An extended model to assess Jeffery–Hamel blood flow through arteries with iron-oxide (Fe₂O₃) nanoparticles and melting effects: Entropy optimization analysis

Nomenclature

v ( r , θ )

radial velocity ( m s − 1 )

V max

maximum velocity ( m s − 1 )

( r , θ , z )

polar coordinates

∇

gradient operator

k nf ( ρ c p ) nf

thermal diffusivity ( m 2 s − 1 )

k nf

thermal conductivity at the free stream ( W m − 1 K − 1 )

( ρ c p ) nf

heat capacitance of the fluid ( J k g − 1 K − 1 )

( ρ c p ) bf

heat capacitance of the nanoparticles ( J k g − 1 K − 1 )

σ

stress tensor

μ

dynamic viscosity ( kg m − 1 s − 1 )

μ ∞ , μ 0

infinite and zero shear rate viscosity ( kg m − 1 s − 1 )

J 1

first kind Rivlin–Erickson tensor

γ ̇

shear rate

λ

material constant

∇ p

pressure differential

n

power index 1 < n < 2

ρ f , ρ P

fluid and nanoparticle density ( kg m − 3 )

μ nf

nanofluid dynamic viscosity

N 1 = = χ 1 v nf α

slip parameter

S ̇ T

dimensional entropy

N gen

dimensionless entropy

T , T w

fluid and wall temperature ( K )

α

channel semi-width

σ r θ

shear stress at the wall

V w

wall velocity ( m s − 1 )

χ 1

slip parameter

φ

volumetric friction of nanoparticles

σ *

Stefan–Boltzmann coefficient ( W m − 2 K − 4 )

k *

mean absorption coefficient

g

gravitational acceleration (9.8 m s − 2 )

γ

air expansion coefficient

δ

latent heat capacity ( J k g − 1 )

C s

heat capacity of the solid surface

C f

skin friction coefficient

Nu

Nusselt number

We = λ 2 V max 2 r 2 α 2

Weissenberg number

Re = α r V m ν f

Reynolds number

Pr = μ f ( C p ) bf κ f

Prandtl number

Rd = 16 σ * T ∞ 3 3 k * ( ρ c p ) f

thermal radiation parameter

Ec = V m 2 T w ( C p ) bf

Eckert number

Gr = g γ T w r 2 ν f V m

Grashof number

Mt = C pf ( T w ) δ + C s ( T w )

melting parameter

1 Introduction

The unprecedented use of nanotechnology in every field, including the environment, medicine, therapeutics, biotechnology, and drug development, has altered the dynamics and expanded the horizons of today’s technological world. Because of the medical studies already described, it is now clear that blockage in arteries is a dangerous and fatal condition, leading researchers to attempt to use nanotechnology to solve the issue. Because nanoparticles can pass through cells, tissues, and organs, researchers are optimistic that nanotechnology may lead to significant advances in the treatment of many disorders. Nanoparticles are used in a wide range of biological processes, including medication administration, biosensing, and imaging. They can be made functional in multilayer channels to make it easier to move materials through various biological tissue layers, improving therapeutic efficacy and minimizing negative effects. Nanoparticles have also improved imaging contrast and function as biosensors to find biological substances. The development of surgical equipment, the usage of X-rays, the killing of cancerous cells, the cure of brain tumors, and other biomedical and health sciences have all benefited significantly from the usage of nanotechnology. Small energy assets have a considerable impact on engineering and industrial products, which is a well-known fact. The thermal characteristics of base liquids are markedly improved by the inclusion of nanoparticles. Choi and Eastman [1] demonstrated in a groundbreaking study on nanofluid that the interaction of nanoparticles enhances the thermal features of the base liquid. The arteries in the body are becoming thinner due to arterial stenoses, which restrict blood flow to the organ structures and can eventually cause mortality. When solid micrometer atoms are added to base fluids, the thermal characteristics and heat system quantity of the fluids are improved compared to the ordinary liquids. To investigate the rheological performance of nanoparticles, Yan et al. [2] conducted an experimental study. Elelamy et al. [3] investigated the arterial flow of a nanofluid through a cardiac valve with the distribution of temperature and slip effects. Shojaie Chahregh and Dinarvand [4] investigated blood-based nanofluid transportation through an artery with the delivery of drugs and the cardiac system in acute respiratory utilization. Shahzadi and Bilal [5] studied the blood flow of nanofluids passing via diverged stenosed arteries when affected by penetration. Through such an arterial section, Abdelsalam et al. [6] provided description of the modifications in blood flow brought on by nanofluid forces using finite difference computation. Although the exact etiology of narrowing arteries is uncertain, its effects on flow characteristics are extensively researched both theoretically and empirically [7,8,9,10]. The physicochemical features of nanoparticles have enabled researchers to fill the gaps and surmount obstacles in a variety of industrial and biological applications. Nanotechnology has attracted a lot of devotion in recent years [11,12,13,14,15,16]. By offering special benefits of imaging, traceability, encapsulation, and drug conjugation, as well as by enhancing drug biodistribution and pharmacokinetics profiles, directing drug delivery to the desired tissues, and having adjuvant and antibacterial properties, nanoparticles assist in overcoming this barrier [17,18].

