ⎡∂ ⎛ 1 ∂F3 − ∂F2 ⎞ − ∂ ⎛ 1 ⎡ F2 + r ∂F2 − ∂F1 ⎤ ⎞⎤
⎢ ⎝⎜ r ∂θ ∂z ⎟⎠ ∂r ⎜ r ⎣⎢ ∂r ∂θ ⎟⎥
⎣ ∂z ⎝ ⎥⎦ ⎠⎦

= 1 ∂ ⎡1 F1 + ∂F1 + 1 ∂F2 + ∂F3 ⎤
r ∂θ ⎢⎣ r ∂r r ∂θ ∂z ⎦⎥

− ⎡ 1 ∂2 F3 − ∂2 F2 − ∂ ⎛ 1 F2 + ∂F2 − 1 ∂F1 ⎞⎤
⎢ r ∂z∂θ ∂z 2 ∂r ⎝⎜ r ∂r r ∂θ ⎟⎠⎦⎥
⎣

= 1 ⎡1 ∂F1 + ∂2 F1 + 1 ∂2 F2 + ∂2 F3 ⎤
r ⎢ ∂θ ∂θ∂r r ∂θ 2 ∂θ∂z ⎥
⎣ r ⎦

− ⎡1 ∂2 F3 − ∂2 F2 − ⎛ − F2 + 1 ∂F2 + ∂2 F2 − −1 ∂F1 − 1 ∂2 F1 ⎞⎤
⎢ ∂z∂θ ∂z 2 ⎜ r2 r ∂r ∂r 2 r 2 ∂θ r ∂r∂θ ⎟⎥
⎣ r ⎝ ⎠⎦

= 1 ∂F1 + 1 ∂2F1 + 1 ∂2F2 + 1 ∂2F3
r2 ∂θ r ∂θ∂r r2 ∂θ 2 r ∂θ∂z

+ ⎡ 1 ∂2 F3 + ∂2F2 − F2 + 1 ∂F2 + ∂2F2 + 1 ∂F1 − 1 ∂2F1 ⎤
⎢− r ∂z∂θ ∂z2 r 2 r ∂r ∂r 2 r2 ∂θ r ∂r∂θ ⎥
⎣⎢ ⎦⎥

= ⎛1 ∂F2 + ∂2 F2 + 1 ∂ 2 F2 + ∂2 F2 ⎞ − F2 + 2 ∂F1
⎜ ∂r ∂r 2 r2 ∂θ 2 ∂z 2 ⎟ r2 r2 ∂θ
⎝ r ⎠

= ⎛1 ∂ ⎛ r ∂F2 ⎞ + 1 ∂2 F2 + ∂2 F2 ⎞ − F2 + 2 ∂F1
⎜ ∂r ⎜⎝ ∂r ⎟⎠ r2 ∂θ 2 ∂z 2 ⎟ r2 r2 ∂θ
⎝ r ⎠

= ∇2 F2 − F2 + 2 ∂F1
r2 r2 ∂θ

c=

= 1 ∂ ⎡1 ⎡∂ (h2h3F1 ) + ∂ ( h3h1F2 ) + ∂ ( h1h2 F3 )⎤⎥ ⎤
h3 ∂x3 ⎢ ⎢ ∂x2 ∂x3 ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦

− 1 ⎡∂ ⎛ h2 ⎡∂ (h1F1 ) − ∂ ( h3 F3 )⎤⎥ ⎞ − ∂ ⎛ h1 ⎡∂ ( h3 F3 ) − ∂ ( h2 F2 )⎤⎥ ⎞⎤
h1h2 ⎢ ⎜⎝⎜ h1h3 ⎢ ∂x1 ⎠⎟⎟ ∂x2 ⎝⎜⎜ h2h3 ⎢ ∂x3 ⎠⎟⎟⎥⎥⎦
⎣⎢ ∂x1 ⎣ ∂x3 ⎦ ⎣ ∂x2 ⎦

