On the net displacement of contact surface centroid in contractile bodies

Abstract

We investigate the motion of the contact surface centroid for contractile bodies on substrates with a viscous friction law and when inertial forces are negligible. We deduce a set of sufficient conditions that ensure that the surface centroid remains still. The conditions are automatically satisfied for linear analysis and homogeneous constant viscous friction parameters. In non-linear analysis additional requirements are necessary: i) the material is incompressible, ii) the material points in contact do not vary, and iii) the surface is flat. These results demonstrate the inability of slender organisms to move under homogeneous viscous contact condition if the contact surface remains constant, regardless of the contractility strategy employed. We numerically simulate some situations that do not comply to these conditions, such as the use of non-homogeneous or anisotropic friction, which illustrate possible strategies for net propulsion.