Numerical simulation of biomagnetic pulsatile flow through a channel

List of Symbols

1. Introduction

Biomagnetic fluid dynamics (BFD) studies the flow and transport of magnetic fluids within living organisms. These fluids, also known as ferrofluids, are composed of tiny magnetic particles suspended in a liquid. BFD is a subfield of biofluid dynamics that focuses specifically on the behavior and control of ferrofluids using magnetic fields. Several biological fluids have biomagnetic properties owing to the existence of ions whose behavior is altered upon the action of an external magnetic field. When oxygenated, blood exhibits diamagnetic characteristics, whereas when deoxygenated, it displays paramagnetic properties.¹ Understanding the dynamics of biomagnetic fluids is essential for applications like magnetic resonance imaging (MRI), selective drug delivery, cancer treatment and medical device design.^2,3,4

The application of numerical methods in BFD has shown to be an invaluable tool in studying the behavior of these fluids in the body. Computational fluid dynamics (CFD) allows for detailed fluid flow analysis, including the impact of the magnetic field. It can provide insight into how the fluids move and behave within the body. Various mathematical models have been developed to investigate the effects of magnetic field on a biomagnetic fluid. The first mathematical model to study BFD was developed by Pai et al.,⁵ Loukopoulos and Tzirtzilakis⁶ studied BFD flow in a channel under the action of a spatially varying magnetic field using the stream function-vorticity formulation. The results indicated that applying a magnetic field has an appreciable influence on the flow by increasing the temperature and skin friction at the lower plate. It is observed that a vortex is developed in the vicinity of the magnetic source. A simple and more stable model was later developed by Tzirtzilakis,⁷ which used a pseudo-transient numerical technique that was able to simulate flows for higher magnetic numbers. The work of Rusli et al.⁸ developed a mathematical model using the finite difference method (FDM). A modified SIMPLE-type algorithm was used to handle the instabilities related to pressure. More recently, a magnetic source has been used to either promote or diminish the formation of vortices according to the need of the study. Mousavi et al.⁹ studied the effects of a magnetic field on a biomagnetic fluid in a constricted channel. The results of their work showed that it is possible to reduce the prominence of the re-circulation region through the application of a magnetic field. Souayeh et al.¹⁰ investigated using magnetic baffles and dimple turbulators. Their results indicated that using magnetic baffles resulted in a decrease in pressure compared to physical baffles and also aided in increasing the heat transfer through turbulence. Increasing the magnetic field created a larger turbulence vortex which resulted in an increase in the Nusselt number. Karimipour et al.¹¹ studied nanofluid flow in a microchannel in the presence of injection and under the influence of the magnetic field. The magnetic field was used to decrease the intensity of the vortices caused by the injection. Increasing the strength of the magnetic field also led to an improvement in heat transfer. Greater efficacy of the magnetic field was observed at higher Reynolds numbers.

The purpose of this research is to examine how a pulsing fluid responds to a magnetic field that is spatially varied. The pulsatile flow for biomagnetic fluids is characterized by the Womersley number ( Wo).¹² It is a nondimensional term related to the oscillation of the fluid. The range of Wo in a typical BFD flow is 2 to 16.¹³ Viscous forces are predominant at a lower Wo while the inertial forces have a more appreciable effect for a Wo above 10.¹² Sankar and Lee¹⁴ studied pulsatile blood flow through a stenosed artery. They considered the core region as a Casson fluid and the peripheral layers as a Newtonian fluid. An increase in the plug flow velocity and flow rate was seen as the Reynolds number increased. Kolke et al.¹⁵ studied pulsatile flow through a channel with a smooth constriction. The pulsating nature is mimicked using a sinusoidally varying pressure gradient, and the numerical model was based on a fractional step-based finite volume method (FVM). The results showed good agreement with previously published data. Reddy et al.¹⁶ investigated pulsatile magnetohydrodynamic flow (MHD) nanofluid flow in a vertically permeable irregular channel. The results indicated that the CuO–Fe₃O₄ hybrid nanofluid had better drug delivery performance when compared to mono-nanofluids.

