Biological interactions between Carreau fluid and microswimmers in a complex wavy canal with MHD effects

  • N. Ali
  • Z. AsgharEmail author
  • M. Sajid
  • O. Anwar Bég
Technical Paper


The efficient magnetic swimming of actual or mechanically designed microswimmers within bounded regions is reliant on several factors: the actuation of these swimmers via magnetic field, rheology of surrounding liquid (with dominant viscous forces), nature of medium (either porous or non-porous), position (either straight, inclined or declined) and state (either active or passive) of the narrow passage. To witness these interactions, we utilize Carreau fluid with Taylor swimming sheet model under magnetic and porous effects. Moreover, the cervical canal is approximated as a two-dimensional complex wavy channel inclined at certain angle with the horizontal. The momentum equations are reduced by means of lubrication assumption, which finally leads to a fourth-order differential equation. MATLAB’s built-in bvp4c function is employed to solve the resulting boundary value problem. The solution obtained via bvp4c is further verified by finite difference method. In both these methods, the refined values of flow rate and cell speed are computed by utilizing modified Newton–Raphson method. These realistic pairs are further utilized to calculate the energy delivered by the microswimmer. The numerical results are plotted and discussed at the end of the article. Our study explains that the optimum speed of the microorganism can be achieved by means of exploiting the fluid rheology and with the suitable application of the magnetic field. The peristaltic nature of the channel walls and porous medium may also serve as alternative factors to control the speed of the propeller.


Magnetic field Complex wavy channel Porous medium Carreau fluid Microswimmer Inclined channel 

List of symbols

Roman symbols

\( a_{S} \)

Wave amplitude in organism surface

\( a_{1} ,\,\,a_{2} \)

Wave amplitude in channel walls

\( C \)

Wave speed

\( U_{S} \)

Swimming speed of the organism

\( X,Y \)

Cartesian coordinates for fixed frame

\( x,y \)

Cartesian coordinates for wave frame

Superscript (+)

Upper half \( \left( {H_{S} \le Y \le H_{1} } \right) \)

Superscript ()

Lower half \( \left( {H_{2} \le Y \le H_{S} } \right) \)


Velocity vector of the fluid

\( V_{1} ,V_{2} \)

Velocity components in fixed frame

\( v_{1} ,v_{2} \)

Velocity components in wave frame

\( B_{0} \)

Strength of magnetic field

\( {\mathbf{F}}_{{\mathbf{C}}} \)

Resultant force on the organism

\( P \)

Pressure in fixed frame

\( p \)

Pressure in wave frame

\( {\text{S}} \)

Extra stress tensor

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

First Rivlin–Ericksen tensor

\( S_{xx} ,\,\,S_{xy} \,\,{\text{and}}\,\,S_{yy} \)

Components of extra stress tensor

\( We \)

Weissenberg number

\( n \)

Power law index

\( Re \)

Reynolds number

\( H \)

Hartmann number

\( g \)

Force due to gravity


Darcy number


Gravitational parameter

\( F \)

Flow rate of the fluid


Body force


Resistance due to porous medium

Greek symbols



\( \Phi \)

Phase difference \( \left( {0 \le \phi \le \pi } \right) \)

\( \rho \)

Fluid density



\( \sigma_{m} \)

Electrical conductivity

\( \alpha \)

Inclination angle

\( \mu_{\,0} \)

Zero-shear-rate viscosity

\( \mu_{\infty } \)

Infinite-shear-rate viscosity

\( \gamma \)

Ratio of infinite- to zero-shear-rate viscosity

\( \Pi \)

Second invariant

\( \Gamma \)

Time constant/relaxation time

\( \delta \)

Dimensionless wave number

\( \Psi \)

Stream function

\( \wp \)

Power delivered by the swimmer

\( \xi_{j} \)

Unit vector normal to the organism

\( \Omega \)

Under-relaxation parameter \( \Omega \in \left. {\left( {0,1} \right.} \right] \)



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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsInternational Islamic UniversityIslamabadPakistan
  2. 2.NUTECH School of Applied Sciences and HumanitiesNational University of TechnologyIslamabadPakistan
  3. 3.Fluid Mechanics and Propulsion, Aeronautical and Mechanical Engineering Department, School of Computing, Science and Engineering (CSE)University of SalfordSalfordUK

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