A mathematical model of the locomotion of bacteria near an inclined solid substrate: effects of different waveforms and rheological properties of couple-stress slime

Morphological mutations in bacterial cell make them the most miscellaneous microscopic group. Their non-flagellated species known as gliding bacteria exhibit self-powered motion and leave an adhesive trail of slime. The self-propelled motion in some gliding bacteria is achieved as a result of backward surface wave in the cell envelope. Motivated by this fact, an undulating surface on a layer of couple-stress fluid is used to model the motion of such gliding bacteria. Five different wave profiles, namely, sawtooth, sinusoidal, triangular, trapezoidal, and square profiles are used to model the waveform of the undulating wave in the outer cell surface. The inclination of the surface is also integrated into the model. The flow equations are set up under the lubrication assumption. Stream function is derived as an elementary function of an organism's speed, undulation amplitude, and couple-stress parameter with its flow rate. Speed of the glider and flow rate ( satisfying equilibrium conditions) are computed by employing modified Newton-Raphson method. These refined values are further utilized to compute the power dissipation. Effects of different waveforms, inclination angle, gravitational and couple-stress parameters on the speed of the microorganism and rate of energy expended are also quantified. Slime velocity is also plotted for fixed glider. In addition, making use of the obtained realistic set of values of the organism's speed, flow rate, occlusion parameter, and couple-stress parameter, streamline patterns of the slime are plotted and discussed in detail.