In the biological sciences, both experimental and theoretical blood flow modeling in constricted arteries is important. Even in developed countries, cardiovascular diseases remain the leading cause of death despite notable advancements and improvements in medical sciences and medicine. Low-density compounds, protein, and cholesterol exacerbate the growth of macrophage white cells, which leads to the chronic inflammatory response identified as atherosclerosis in the arterial walls. As a result, an incision forms on the arterial blood vessels inside the walls. Blood flow decreases or may even stop because of a thrombus, which is created when a squashy plaque bursts. The behavior of blood flow in stenosis arteries and normal arteries varies dramatically. The irregularity of the blood flow and the spontaneous characteristics of the artery play a big role in the malfunctioning of the cardiovascular system. Blood flow is hampered by stenosis, which narrows the blood flow region. Blood flow disorder in the artery can lead to blockage, shear stress, and uneven volumetric flow rate. Mathematical analysis is necessary to solve blood flow problems because it provides comprehensive information on all geometrical and flow features. The binary nature of human blood viscoelasticity has been discovered through experiments. Large shear rates give blood a Newtonian character, whereas moderate shear rates or illness give blood a non-Newtonian character that predominates. Liu et al. [19] computed a blood barge pulsating blood flow. Layek et al. [20] used Cartesian coordinates to simulate the pulsating blood flow. They used blood as a model fluid for their experiment. Shear stress reduces due to the formation of contraction, and rheological fluid is used to predict blood viscoelasticity in the narrowing domain. Sankar and Hemalatha [21] employed perturbation techniques to carry out a thorough analysis of the Herschel–Bulkley fluid (HBF) via a special contraction artery. Abbas et al. [22] examined the flow of HBF through a blood vessel with time-invariant interlaced stenosis. Sharma et al. [23] investigated one-directional MHD blood flow through an artery with a single gentle shrinkage. Sarwar and Hussain [24] deliberated on human blood flow performance in response to stenosis claims. Recent studies proposed broad study inquiries on blood flow utilizing diverse fluid models [25,26].

In liquid form, blood is a connective tissue (see, e.g., Sembulingam and Sembulingam [27]). In the medical field, blood flow using nanoparticles is crucial for drug delivery and cancer treatment. Koriko et al. [28] studied the Carreau nanoliquid flow along the horizontal plate of a paraboloid of rotation. They noticed that minor values of the Deborah number displayed the highest values for surface drag and the rate of heat transfer. The blood temperature increased as the volumetric percentage of carbon nanotubes increased (see Koriko et al. [28]). According to Dinarvand et al. [29], the usage of CuO and Cu hybrid nanoparticles decreased the capillary hemodynamic effect in comparison to pure copper. Recently, Ashraf et al. [30] investigated the peristaltic flow of Casson blood-based nanomaterial incorporating platelet-shaped magnetic nanoparticles using the generalized differential quadrature method.

The term “entropy production” refers to the amount of entropy created by irreversible processes such as fluid drift through flow confrontation, Joule heating, heat flow via resistance to heat, interaction among solid materials, dispersion, and fluid viscosity inside a system. According to the second law of thermodynamics, the total entropy of the entire system does not change during a reversible operation. The fundamental law of thermodynamics states that the amount of energy is always conserved, while the second law indicates that temperature gradients are always propagated. In other words, since the quantity of energy is conserved, energy gradients gradually dissipate, reducing the quantity ability to perform work. An increase in entropy measures the loss of the energy gradient. For reducing the irreversibilty rate in any closed system, Bejan [31,32] focused on the significance of irreversibility in four crucial heat transport pathways. Akbar and Butt [33] studied entropy properties in porous walled arteries. In order to evaluate the entropy formation of blood flow through a stenosed artery by means of a catheter, Mekheimer et al. [34] used entropy hemodynamics. Zhang et al. [35] studied the entropy of blood flow through constricted arteries while suspending ZnO nanoparticles. Zidan et al. [36] used a multi-stenosed artery with a thrombus to conduct an entropy generation analysis. Sharma et al. [37] inspected the influence of hybrid nanoparticles on entropy formation on blood flow through a narrowing artery. Kumawat et al. [38] presented entropy creation using a curved permeable stenosed artery with a chemical reaction and heat source. Sharma et al. [39] scrutinized the entropy production for higher-order exothermic–endothermic chemical processes. Gandhi et al. [40] investigated the impacts of entropy using a temperature-dependent viscosity artery with hybrid nanoparticles.

