Analysis of hydromagnetic hybrid nano fluid flow with iron(II)oxide and iron(III)oxide suspension in human blood

Introduction

The background of the study

Hydromagnetic combines the word "hydro,"which refers to fluid dynamics and "magnetic," indicating the presence of the magnetic field. Alfvén (1942) [1] identified a branch of science focused on studying the behavior of electrically conducting fluids when they traverse a magnetic field. He designated this field as magnetohydrodynamics (MHD), hydro magnetics, or magnetofluid dynamics. This branch of fluid mechanics became significant due to its role in various industrial applications, such as bioreactors, electricity generation, and designing electromagnetic pumps and flowmeters. Fluid is defined as a substance that deforms steadily in response to the influence of shear stress (lateral force per unit area) or external force. Often categorized as a state of matter, including liquids, vapors, and ionized gases. Flowing substance can be grouped into two fundamental varieties: Newtonian (Linear-viscosity fluids) and non-Newtonian (Fluids with non-linear viscosity). For Linear-viscosity fluids, there is a proportional linear relationship with regard to shear stress and shear strain rate; examples are oil, gasoline, alcohol, water, and air. Fluids with non-linear viscosity do not exhibit a simple linear relationship with regard to shear stress and shear strain rate. Blood is an example of a fluid with non-linear viscosity that shows shear-thinning behavior (i.e., its viscosity falls as the shear rate goes up). Fluids with non-linear viscosity have broad applications in manufacturing engineering, such as in food processing, the movement of chyme in the digestive tract, and plasma flow, mercury amalgams, and lubrication with heavy oils and greases. According to Krishnamurthy et al. (2016) [2] , these fluids are divided into categories including liquid metals (e.g., melted iron, mercury, gallium; water; and electrically charged gases such as those present in the sun's outer layers), together with other types, such as saline solutions. Nano fluid is a term used to describe a suspension of nanoparticles in a base fluid. It has gained substantial emphasis as a result of its one-of-a-kind properties and amidst the context of a nano fluid. The base fluid can be a liquid such as water, oil, or ethylene glycol, while the nanoparticles are solid particles with sizes typically ranging from 1 to 100 nanometers. These nanoparticles are commonly composed of materials such as metals (e.g., iron, silver, copper), metal oxides (e.g., iron oxide, titanium dioxide), carbon nanotubes, or graphene. Because the nanofluid alters the hydraulic and thermal properties of the base fluid and enhances heat transfer. Many researchers have investigated the thermophysical properties of individual types of nanofluids. In this context, the objective of this project is to investigate the attributes of the newly formulated hybrid nano fluid. The enhanced heat transfer capabilities of nanofluids have led to their use in diverse applications such as cooling systems, electrical and industrial devices, automotive technology, nuclear reactors, and the biomedical industry. The varied applications led to sparking a curiosity about the existence of a fluid with higher thermal conductivity. Scientists examined several techniques for enhancing thermal conductivity in nanofluids, Suresh et al. (2011) [3] introduced a novel category of fluid known as hybrid nanofluids, which established an entirely new domain of fluid studies. Researcher conducted by Hayat et al. (2017) [4] , Subhani et al. (2018) [5] , Sheikholeslami et al. (2019) [6] , Waini et al. (2019), Tassaddiq et al. (2020) [7] , and Othman et al. (2021) [8] began to formulate the core understanding in this discipline area. Potential applications in industries include energy, automotive, and manufacturing. The findings suggest that hybrid nano fluids have significant potential for improving the heat rate of transfer efficiency and reducing energy consumption in industrial processes. The literature review expands upon their research findings. In order to substantiate the project's validity, a series of deterministic experiments were undertaken. The results of these experiments clearly indicate that the hybrid nano fluid outperforms the standard fluid in terms of both thermal and electrical characteristics. Despite the innovative advancements in the field of hybrid fluids, a substantial challenge also surfaced. The body dealt with nanoparticles as foreign entities and this often resulted in adverse effects on healthy cells and tissues.

