#
Biomagnetic Flow with CoFe_{2}O_{4} Magnetic Particles through an Unsteady Stretching/Shrinking Cylinder

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{2}O

_{4}as magnetic particles. Where blood acts as an electrically conducting fluid along with magnetization/polarization. The main concentration is to study a time-dependent biomagnetic fluid flow with magnetic particles that passed through a two dimensional stretching/shrinking cylinder under the influence of thermal radiation, heat source and partial slip condition which has not been studied yet as far as best knowledge of authors. This model is consistent with the principles of magnetohydrodynamic and ferrohydrodynamic. The flow equations, such as momentum, energy which is described physically by a system of coupled, nonlinear partial differential equation with appropriate boundary conditions and converted into a nonlinear system of ordinary differential equations by using suitable similarity transformations. The resultant ODEs numerically solved by applying by applying an efficient numerical technique based on a common finite differencing method along with central differencing, tridiagonal matrix manipulation and an iterative procedure. The values assigned to the parameters are compatible with human body conditions. The numerous results concerning velocity, temperature and pressure field, as well as the skin friction and the rate of heat transfer, are presented for the parameters exhibiting physical significance, such as ferromagnetic interaction parameter, magnetic field parameter, volume fraction, unsteady parameter, curvature parameter, etc. The main numerical findings are that the fluid velocity is decreased as the ferromagnetic number is enhanced gradually in both stretching or shrinking cases whereas, the opposite behavior is found for the skin friction coefficient. The rate of heat transfer with ferromagnetic interaction parameter was also monitored and found that opposite behavior occurs for stretching and shrinking cases. Comparisons were made to check the accuracy of the present numerical results with published literature and found to be in excellent agreement. Hopefully, this proposed model will control the blood flow rate, as well as the rate of heat transfer, such as magnetic hyperthermia.

## 1. Introduction

_{3}O

_{4}) and the other one was non-magnetic particles (Al

_{2}O

_{3}) in presence of prescribed heat flux. In that study, they showed that the rate of heat transfer gets higher for Al

_{2}O

_{3}comparable to that with Fe

_{3}O

_{4}in the case of when the magnetic field is absent. Singh et al. [20] investigated the behavior of Cu-H

_{2}O flow and heat transfer through a stretchable porous cylinder and numerical solutions obtained with aid of the Keller box method. Note that in numerical computation they considered the value of volume fraction of nanoparticles up to 25%. Malik et al. [21] studied the model of sisko fluid flow over a non-linear stretchable cylinder with existing Cattaneo-Christov heat flux. Abbas et al. [22] demonstrated a mathematical model of unsteady flow and heat transfer through a stretching/shrinking cylinder along with partial slip condition and suction. Further, the effect of joule heating and heat source/sink on radiative mixed convection maxwell nanofluid flow through a stretching cylinder was studied by Islam et al. [23]. In this manuscript, the authors solved the system of ODEs with the help of the HAM method. Salahuddin et al. [24] analyzed MHD carreau nanofluid over a stretching cylinder in the presence of reactive species and slip effect and later on this model was numerically solved by using the implicit finite difference method which is also known as the Keller box method. The flow of hydromagnetic nanofluid over a stretching cylinder was conducted by Zeeshan et al. [25]. Alsorynejad et al. [26] applied the fourth-order Runge-Kutta method in order to investigate the effect of MHD Cu/Ag/Al

_{2}O

_{3}/TiO

_{2}-water nanofluid flow over a stretching sheet. Gangadhar et al. [27] examined the Cattaneo-Christov flux model through a stretching cylinder with slip effects where water was assumed as the base fluid and Cu, Ag, Al

_{2}O

_{3}, TiO

_{2}were taken as nanoparticles and the governing set of ODEs of this problem numerically solved by utilizing the SRM technique.

