Electroosmotic flow of Jeffrey ternary hybrid nanofluids in converging–diverging ciliary microvessels

Abstract

The intricate behavior of blood flow conveying multiple nanoparticles through micro-scale biological channels remains insufficiently understood, particularly under physiological conditions involving diverging and converging ciliary microvessels. A key challenge lies in capturing the combined effects of complex body forces and non-Newtonian fluid characteristics when ternary-hybrid nanoparticles specifically tricalcium phosphate Ca₃(PO₄)₂,TiO₂ (titanium dioxide), and Cu (copper) are introduced into the bloodstream. Existing models often fail to represent the synergistic interactions among electrokinetic, magnetic, and elastic influences in such flows. This study addresses the gap by formulating a mathematical model for the dynamics of a Jeffrey fluid representing blood, embedded with Ca₃(PO₄)₂, TiO₂, and Cu ternary-hybrid nanoparticles, flowing through a diverging/converging ciliary micro-vessel. The model accounts for electroosmosis forces, Lorentz forces, buoyancy, heat generation, and ciliary movement. The governing nonlinear equations are non-dimensionalized and solved using the homotopic perturbation method (HPM) to derive analytical approximations. Findings reveal that electro-osmosis and the Helmholtz–Smoluchowski slip velocity significantly enhance fluid motion in the central region of the vessel, whereas resistance dominates at the periphery. Cilia elongation reduces the circulation efficiency of the nanofluid. In addition, diverging vessels facilitate higher heat dissipation than converging ones, making them preferable for biomedical procedures requiring precise thermal control.

Introduction

Peristalsis is a fundamental method for moving fluids that can be observed in various natural systems and technological applications. Essentially, this process involves the transmission of wave-like movements along a flexible boundary or channel, which moves the fluid by pushing it forward, rather than by traditional pumping or suction methods. One of the major components of physiological processes in biology is peristalsis, especially in the gastrointestinal tract, where its function is fully explored. The muscles of the esophagus, stomach, and intestines coordinate their actions to move food and liquids efficiently and in a single direction through the digestive system¹. Also, the peristalsis is used for accurate control of fluid movement while preventing any backward flow. This helps maintain a sterile environment by keeping the fluid isolated from external sources. Additionally, it is effective for handling fluids that contain different levels of thickness (viscosity) or suspended particles. Similar mechanisms are found in many biological contexts, including the lymphatic and reproductive systems, which point to the significance of the movement of the fluid, which reduces the shearing forces². The peristaltic motion underlying mechanism has also led to their inspiration for various industrial and engineering applications. Peristaltic mechanisms are used when designing peristaltic pumps used in biomedical engineering, such as drug delivery systems, extracorporeal circulation devices. Peristalsis also ensures precise control of fluid movement and prevents any backward flow, which helps maintain a sterile environment by keeping fluids separate from external contaminants³. This mechanism is especially useful for moving fluids with varying thickness (viscosity) or those containing suspended particles. Similar wave-driven fluid transport occurs in other biological systems, such as the lymphatic and reproductive systems, highlighting the importance of peristalsis in reducing harmful shear forces during fluid movement. Peristaltic transport offers unique advantages for mechanical and process engineering purposes that make the special transport very applicable for systems where sanitary, leak-free, and small-maintenance pumping systems are required. The strong treatment of abrasive, viscous, or corrosive material has embedded peristaltic technology into numerous industrial procedures, including the production of food, the management of chemicals, and the movement of slurry⁴. Peristaltic approaches are also useful in energy systems, most especially in the improvement of microscale and renewable energy processes⁵. In particular, the peristalsis technique of mechanics is the integral principle in microfluidic devices, i.e., in lab-on-chip applications, to appropriately route and mix minuscule quantities of liquid. Wu et al.⁶ explored the transport of a non-Newtonian Ree-Eyring fluid by peristalsis, considering variations in fluid properties and the impact of heat transfer. They dealt with the problem by using long-wavelength and low-Reynolds-number approximations and solved the nonlinear equations by applying regular perturbation techniques. Qian et al.⁷ designed an approach to study blood movement in a porous artery due to the flow of ZrO₂ nanoparticles, considering electro-osmosis, nanoparticle aggregation, and changes in the fluid’s viscosity using a third-grade fluid model. Their research suggests that higher electroosmosis factors result in better flow control, lower convection causes poorer heat transmission, and the position of arteries influences shear stress.