Due to the lack of literature on pulsating biomagnetic flow, the objective of this work is to investigate the same. This study investigates the effect of a spatially varying magnetic field on a pulsatile biomagnetic fluid flowing through a straight channel. Numerical simulations are performed using the FVM. The work studies the effect of the Womersley number ( Wo), Reynolds number ( Re) and Magnetic number ( Mn) on a pulsating biomagnetic fluid, which will have applications that aid the development of medical devices.

2. Numerical Procedure

2.1. Mathematical model

This work assumes that the flow is two-dimensional, incompressible, Newtonian and isothermal. The governing fluid flow equations used are the continuity equation, x-momentum equation and y-momentum equations as given in Eqs. (1)–(3), respectively. The magnetic force is added as a body force in the momentum equations. The formulation for this has been taken from Refs. 7 and 18.

where $\rho$ is the density of the fluid, $\nu$ is the kinematic viscosity of the fluid, ${\mu }_0$ is the fluid magnetic permeability, given by Eq. (4). While $\frac{M{\mu }_0}{\rho }\frac{\partial H_x^{\ast }}{\partial x^{\ast }}$ and $\frac{M{\mu }_0}{\rho }\frac{\partial H_y^{\ast }}{\partial y^{\ast }}$ describe the force acting per unit mass, which arises due to the magnetization of the fluid particles. The intensity due to the magnetic effect in the channel length $(x)$ and channel height $(y)$ directions is given by Eqs. (5) and (6). The above equations are converted to their nondimensional form by selecting the inlet velocity ( U) as the reference velocity and the channel height ( D) as the reference length. The nondimensional parameters obtained are The nondimensional continuity, x-momentum and y-momentum equations are given in Eqs. (7)–(9), respectively : The Magnetic number, Mn, and the Reynolds number, Re, are given by Eqs. (10) and (11), respectively. For the implementation of the pulsatile nature of the flow, an external pressure gradient is applied, which is given by the following equation : where f is the oscillating frequency described by Womersley number ( Wo) and Strouhal number ( St) as shown in Eqs. (13) and (14), respectively.

2.2. Geometry and numerical methods

Consider a straight, rectangular channel with length L and height D, as shown in Fig. 1. The top and bottom of the channel are enclosed by walls, at which the no-slip boundaries are imposed. Periodic boundary conditions are imposed in the x-direction. The pressure boundary conditions at the inlet, top and bottom are set with zero gradients, and no condition is specified for the outlet.

The numerical simulation is performed on a staggered mesh in a Cartesian coordinate system. An FVM-based semi-implicit fraction step method is used.^18,19 The velocity and pressure terms at the next time level, the (n+1)th level, are obtained by introducing an intermediate velocity and pseudo-pressure to satisfy the flow-incompressibility constraint. The equations used are depicted as follows :

where $û$ is the intermediate velocity, $\mathrm{\Delta }t$ is the time step-size, and $\varphi$ is the pseudo-pressure used to satisfy the flow-incompressibility constraint and $F_M$ is the magnetic force. The discretized momentum equations are then iteratively solved using the Successive Over-Relaxation (SOR) scheme.

3. Results and Discussion

3.1. Validation

First, the numerical model for a regular two-dimensional, $5\times 1$ rectangular channel with Re = 10, and no magnetic field is developed. The results obtained are validated by comparing the centerline velocity of the numerical model with the analytical solution for plane Poiseuille flow. Good agreement is obtained between the two as shown in Fig. 2.

Further, to observe the influence of the magnetic field on the flow, a magnetic source ( Mn = 100) is introduced at (2,-0.05). The streamlines in Fig. 3 show the formation of a primary vortex near the source due to the presence of the magnetic field.