Flows in convergent–divergent channels are extensively studied due to its importance in numerous applications in aerospace, chemical, industrial, and biomechanical engineering (arteries and veins). Such flows are well known to be unstable [41]. These flow forms describe the exit and inlet of an incompressible Newtonian fluid in a consistently expanding conduit (wedge) with a specific angle among the walls. We consider that the angle among the wedge walls is 2 α ; therefore, we may consider a two-dimensional (2D) sink or source flow as a good example for comprehending canal flow. The Jeffery–Hamel flow is named after Jeffery [42], who discovered it in 1915, and Hamel [43], who identified it simultaneously in 1916. Jeffery’s goal was to find analytical solutions to the velocity equation of a Navier–Stokes fluid. After that, Hamel examined the 2D radial flow of a viscous fluid. He noticed that certain steady flows differ significantly from radial flows. Dean [41] later investigated similar solutions and studied their impact on the stability analysis of radial motion; he also discovered additional innovative solutions for particular divergent flows. Many manufacturing processes rely heavily on flows through converging–diverging channels, flows via diffusers, nozzles, and reducers. Convergent/divergent flows have also been successfully employed to replicate dilute polymer solution flows via porous media. In the polymer sector, flows over converging conduits are also used to advance the mechanical qualities of goods, such as plastic sheets and rods. Another notable potential for this type of flow is molten polymer extrusion via convergent dies. Significant advances in the analysis of nonlinear flows in convergent and divergent conduits can be found in previous studies [44,45,46].

The current investigations demonstrate that several investigators explored the blood flow of several rheological fluids under certain physical circumstances considering different morphological structures of blood vessels [47,48,49,50]. They considered the arteries as a stretchable surface, pipe-like structure, and contracting/expanding channels. However, the transport of blood nanofluid with iron oxide nanoparticles through the segment of the arteries or veins that are converging and divergent in nature is not deliberated so far. The finite segment of the blood vessel is convergent/divergent based on its non-uniform and cylindrical cross section throughout the body. With these objectives in mind, the current investigation analyzes the heat transport and irreversibility mechanism due to energy losses with blood flow in the arterial network, whose structure are long conduit with cylindrical cross section. The Carreau fluid model reveals that the shear thinning behavior limitations under elevated shear stress are considered blood. The heart exerts pressure on the blood to force it through the arteries. The fully developed flow of nanofluid through an arterial wall in the presence of mixed convection, viscous dissipation, radiative heat flux, and melting heat is analyzed. The iron oxide ( F e 2 O 3 ) nanoparticle is considered because of its significant biomedical applications, such as magnetized bioseparation, magnetofection medication, DNA molecule hyperthermia, identification, and targeted medication transport. The fluid viscous dissipation, in conjunction with a radiative heat flux, is used to model the entropy equation for an irreversible procedure using the second law of thermodynamics. The similarity mechanism is used to convert the controlling equations into a dimensionless framework. These modified equations are solved numerically using the fourth- and fifth-order Runge–Kutta (RK) approach together with the shooting technique. The simulated results are used to account for the effects of skin drag coefficient, heat transfer, and entropy analysis. This perspective is particularly valuable in fields such as the treatment of cancer, nano-drug transport in blood vessels, cardiovascular disease, and infusing medication, blood, and drugs into the circulatory system.

2 Description of the physical model

Blood is considered a super-natural fluid that flows through living beings, attracting a great deal of magical and mythological attention. According to the literature, blood has been researched as both a quantitative and qualitative entity since the beginning of time, although our method of study is quantitative.

The mathematical framework contracts through the implementation of several assumptions on the blood flow in the human vascular system.

1. The 2D Navier–Stokes equation and continuity equation for an incompressible non-Newtonian fluid in cylindrical coordinate ( r , θ ) were utilized to develop the mathematical modeling of arterial blood flow: r denotes the radial direction, and θ is the axial flow direction (Figure 1).

2. We evaluate an artery with an arbitrary cross-sectional area (which is comparable to a pipe), and the viscoelastic effect is ignored.

3. Iron oxide ( F e 2 O 3 ) nanoparticles are suspended in blood.

4. In order to inspect the heat transport, the melting properties are taken into account.

5. Blood flow exerts stresses on the channel wall, and because of the strain, the channel expands, and fluid travels in a diverging section at its fastest possible rate.

6. It is presumed that the blood temperature is higher than the arterial wall temperature, i.e., T > T w .

7. We assume that the boundary conditions are established not only by stress at the arterial walls but also by constant pressure differentials at the inlet and outflow of the artery.

