(1) Solutions of equation (2.1.1) and of the continuity equation whose associated pressure or stress vanish over the boundaries of the flow are called Green's functions of the second kind or Neumann's functions, in accord with the corresponding nomenclature for potential flow (Garabedian 1964, p. 240).

(2) In general, the stress tensor of a periodic Green's function for internal flow satisfies the identity

*T _{ijk}(x, x_{0})
= T_{ijk}(x+x_{p},
x_{0})
+ d_{ik}D
p_{j
}*

where D*p*j is
the pressure drop across one period. The value of
D*p*_{j}
depends upon the topology of the domain of flow as well as the particular
choice of Green's function.

Examples
of Green's functions with D*p*_{j}=0 are the
Green's function representing a one-dimensional array of three-dimensional
point forces in an infinite or semi-infinite domain of flow, and the Green's
function representing a two-dimensional array of three-dimensional or two-dimensional
point forces. Examples of Green's functions with D*p*j
that is not necessarily equal to zero are those representing a three-dimensional
array of three-dimensional point forces, a two-dimensional array of two-dimensional
point forces, a one-dimensional array of three-dimensional point forces
within a cylindrical tube, and a one-dimensional array of two-dimensional
point forces between two flat plates. In all of these cases, D*p*j
may be annihilated by enhancing the Green's function with a proper regular
flow with a finite pressure gradient. For instance, in the case of
the one-dimensional array of three-dimensional point forces within a cylindrical
tube, D*p*j is annihilated by adding a
Poiseuille flow along the tube. Note, however, that in that case
the flow rate along the tube will have a finite value. The periodic
Green's functions discussed on pages 91 and 99 have a vanishing flow rate
and a finite value of D*px* where *x
*is
the axial direction of the tube or channel.

(3) Note that the ff
component of Q^{R}_{ff1}
is equal to *I*_{32}+ss_{0}(*I*_{31}-*I*_{33})
but is not equal to 2*I*_{12} as the dual expression in (2.4.21)
might imply.

(4) The principal value integrals on the right-hand
sides of (4.1.7) and (4.1.8) are genuine Cauchy principal value integrals.
Thus, they must be computed by excluding from the domain *D *a small
circular disk centered at the singularity, and taking the limit as the
radius of the disk tends to zero.

(5) Combining (2.3.30) and (4.1.7) we derive the
following integral equation of the second kind for the surface force on
a rigid body immersed in an incident flow (Liron & Barta, *J. Fluid
Mech. ***238**, 597-598)

f

(6) The normal vector may be extended off the
interface by setting ** n=** ÑF/|
ÑF| where