Influence of thermal jump and inclined magnetic field on peristaltic transport of Jeffrey fluid with silver nanoparticle in the eccentric annulus.

This study investigates the impacts of thermal jump and inclined magnetic field on the peristaltic transport of Jeffrey fluid containing silver nanoparticles in the eccentric annuls under the long wavelength and low Reynolds number assumption. In medical studies, the impact of thermal jumps and slanted magnetic fields on public health is of interest. Peristaltic motion's ability to transmit heat and create a magnetic field has several uses in biomedical and bioengineering. The non-Newtonian Jeffrey fluid with silver nanoparticles is considered in the space between two cylindrical tubes that are eccentrically aligned. The homotopic perturbation method is semi-analytical for modeling and nonlinear partial differential equations (HPM). Analytical solutions for velocity, pressure gradient, and pressure rise were found. To show how physical parameters affect temperature, velocity, concentration, frictional force, and pressure rise of inner and outer tubes were plotted. A comparison of the present method with the exact solution for temperature and nanoparticle concentration profile is shown graphically. The present analysis of analytical solution approaches to the exact solution. The most significant thing in the current investigation is that the Hartmann number and thermophoresis number make the velocity profile decline. Jeffrey fluid parameter and magnetic field angle make the velocity rise. The nanofluid's temperature rises as a result of the thermal jump. In addition, the Jeffrey nanofluid has a higher momentum and temperature than the Jeffrey fluid. This analysis can better evaluate the syringe's injection speed and fluid flow features during cancer treatment, artery blockage removal, and reduced bleeding throughout the surgery.

Introduction

Nowadays, there has been a lot of significance in peristaltic movements of non-Newtonian fluids in the presence of a magnetic field in physiology, notably in the form of a device known as Magnetic Resonance Imaging (MRI). The MRI is widely used to diagnose vascular, brain, and total human body diseases. For example, blood circulation in arteries with arterial disease, such as arterial stenosis or arteriosclerosis, can be improved by utilizing a magnetic field as a blood pump during cardiac surgery. The red blood cells are the predominant biomagnetic component in biomedicine. Therefore, the existence of magnetic fields may alter blood transportation. The rate of blood flow is lowered by increasing resistance to blood flow or decreasing blood pressure. It is observed that when exposed to the magnet field, the flow properties of an electrically conducting fluid are improved. Nanofluid magnetic fields are also widely used in medical applications, including cancer treatment and medicinal delivery. In cancer patients, titanium nanoparticles and nanofluids have been utilized to deliver radiation therapy because of their magnetic properties, which allow them to attach to tumor cells without hurting healthy cells. The recent investigation of MHD [1], [2], [3], [4], [5], [6], [7].

It is a crucial research topic for prospective researchers with the several uses of heat transmission. Heat transfer usage for nuclear reactors, processing materials, energy generation, thermal rocket ablation, etc. The heat transmission resulting from peristaltic motion has a wide range of uses in biomedical and bioengineering and oxygen, hypothermia, and hemodialysis therapy. Blood conveys a significant amount of heat to the various portions of the body when passing through the blood vessels. Heat transmission occurs by one of the four mechanisms on the skin surface convection, radiation, conduction, and evaporation. The energy transfer in biological living systems can be detected in metabolic heat generation, fever, skin burning, blood perfusion, hypothermia, and convectional blood and tissue heat exchanges. The temperature change significantly influences the non-Newtonian blood behavior in several regions of the circulatory system. Fuzhang et al. [8] have analyzed that heat transfer has a high dependence on the Prandtl number. Many scientists have concentrated on thermal jump peristaltic flow [9], [10], [11], [12], [13].

Care should be given to managing the excess heat created in production processes. Several approaches have been endeavored in this fashion. One of these measures is to improve the colloidal suspension of nanoparticles to the coolant's thermal conductivity. Applications in the medical field, biology, energy, and process system engineering have attracted considerable interest in nanofluid dynamics. Nanofluids are specific types of classic fluids that magnify the thermophysical properties of fluids. These fluids also improve heat transmission efficiency, creating more efficient equipment for improved performance. The nanofluid mixture comprises the base fluid and is a homogenous mixture of nanoparticles. Nanofluid is usually a liquid suspension with a diameter of small particles between 1-100 nm. Oxides, carbon nanotubes, metals, and carbides may comprise nanoparticles. The base fluid can include oil, water, and ethylene glycol. Choi [14] pioneered the study of nanoparticles. As Mekheimer et al. [15] studied, mixing blood with synovial fluid, resulting in a mixture, might modify the rheological characteristics of the blood and the mechanical qualities of the stenosis produced. In addition, several additional researchers [16], [17], [18], [19], [20], [21], [22], [23], [24] have worked on the nanoparticles mentioned in the references. The FDA (Food and Drug Administration) has approved coated magnetic nanoparticles in medical processes such as blood pressure regulation, medication, surgery, and alcohol detoxification.