= ∂ ⎡ 1 ⎡ ∂ ( rF1 ) + ∂ ( F2 ) + ∂ ( rF3 )⎤⎦⎥ ⎤
∂z ⎣⎢ r ⎢⎣ ∂r ∂θ ∂z ⎦⎥

− 1 ⎡∂ ⎛ r ⎡∂ ( F1 ) − ∂ ( F3 )⎤⎦⎥ ⎞ − ∂ ⎛ 1 ⎡∂ ( F3 ) − ∂ ( rF2 )⎦⎥⎤ ⎞⎤
r ⎢ ⎝⎜ ⎢⎣ ∂z ∂r ⎠⎟ ∂θ ⎜⎝ r ⎣⎢ ∂θ ∂z ⎠⎟⎦⎥
⎣ ∂r

= ∂ ⎡1 ⎡ F1 + r ∂F1 + ∂F2 + r ∂F3 ⎤⎤
∂z ⎢⎣ r ⎢⎣ ∂r ∂θ ∂z ⎦⎥ ⎦⎥

− 1 ⎡∂ ⎛ r ∂F1 − r ∂F3 ⎞ − ∂ ⎛ 1 ∂F3 − ∂F2 ⎞⎤
r ⎣⎢ ∂r ⎝⎜ ∂z ∂r ⎠⎟ ∂θ ⎝⎜ r ∂θ ∂z ⎠⎟⎦⎥

= ∂ ⎡ F1 + ∂F1 + 1 ∂F2 + ∂F3 ⎤
∂z ⎣⎢ r ∂r r ∂θ ∂z ⎥⎦

− 1 ⎡⎛ ∂F1 + r ∂2 F1 − ∂F3 − r ∂2 F3 ⎞ − ⎛ 1 ∂2 F3 − ∂2 F2 ⎞⎤
r ⎢⎜ ∂z ∂r∂z ∂r ∂r 2 ⎟ ⎜ r ∂θ 2 ∂θ∂z ⎟⎥
⎣⎝ ⎠ ⎝ ⎠⎦

= 1 ∂F1 + ∂2F1 + 1 ∂2F2 + ∂2F3
r ∂z ∂z∂r r ∂z∂θ ∂z2

+ ⎡ 1 ∂F1 − ∂2 F1 + 1 ∂F3 + ∂2F3 +1 ∂2 F3 − 1 ∂2F2 ⎤
⎢− ∂r∂z r ∂r ∂r 2 r2 ∂θ 2 r ∂θ∂z ⎥
⎢⎣ r ∂z ⎥⎦

= 1 ∂F3 + ∂2F3 + 1 ∂2F3 + ∂2F3
r ∂r ∂r2 r2 ∂θ 2 ∂z2

= 1 ∂ ⎛ r ∂F3 ⎞ + ∂2 F3 + 1 ∂ 2 F3 + ∂2 F3
r ∂r ⎝⎜ ∂r ⎟⎠ ∂r 2 r2 ∂θ 2 ∂z 2

= ∇2F3

3. Derivatives of the unit vectors in orthogonal curvilinear coordinate systems

The last topic to be discussed concerning curvilinear coordinates is the procedure to obtain the

derivatives of the unit vectors, i.e. ∂ eˆ i = eˆij
∂x j

Such quantities are required, for example, in obtaining the rate-of-strain and rotation tensor
eij = ∇V

( )εij 1 = 1 (∇V + V∇)
= 2 eij + eiTj
2

( )ωij 1 = 1 (∇V − V∇)
= 2 eij − eiTj
2

To simplify the notation we define:

R = ri , R = hieˆi = ri
xi xi

∂ R xi = rij and ∂ hi = hij
∂x j ∂x j

Note that rij is symmetric, i.e. rij = rji

r1 = h1eˆ1

r2 = h2eˆ2

r = h3eˆ 3
3

r11 = aeˆ1 + beˆ2 + ceˆ3 = h11eˆ1 + h1eˆ11

r12 = h12eˆ1 + h1eˆ12

r13 = h13eˆ1 + h1eˆ13

3.1 Derivation of eˆ11 = − h12 eˆ 2 − h13 eˆ 3
h2 h3

r1 ⋅ r1 = h12

r1 ⋅ r11 = h1h11

r ⋅ r = h1h12
1 12

r1 ⋅ r13 = h1h13

r1 ⋅ r2 = 0

→ ∂ (r1 ⋅r2 ) = 0

∂x1
→ r11 ⋅ r2 + r1 ⋅ r21 = 0
→ r11 ⋅ r2 = −r1 ⋅ r12

→ r11 ⋅ r2 = −h1h12

r1 ⋅ r3 = 0

→ ∂ (r1 ⋅r3 ) = 0

∂x1
→ r11 ⋅ r3 + r1 ⋅ r31 = 0
→ r11 ⋅ r3 = −r1 ⋅ r13
→ r11 ⋅ r3 = −h1h13

r11 = h11eˆ1 − h1h12 eˆ 2 − h1h13 eˆ 3 = h11eˆ1 + h1eˆ11
h2 h3

→ eˆ11 = − h12 eˆ 2 − h13 eˆ 3
h2 h3

3.2 Derivation of eˆ 22 =− h21 eˆ1 − h23 eˆ 3
h1 h3

r2 ⋅ r2 = h22

r2 ⋅ r22 = h2h22

r2 ⋅ r21 = h2h21

r ⋅ r = h2h23
2 23

r1 ⋅ r2 = 0

→ ∂ (r1 ⋅r2 ) = 0

∂x2

→ r ⋅ r + r ⋅ r = 0
12 2 1 22

→ r22 ⋅ r1 = −r2 ⋅ r21

→ r22 ⋅ r1 = −h2h21

r ⋅ r = 0
2 3

→ ∂ (r2 ⋅r3 ) = 0

∂x2

→ r22 ⋅ r3 + r2 ⋅ r32 = 0

→ r ⋅ r = −r2 ⋅ r
22 3 23

→ r22 ⋅ r3 = −h2h23

r22 = − h2h21 eˆ1 + h22eˆ 2 − h2h23 eˆ 3 = h22eˆ 2 + h2eˆ 22
h1 h3

→ eˆ 22 = − h21 eˆ1 − h23 eˆ 3
h1 h3

3.3 Derivation of eˆ 33 = − h31 eˆ1 − h32 eˆ 2
h1 h2

r ⋅ r = h32
3 3

r3 ⋅ r31 = h3h31

r3 ⋅ r32 = h3h32

r3 ⋅ r33 = h3h33

r1 ⋅ r3 = 0

→ ∂ (r1 ⋅r3 ) = 0

∂x3

→ r13 ⋅ r3 + r1 ⋅ r33 = 0

→ r ⋅ r = −r3 ⋅ r
33 1 31

→ r33 ⋅ r1 = −h3h31

r2 ⋅ r3 = 0

→ ∂ (r2 ⋅r3 ) = 0

∂x3

→ r23 ⋅ r3 + r2 ⋅ r33 = 0

→ r33 ⋅ r2 = −r3 ⋅ r32

→ r ⋅ r = −h3h32
33 2

r33 = − h3h31 eˆ1 − h3h32 eˆ 2 + h33eˆ 3 = h33eˆ 3 + h3eˆ 33
h1 h2

→ eˆ 33 = − h31 eˆ1 − h32 eˆ 2
h1 h2

3.4 Derivation of eˆ 32 = h23 eˆ2 , eˆ 23 = h32 eˆ 3
h3 h2

r1 ⋅ r2 = 0 → ∂ (r1 ⋅ r2 ) = r13 ⋅ r2 + r1 ⋅ r23 = 0 (1)
∂x3 (2)
(3)
r2 ⋅ r3 = 0 → ∂ (r2 ⋅ r3 ) = r21 ⋅ r3 + r2 ⋅ r31 = 0
∂x1

