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Mathematical modeling related to bacterial gliding mechanism at low Reynolds number with Ellis Slime

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Abstract

The self-propelling mechanism of rod-shaped bacteria over complex rheological slime is vital. These bacteria glide near the surface by producing waves in their body and secrete an extracellular polymeric substance (EPS) that allows them to move without flagella. During gliding motion, bacteria experience hydrodynamic interactions with complex rheological fluid attached to the surfaces which are either rigid or soft. Their natural response to such interactions affects their gliding speed and power required for propulsion. Motivated by this fact, we investigate the fundamental mechanics of bacterial locomotion by utilizing an undulating surface model combined with the Ellis fluid model. The substrate beneath the organism is considered a wavy (soft substrate) or a rigid surface. The equations of motion are reduced into a single ordinary differential equation under the lubrication approximation. A numerical solution of this equation is computed via the MATLAB bvp5c routine. The code is adjusted in such a way that it refines the unknowns, i.e., flow rate and gliding speed by employing a modified Newton–Raphson algorithm until the satisfaction of equilibrium conditions. The computed pairs of speed and flow rate are then utilized to compute the energy consumed by the glider. Work done by the glider, gliding speed, flow rate, velocity of the slime, and streamlines patterns are also visualized by graphs.

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Abbreviations

X, Y :

Cartesian coordinate in the fixed frame

x, y :

Cartesian coordinate in the wave frame

\(V_{g}\) :

Gliding speed of the organism

\(c' \) :

Wave speed

\({\tilde a_G}\) :

Amplitude of the wave in glider

\({\tilde a_S}\) :

Amplitude of the wave in substrate

\(\hbar_{0}\) :

Mean distance between wavy sheet to the soft substrate

\(\phi_{G} ,\,\,\phi_{S}\) :

Occlusion parameters

\(\bar U\) :

Velocity vector

\({\bar U_1},\,{\bar U_2}\) :

Components of velocity field in wave frame

\({\text{A}}_{1}\) :

First Rivlin–Ericksen tensor

\(\frac{d}{dt}\) :

Material derivative

t :

Time

\({\mathbf{T}}\) :

Extra stress tensor

\(Re\) :

Reynolds number

\(Q\) :

Flow rate of the fluid

:

Work done by the glider

\(\rho\) :

Density

\(\nabla\) :

Gradient operator

\(\psi\) :

Stream function

\(\mu\) :

Dynamic viscosity

\(\delta\) :

Dimensionless wave number

\(\lambda\) :

Wavelength

\(\alpha\) :

Material constant

\(\beta\) :

Dimensionless material parameter

\(\Pi\) :

Second-order invariant

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Acknowledgements

The constructive comments of the worthy reviewers are greatly appreciated. N. Ali and R. A Shah are thankful to the HEC Pakistan for financial assistance through Grant No: 7671/ Federal/NRPU/ R&D/HEC/2017.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    Rehman Ali Shah & Nasir Ali

  2. NUTECH, School of Applied Sciences and Humanities, National University of Technology, Islamabad, 44000, Pakistan

    Zeeshan Asghar

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  1. Rehman Ali Shah
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Correspondence to Zeeshan Asghar.

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Shah, R.A., Asghar, Z. & Ali, N. Mathematical modeling related to bacterial gliding mechanism at low Reynolds number with Ellis Slime. Eur. Phys. J. Plus 137, 600 (2022). https://doi.org/10.1140/epjp/s13360-022-02796-3

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