In their 2016 study, Kameswaran et al. examined convective heat transfer by evaluating the effects of the non-linear Boussinesq approximation, thermal stratification, and convective boundary conditions on the non-Darcy nano fluid flow over a vertical wavy surface. They noted that an increase in the amplitude-to-wavelength ratio resulted in an augmentation in the average Nusselt number. Given the diverse applications and numerous cancer cases, comprehending the dynamics of analyzing the hydromagnetic hybrid nano fluid flow with FeO and Fe 2 O 3 suspension in human blood is vital for medical applications. Therefore, the objective of this research is to explore the effects of pertinent parameters on the nano fluid flow, the heat transfer rate, and the skin friction.

Statement of the problem

Hybrid nano fluid flow which is the dispersion of nano particles in a base fluid, which has demonstrated significant potential as a power law the non-Newtonian fluid flow with assumptions of steady and incompressible flow with potential applications in the treatment of cancer due to their unique properties such as enhanced thermal conductivity and increased surface area. In cancer treatment, medical practitioners frequently utilize magnetic nanoparticles; by applying a magnetic field, one can direct iron II oxide and iron III oxide toward cancer cells. Researchers use computational simulations to analyze the MHD flow behavior of the hybrid nano fluid in a micro fluidic channel under the influence of a magnetic field. The research outcomes demonstrated that the addition of the hybrid nano fluid to human blood significantly increase the magnetic susceptibility of the blood, which could potentially enhance the targeting of cancer cells with magnetic nanoparticles. The study also showed the addition of the hybrid nano fluid could affect the flow dynamics of the blood, which could potentially improve the delivery of drugs and other therapeutic agents to cancer cells. Several studies have been done on MHD flow especially on the nano fluid exhibiting properties like higher thermal conductivity, critical heat flux, and efficient heat transfer, establishing it as a promising coolant. This enhances the current study to analyze the treatment of cancer by suspension of FeO and Fe 2 O 3 in human blood on MHD flow of hybrid nano fluid on energy dissipation effects in this project.

The justification for the study

Most studies have focused on analyzing non-Newtonian fluids, particularly in the context of Casson fluids, air, and nano fluids. For instance, they have explored the motion of air over the pointed edges of bullets or car bonnets. However, none of these studies has addressed the suspension of FeO and Fe 2 O 3 in human blood. There is a gap in the literature when it comes to researching the treatment of cancer through the suspension of FeO and Fe 2 O 3 in human blood using MHD flow analysis of hybrid nano fluid with the Runge-Kutta-Gill's Method.

Objectives of the study 1.5 General objectives

This research is designed to analyze the hydromagnetic hybrid nano fluid flow with FeO and Fe 2 O 3 suspension in human blood.

Specific objectives

The distinct goals of this research are to: i. To formulate the fundamental equations governing the behavior of hybrid nanofluids.

ii. Convert the control equations to dimensionless form, and the boundary conditions to convert PDE to ODE.

iii. Evaluate how relevant parameters affect nanofluid flow, temperature, and skin friction.