_{2}O

_{4}as magnetic particles (assumed in a spherical shape). The governing set of nonlinear partial differential equations is simplified by using suitable transformations in order to obtain a set of ordinary differential equations (ODEs). Later on, the resultant ODEs are solved by applying an efficient numerical technique based on a common finite differencing method along with central differencing, tridiagonal matrix manipulation and an iterative procedure in order to obtain the numerical results. In the whole numerical process, the value of volume fraction used for magnetic particles was up to 20% and also neither the electrical conductivity nor polarization was neglected for the base fluid in numerical calculations. The graphs and tables are presented and discussed with numerical outcomes of various parameters. The results presented concerning the velocity, temperature, pressure distributions, skin friction coefficient and rate of heat transfer, show that the blood-CoFe

_{2}O

_{4}flow is appreciably influenced by the application of magnetic field, where suction parameter plays a vital role. Such obtained results encouraged us and indicated that the application of a magnetic field could be useful in biomedical and bioengineering sectors.

## 2. Mathematical Flow Equations with Flow Geometry

_{2}O

_{4}magnetic particles as a spherical shape, which is incompressible, electrically conducting passed through a two dimensional stretched/shrinking cylinder with of radius $R$ is considered in this study. The cylinder is also considered to be stretched with velocity ${U}_{w}\left(t,z\right)=\frac{az}{1-\alpha t}$, where $a$ and $\alpha $ are positive constants along axial $z$-axis and $r$-axis is the radial direction of the cylinder as shown in Figure 1. The temperature of the cylinder surface and ambient fluid is assumed to be ${T}_{w}$ and ${T}_{c}$ respectively, where ${T}_{w}<{T}_{c}$. The magnetic field of intensity $H$, is generated by a magnetic dipole located below the sheet at steady distance $c$.

_{2}O

_{4}defined from earlier studies [45,46] relevant to this field and shown in Table 1. Where $\varphi $ indicates the volume friction of magnetic particles while $\varphi =0$ corresponds to regular fluid. Additionally, note that the subscript symbol ${\left(\right)}_{mf}$ is used to signify that the quantity is that of the magnetic fluid and the symbols ${\left(\right)}_{f}$ and ${\left(\right)}_{s}$ indicate corresponding quantities for the base fluid and magnetic particles themselves.

## 3. Transformation Analysis

## 4. Physical Quantities of Skin Friction Coefficient and Rate of Heat Transfer (Local Nusselt Number)

## 5. Numerical Procedure

## 6. Numerical Code Validation with Previous Published Literature

## 7. Parameter Estimation and Values of Thermophysical Properties of Blood and CoFe_{2}O_{4}

_{w}= 37 °C = 310 K [50] while body Curie temperature is T

_{c}= 41 °C = 314 K. For the above values, the dimensionless temperature is $\epsilon =\frac{{T}_{c}}{{T}_{c}-{T}_{w}}=\frac{314}{314-310}=78.5$ [10], the viscous dissipation number ${\mathit{\lambda}}_{1}=\frac{a{\mu}_{f}^{2}}{{\rho}_{f}{\kappa}_{f}\left(1-\alpha t\right)\left({T}_{c}-{T}_{w}\right)}=\frac{1.28\times {10}^{-5}\times {\left(3.2\times {10}^{-3}\right)}^{2}}{1050\times 0.5\times \left(1-0\right)\times \left(314-310\right)}=6.4\times {10}^{-14}$ at initial moment of fluid flow (t = 0) [10] and the dimensionless distance ${\alpha}_{1}=1$ [41]. Now the Prandtl number for human body is $\mathrm{Pr}=\frac{{\left(\mu {C}_{p}\right)}_{f}}{{\kappa}_{f}}=\frac{3.2\times {10}^{-3}\times 3.9\times {10}^{3}}{0.5}\approx 25$. The values of thermo-physical properties of base fluid (blood) and magnetic particles (CoFe

_{2}O

_{4}) are shown in Table 4. The remaining values of appearing parameters in this problem are utilized in the numerical procedure by considering:

- Volume fraction $\varphi =0.001,0.01,0.05,0.1,0.15,0.2$ as in [46]
- Prandtl number $\mathrm{Pr}=21,25$ as in [51]
- Suction parameter $S=1$ as in [46]
- Heat source parameter $Q=0.5,1,1.5,2$ as in [52]
- Radiation parameter $Nr=0.1,1,2,3$ as in [46]
- Eckert number $Ec=1,1.5,2$ as in [46]
- Velocity slip parameter $B=0.5,1$ as in [46]
- Unsteady parameter $A=0.1,0.3,0.5,0.7,1,2,3$ as in [34]
- Thermal conductivity parameter $b=1,1.5,2,3$ as in [53]

