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The biological fluids encountered by self-propelled cells display complexmicrostructures and rheology. We consider here the general problem oflow-Reynolds number locomotion in a complex fluid. {Building on classical workon the transport of particles in viscoelastic fluids,} we demonstrate how tomathematically derive three integral theorems relating the arbitrary motion ofan isolated organism to its swimming kinematics {in a non-Newtonian fluid}.These theorems correspond to three situations of interest, namely (1) squirmingmotion in a linear viscoelastic fluid, (2) arbitrary surface deformation in aweakly non-Newtonian fluid, and (3) small-amplitude deformation in anarbitrarily non-Newtonian fluid. Our final results, valid for a wide-class of{swimmer geometry,} surface kinematics and constitutive models, at most requiremathematical knowledge of a series of Newtonian flow problems, and will beuseful to quantity the locomotion of biological and synthetic swimmers incomplex environments.

Locomotion in complex fluids: Integral theorems