Recent advances in microfluidic technology and bioinspired engineering have produced highly sophisticated fluid transport systems that closely mimic complex biological processes at the microscopic level⁸. There is considerable interest in studying ciliated micro-vessels with diverging and converging shapes, primarily because their geometry, the movement of their cilia, and their fluid flow patterns interact in complex ways. These designs, which are typically depicted as channels with variable cross-sections, make use of coordinated cilia-like action on inner surfaces to control fluid flow to high precision^9,10. The body contains analogous systems, represented by the respiratory tract and fallopian tubes, wherein cilia guide the flow of fluid and particles through a network of single-ended branching and tapering channels. Designing synthetic analogues of these mechanisms aids the creation of diagnostic and therapeutic systems that can better simulate realistic physiological conditions than static benchtop models¹¹ Ciliated micro-vessels play a crucial role in tissue engineering and regenerative medicine because they help create dynamic microenvironments that support cell survival and function. Furthermore, these systems are essential for personalized medicine because they can precisely control the movement of fluids at very small scales. Accurate control over how fluids flow is crucial for delivering treatments to targeted areas within complex biological structures. Ciliated microvessels play a critical role as components for engineers in the pursuit of auto-regulated microfluidic circuitry. Lab-on-a-chip technologies are the systems where these advantages are especially important since conventional pump solutions are too bulky and power-consuming for the micro- and autonomous devices needed. Engineers have developed a device that can provide unidirectional flow, phase separation, or better chemical reactions from mixing only through the integration of artificial cilia into custom channel layouts without the need for any external power^12,13. By changing the shape of the vessels to become either divergent or convergent, the engineers can magnify either selectively or dampen ciliary-driven flow selectively, optimizing the performance metrics. Ajithkumar et al.¹⁴ study the movement of blood and the transfer of heat in a ciliated artery that has been affected by both stenosis and dilatation, using a fractional second-grade fluid model to capture the actions of blood containing modified trihybrid nanoparticles (gold, copper, titania, and SWCNTs). The research examines both electromagnetic influences and the presence of thin layers at the interface by applying the homotopy perturbation method. Jagadesh et al.¹⁵ research the movement of fractional second-grade blood containing trihybrid nanoparticles (gold, copper, MWCNTs) through a diverging ciliated micro-vessel, while considering effects at the vessel interface. It describes impacts from cilia, buoyant forces, heating and cooling effects, viscous dissipation, movement caused by electric currents, and areas with strong magnetic fields.

The study of how electromagnetic fields interact with fluid flow in both biological and engineering systems has given rise to a specialized scientific field called Electro-magneto-Hemodynamics (EMHD). Electromagnetic hydrodynamics (EMHD) combines principles from classical electrodynamics and fluid mechanics to study how electrically conducting fluids, such as blood, respond to surrounding electric and magnetic fields. The synthesis of these scientific realms in EMHD yields a robust model explaining and controlling the movement of fluids in situations dominated by electromagnetic forces. The basis of EMHD is the Lorentz force, which explains how charged particles behave in electric and magnetic fields. In electrically conductive fluids, such as ionized plasmas or blood containing electrolytes, these forces have a significant influence on how the fluid flows¹⁶. The EMHD effects are especially significant for the analysis or control of the fluid flow when mechanical approaches cannot be applied. Non-invasive control of the flow of blood is being achieved via the use of electromagnetic fields in biomedical systems has opened the possibility for innovative aspects in targeted drug delivery, artificial organs, and advanced diagnostic imaging. By taking advantage of blood’s electrical conductivity, researchers can measure and control blood flow characteristics precisely without needing to insert instruments into the body¹⁷. In engineering, phenomena of electromagnetic hydrodynamic (EMHD) offer new opportunities for controlling coolant flow in nuclear reactors, enhancing heat transfer properties, and handling fluid transport in hazardous environments. The benefits of EMHD cause mechanical designs for EMHD to be less complicated and increase reliability, especially in closed-loop systems and spaces suitable for EMHD applications¹⁸. Electromagnetic hydrodynamics (EMHD) plays an important role in energy systems because it is central to the operation of magnetohydrodynamic (MHD) generators. In these generators, the kinetic energy from a moving, electrically conductive plasma interacts with magnetic fields, allowing this kinetic energy to be converted directly into electrical power¹⁹. This provides a great opportunity to design more effective energy conversion systems. Karmakar and Das²⁰ recently investigated how blood, which contains different types of nanoparticles, modulates under electrokinetic effects within an arterial vessel that is both electrically charged and exposed to a magnetic field, while also considering the vessel’s local curvature. Ponalagusamy et al.²¹ examine how solute mixing occurs on electro-magneto-hydrodynamic peristaltic flow of a Jeffrey fluid through a porous channel with flexible walls, with both uniform (homogeneous) and non-uniform (heterogeneous) chemical reactions analyzed using the long wavelength approximation method. Das et al.²² investigate the blood flow carrying silver and aluminium oxide nanoparticles inside a tubular section and an arterial wall. The model considers electroosmosis, Hall and ion-slip effects, Joule heating, creation of internal energy, blood coagulation, and variations in nanoparticle shape.

