A parallel solver for incompressible fluid flow simulation, used in biomedical device design among other applications, is discussed. The major compute- and communication-intensive portions of the code are described. Using unsteady flow in a complex implantable axial blood pump as a model problem, scalability characteristics of the solver are briefly examined. The code that exhibited so far good scalability on typical PC clusters is shown to suffer from a specific bottleneck when thousands of processors of a Blue Gene are employed. After resolution of the problem, satisfactory scalability on up to four thousand processors is attained.