The relevance of the research

Given the numerous potential applications of hybrid nano fluids, the aim of this research results will be of help in the medical field especially the treatment of cancer. Knowledge on the behavior of iron particles in human blood will be instrumental in understanding how cancer medicine can be supplied to the affected area without harming other useful cells in the body. [9] studied the MHD flow and temperature effects dissipation characteristics of a hybrid nanofluid in a lid-driven cavity with a magnetic field using numerical simulations. They reported that the heat dissipation rate of the hybrid nanofluid increases with the addition of magnetic fields and the nanoparticle volume fraction. The research conducted by Gireesha et al. (2020) [10] experimentally investigated the MHD flow and thermal dynamics of the hybrid nanofluid in a circular pipe under the effect of a magnetic field. The researchers discovered that nanoparticles when combined with magnetic fields produce a hybrid nanofluid which enables enhanced heat transfer rates. Research studies about MHD flow of hybrid nano fluids have used multiple methods which include theoretical approaches and numerical simulations and experimental testing. In another study, Jain et al. (The research conducted by (2020) evaluated Fe 2 O 3 nanoparticles as potential cancer-fighting agents. The researchers discovered that nanoparticles showed strong cancer cell toxicity which led to cell death through apoptosis thus making them suitable for medical treatment. The research proved that Fe 2 O 3 nano particles showed high biocompatibility which makes them suitable for use in biological systems. The research by Tariq et al. (The research by Khaled and Al-Najem (2021) used numerical methods to study MHD hybrid nanofluid flow through a rotating channel which received magnetic field application. The researchers discovered that hybrid nano fluids experience changes in their flow behavior and heat transfer performance because of the magnetic field presence. The researchers found that hybrid nano fluids showed higher velocity and temperature gradient values when magnetic fields were applied. The Lorentz force which acts on charged particles in nano fluids produces modified fluid flow patterns that produce increased gradient values. The researchers found that the magnetic field produced an increase in nano fluid heat transfer rates. A recent investigation by Ahmad et al. (2021) examined the effects of MHD flow on the transport of the hybrid nanofluid in human blood vessels for the cancer treatment. The authors used the finite element approach to analyze the MHD flow and found that both the magnetic field strength and the nanoparticle volume fraction significantly affect velocity with temperature distribution on the nano fluid in the blood vessels. By using the finite element method, the researchers were able to create a computational model of the MHD flow and examine the behavior of the nano fluid under different conditions. The research results show that blood vessel nano fluid transport strongly depends on both the strength of the magnetic field and the quantity of nanoparticles used. Specifically, the researchers found out that the higher magnetic field strengths and volume fractions of the nanoparticles led to changes in the velocity and temperature profile of the nano fluid. The research established that MHD flow applications would improve the process of delivering therapeutic agents to human cancer cells throughout the body. Based on the formulation literature reviews,it is evident that a research gap exists which shows that there is need to carry out analysis of hydromagnetic hybrid nano fluid flow with FeO and Fe 2 O 3 suspension in human blood.This project is motivated to fill this research gap.

Literature review

Materials and Methods

Introduction

In this chapter equations governing fluid flow are discussed, the equations are; transport, momentum, energy, continuity and maxwell's. The research has the following assumptions.

Assumptions

To simplify the analysis, the following assumptions are made: i. The flow is steady and laminar.

ii. The fluid is Newtonian fluid and incompressible.

iii. The properties of the hybrid nanofluid are constant except for the density variation in the buoyancy term (Boussinesq approximation).

iv. The magnetic Reynolds number is small, so the induced magnetic field is negligible.

v. The effects of viscous dissipation and Joule heating are negligible.

Model description

The continuity equation for incompressible MHD flow ensures mass conservation and is expressed as:

and the momentum equation is 𝜌( ∂𝑣 ∂𝑡 + ( 𝑣.∇ ) 𝑣) = -∇𝑝 + 𝜇∇ 2 𝑣 + ( 𝐽 × 𝐵 ) where ρ is the Density of the fluid, t is the Time, v is the Velocity vector field, representing fluid flow velocity, p is the Pressure of the fluid, J is the Electric current density vector and B = B 0 +b is the total magnetic field which is the sum of the applied magnetic field B 0 and the induced magnetic field b. The Cauchy stress tensor τ within the fluid is

where p is pressure, I is the identity tensor, µ is the dynamic viscosity, ∇𝑣 is the velocity gradient tensor and T is the time constant.

The components of the extra stress tensor are

Governing equations

The following equations governed the flow;

𝜇 𝑛𝑓 = (1 + 8.0𝜙 + 130𝜙 2 )𝜇 𝑏𝑓 .