r3 ⋅ r1 = 0 → ∂ (r3 ⋅r1 ) = r32 ⋅ r1 + r3 ⋅ r12 = 0
∂x2

(3) − (2) = 0

→ r32 ⋅ r1 − r2 ⋅ r31 = 0

→ r ⋅ r = r ⋅ r
32 1 2 31

→ r32 ⋅ r1 = −r1 ⋅ r23

→ r32 ⋅ r1 = 0

r = aeˆ1 + beˆ2 + ceˆ3 = h32eˆ 3 + h3eˆ 32
32

r = aeˆ1 + beˆ2 + ceˆ3 = h23eˆ 2 + h32eˆ3
32

eˆ 32 = h23 eˆ 2
h3

r23 = h23eˆ 2 + h2eˆ 23

r = r = aeˆ1 + beˆ2 + ceˆ3 = h23eˆ 2 + h32eˆ 3
23 32

eˆ 23 = h32 eˆ 3
h2

3.5 Derivation of eˆ 21 = h12 eˆ1 , eˆ12 = h21 eˆ 2
h2 h1

(2) − (1) = 0

→ r21 ⋅ r3 − r1 ⋅ r23 = 0

→ r21 ⋅ r3 = r1 ⋅ r23

→ r21 ⋅ r3 = r3 ⋅ r12

→ r ⋅ r = 0
21 3

r21 = h21eˆ 2 + h2eˆ 21

r21 = h12eˆ1 + h21eˆ 2

eˆ 21 = h12 eˆ1
h2

r12 = h12eˆ1 + h1eˆ12

r12 = r21 = h12eˆ1 + h21eˆ 2

eˆ12 = h21 eˆ 2
h1

3.6 Derivation of eˆ13 = h31 eˆ3 , eˆ 31 = h13 eˆ1
h1 h3

(3) − (1) = 0

→ r3 ⋅ r12 − r13 ⋅ r2 = 0
→ r2 ⋅ r13 = r3 ⋅ r12
→ r2 ⋅ r13 = −r2 ⋅ r31
→ r2 ⋅ r13 = 0

r13 = h13eˆ1 + h1eˆ13

r13 = h13eˆ1 + h31eˆ3

eˆ13 = h31 eˆ 3
h1

r31 = h31eˆ3 + h3eˆ31

r31 = r13 = h13eˆ1 + h31eˆ3

eˆ 31 = h13 eˆ1
h3

4. Incompressible N-S equations in orthogonal curvilinear coordinate systems

4.1 Continuity equation ∇ ⋅ V = 0

Since ∇⋅F = 1 ⎡∂ (h2h3F1 ) + ∂ (h3h1F2 ) + ∂ ( h1h2 F3 ) ⎤
h1h2h3 ⎢ ∂x2 ∂x3 ⎥
⎣ ∂x1 ⎦

and V = v1eˆ1 + v2eˆ2 + v3eˆ3

∇ ⋅V = 1 ⎡∂ (h2h3v1 ) + ∂ ( h3h1v2 ) + ∂ ( h1h2v3 )⎥⎤ = 0
h1h2h3 ⎢ ∂x2 ∂x3
⎣ ∂x1 ⎦

4.2 Momentum equation ∂V + (V ⋅∇) V = − 1 ∇p +ν∇2V , (where p piezometric pressure)

∂t ρ
Since V = v1eˆ1 + v2eˆ2 + v3eˆ3 , we can expand the momentum equation term by term

Local derivative ∂V = ∂v1 eˆ1 + ∂v2 eˆ 2 + ∂v3 eˆ 3
∂t ∂t ∂t ∂t

Convective derivative (V ⋅∇) V

Since V = v1eˆ1 + v2eˆ2 + v3eˆ3 and V⋅∇ = v1 ∂ + v2 ∂ + v3 ∂
h1 ∂x1 h2 ∂x2 h3 ∂x3