## 8. Results and Discussion

## 9. Conclusions

_{2}O

_{4}. Hence, the results are summarized as follows:

- The velocity and pressure profiles of blood-CoFe
_{2}O_{4}are decreased for both stretching and shrinking cases with the enhancement of the values of ferromagnetic interaction parameter, thermal conductivity parameter and radiation parameter. - Increasing values of curvature parameter, volume fraction of magnetic particles and/or heat source are causing a rise in the velocity profile.
- The velocity profile is reduced when the values of the magnetic parameter and unsteady parameter are increased gradually for the stretching case, whereas the opposite behavior is observed for the shrinking case. Similar behavior is also observed for the pressure profile.
- The blood pressure is enhanced for larger values of the curvature parameter and volume fraction for the stretching case, whereas the opposite is true for the shrinking case.
- For both stretching and shrinking cases the temperature profile exacerbates when the values of the unsteady parameter, radiation parameter and thermal conductivity parameter are increased; while the contrary behavior is found for the heat source parameter.
- With increasing values of ferromagnetic interaction parameter, magnetic field parameter, curvature parameter and the temperature profile are increased for the stretching cylinder while they are decreased in the cylinder surface for the shrinking mode.
- It is obtained temperature profile diminution with the volume fraction of the nanoparticles for the stretching mode, whereas it is raised for the shrinking case.
- The skin friction coefficient onward in stretching mode while it is decreased for the shrinking mode when the values of magnetic field parameter, curvature parameter and unsteady parameter are large.
- For increasing values of the ferromagnetic interaction parameter, the skin friction coefficient is enhanced for both cases, while the reverse is observed for the values of the magnetic particles volume fraction.
- The rate of heat transfer is diminished for both stretching and shrinking cases due to the enhancing values of the unsteady parameter while, interestingly, the reverse attitude is observed for the curvature parameter, magnetic field parameter and volume fraction.
- The rate of heat transfer is increased with increasing values of the ferromagnetic number as in the stretching case while the opposing trend is found for the shrinking case.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Graphical representation and coordinate system of given flow problem (

**a**) for stretching cylinder and (

**b**) for shrinking cylinder.

**Figure 5.**Variation of $\beta $ on (

**a**) velocity profile, (

**b**) Pressure profile, (

**c**) Temperature profile.

**Figure 8.**Variation of $\varphi $ on (

**a**) Velocity profile, (

**b**) Pressure profile, (

**c**) Temperature profile.

**Figure 11.**Variation of $Nr$ on (

**a**) Velocity profile, (

**b**) Pressure profile, (

**c**) Temperature profile.

**Figure 13.**Influence of $\beta $ against $M$ on (

**a**) Skin friction coefficient, (

**b**) Rate of heat transfer.

**Figure 14.**Influence of $D$ against $M$ on (

**a**) Skin friction coefficient, (

**b**) Rate of heat transfer.

**Figure 15.**Influence of M against $\mathit{\beta}$ on (

**a**) Skin friction coefficient, (

**b**) Rate of heat transfer.

**Figure 16.**Influence of $\mathit{\phi}$ against $\mathit{\beta}$ on (

**a**) Skin friction coefficient, (

**b**) Rate of heat transfer.

**Figure 17.**Influence of $A$ against $\mathit{\beta}$ on (

**a**) Skin friction coefficient, (

**b**) Rate of heat transfer.

Magnetic Fluid Properties | Applied Model |
---|---|

Density | ${\rho}_{mf}=(1-\varphi ){\rho}_{f}+\varphi {\rho}_{s}$ |

Dynamic viscosity | ${\mu}_{mf}={\mu}_{f}{(1-\varphi )}^{-2.5}$ |

Heat capacitance | ${(\rho {C}_{p})}_{mf}=(1-\varphi ){(\rho {C}_{p})}_{f}+\varphi {(\rho {C}_{p})}_{s}$ |

Electrical conductivity | $\frac{{\sigma}_{mf}}{{\sigma}_{f}}=1+\frac{3\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\varphi}{\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}+1\right)-\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\varphi}$ |

Thermal conductivity | $\frac{{k}_{mf}^{*}}{{k}_{f}}=\frac{({k}_{s}+2{k}_{f})-2\varphi ({k}_{f}-{k}_{s})}{({k}_{s}+2{k}_{f})+\varphi ({k}_{f}-{k}_{s})}$ |

**Table 2.**Comparison of the skin friction coefficient with Vajravelu et al. [48] for different values of magnetic field parameter and curvature parameter when $\varphi =A=\beta =Nr=Ec=S=B=0,\mathrm{Q}=\mathrm{b}=\mathit{\lambda}={\kappa}_{f}={\kappa}_{mf}^{*}=1$.