The ongoing effort to enhance thermal efficiency has led researchers to explore and develop new types of heat transfer fluids over the past several decades. Many manufacturing industries have employed water, ethylene glycol, and engine oils as heat transfer fluids^23,24. However, their thermal efficiency is limited by a large variety of modern processes where rapid and effective heat transfers are required, owing to their low thermal conductivities. The development of nanotechnology made it possible to create nanofluids, which are produced by adding extremely small particles (typically smaller than 100 nm in diameter) to regular base fluids. The remarkable thermal properties associated with nanofluids have made it possible for many applications that require high transfer of energy to embrace nanofluids^25,26. Advancements in scientific research have led to the development of hybrid nanofluids, which are fluids that contain two distinct kinds of nanoparticles. The class of ternary-hybrid nanofluids is distinguished by its complex formulation, which involves the simultaneous addition of three different types of nanoparticles typically selected from metals, metal oxides, or carbon-based materials. Metal oxide nanoparticles (alumina, titania etc.) will facilitate enhanced stability and compatibilities with the base fluid, whereas application of materials based on carbon: graphene, carbon nanotubes, etc.²⁷. Ternary hybrid nanofluids are studied for application in radiators and thermal exchangers in automotive and aerospace applications, which could improve fuel efficiency and increase operational safety. The same is true for applications of metal cutting and welding in manufacturing, where ternary-hybrid nanofluids enhance heat dissipation and result in reduced thermal stress²⁸. The use of hybrid nanofluids is becoming more important in the energy sector, especially for solar thermal and geothermal applications. For example, in solar collectors, the working fluids are critically involved in effective solar heat absorption and subsequent transportation²⁹. The enhanced absorption and thermal characteristics of ternary-hybrid nanofluids lead directly toward improved robustness in energy capture and storage. Ali et al.³⁰ studied the electromagnetic influences on cilia-actuated peristaltic transport of hybrid nano-blood in an arterial conduit under a regnant magnetic field with the Jeffrey model. Faisal et al.³¹ perform numerical studies on heat and mass transfer in a Maxwell slip three-dimensional flow between a mixture of nanoparticles and fluids governed by Smoluchowski–Nield. Nanoparticles are mixed into water to enhance its ability to transfer heat. The model takes into account Brownian motion, thermophoresis, and thermal radiation factors that play important roles in accurately cooling devices used in microelectronics and biological applications. Kumar and Tripathi³² study the heat transfer and flow analysis in tumor-obstructed propagating microchannel. Vaidya et al.³³ studied the surface roughness impact on bioinspired rhythmic contractile microfluidic membrane pumping mechanism: A computational analysis. Kumar et al.³⁴ explored the magnetic field modulation of electroosmotic-peristaltic flow in tumor microenvironment. Choudhari et al.³⁵ modeled the heat transfer in peristaltic flow of electrokinetically modulated Carreau fluid under the influence of magnetic field. Jangid et al.³⁶ investigated the heat transfer analysis in membrane-based pumping flow of hybrid nanofluids. Kumar et al.³⁷ studied the non-Newtonian lacuno-canalicular fluid flow in bone altered by mechanical loading and magnetic field. Tripathi et al.³⁸ explored a critical review of micro-scale pumping based on insect-inspired membrane kinematics. Narla et al.³⁹ investigated the electrokinetic insect-bioinspired membrane pumping in a high aspect ratio bio-microfluidic system. Bhandari et al.⁴⁰ examines the electro-osmosis modulated periodic membrane pumping flow and particle motion with magnetic field effects.

The accurate modeling of blood flow infused with nanoparticles within microvascular networks remains a complex and unresolved challenge in fluid dynamics and biomedical engineering. Although nanofluids, especially hybrid and ternary-hybrid nanofluids, have demonstrated enhanced thermal and transport properties, their behavior in non-Newtonian physiological fluids, particularly Jeffrey-type fluids, within diverging/converging ciliary micro-vessels, has not been sufficiently investigated. Existing literature lacks a comprehensive analysis that accounts for the simultaneous effects of electro-osmosis, Lorentz forces, buoyancy, internal heat generation, and viscoelasticity on ternary-hybrid nanofluid transport. Without this understanding, the effective design of biomedical microdevices and targeted therapeutic applications remains hindered. This study aims to develop and analyze a robust mathematical model that captures the behavior of blood flow carrying ternary-hybrid nanoparticles specifically Ca₃(PO₄)₂, TiO₂, and Cu ternary-hybrid nanoparticles through a diverging or converging ciliary micro-vessel using the Jeffrey fluid model. The model seeks to account for the interplay of several physical phenomena, including electroosmosis effects, Lorentz forces, buoyancy, internal heat generation, and the coordinated motion of cilia lining the vessel wall. This integrated approach enables a more accurate representation of real physiological conditions, particularly in complex microvascular environments. To achieve this aim, the study formulates a non-dimensionalized set of governing equations that describe the fluid’s momentum and thermal characteristics under the influence of the aforementioned forces. Analytical solutions are obtained using the Homotopic Perturbation Method (HPM), enabling the examination of how key parameters such as nanoparticle concentration, magnetic field intensity, electrokinetic velocity, cilia length, and heat source strength affect the flow and temperature profiles. The study also investigates the comparative performance of ternary-hybrid nanofluids against binary and mono nanofluids in enhancing blood flow and heat transfer. Ultimately, this work provides critical insights that support the optimization of nanoparticle-assisted transport in physiological systems, with direct relevance to the development of advanced biomedical microdevices.