Similarity variables

The nonlinearity in the governing equations, coupled with the numerous parameters, makes the equations difficult to solve. The number of parameters can be reduced by parametrizing the equations. Hence, adopting the similarity variables, the number of parameters can be reduced and the system can be reduced to ordinary differential equation. Consider the similarity variables 𝜂 = 𝑦𝑎 𝐶 -𝐶 ∞ = ΦΔ𝐶.

Noting that the similarity variable 𝜂 is dependent only on 𝑦, then the derivative with respect to 𝑦 is found as

The derivatives of the remaining variables with respect to 𝑥 are found below. Then,

The derivatives with respect to 𝑦 are;

Then equation (3.2.36) become

The dimensionless form of the concentration equation is

Now, turn to the boundary conditions at 𝑦 = 0. It is clear that 𝜂 = 0 when 𝑦 = 0. So, we consider the case 𝜂 = 0:

and the final the wall condition is,

Analysis and discussion of results

Numerical solution

The dimensionless form of the governing equations is

The dimensionless form of the governing equations is

𝑢 2 → 0, 𝑢 4 → 0, 𝑢 6 → 0 𝑎𝑠 𝜂 → ∞ .

The system of dimensionless ordinary differential equations derived from the governing equations was solved using the MATLAB bvp4c function, which employs a collocation method for boundary value problems. The boundary conditions at 𝜂 = 0 and 𝜂 → ∞ were incorporated to ensure the accuracy of the solution. Initial guesses for the solution profiles were carefully chosen to facilitate convergence, given the nonlinearity of the equations. The solutions for primary ve-locity (f^' ) and secondary velocity (f) and temperature (Θ) and concentration (Φ) were derived based on the magnetic field parameter (M) and Eckert number (Ec) and nanoparticle volume fraction (Φ). The computational process required multiple iterations to achieve the minimum residuals of both governing equations and boundary conditions which produced stable numerical results. The default values for the parameters are as follows; 𝛾 = 0.10; 𝐺𝑡 = 1.00; 𝐺𝑠 = 1.00; 𝑀 = 2.00; 𝑁𝑏 = 0.10; 𝑁𝑡 = 0.10; 𝐸𝑐 = 0.10; Pr = 4.00; 𝑆𝑐 = 0.62; 𝐵𝑖 = 0.10; 𝜂 ∞ = 10.00;

The thermophysical properties of the nanofluid are also provided in Table 1.

Discussion of results

The numerical simulation produced essential data which showed how various essential parameters influence hydromagnetic hybrid nanofluid flow. The research investigates how three parameters which include magnetic field parameter (M) and Eckert number (Ec) and nanoparticle volume fraction (ϕ) affect the flow patterns. The velocity profiles and temperature distributions and concentration levels in the fluid become determined by these essential factors.

The magnetic field parameter (M) produces a major impact on fluid dynamics because it creates a Lorentz force which works against fluid flow. The system experiences resistance which affects both the fluid velocity and temperature distribution patterns. The magnetic field strength increase leads to a significant decrease in both primary velocity and secondary velocity according to Figures 4.1 and 4.2. The magnetic field creates two opposing forces which reduce fluid motion through its resistive and damping effects. The Lorentz force appears because charged particles in the fluid experience magnetic field interactions which generate a force that resists fluid movement and decreases its kinetic energy. The observed effect exists in actual situations which generate beneficial results but simultaneously generates detrimental effects. The decreased fluid velocity in cooling systems allows the fluid to stay longer near heat exchangers which could lead to better heat transfer performance. The magnetic field strength creates obstacles for processes which need fast fluid movement because it affects their operational efficiency in chemical reactors and mixing systems. The magnetic field strength causes fluid velocities to decrease but the temperature and concentration profiles show opposite behavior according to Figures 4.3 and 4.4. The increase in temperature can be attributed to enhanced thermal energy retention. The magnetic field creates opposition to fluid movement which produces increased heat through the process of viscous dissipation that occurs within the fluid. The thermal energy which builds up from this process becomes useful for medical procedures which need exact temperature control in their thermal management systems. The magnetic field generates a concentration gradient which produces higher solutal retention across the entire fluid surface area. The opposition to movement enables solutes to spread evenly which could enhance reaction speed in mass transfer based processes. The behavior reaches its peak intensity at this particular time and significant in applications like electrochemical cells and various forms of chemical synthesis, where localized concentration gradients can critically influence efficiency and yield. The Eckert number (Ec) shows how fluid viscosity affects the distribution of temperature and velocity which occurs in fluid systems. The graph illustrated in Figures 4.5 The concentration gradient becomes smaller because nanoparticle concentrations rise which leads to this decrease in concentration. The presence of a higher volume fraction of nanoparticles tends to favour thermal diffusion over solutal diffusion, thereby disrupting the conventional concentration distribution.