(V ⋅∇) V = (V ⋅∇)(v1eˆ1 ) + (V ⋅∇)(v2eˆ2 ) + (V ⋅∇)(v3eˆ3 )

(V ⋅∇)(v1eˆ1 ) =

= v1 ∂ (v1eˆ1 ) + v2 ∂ (v1eˆ1 ) + v3 ∂ (v1eˆ1 )
h1 ∂x1 h2 ∂x2 h3 ∂x3

= v1 ∂v1 eˆ1 + v1v1 ∂eˆ1 + v2 ∂v1 eˆ1 + v2v1 ∂eˆ1 + v3 ∂v1 eˆ1 + v3v1 ∂eˆ1
h1 ∂x1 h1 ∂x1 h2 ∂x2 h2 ∂x2 h3 ∂x3 h3 ∂x3

= ⎛ v1 ∂v1 + v2 ∂v1 + v3 ∂v1 ⎞ eˆ1 + ⎛ v1v1 eˆ11 + v2v1 eˆ12 + v3v1 eˆ13 ⎞
⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟ ⎜ h1 h2 h3 ⎟
⎝ ⎠ ⎝ ⎠

= ⎛ v1 ∂v1 + v2 ∂v1 + v3 ∂v1 ⎞ eˆ1 + v1v1 ⎛ − h12 eˆ 2 − h13 eˆ 3 ⎞
⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟ h1 ⎜ h2 h3 ⎟
⎝ ⎠ ⎝ ⎠

+ v2v1 ⎛ h21 eˆ 2 ⎞ + v3v1 ⎛ h31 eˆ 3 ⎞
h2 ⎜ h1 ⎟ h3 ⎜ h1 ⎟
⎝ ⎠ ⎝ ⎠

= ⎛ v1 ∂v1 + v2 ∂v1 + v3 ∂v1 ⎞ eˆ1 + ⎛ v2v1h21 − v1v1h12 ⎞ eˆ 2 + ⎛ v3v1h31 − v1v1h13 ⎞ eˆ 3
⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟ ⎜ h1h2 h1h2 ⎟ ⎜ h3h1 h3h1 ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(V ⋅∇)(v2eˆ2 ) =

= v1 ∂ (v2eˆ2 ) + v2 ∂ (v2eˆ2 ) + v3 ∂ (v2eˆ2 )
h1 ∂x1 h2 ∂x2 h3 ∂x3

= v1 ∂v2 eˆ 2 + v1v2 ∂eˆ 2 + v2 ∂v2 eˆ 2 + v2v2 ∂eˆ 2 + v3 ∂v2 eˆ 2 + v3v2 ∂eˆ 2
h1 ∂x1 h1 ∂x1 h2 ∂x2 h2 ∂x2 h3 ∂x3 h3 ∂x3

⎛ v1 ∂v2 + v2 ∂v2 + v3 ∂v2 ⎞ eˆ 2 + v1v2 eˆ 21 + v2v2 eˆ 22 + v3v2 eˆ 23
=⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟ h1 h2 h3
⎠
⎝

= ⎛ v1 ∂v2 + v2 ∂v2 + v3 ∂v2 ⎞ eˆ 2 + v1v2 ⎛ h12 eˆ1 ⎞
⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟ h1 ⎜ h2 ⎟
⎝ ⎠ ⎝ ⎠

+ v2v2 ⎛ − h21 eˆ1 − h23 eˆ 3 ⎞ + v3v2 ⎛ h32 eˆ 3 ⎞
h2 ⎜ h1 h3 ⎟ h3 ⎜ h2 ⎟
⎝ ⎠ ⎝ ⎠

= ⎛ v1v2h12 − v2v2h21 ⎞ eˆ1 ⎛ v1 ∂v2 + v2 ∂v2 + v3 ∂v2 ⎞ eˆ 2 ⎛ v3v2h32 − v2v2h23 ⎞ eˆ 3
⎜ h1h2 h2h1 ⎟ +⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟ +⎜ h2h3 h2h3 ⎟
⎝ ⎠ ⎠ ⎠
⎝ ⎝