Research questions

This study seeks to address the absence of a clearly articulated research focus in prior investigations of electrokinetically driven, non-Newtonian nanoparticle-enhanced blood flow in ciliary micro-vessels. The following research questions guide the theoretical and numerical analysis:

• How do electroosmosis forces, Lorentz forces, and buoyancy collectively influence the velocity profile of a Jeffrey fluid containing Ca₃(PO₄)₂–TiO₂–Cu ternary-hybrid nanoparticles in diverging and converging ciliary micro-vessels?

• How does geometry (diverging vs. converging) affect heat dissipation when ternary-hybrid nanoparticles are dispersed in a non-Newtonian bloodstream subjected to electrokinetic and magnetic fields?

• What is the quantitative impact of the Helmholtz–Smoluchowski slip velocity on core–periphery flow differences in micro-vessels carrying ternary-hybrid nanoparticle-enhanced blood?

• How can the combined effects of electroosmosis, ternary-hybrid nanoparticles, and vessel geometry be leveraged for targeted drug delivery, hyperthermia treatment, or microfluidic diagnostic applications?

Assumptions and model constraints

We consider a mathematical model describing the electroosmosis flow of an electrically conductive, incompressible THNFs comprising , TiO₂, and Cu nanoparticles driven by peristaltic motion through a converging/diverging ciliated microcirculatory channel. The model accounts for internal heat generation and the influence of external magnetic (Lorentz) forces. The following assumptions and boundary limitations are applied:

• The flow is modeled in a cylindrical coordinate system (), where is the radial axis and is the axial direction, respectively. The geometry describes the capillary vessel of length L, which exhibits convergence or divergence, and is displayed in Fig. 1.

• At a constant temperature , the wall of the ciliated micro-vessel is maintained and fixed.

• The interaction of the induced electric current with the applied magnetic field results in a Lorentz force that affects the flow dynamics.

• Each kind of nanoparticle (Ca₃(PO₄)₂, TiO₂, and Cu) are considered in four geometrical shapes: spherical, cylindrical, brick-shaped, and platelet-shaped forms.

• The electro-kinetic body force is expressed as where represent the is the local net charge density and denotes the electric field.

• The micro-vessel wall is ciliated, and the cilia beating mechanism is accounted for as a time-averaged slip velocity at the wall.

Fig. 1.

Geometrical representation of the physical problem.

The ciliary motion is represented as a combination of traveling waves, each with sinusoidal deformation components. Using this mechanical model, the equation below describes the pattern at the tip of the cilia¹⁰.

1

Here, and represent instantaneous positions in the deformed configuration, not the basic coordinate definitions. This ensures consistency with the cylindrical coordinate framework and dimensionality.

The horizontal and vertical velocity components velocity field components () of the blood ternary-hybrid nanofluid circulation can be mathematically defined in the () frame, which represents the position at the tips of the cilia^10,12.

2

Equation (2) can be rewritten in a modified form by applying the relationships established in Eq. (1). Thus,

3

4

Jeffrey fluid

The Jeffrey fluid model explains the properties of specific non-Newtonian fluids like blood in different flow environments. This type of fluid behaves like others but also accounts for viscoelastic actions and relaxation, alongside the fluidity effects found in Newton and power-law fluids. Many researches use it to describe materials that have properties in between those of solids and liquids. The Jeffrey fluid constitutive equation can be defined as¹¹:

5

6

7

8

Electroosmotic flow

The concept of direct electrostatic interaction is used by the Poisson equation to describe how electrostatic potential behaves in an electrolytic solution. As demonstrated in References^6,41, the electromotive potential within the electrical double layers that supports this hypothesis is defined as:

9

Here, the total charged density of the ionized () can be expressed as:

10

The densities of negative ions () and positive () in the ionic system, can be expressed as follows:

11

In the context of ionic blood flow, the amount of electric charge contained within a unit volume is commonly referred to as the charge density and can be expressed as follows:

12

By substituting Eq. (12) into Eq. (9), the resulting form of the Poisson equation can be written as follows:

13

In the electrical double layers, the function describes how the electrical potential is distributed. Based on the formulation of the governing equations, we consider the effects of both electroosmosis and Lorentz forces acting on the unsteady flow of ternary-hybrid nanoblood through a non-uniform ciliated microtube. By applying the Boussinesq approximation and assuming the previously stated hypotheses, and under the methodologies presented in Refs^10–12., the equations governing the flow behavior can be established as follows:

14

15

16

17

The boundary conditions are:

18

Nanoparticles and thermophysical properties

Table 1 shows the important characteristics of blood and the three nanoparticles utilized in the present work. Meanwhile, the correlations of these thermo-physical properties (TPPs) of the ternary hybrid nanofluid are given in Table 2.

