A3
General wall-jet co-ordinate transformations

The transformations presented in this appendix are for general functions, allowing various expanding grids and conformal mapping functions to be incorporated.

A3.1 Curvilinear transformation of z: (r,z) => (r,y)

The steady-state mass transport equation for a wall-jet electrode is given by

(A3.1)

We seek a transformation from cylindrical polar co-ordinates:

=> (A3.2)

to a transformed space

=> (A3.3)

based on some curvilinear co-ordinate, y, replacing z :

=> (A3.4)

Substituting (A3.4) into (A3.3) and comparing coefficients with (A3.2) gives the first partial derivatives and their associated operators:

=> (A3.5)

and

=> (A3.6)

To get the second partial derivative in z, the 1st order operator may be applied to its associated derivative:

=
=
(A3.7)

Similarly in r:

(A3.8)
First Term =
=
(A3.9)
Second Term = (A3.10)

For conciseness the notation is introduced:


(A3.11)

This gives, for the first derivatives:

and (A3.12)

and for the second derivatives:

(A3.13)
(A3.14)

Therefore the mass transport equation in (r,y) space is:

(A3.15)

A3.2 Transformation of both z (curvilinear) and r : (r,z) => (r,y)

We seek a second transformation from cylindrical polar co-ordinates:

=> (A3.16)

to a transformed space

=> (A3.17)

based, as before, on a curvilinear co-ordinate, y, replacing z :

=> (A3.18)

but in addition a transformation of r:

=> (A3.19)

Substituting both (A3.18) and (A3.19) into (A3.17) and comparing coefficients with (A3.16) gives the first partial derivatives and their associated operators:

=> (A3.20)

and

=> (A3.21)

The second partial derivative in z, is analogous to the previous transformation:

=
=
(A3.22)

The second derivative in r is more complex, reflecting the dependence of y on r:

(A3.23)
First Term =
=
(A3.24)
Second Term =
=
(A3.25)

Again the concise notation is now used:

(A3.26)

This gives, for the first derivatives:

and (A3.27)

and for the second derivatives:

(A3.28)
(A3.29)

Therefore the mass transport equation in (r,y) space is:

(A3.30)

A3.3 Two-step transformation : (r,z) => (r,y) => (r,y)

In order to check the algebra, we may confirm that applying the transformations in succession gives the same result. This algebra is also useful for the computational implementation as it allows the grid functions to be applied independently or together. In order to confirm that the previous transformation:

=> (A3.31)

gives the same result as the first transformation, followed by transformation of r:

=> => (A3.32)

we note that the transformed radial co-ordinate depends only on the original radial co-ordinate?

=> (A3.33)

The first derivative therefore transforms as:

(A3.34)

and the second as:

(A3.35)

We must also consider the cross-derivatives:

(A3.36)

The other cross derivative is similar, noting that dr/dr is constant at constant r:

(A3.37)

As expected, the successive transformations give the same result as the direct transformation, which can be seen by substituting equations (A3.34), (A3.35), (A3.36) and (A3.37) into equation (A3.15) giving equation (A3.30).