3.2. Pulsatile biomagnetic flow

The study is further extended to simulate biomagnetic pulsatile flow for different values of Re, Mn and Wo. These conditions were simulated by running through multiple time periods to allow the oscillatory flow to stabilize, as shown in Fig. 4.

In the first case, Reynolds number ( Re = 10) and Womersley number ( Wo = 10) are held constant, and the Magnetic number ( Mn = 100,150,250,500) is varied. The results comparing the streamlines are shown in Fig. 5. The formation of the primary vortex near the magnetic source can be noticed in all cases. The pulsatile nature of the flow results in considerable changes to the re-circulation region. In Fig. 5(a), we can see the periodic growth and decay of the secondary vortex beside the primary vortex. The formation of a tertiary vortex can be seen at the end of the time step in Figs. 5(b) and 5(c). When the magnetic number is increased to 500, the results shown in Fig. 5(d) reveal the presence of the tertiary vortex throughout the time period and also show the presence of additional vortices near the magnetic source. Further, the maximum vorticity of the fluid over a time period has been tabulated in Table 1. The results show that an increase in the magnetic number produces a proportional increase in the vorticity. Therefore, it can be quantitatively inferred that an increase in the magnetic number results in the intensity of the re-circulations.

In the second study, the Reynolds number ( Re = 10,20) is varied while the Magnetic number ( Mn = 100) and Womersley number ( Wo = 10) are fixed. From the streamlines and velocity contours shown in Figs. 6 and 7, it is clear that when Re increases, there is a noticeable effect on the flow. In Fig. 6(a), the periodic growth and decay of a secondary vortex can be observed, but when the Reynolds number is increased to 20, the results displayed in Fig. 6(b) show the formation of additional vortices. Further, the results in Table 2 show that there is a proportional increase in the maximum vorticity with respect to the Reynolds number.

In the third case, the Womersley number ( Wo = 5,10) varies while keeping the Magnetic number ( Mn = 100) and Reynolds number ( Re = 10) constant. From the streamlines, shown in in Figs. 8(a) and 8(b), and velocity contours, shown in Figs. 9(a) and 9(b), it can be observed that the average flow velocity over one time period is lower when the Womersley number is higher. While this does not have a pronounced effect on the maximum vorticity (Table 3), a difference in the size of the re-circulation regions can be seen. In Fig. 8(a), a formation of the primary vortex can be seen through most of the time period, while in Fig. 8(b), a periodic growth and decay of the secondary vortex can be observed. This is because the magnetic field has a more pronounced effect when the fluid velocity is lower.

4. Conclusion

This study discusses the implementation of a FVM-based numerical model to study two-dimensional biomagnetic fluid flow under pulsatile boundary conditions through a straight channel. Through the results of the simulations, it is found that the pulsatile behavior of the flow results in periodic growth and decay of multiple re-circulation zones near the source of the magnetic field. Further, it is seen that the Magnetic number ( Mn), Reynolds number ( Re) and Womersley number ( Wo) all have a significant effect on the flow dynamics. Increasing any of the three mentioned parameters increased the number of re-circulations. An increase in the Reynolds number and magnetic number results in a proportionate increase in the maximum vorticity of the fluid. With an increase in the magnetic number from 100 to 150, 250 and 500, the maximum vorticity increases by 48.04%, 149.84% and 402.68%. When the Reynolds number was increased from 10 to 20, a 102.52% increase in the maximum vorticity was observed, while almost no change in vorticity was found when varying the Womersley number, but it had a prominent effect on the size of the re-circulation zone. This results in high-energy losses, which is why a combination of the three parameters must be carefully chosen when developing devices involving biomagnetic microchannel flows. Future researchers have the opportunity to extend this work by exploring thermal effects, irregular geometries and the influence of incorporating additional magnetic sources.

ORCID

Kaustubh   https://orcid.org/0009-0005-0140-9654

Ranjith Maniyeri   https://orcid.org/0000-0001-5997-473X