Table 1.

Physical characteristics of blood-based nanofluids, tricalcium phosphate Ca₃(PO₄)₂, TiO₂ (titanium dioxide), and Cu (copper).

Physical property	(	()	k(W/mK)	σ(Sm−1)
Blood	1060	3770	0.55	0.7
Ca3(PO4)2	3140	840	0.5	0.02418
TiO2	4250	690	8.953	2.4 × 106
Cu	8933	385.0	401.0	5.96 × 107

Table 2.

Thermal properties of the ternary hybrid nanofluid^28,31.

Ternary-hybrid nanofluid	Properties
	Thermal conductivity
	Thermal expansion coefficient
	Electrical conductivity
	Viscosity
	Heat capacity
	Density

Flow analysis in wave frame

Similarity transformation is well used in several branches of science and engineering, such as thermal transport, computational fluid dynamics, and so on. This method is useful as a means to elaborate complex concerns and gather essential knowledge concerning the function of physical systems. This technique can be used as a way of making complex concepts easier to understand and to gain a crucial insight into the functioning of physical systems. As a result, we derive the equations that describe how physical quantities measured in the moving wave frame are related to those measured in a stationary reference frame^10,12:

19

The Eqs. (14–18) can be rewritten in dimensionless form by applying Eqs. (19) as follows:

20

21

22

23

Boundary conditions (BCs) are:

24

The Poisson Eq. (20), along with the boundary conditions (24), was solved to obtain³⁰:

25

Here, is the non-dimensional electric potential, is the time, is the magnetic parameter, Jeffrey parameter, , heat source parameter, is the wave number is the electroosmosis factor, is the Reynolds number, is the Grashof number, is the Brinkman number, is the non-uniform parameter, = , = , = NPs volume fractions.

Homotopy perturbation method (HPM)

The HPM was designed to obtain approximate solutions for linear and nonlinear differential equations. Although this approach uses concepts from topology, specifically Homotopy, as well as perturbation techniques, it does not depend on the presence of a small parameter. As a result, it can be conveniently applied to a wide range of scientific and engineering problems. This section examines the method employed to solve the expression (21–24), the limitations set via the interface conditions specified in expression (24). Consequently, the converging power series solutions of these equations are expressed using the HPM. The HPM is used to determine solutions to this expression, and these solutions are expressed as series that converge. By using Eq. (25) to replace the terms in Eq. (22), then Eqs. (22 and 23) are obtained as.

26

27

where the constants are It is provided as follows:

28

For this purpose, the homotopy representations of Eqs. (26 and 27) are given below:

29

30

here, . The initial approximations for and are assumed below:

31

32

By setting in Eq. (31) and applying the HPT, we derive the following result:

33

34

To establish a system of linear equations, we first replace Eqs. (31 and 32) into reliability functions (29 and 30). Next, make sure that the coefficients of the same powers of p ˇ on each side are equal. By following these steps, you get a set of linear equations. The following equations also include slight modifications to describe the distributions of velocity and temperature³⁰:

35

36

We got excellent results that agree with the results from Ref³⁰., as demonstrated in Fig. 2. This comparison serves to assess the accuracy of our present simulation outcomes.

Fig. 2.

Validation of the results.

Volumetric flow rate

The volumetric flow rate Q is defined as

37

This results in a pressure gradient, which is defined as:

38

The average volume of fluid transported by each individual peristaltic wave can be expressed as follows:

39

The stream function is mathematically defined as:

40

and

41

By substituting Eq. (35) into the integral expression The resulting form of the stream function is obtained as follows:

42

The expression for the tangential stress acting on the wall is given by:

43

The heat transport coefficient for the wall that is coated with cilia and has a diverging or converging shape is given by:

44

Results and discussion

This section presents and interprets the analytical results derived from the homotopy perturbation method (HPM) for the formulated model of Jeffrey fluid flow embedded with Cu–Au–TiO₂ ternary-hybrid nanoparticles through ciliary microvessels. The analysis focuses on the influence of electro-osmosis, Lorentz forces, ciliary motion, thermal effects, and vessel geometry (diverging versus converging). Table 3 shows a comparison between how conventional, Ca₃(PO₄)₂–TiO₂/Blood hybrid nanofluid and Ca₃(PO₄)₂–TiO₂–Cu/Blood ternary hybrid nanofluid solutions (NFs, HNFs and THNFs) respond to different values of the parameter on heat flow. The performance measure for the NFs is found to be 0.47215. When (HNFS) is used, the value increases to 0.51928, resulting in a 9.98% increase over the conventional nanofluid. For Ca₃(PO₄)₂–TiO₂–Cu/Blood ternary hybrid nanofluid, the value further upsurges to 0.54567, signifying a 15.58% enhancement compared to the conventional case. When is enhanced to 2.0, the nanofluid value becomes 0.48096. This value again upsurges significantly as Ca₃(PO₄)₂–TiO₂/Blood hybrid nanofluid are employed, reaching 0.52840 increment, which corresponds to a 9.86% upsurge. The Ca₃(PO₄)₂–TiO₂–Cu/Blood ternary hybrid nanofluid performs even better at this parameter level, yielding a value of 0.55487, which indicates a 15.37% enhancement. At  = 3.0, the traditional nanofluid yields a value of 0.49523. In comparison, the Ca₃(PO₄)₂–TiO₂/Blood hybrid nanofluid achieves 0.54071, offering a 9.19% increase over the NFs. The Ca₃(PO₄)₂–TiO₂–Cu/Blood ternary hybrid nanofluid again leads to further improvement, attaining a value of 0.56934, which marks a 14.95% gain over the NFS. As upsurges to 4.0, the conventional nanofluid records a value of 0.50389. The corresponding HNFS value is 0.55219, representing a 9.56% increase, while the Ca₃(PO₄)₂–TiO₂/Blood hybrid nanofluid reaches 0.58098, showcasing a 15.29% enhancement over the conventional fluid. Finally, at  = 5.0, the NFSvalue reaches 0.51504. This is further enhanced to 0.56378 inthe case of HNFS, a 9.47% increase, and 0.59231 inthe case of THNFS, yielding a 15.00% improvement over the conventional nanofluid.

Table 3.

The comparison of distinct kinds of fluids at varying values.

	Conventional (NFS)	% increase	(HNFS)	% increase	(THNFS)	% increase
1.0	0.47215	–	0.51928	9.98	0.54567	15.58
2.0	0.48096	–	0.52840	9.86	0.55487	15.37
3.0	0.49523	–	0.54071	9.19	0.56934	14.95
4.0	0.50389	–	0.55219	9.56	0.58098	15.29
5.0	0.51504	–	0.56378	9.47	0.59231	15.00

Magnetic parameter

As illustrated in Figs. 3 and 4, it can be observed that for both the diverging ciliary flow tube and the converging ciliary flow tube, the fluid velocity noticeably decreases when the magnetic parameter has a stronger effect on the flow for two different geometries. Magnetohydrodynamics (MHD) illustrates the performance of electrically conducting fluids, such as plasmas or fluid metals, in the presence of magnetic fields. In this context, when a magnetic field moves through such a liquid, it produces a force known as the Lorentz force. Magnetic forces in the liquid cause the electrically conducting particles to resist moving. Lorentz force impedes the momentum of the fluid, which brings about a weakening of the flow speed throughout the fluid system. As the Magnetic parameter increases, inertial and viscous forces are overcome by electromagnetic effects, which make the flow much more resistant. Because of this, the speed of the flowing fluid decreases and is slowed down the most near the central section of the boundary layer. This kind of flow is observed in Magneto-hydrodynamics, and controlling the damping effect plays an important role in applications with high-conductivity fluids.

Fig. 3.

Effect of on fluid velocity for the case of CCT.

Fig. 4.

Effect of on fluid velocity for the case of DCT.

Figures 5 and 6 clearly show that the temperature increases noticeably when the magnetic parameter becomes larger for both the diverging ciliary flow tube and the converging ciliary flow tube. This effect can be understood using magnetohydrodynamics (MHD) principles: when a magnetic field is applied to an electrically conducting fluid, it creates an extra force called the Lorentz force. This force acts on the fluid, leading to a rise in temperature. An increase in the magnetic parameter leads to a stronger magnetic field, which causes better interaction between the magnetic field and the traveling charged particles in the fluid. Due to this interaction, Joule heating (also called Ohmic heating) increases, meaning the electrical energy is changed into thermal energy when the moving charged particles experience resistance. The magnetic parameter, when seen physically, represents how the viscous forces in the flow change in comparison to the electromagnetic forces. If this parameter increases, there is a higher loss of energy through the electromagnetic field, and that loss means the system gets hotter. Because of this, temperature increases throughout the entire fluid domain. Also, the Lorentz force leads to a slower-moving fluid which lowers the ability to transfer heat by convection. Thus, heat gathers in the fluid near the edges which makes the temperature distribution rise.

Fig. 5.

Influence of on fluid temperature for the case of CCT.

Fig. 6.

Influence of on fluid temperature for the case of DCT.

