The present study is focused on the unsteady two-phase flow of blood in a cylindrical region. Blood is taken as a counter-example of Brinkman type fluid containing magnetic (dust) particles. The oscillating pressure gradient has been considered because for blood flow it is necessary to investigate in the form of a diastolic and systolic pressure. The transverse magnetic field has been applied externally to the cylindrical tube to study its impact on both fluids as well as particles. The system of derived governing equations based on Navier Stoke’s, Maxwell and heat equations has been generalized using the well-known Caputo–Fabrizio (C–F) fractional derivative. The considered fractional model has been solved analytically using the joint Laplace and Hankel (L&H) transformations. The effect of various physical parameters such as fractional parameter,

Biomagnetic fluid dynamic (BFD) is a new area in fluid mechanics. It focuses on the usage of the magnetic particles as drug carriers in magnetic drug targeting, cancer tumor treatment and many more [

Since blood is a biological fluid, biological heating is significant for metabolic heat generation [

The consideration of Two-phase flow is due to the presence of numerous interfaces separating two immiscible phases. The blood flow through a tiny tube at a very low shear is responsible for the two-phase flow surrounded by a cell-depleted peripheral layer. Different types of particles have been considered as the second phase in blood flow, but the most recommended and suitable particles are magnetic particles. The magnetic particles in blood have a vital role in numerous medical applications [

Due to multidimensional features, the non-integer order calculus is attracting the attention of scientists and researchers [

There is no attempt found in the literature relevant to Caputo–Fabrizio fractional approach to find the closed-form solution for magnetite particles-based blood flow with thermal concentration. Hence, in the present article, the work of Saqib et al. [

The blood flow is considered in a vertical cylinder having a radius

The magnetic particles are equally distributed throughout the blood flow. The cylinder has been considered along the z-axis and

where

The unsteady Brinkman-type blood flow in a cylinder is specified by:

the oscillating pressure gradient [

where

The thermal equation is specified by:

subjected to the following IBCs

By incorporating the Non-dimensional variables

into

For a generalized fractional model, the newly developed CF time-fractional derivative has been used to covert the linear model to the fractional model, therefore

where

For the solution of

Applying the joint L&H transforms using

where

and

Applying inverse L&H transformations to

The Non-dimensional Nusselt number is given by

To obtain the solution for the blood velocity and Magnetic particles velocity, the Laplace and Hankel transforms have been applied on

Now for the blood velocity,

The simplified form of

After further simplification the

where

In component form

where

and

By applying inverse Laplace transform to

we get

where

Applying the finite Hankel transform of order zero to

where

Now for the solution of magnetic particles velocity applying the inverse L&H transformations to

From

Nu | ||
---|---|---|

0.3 | 2 | 2.503 |

0.5 | 2 | 2.684 |

0.7 | 2 | 3.012 |

0.9 | 2 | 3.412 |

1 | 2 | 3.576 |

The considered work aims to study the generalized two-phase blood flow of Brinkman type fluid in a cylindrical tube. The analytical solutions have been attained for energy, velocity as well as for the magnetic particles contained in the blood. Various parameters have been discussed physically on velocities of the blood, particles and temperature.

The Caputo–Fabrizio time-fractional derivative has been used. The effect of relative parameters has been shown graphically. Closed-form expressions have been obtained by using the Laplace transform and Hankel transform techniques. Based on the graphical study, it has been concluded that the velocity profile decreases in the response of an external applied magnetic field and Brinkman parameter. This phenomenon might play an important role in Magnetic wounds. Furthermore, by increasing the fractional parameter, the fluid memory becomes thicker.