Conclusions

The research investigated hydromagnetic flow patterns which occur when iron (II, III) oxide ( 𝐹𝑒 3 𝑂 4 ) and iron (III) oxide ( 𝐹𝑒 2 𝑂 3 )

nanoparticles are dispersed in human blood. The research studied how magnetic field strength and Eckert number and nanoparticle volume fraction affect both flow patterns and temperature and concentration distribution. The research findings effectively solved all the particular research questions which were established at the beginning of the investigation. The fundamental equations which describe hybrid nanofluids behavior were developed through applications of fluid mechanics and thermodynamics principles. The model consisted of four basic equations which handled continuity and momentum and energy and concentration behavior and incorporated magnetic field effects and buoyancy force impacts. The research achieves its first specific goal through the creation of a strong theoretical model which explains hybrid nanofluid behavior. The governing equations underwent dimensionless transformation through the application of similarity variables. The method successfully transformed the system of partial differential equations (PDEs) into ordinary differential equations (ODEs) which allowed for quick numerical solution methods. The boundary conditions were also converted to a dimensionless form to align with the reduced system. The solution fulfills the second particular goal by showing how similarity transformations help users solve complicated fluid flow problems through simplification. Finally, the influence of key parameters on the flow of the velocity, temperature, and concentration of the hybrid nanofluid flow was analysed. The results showed that increasing the magnetic field parameter reduced both primary and secondary velocities while enhancing the temperature and concentration profiles due to the resistive Lorentz force. The Eckert number produced higher velocities and temperatures because of viscous dissipation but it also increased concentration levels. The nanoparticle volume fraction led to better velocities and temperature results but it caused the concentration profile to deteriorate because thermal diffusion became the dominant factor. These findings satisfy the third specific objective by providing a detailed evaluation of the parameter effects on the hybrid nanofluid flow. The numerical results

were obtained using MATLAB's bvp4c solver, which proved to be a reliable tool for solving the nonlinear ODEs. The research findings from this study will produce major effects which will benefit both biomedical and industrial sectors. The research results can help developers create better drug delivery systems and improve magnetic hyperthermia treatments and electromagnetic pump and cooling system performance.

Acknowledgement

Utmost gratitude to my Lord Jesus Christ for the support in the entirety of the course of my study. Special recognition is given to my supervisor, Prof. Winifred Mutuku of Kenyatta University, Senior Lecturer and Chairperson, Department of Mathematics and Actuarial Science, for her insight, guidance, and suggestions that brought this research project to a successful and prompt completion. I also highly appreciate the students and lecturers of the Department of Mathematics in Kenyatta University for giving me a place in the facilities, resources, and academic support services that would aid me in the completion of this project successfully.

To my family and friends: thank you for all the encouragement, endless support, patience, and understanding you showered me with in the course of this journey. Your encouragement, prayer, and financial support gave me strength and motivation in moments that seemed hardest.

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20. How to cite this article: Mnang'at Ariangatom D. Analysis of hydromagnetic hybrid nano fluid flow with iron(II)oxide and iron(III)oxide suspension in human blood. International Journal of Statistics and Applied Mathematics. 2026;11(9):148-162. doi: https://www.doi.org/10.22271/maths.2026.v11.i9b.2698