Nanoparticle volume fraction

When the proportion of nanoparticles in the mixture increases, the total amount of liquid that flows through both the diverging and converging ciliary flow tubes decreases. This trend is observed for both types of tube geometries, as shown in Figs. 7 and 8. When solid nanoparticles are introduced into a liquid, it results in what is known as a nanofluid. As the proportion of nanoparticles in the fluid gets bigger, the fluid becomes thicker. Due to the greater friction, there is not as much speed in the flow, which spreads the fluid particles more slowly through the whole system. Since the liquid has a greater concentration of nanoparticles, its internal structure becomes more disordered. As a result, it behaves more like a colloidal suspension, rather than displaying only the typical properties of a Newtonian fluid. This alteration brings a bigger force of viscous drag, preventing the different layers from traveling over each other, mainly in regions influenced by shearing. If there are many nanoparticles, they can gather into clusters that lower the fluid flow in certain parts and cause resistance. Combined, these effects cause the velocity of the nanofluid to drop nearer the boundaries and in the central part of the flow due to the difficult transfer of momentum from one component to another because of the additional particles. These figures demonstrate that a larger volume fraction of nanoparticles leads to a lowered velocity distribution directly due to the higher viscosity and flow resistance caused by the added nanoparticles. It occurs because there is more friction, stronger interactions between particles, and more energy being used up in the fluid. Additionally, a greater number of nanoparticles mixed into the fluid leads to a noticeable rise in the fluid’s thermal conductivity. Because the fluid can conduct heat more effectively, heat transfer, viscous dissipation, and fluid motion all become more significant. This combined effect causes the temperature to increase throughout the whole fluid system, as illustrated in Figs. 9 and 10.

Fig. 7.

Effect of on fluid flow for the case of CCT.

Fig. 8.

Impact of on fluid flow for the case of DCT.

Fig. 9.

Influence of on fluid temperature for the case of CCT.

Fig. 10.

Influence of on fluid temperature for the case of DCT.

Jeffrey parameter

Rising values of the Jeffrey parameter exhibit a rising profile of the velocity distribution, as shown in Figs. 11 and 12. This occurs because the Jeffrey fluid model, which takes into account both viscosity and elasticity, goes beyond the traditional Newtonian fluid framework by incorporating the effects of fluid relaxation and retardation. Rising Jeffrey parameter values show that the fluid takes longer to adjust after a change in its properties. A rise in the Jeffrey parameter makes the internal friction of the fluid decrease as its rheology is affected. Because of this modification, the viscosity is lower, and less shell force is needed to maintain the flow. Hence, the fluid speed within the boundary layer is particularly high since shear forces are greatest there. Physically, engineers in biomedical engineering can use the Jeffrey-type equations to model viscoelastic materials like blood and synovial fluid. To avoid injury to delicate living tissues, using Jeffrey value information to alter flow rates can enhance new medical devices interacting with blood and other internal liquids. Also, in the processing of polymers which is full of fluids with viscoelastic features, changing the Jeffrey parameter by adjusting materials or the way the fluid flows helps improve productivity with faster movement and less power used. In the food industry, the model allows precise management of the flow of sauces, gels and pastes for packaging and processing. In summary, as the Jeffrey parameter rises, fluid velocity raises because of the viscoelastic properties of the Jeffrey fluid which help with flow by lowering its viscosity.

Fig. 11.

Effect of on fluid flow for the case of CCT.

Fig. 12.

Impact of on fluid flow for the case of DCT.

Helmholtz–Smoluchowski velocity

Figures 13 and 14 clearly show that the flow velocity decreases due to increasing values of the Helmholtz–Smoluchowski velocity parameter. This is possible because, at the micro- and nanoscale, fluid movement is governed by electrokinetic principles. This is particularly true for electroosmotic flow, where the motion of the fluid is controlled by electric fields acting on charged particles within the fluid. An applied electric field on a charged surface causes fluids in microchannels to reach their highest velocity, which is known as the Helmholtz–Smoluchowski. The way it works physically is through the push by the electrical body force on mobile ions in the EDL, which is balanced out by the fluid’s viscosity. As the velocity of Helmholtz–Smoluchowski increases, it generally means the electroosmotic drive at the edges has grown stronger, which changes how fluid flows within the microchannel. Although increasing the Helmholtz–Smoluchowski velocity can enhance fluid movement near charged particles, it often leads to a reduction in the total flow velocity in many practical situations, especially when other factors such as pressure gradients or magnetohydrodynamic (MHD) forces are also present. Because of this, a stronger electroosmotic force near boundaries can create faster velocity changes and thinner edges, which cause higher shear and energy loss. The increase in temperature observed with higher Helmholtz–Smoluchowski velocity (see Figs. 15 and 16) is primarily caused by more intense electroosmotic flow. This stronger flow leads to greater viscous losses, which generate additional heat within the ion channel. As the fluid moves at a higher speed, there is increased shear and friction, resulting in more heat production. If the system is not able to dissipate this heat efficiently, it accumulates inside the channel.