(V ⋅∇)(v3eˆ3 ) =

= v1 ∂ (v3eˆ3 ) + v2 ∂ (v3eˆ3 ) + v3 ∂ (v3eˆ3 )
h1 ∂x1 h2 ∂x2 h3 ∂x3

= v1 ∂v3 eˆ 3 + v1v3 ∂eˆ3 + v2 ∂v3 eˆ 3 + v2v3 ∂eˆ 3 + v3 ∂v3 eˆ 3 + v3v3 ∂eˆ3
h1 ∂x1 h1 ∂x1 h2 ∂x2 h2 ∂x2 h3 ∂x3 h3 ∂x3

= ⎛ v1 ∂v3 + v2 ∂v3 + v3 ∂v3 ⎞ eˆ 3 + v1v3 eˆ31 + v2v3 eˆ 32 + v3v3 eˆ 33
⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟ h1 h2 h3
⎝ ⎠

= ⎛ v1 ∂v3 + v2 ∂v3 + v3 ∂v3 ⎞ eˆ 3 + v1v3 ⎛ h13 eˆ1 ⎞ + v2v3 ⎛ h23 eˆ 2 ⎞
⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟ h1 ⎜ h3 ⎟ h2 ⎜ h3 ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠

+ v3v3 ⎛ − h31 eˆ1 − h32 eˆ 2 ⎞
h3 ⎜ h1 h2 ⎟
⎝ ⎠

= ⎛ v1v3h13 − v3v3h31 ⎞ eˆ1 + ⎛ v2v3h23 − v3v3h32 ⎞ eˆ 2 + ⎛ v1 ∂v3 + v2 ∂v3 + v3 ∂v3 ⎞ eˆ 3
⎜ h1h3 h3h1 ⎟ ⎜ h2h3 h3h2 ⎟ ⎜ h1 ∂x1 h2 ∂x2 h3 ∂x3 ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Pressure gradient ∇p = 1 ∂p eˆ1 + 1 ∂p eˆ 2 + 1 ∂p eˆ 3
h1 ∂x1 h2 ∂x2 h3 ∂x3

Viscous term ∇2V
∇2V =

= 1 ∂ ⎡1 ⎡∂ (h2h3v1 ) + ∂ ( h3h1v2 ) + ∂ ( h1h2v3 )⎥⎤ ⎤ eˆ1
h1 ∂x1 ⎢ ⎢ ∂x2 ∂x3 ⎥
⎣ h1h2 h3 ⎣ ∂x1 ⎦⎦

− 1 ⎡∂ ⎛ h3 ⎡∂ ( h2v2 ) − ∂ (h1v1 )⎥⎤ ⎞ − ∂ ⎛ h2 ⎡∂ (h1v1 ) − ∂ ( h3v3 )⎥⎤ ⎞⎤ eˆ1
h2h3 ⎢ ⎜⎜⎝ h1h2 ⎢ ∂x2 ⎟⎟⎠ ∂x3 ⎜⎜⎝ h1h3 ⎢ ∂x1 ⎟⎟⎠⎥⎥⎦
⎢⎣ ∂x2 ⎣ ∂x1 ⎦ ⎣ ∂x3 ⎦

+ 1 ∂ ⎡1 ⎡∂ (h2h3v1 ) + ∂ (h3h1v2 ) + ∂ ( h1h2v3 ) ⎤ ⎤ eˆ 2
h2 ∂x2 ⎢ ⎢ ∂x2 ∂x3 ⎥ ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦

− 1 ⎡∂ ⎛ h1 ⎡∂ (h3v3 ) − ∂ ( h2v2 )⎥⎤ ⎞ − ∂ ⎛ h3 ⎡∂ ( h2v2 ) − ∂ ( h1v1 )⎥⎤ ⎞ ⎤ eˆ 2
h1h3 ⎢ ⎜⎜⎝ h2h3 ⎢ ∂x3 ⎟⎟⎠ ∂x1 ⎜⎝⎜ h1h2 ⎢ ∂x2 ⎟⎟⎠ ⎥
⎢⎣ ∂x3 ⎣ ∂x2 ⎦ ⎣ ∂x1 ⎦ ⎥⎦

+ 1 ∂ ⎡1 ⎡∂ (h2h3v1 ) + ∂ (h3h1v2 ) + ∂ ( h1h2v3 )⎥⎤ ⎤ eˆ 3
h3 ∂x3 ⎢ ⎢ ∂x2 ∂x3 ⎥
⎣ h1h2 h3 ⎣ ∂x1 ⎦⎦

− 1 ⎡∂ ⎛ h2 ⎡∂ (h1v1 ) − ∂ ( h3v3 )⎤⎥ ⎞ − ∂ ⎛ h1 ⎡∂ ( h3v3 ) − ∂ ( h2v2 )⎤⎥ ⎞⎤ eˆ 3
h1h2 ⎢ ⎜⎜⎝ h1h3 ⎢ ∂x1 ⎟⎟⎠ ∂x2 ⎜⎜⎝ h2h3 ⎢ ∂x3 ⎟⎟⎠⎥⎥⎦
⎣⎢ ∂x1 ⎣ ∂x3 ⎦ ⎣ ∂x2 ⎦

4.2.1 Combine terms in eˆ1 direction to get momentum equation in eˆ1 direction

∂v1 + v1 ∂v1 + v2 ∂v1 + v3 ∂v1 + v1v2h12 − v2v2h21 + v1v3h13 − v3v3h31
∂t h1 ∂x1 h2 ∂x2 h3 ∂x3 h1h2 h2h1 h1h3 h3h1
= − 1 1 ∂p

ρ h1 ∂x1

+ν 1 ∂ ⎡1 ⎡∂ (h2h3v1 ) + ∂ (h3h1v2 ) + ∂ ( h1h2v3 ) ⎤ ⎤
h1 ∂x1 ⎢ ⎢ ∂x2 ∂x3 ⎥ ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦

−ν 1 ⎡∂ ⎛ h3 ⎡∂ (h2v2 ) − ∂ ( h1v1 )⎥⎤ ⎞ − ∂ ⎛ h2 ⎡∂ (h1v1 ) − ∂ ( h3v3 )⎥⎤ ⎞ ⎤
h2h3 ⎢ ⎜⎜⎝ h1h2 ⎢ ∂x2 ⎟⎟⎠ ∂x3 ⎜⎜⎝ h1h3 ⎢ ∂x1 ⎟⎠⎟ ⎥
⎣⎢ ∂x2 ⎣ ∂x1 ⎦ ⎣ ∂x3 ⎦ ⎥⎦

4.2.2 Combine terms in eˆ2 direction to get momentum equation in eˆ2 direction

∂v2 + v2v1h21 − v1v1h12 + v1 ∂v2 + v2 ∂v2 + v3 ∂v2 + v2v3h23 − v3v3h32
∂t h1h2 h1h2 h1 ∂x1 h2 ∂x2 h3 ∂x3 h2h3 h3h2

= − 1 1 ∂p
ρ h2 ∂x2

+ν 1 ∂ ⎡1 ⎡∂ (h2h3v1 ) + ∂ (h3h1v2 ) + ∂ ( h1h2v3 )⎤⎥ ⎤
h2 ∂x2 ⎢ ⎢ ∂x2 ∂x3 ⎥
⎣ h1h2h3 ⎣ ∂x1 ⎦⎦