The relationship between peristalsis and silver (Ag) nanoparticles has excellent medical significance, focusing on antibacterial and anti-cancer properties and possible processes such as vaccine adjuvant, biosensor, bone curative therapy, denture application, injury repair, and an antidiabetic agent. Previous investigations on peristaltic flow using Ag nanoparticles have been discussed [25], [26].

Peristalsis is an underlying phenomenon in which the wave's flexible tube/canal walls may be constricted and relaxed. In many physiological and industrial systems, the uses of peristaltic motion may be discovered. The many approaches to real-life include the urinary transfer from the renal to a bladder through a ureter, transmission of the lymph by the lymphatic vessels, foods swallowing by the oesophagus, transportation of the chyme in the gastrointestinal tract, targeted transportation of medicines, etc. The peristalsis research of fluid motion was pioneered in 1969 by Latham [27]. Elmaboud [28] explored how pressure peaks change over time and reach various levels with non-integral frequencies of peristaltic waves. Peristaltic transportation has been a prominent subject of investigation, and various investigators have explored conceptual as well as practical facets of the mechanism [29], [30], [31], [32], [33], [34].

Non-Newtonian materials play a role in various processes, including physiology, technology, and industry. In terms of their rheological properties, these materials are incredibly different. Cosmetics, jelly, ketchup, glue, organic solvents, and honey are examples of these fluids. The majority of physiological fluids are known to be non-Newtonian fluids. Peristaltic flow in Jeffrey fluids is significant in medicine and industry because of its wide range of exertions and mathematics because of its geometry and nonlinear equation solutions. Raju and Devanathan [35] may have been the first to examine this topic. Several models have been studied [36], [37]. The eccentric annulus is a little trickier to work efficiently (without peristaltic). The flow between eccentric cylinders intrigues investigators due to its enormous theoretical and technical relevance. Mekheimera et al. [38] were among the first to investigate peristaltic flow via eccentric cylinders. Many scholars addressed the use of peristaltic flow through an eccentric cylinder, as seen in references [39], [40].

He solved linear, nonlinear, initial value, and boundary value problems using the Homotopy and Perturbation techniques [41], [42], [43]. By integrating the classic Homotopy and Perturbation procedures, the Homotopy Perturbation methodology was created. Most of the time, this HPM leads to a solution series that converges quickly, with few iterations yielding very accurate answers. As a result, He's HPM is comprehensive and capable of solving nonlinear equations.

The topic discussed in the present study effort promises biomedical, engineering, and industrial applications because of all objectives mentioned earlier. Thermal jump and inclined magnetic field on the peristaltic motion of Jeffrey fluid containing Ag nanoparticles in an eccentric cylinder have been utilized for the investigation. We have noticed that many researchers investigated peristaltic flow throughout our literature examination. However, no research has yet been conducted to apply and illustrate the impact of thermal jump and inclined magnetic field with silver nanoparticles in eccentric cylinders. The non-Newtonian Jeffrey fluid with silver nanoparticles has been considered in the space between two eccentric cylindrical tubes. This analysis provides a better evaluation of the injection speed and the fluid flow properties of the syringe. The homotopic perturbation technique (HPM) is a semi-analytical approach for solving partial differential equations and nonlinear equations. After much deliberation, we arrived at analytical solutions for the variables of velocity, pressure gradient, and pressure rise. After that, a graphic representation of the impacts of various physical parameters on temperature, nanoparticle concentration, velocity, pressure rise, and frictional force of the inner and outer tubes were depicted. Also, they compared the analytical solution with the present exact solution.

Mathematical formulation

Contemplate the peristaltic motion of an incompressible nanofluid amid two eccentric cylinders in a three-dimensional motion. The inner tube is stiff, and the sinusoidal wave propagates along the length of the outer tube, as indicated by the flow geometry. The inner tube's radius is ; however, we want to talk about the motion to the outer tube's center. The inner tube's center is now at , , where r and z are coordinates in the pipe's cross-section as indicated in Figs. 1(a) and 1(b). The velocity field is interpreted as depending on the type of flow. Here ε is the inner tube position's eccentricity parameter. Furthermore, we assume that the inner tube's boundary is retained at temperature . In contrast, the outer tube is kept at , and are the nanoparticle concentrations at the inner and outer cylinder walls, respectively.

Figure 1

Geometry of the problem.

The two boundary equations are as follows: The inner tube's boundary given in equation (1), thus specified to order by . (As determined by the cosine method), where () is the parameter controlling the eccentricity of the inner tube location. The outer tube is specified by the equation (2).