Fig. 13.

Effect of on fluid flow for the case of CCT.

Fig. 14.

Impact of on fluid flow for the case of DCT.

Fig. 15.

Influence of on the temperature for the case of CCT.

Fig. 16.

Influence of on the temperature for the case of DCT.

Electroosmosis factor parameter

Based on the results revealed in Figs. 17 and 18, the temperature decreases significantly as the electroosmosis factor increases. This pattern of flow commonly occurs when electroosmotic flow is present in fluids moving through microchannels or porous materials. As shown in Figs. 19 and 20, when the electroosmotic velocity parameter increases, the fluid velocity rises as well. This behavior is controlled by electroosmosis, a process where a fluid moves due to an external electric field on the total positive or negative charge in the electric double layer (EDL) near the charged surface. The electroosmotic velocity parameter describes how strong the electrokinetic effects. As this value increases, the electric force acting on the mobile ions becomes stronger. This stronger force draws the fluid molecules closer together, which causes the fluid to move faster within the flow area. The velocity of ions increases because a stronger electric field applies a greater force on the ions as they pass through the edge of the pore. Increasing the electroosmotic velocity in systems where it is important, such as microfluidic channels, significantly improves the total fluid flow. This improvement occurs without relying on mechanical devices or higher pressure.

Fig. 17.

Effect of on fluid flow for the case of CCT.

Fig. 18.

Impact of on fluid flow for the case of DCT.

Fig. 19.

Influence of on temperature for the case of CCT.

Fig. 20.

Influence of on temperature for the case of DCT.

Debye length parameter

Figures 21 and 22 illustrate that as the Debye length increases, the fluid’s velocity becomes lower. This trend is related to the fluid’s electrokinetic characteristics, which are especially significant in micro- and nanoscale systems because electric double layers (EDLs) have a strong influence in these environments. The Debye length shows the thickness of the EDL that exists near electrically charged surfaces because of the electrolyte. It determines how far electrostatic attractions act, which has an immediate impact on how charges impact fluid motions. As the Debye length is small, the EDL gets thin, remains close to the wall, and enables electroosmotic forces to push the fluid well into the channel. Still, as the Debye length enlarges, more of the EDL appears in the channel, adjusts the fluid profile, and results in stronger viscosity that interferes with the flow. As the Debye length is higher, this means there is more room in the channel for the ion-rich layer, which can cause the EDLs to flow more easily.

Fig. 21.

Effect of on fluid flow for the case of CCT.

Fig. 22.

Impact of on fluid flow for the case of DCT.

Conclusions

This study has developed a comprehensive mathematical model to explore the behavior of a Jeffrey blood-based nanofluid embedded with copper (Cu), gold (Au), and titania (TiO₂) ternary-hybrid nanoparticles within diverging and converging ciliary micro-vessels. By incorporating critical physiological forces, including electroosmosis, Lorentz, buoyancy, and heat generation effects as well as ciliary motion, the model addresses a significant gap in understanding the complex interplay of body forces and non-Newtonian rheology in microcirculatory environments.

• The Lorentz-induced resistive heating (Joule heating) elevates the fluid’s temperature distribution.

• Increased elasticity (higher Jeffrey parameter) allows fluid elements to store and release mechanical energy more effectively, elevating overall velocity.

• Raising the Jeffrey parameter enhances transport efficiency in viscoelastic nanofluid systems, with minor impact on thermal profiles through improved convection.

• A stronger Helmholtz–Smoluchowski (electroosmotic slip) velocity at the wall leads to reduced bulk velocity, as the slip layer shifts shear toward the channel interior and dampens central flow.

• The shift in shear distribution induces higher viscous dissipation near the wall, marginally elevating temperature close to boundaries.

• A larger Debye length (thicker electric double layer) weakens the electric field penetration into the fluid core, reducing electroosmotic drive and thereby lowering velocity.

• Adding more nanoparticles increases effective viscosity, heightening resistance to flow and thus reducing velocity.

• Higher nanoparticle loading improves thermal conductivity and Brownian motion, elevating the temperature distribution.

• Reduced convective mixing from diminished flow leads to a slight accumulation of thermal energy, raising the overall temperature profile

• A higher electroosmotic factor intensifies wall–driven slip, boosting the overall velocity distribution in the channel.

Author contributions

A.M.O; U.K: Conceptualization, Methodology, Software, Formal analysis, Validation; Writing—original draft. A.F.I: Writing—original draft, Data curation, Investigation, Visualization, Validation. P.T: Conceptualization, Writing—original draft, Writing—review & editing, Supervision, Resources. G.A.A: Validation, Investigation, Writing—review & editing, Formal analysis; Project administration; Funding acquisition, software.

Funding

This research work received funding from the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R820), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.