The use of flagella by sessile organisms to generate feeding currents is analysed. The organism consists of a spherical cell body (radius A) to which a smooth flagellum (radius a, length L) is attached radially. The cell body is a height H above the plane substrate to which it is rigidly attached via a stalk. The organism propagates plane sinusoidal waves (amplitude α, wavenumber k) from base to tip. The flagellum is represented by distributions of stokeslets and dipoles along its centre-line. The cell body and substrate are modelled by employing an approximate form of the Green's function for the sphere in the half space. The error terms in the model are O(a/L) and O(A2/H2). The analysis and method of solution are adapted from Higdon (1979).

The mean flow rate and power consumption are calculated for a wide range of parameters. Optimal motions are determined with the criterion of minimizing the power required to achieve a given flow rate. The optimum wave has maximum slope in the range 2 < αk < 2·5 (compared to the optimum value αk = 1 for swimming). The optimum number of waves Nλ increases linearly with flagellar length for L/A > 10 and is approximately constant, Nλ = 1, for shorter flagella. The optimum flagellar length is in the range 5 < L/A < 10. There is no optimum flagellar radius a/A. For optimal efficiency, the height H should be greater than or equal to the length of the flagellum.

The optimum values of the parameters are compared to the values for the choano-flagellates described by Lapage (1925) and Sleigh (1964). Excellent agreement is found between the predicted optima and the observed values. The calculated velocity field closely resembles the flow described by Sleigh and Lapage.