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Gliding motion of bacteria on power-law slime. (English) Zbl 1075.35058

“The present investigation deals with an undulating surface model for the motility of bacteria gliding on a layer of a non-Newtonian slime. The slime being the viscoelastic material is considered as a power-law fluid”. “Most motile bacteria swim or glide. Mechanism for swimming are known, but gliding remains ‘curious’, strange, and at present still unexplained”. “Although a considerable amount of research has been undertaken on an undulating-surface model for the motility of certain gliders that adhere to surfaces by means of slime, little is known about the non-Newtonian nature of slime”. “Exuded slime is pushed backward which produces the force to propel the bacterium in the direction opposite to the direction of the travelling wave”. (From the summary and the text.)
The authors present a two-dimensional analysis of the model. Using a dimensionless framework, after some simplifying assumptions, they consider the following nonlinear fourth order equation \[ \frac{\partial^2}{\partial y^2}\left\{\left[\left(\frac{\partial^2\psi}{\partial y^2}\right)^2\right]^m\frac{\partial^2\psi}{\partial y^2}\right\}=0, \] where \(y\) is the coordinate perpendicular to the surface, \(\psi(x,y)\) is the stream function (with \(x\) as the coordinate parallel to the motion), and \(m\) is the power-law exponent (\(m<0\) shear-thinning, \(m>0\) shear-thickening, \(m=0\) Newtonian fluid). Boundary conditions are given at \(y=0\) and at \(y=h(x)=1+\varphi\sin x\), where \(h\) is the form of the wave. The authors directly solve such a boundary value problem numerically, by a finite difference method with an iteration technique. In particular, they compare the numerical results for various values of the power-law exponent \(m\).

MSC:

35Q35 PDEs in connection with fluid mechanics
92C35 Physiological flow
76A05 Non-Newtonian fluids
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