A fluid flow stabilizing method and a pump having a stabilized fluid flow passage where a process fluid contains suspended solid particles and is subject to pressure oscillations. The method comprises positioning an expandable membrane along the lower transverse surface, inflating the membrane with a compressible fluid, and alternatingly expanding and contracting the membrane to dampen the pressure oscillations and inhibit settling of the solid particles at the lower transverse surface. The pump comprises a stabilizer comprising a compressible fluid in an expandable membrane elongated along the lower transverse surface of a manifold in fluid communication with the fluid end. Another method comprises reciprocating a plunger in the pump to pass a slurry through the manifold, and alternatingly expanding and contracting a stabilizer in the manifold.