−ν 1 ⎡∂ ⎛ h1 ⎡∂ (h3v3 ) − ∂ ( h2v2 )⎤⎥ ⎞ − ∂ ⎛ h3 ⎡∂ (h2v2 ) − ∂ ( h1v1 ) ⎤ ⎞⎤
h1h3 ⎢ ⎜⎜⎝ h2h3 ⎢ ∂x3 ⎟⎠⎟ ∂x1 ⎜⎝⎜ h1h2 ⎢ ∂x2 ⎥ ⎠⎟⎟⎥⎦⎥
⎢⎣ ∂x3 ⎣ ∂x2 ⎦ ⎣ ∂x1 ⎦

4.2.3 Combine terms in eˆ3 direction to get momentum equation in eˆ3 direction

∂v3 + v3v1h31 − v1v1h13 + v3v2h32 − v2v2h23 + v1 ∂v3 + v2 ∂v3 + v3 ∂v3
∂t h3h1 h3h1 h2h3 h2h3 h1 ∂x1 h2 ∂x2 h3 ∂x3

= − 1 1 ∂p
ρ h3 ∂x3

+ν 1 ∂ ⎡1 ⎡∂ (h2h3v1 ) + ∂ (h3h1v2 ) + ∂ ( h1h2v3 )⎥⎤ ⎤
h3 ∂x3 ⎢ ⎢ ∂x2 ∂x3 ⎥
⎣ h1h2 h3 ⎣ ∂x1 ⎦⎦

−ν 1 ⎡∂ ⎛ h2 ⎡∂ (h1v1 ) − ∂ ( h3v3 )⎤⎥ ⎞ − ∂ ⎛ h1 ⎡∂ (h3v3 ) − ∂ ( h2 v2 ) ⎤ ⎞ ⎤
h1h2 ⎢ ⎜⎜⎝ h1h3 ⎢ ∂x1 ⎟⎟⎠ ∂x2 ⎜⎜⎝ h2h3 ⎢ ∂x3 ⎥ ⎟⎟⎠ ⎥
⎢⎣ ∂x1 ⎣ ∂x3 ⎦ ⎣ ∂x2 ⎦ ⎥⎦

5. Example: Incompressible N-S equations in cylindrical polar systems

5.1 Continuity equation ∇ ⋅ V = 0 in cylindrical coordinates (r,θ , z)

For cylindrical coordinates (r,θ , z) , h1 = hr = 1, h2 = hθ = r, h3 = hz = 1

∇ ⋅ V = 1 ⎡∂ ( rvr ) + ∂ ( vθ ) + ∂ ( rvz )⎦⎥⎤ = 0
r ⎢⎣ ∂r ∂θ ∂z

= 1 ∂ ( rvr ) + 1 ∂ ( vθ ) + ∂ (vz ) = 0
r ∂r r ∂θ ∂z

5.2 Momentum equation ∂V + (V ⋅∇) V = − 1 ∇p +ν∇2V in cylindrical coordinates (r,θ , z)

∂t ρ

For cylindrical coordinates (r,θ , z) , h1 = hr = 1, h2 = hθ = r, h3 = hz = 1 and only h21 = hθr = 1, all

others are zero.

5.2.1 The r-momentum equation:

∂vr + vr ∂vr + vθ ∂vr + vz ∂vr − vθ2 =− 1 ∂p +ν ⎛ ∇ 2vr − 1 vr − 2 ∂vθ ⎞
∂t ∂r r ∂θ ∂z r ρ ∂r ⎜⎝ r2 r2 ∂θ ⎟⎠

5.2.2 The θ -momentum equation:

∂vθ + 1 vr vθ + vr ∂vθ + vθ ∂vθ + vz ∂vθ =− 1 ∂p +ν ⎛ ∇ 2 F2 − F2 +2 ∂F1 ⎞
∂t r ∂r r ∂θ ∂z ρr ∂θ ⎝⎜ r2 r2 ∂θ ⎟⎠

5.2.3 The z-momentum equation:

∂vz + vr ∂vz + vθ ∂vz + vz ∂vz = −1 ∂p +ν∇2vz
∂t ∂r r ∂θ ∂z ρ ∂z