Tracer Dispersion due to Pulsatile Casson Fluid Flow in a Circular Tube with Chemical Reaction Modulated by Externally Applied Electromagnetic Fields

Abstract

The present article describes a detailed mathematical investigation of electro-magneto-hydrodynamic dispersion in the pulsatile flow of a Casson viscoplastic fluid in a tube packed with a porous medium. Using appropriate transformations, the model is rendered non-dimensional. Via the generalized dispersion method and finite Hankel transforms, analytical solutions for the solute concentration dispersion and convection coefficients have been obtained. The impact of the Hartmann (magnetic) number, Debye–Hückel(electrokinetic) parameter, Darcy number, and chemical reaction parameter with regard to dispersion phenomena has been studied. The evolution in velocity and concentration profiles are investigated graphically for realistic ranges of the various physical parameters. The present investigation, highlights the dual nature of the Debye–Hückel parameter in the dispersion process. Increment in the lower or higher magnitudes of Debye–Hückel parameter induces or decreases the magnitudes of effective dispersion coefficient, whereas it induces a reverse dual mechanism in the zenith of the average concentration profile. The present simulations are relevant to enhancing the performance of diagnostic tools in biochemical engineering, pumping of intelligent rheological working fluids in biomedicine, and also soft robotics.

Appendix

Magnetic and Electric field Expressions

Magnetic field in Eq. (4) and Poisson–Boltzmann Eq. (5), which are expressed as follows:

Maxwell’s equations are a set of four partial differential equations that describe the force of electromagnetism in an electromagnetic field. The electromagnetic theory depends on Gauss’ law, Faraday’s law, Ampere’s law, and the current continuity equation. These equations are stated mathematically as:

$$ \nabla .B_{0}^{ * } = 0 $$

(55)

$$ \nabla \times E^{ * } = - \frac{{\partial B_{0}^{ * } }}{{\partial t^{ * } }} $$

(56)

$$ \nabla \times M^{ * } = J^{ * } $$

(57)

$$ \nabla .J^{ * } = 0 $$

(58)

where $B_{0}^{ * }$ denotes the magnetic flux, $E^{ * }$ indicates the electric field, $M^{ * }$ represents the magnetic field and $J^{ * }$ designates the current density.

Here $B_{0}^{ * } = B_{1}^{ * } + B_{2}^{ * }$ (Sum of external and induced magnetic field).

Under a small magnetic Reynolds number induced magnetic field $\left( {B_{2}^{ * } } \right)$ is negligibly small compared to the external magnetic field $\left( {B_{1}^{ * } } \right)$.

Further, the electric field due to the polarization change is also negligible. By Ohm’s law

$$ J^{ * } = \sigma^{ * } \left( {E^{ * } + u^{ * } \times B_{0}^{ * } } \right) $$

(59)

where $\sigma^{ * }$ denotes the electrical conductivity.

The electric fields imposed and induced are presumed to be negligible.

Hence, the term $J^{ * } \times B_{0}^{ * }$ is simplified as $- \sigma^{ * } B_{0}^{{^{{ *^{2} }} }} u^{ * }$.

According to the theories of Electrohydrodynamics and Navier–Stokes equations for an incompressible viscous dielectric fluid, the Electrohydrodynamics equations can be summarized as.

$$ \nabla .E^{ * } = \frac{{\rho_{e}^{ * } }}{\kappa } \, \left( {\text{Gauss law}} \right) $$

(60)

$$ E^{ * } = - \nabla \Phi^{ * } \, \left( {\text{Relation between irrotational field and scalar potential}} \right){.} $$

(61)

$$ \nabla .J^{ * } + \frac{{\partial \rho_{e}^{ * } }}{{\partial t^{ * } }} = 0 $$

(62)

$$ \nabla .u^{ * } = 0 $$

(63)

$$ \rho^{ * } \frac{{du^{ * } }}{{dt^{ * } }} = - \nabla p^{ * } + \mu^{ * } \nabla^{2} u^{ * } + \rho_{e}^{ * } E_{z}^{ * } $$

(64)

Substituting Eqn. (61) into Eqn. (60), we get the following Poisson’s equation

$$ \nabla^{2} \Phi^{ * } = - \frac{{\rho_{e}^{ * } }}{\kappa } $$

(65)

where $\rho_{e}^{ * }$ denotes the net charge density, $\kappa$ designates the permittivity of the free space (dielectric constant), $\Phi^{ * }$ represents the electric potential and $E_{z}^{ * }$ indicates the component of external electric field applied in the axial direction (Table

Table 3 List of variable and parameters

3).