M | D | Present Results | Vajravelu et al. [48] |
---|---|---|---|

0.0 | 0.0 | 1.069 | 1.00000 |

0.25 | 1.087 | 1.091826 | |

0.5 | 1.189 | 1.182410 | |

0.75 | 1.271 | 1.271145 | |

1.0 | 1.361 | 1.358198 | |

0.5 | 0.0 | 1.279 | 1.224745 |

0.25 | 1.328 | 1.328505 | |

0.5 | 1.412 | 1.42715 | |

0.75 | 1.523 | 1.521975 | |

1.0 | 1.623 | 1.613858 | |

1.0 | 0.0 | 1.461 | 1.414214 |

0.25 | 1.521 | 1.523163 | |

0.5 | 1.622 | 1.626496 | |

0.75 | 1.718 | 1.725576 | |

1.0 | 1.822 | 1.821302 |

**Table 3.**Comparison of Local Nusselt number (rate of heat transfer) with Bhattacharyya et al. [49] for several values of curvature parameter, suction parameter and Prandtl number when $\varphi =A=\beta =Nr=M=B=b=Q=Ec=0,\mathit{\lambda}=-1,{\kappa}_{f}={\kappa}_{mf}^{*}=1$.

D | S | Pr | − ${\mathit{\theta}}^{\prime}$ (0) | |
---|---|---|---|---|

Present Results | Bhattacharyya et al. [49] | |||

0.1 | 2.6 | 0.5 | 1.117 | 1.1198103 |

0.2 | 2.6 | 0.5 | 1.12 | 1.1225730 |

0.3 | 2.6 | 0.5 | 1.132 | 1.131007 |

0.1 | 2.5 | 0.5 | 1.059 | 1.0671973 |

0.1 | 2.7 | 0.5 | 1.268 | 1.2746036 |

0.1 | 2.6 | 0.3 | 0.7143 | 0.7119983 |

0.1 | 2.6 | 1.0 | 2.079 | 2.0825834 |

Properties | Base Fluid | Magnetic Particles |
---|---|---|

Blood | CoFe_{2}O_{4} | |

${C}_{p}\left(jk{g}^{-1}{K}^{-1}\right)$ | $3.9\times {10}^{3}$ | $700$ |

$\rho \left(kg{m}^{-3}\right)$ | $1050$ | $4907$ |

$\sigma \left(s{m}^{-1}\right)$ | $0.8$ | $1.1\times {10}^{7}$ |

$\kappa \left(W{m}^{-1}{K}^{-1}\right)$ | $0.5$ | $1.3\times {10}^{-5}$ |

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**MDPI and ACS Style**

Ferdows, M.; Alam, J.; Murtaza, G.; Tzirtzilakis, E.E.; Sun, S.
Biomagnetic Flow with CoFe_{2}O_{4} Magnetic Particles through an Unsteady Stretching/Shrinking Cylinder. *Magnetochemistry* **2022**, *8*, 27.
https://doi.org/10.3390/magnetochemistry8030027

**AMA Style**

Ferdows M, Alam J, Murtaza G, Tzirtzilakis EE, Sun S.
Biomagnetic Flow with CoFe_{2}O_{4} Magnetic Particles through an Unsteady Stretching/Shrinking Cylinder. *Magnetochemistry*. 2022; 8(3):27.
https://doi.org/10.3390/magnetochemistry8030027

**Chicago/Turabian Style**

Ferdows, Mohammad, Jahangir Alam, Ghulam Murtaza, Efstratios E. Tzirtzilakis, and Shuyu Sun.
2022. "Biomagnetic Flow with CoFe_{2}O_{4} Magnetic Particles through an Unsteady Stretching/Shrinking Cylinder" *Magnetochemistry* 8, no. 3: 27.
https://doi.org/10.3390/magnetochemistry8030027