Figure 1

Diagrammatic sketch of the flow model.

The mass, momentum, and energy conservation are the balanced equation for the laminar incompressible Carreau fluid in the presence of gravitational forces [51]:

where is the body force due to mixed convection.

3 Rheological equation

Non-Newtonian fluid problems have recently attracted a lot of attention, especially because of their potential medical uses. So far, no one constitutive equation can account for all the characteristics of the blood. However, the Carreau model exhibits the phenomenological behavior of blood-like shear thinning and shear thickening. We consider the generalized non-Newtonian fluid that is steady and adheres to the rheological characteristics of the Carreau fluid model. The temperature and velocity boundary conditions at the wall of the channel were set to the Neumann conditions . The equation for the Cauchy stress tensor in this fluid that follows the rheological model [45,52] is as follows:

where the phenomenological constitutive equations for the Carreau liquid are defined as

Here,

The shear rate γ ̇ is defined as

where J 1 is the second invariant or first Rivlin–Ericksen tensor defined as

We consider μ ∞ ≪ μ 0 ; therefore, μ ∞ = 0 . Thus, we attain the physical situation. Equation (5) can thus be written as

When nanoparticles are suspended in the working liquid, the rest of the viscosity μ 0 of the Carreau liquid is upgraded with the Carreau nanofluid viscosity μ nf , and other physical features turn into physical attributes of nanofluid physical characteristics:

The dimensional L , J 1 and shear rate γ ̇ in view of the velocity field V = v ( r , θ ) are

Eq. (4) in the context of Eqs. (9)–(11) takes the following form:

Thus, Eq. (4) in component notation is as follows:

4 Governing equations

Under the above formulation, the system of equations governing mass, momentum conservation, and energy becomes [53]

Using Eqs. (13)–(15) in Eqs. (17)–(19), we have

5 Boundary conditions

The initial and boundary conditions included outlet, inlet, arterial wall, and temperature and are defined as follows:

Inlet: At the arterial entrance, the blood circulation rate was estimated. The blood velocity at the opening and core region may regulate the blood volume. Figure 1 shows how the model intake was determined.

Outlet: In order to provide a realistic portrayal of the model, we depicted the flow model velocity at the outflow. The outlet, shown in Figure 1, was located on the opposite side of the entry and served as the point at which blood departed to the capillaries.

The topological conditions on the channel symmetry line (i.e., on θ → 0 ) are [54,55,56]

When the artery wall is treated rigidly, the velocity boundary conditions are the typical velocity slip conditions provided by

The flow is considered to be entirely developed and laminar at the inlet; hence, the inlet velocity conditions are thought to have a parabolic velocity profile similar to a long circular cross-sectional channel of arbitrary radius.

And

The constitutive relations for fluid density ρ nf , viscosity, μ nf , heat capacitance ( ρ C p ) np , k bf , and k np are the thermal conductivity of the base liquid and nanoparticles, as displayed in equation (25), respectively. In the above relations, ρ bf and ρ np are the density of the base fluid and nanoparticles, ( C p ) nf and ( C p ) bf are the heat capacitances, φ is the volumetric friction quantity, δ is the latent heat capacity, C s denotes the heat capacity of the solid surface, and γ is the air expansion coefficient. Furthermore, σ * is the Stefan–Boltzmann coefficient, k * is the coefficient of mean absorption, and g is the gravitational acceleration (9.8 m s − 2 ). Furthermore, V m is the maximum velocity at the central track, and θ → 0 and θ → α indicate the central region of the channel and wall surface, respectively.

To admit the similarity solution of the governing equation, the following variables are introduced [54,57]:

The normalized boundary conditions are as follows (Table 1):

6 Entropy assessment

The subsequent equations are utilized to compute the overall entropy generation, and the volume integral symbolizes entropy generation across the entire fluid domain. The following mathematical equations can be used to determine the thermal and frictional entropy production [58]:

where

The total entropy can be optimized by computing the volume integral over the whole fluid domain:

Thus,

In this context, the total entropy generation for the nanofluid can be computed as

In the context of dimensionless variables (equation (26)),

7 Computational algorithm

The nonlinear governing differential equations (27) and (28) are decoded using the RK Fehlberg [59] integration method with boundary constraints (29) and (30) numerically. The RK45 methodology uses the shooting technique and is composed of a number of repeating procedures for estimating the solutions. While computationally integrating ordinary differential equations, the RK approach uses a testing step in the middle of an interval to eliminate lower-order error factors. A series of first-order differential equations are created by combining the boundary restrictions and the guided equations. The following steps are adopted in this computational mechanism [60].

Step I:

where