E n d n o t e s

(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

Tijk(x, x0) = Tijk(x+xp, x0) + dikD pj

where Dpj is the pressure drop across one period.  The value of Dpj 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 Dpj=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 Dpj 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, Dpj 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, Dpj 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 Dpx where x is the axial direction of the tube or channel.

(3) Note that the ff component of QRff1 is equal to I32+ss0(I31-I33) but is not equal to 2I12 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)

fi+(x0) = 2 fi¥(x0) - (1/(4p)) nk (x0òD      Tijk(x0, x) fj+(x) dS(x)

(6) The normal vector may be extended off the interface by setting n= ÑF/| ÑF| where F(x, y, z)=0 describes the location of the interface.