For an incompressible non-Newtonian nanofluid, the mass, momentum, energy, and nanoparticle concentration equations have been written as [38], [39], [40], Using modifications as given in equation (9), The boundary criteria are shown in equation (10) [10], [11], [12], [38], [39], [40]. The dimensionless specifications are as follows in equation (10): S is a stress tensor for incompressible non-Newtonian Jeffrey fluid is given in (12), Non-dimensional Jeffrey fluids stress tensor components are provided in (13),

Thermal jump

Experimental findings indicated that a fluid's temperature might not be identical to the surface temperature at its solid-fluid interface. This difference is called a temperature or thermal jump, and the normal thermal flux to the interface can be established and expressed by the equation (14). The coefficient is a function of the surface accommodation coefficient and is determined by the characteristics of the fluid's interaction with the surface. Their expression might arise from considerations in kinetic theory.

By incorporating dimensionless requirements (11), modifications (9) and using the supposition of a low Reynold number and a long wavelength, equations (3) to (8) were simplified to the following form. Therefore (15) to (20) are non-dimensional governing equations. Where .

Similarly (10) reduced to non-dimensional boundaries, and the required conditions are listed below equation (21). The Thermophysical characteristics of nanofluids, including density, thermal conductivity, thermal diffusivity, and heat capacitance, are shown in Table 1. The Thermophysical properties of blood and silver nanoparticles are shown in Table 2.

Table 1

Thermo-Physical properties [16], [17], [18], [19].

Properties	Nanofluid
Density	ρ_nf=(1−Φ)ρ_f_0+Φρ_p
Heat Capacitance	(ρC_p)_nf=(1−Φ)(ρC_p)_f_0+Φ(ρC_p)_p
Thermal diffusivity	α_nf=k_nf(ρC_p)_nf
Thermal conductivity of nanofluid	k_nf=k_f_0(k_p+2k_f_0+2(k_p−k_f_0)Φk_p+2k_f_0−(k_p−k_f_0)Φ)

Table 2

Thermo-Physical properties of base fluid and nanoparticles [25], [26], [29].

Properties	Blood	Ag
ρ (kg m−3)	1063	10500
Cp (J kg−1 K−1)	3594	235
k (W m−1 K−1)	0.492	429
β × 10−5 (k−1)	0.18	1.89

Analytical determination

In order to resolve the aforementioned nonlinear, non-homogenous, and coupled partial differential equations of the second order, we employ the Homotopy Perturbation Technique [41], [42], [43]. For the equations (18)–(20), the deformation equations are manipulated in the following way. We'll employ the linear operator for the sake of convenience. The following definitions can be used to define the initial assumptions for the equations mentioned above that fulfill the boundary conditions, which is expressed by the equation (25). Let us define The (26)–(28) series tends to be converging in most circumstances. On the other hand, the convergence depends upon the equation's nonlinear component being estimated. Using the symbolic program Mathematica 11.0, the series solutions for , , and are determined, and the constants are obtained from the specified boundary conditions.

The dimensionless flow rate is given by (29): The pressure rise over a wavelength is defined as equation (30), The time-averaged flow rate of volume [40] is defined as equation (31),

Results and discussions

The current paper examines the influence of thermal jump and inclined magnetic field on the peristaltic transport of Jeffrey fluid with silver nanoparticles in the eccentric annulus. Utilizing the MATHEMATICA 11.0 program, velocity, nanoparticle concentration, temperature, and pressure gradient results are obtained. This section explains the results, which are graphically depicted using ORIGIN software. The comparison of current work with Meikhemer [38] is detailed in Table 3. The impact of Jeffrey fluid and Jeffrey nanofluid on velocity field and temperature profile are shown in Table 4.

Table 3

Comparison of the velocity field.

r	Mekheimer et al. [38]	Present work
	w(r,z)	w(r,z) for _λ_1=0	w(r,z) for _λ_1=0.01
0.2	0.1	0.1	0.1
0.3	0.1119	0.1119	0.1111
0.4	0.1096	0.1098	0.1118
0.5	0.0995	0.1016	0.1041
0.6	0.0829	0.0841	0.0886
0.7	0.0606	0.0617	0.0658
0.8	0.0329	0.0339	0.0361
0.9	0	0	0

Table 4

Velocity field and temperature profile for Jeffrey fluid and Jeffrey Nanofluid.

R	w(r,z)		Θ(r,z)
	Jeffrey fluid	Jeffrey Nanofluid	Jeffrey fluid	Jeffrey Nanofluid
		(Jeffrey fluid + Ag nanoparticles)		(Jeffrey fluid + Ag nanoparticles)
0.2	0.1	0.1	0	0
0.4	0.173518	0.183531	0.400718	0.404861
0.6	0.188046	0.208195	0.689296	0.695457
0.8	0.155618	0.181347	0.885201	0.891253
1	0.0884418	0.109512	0.999964	1.00378
1.2	0	0	1.04101	1.04047

Velocity field

Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 show the velocity field for different parameters. Fig. 2 shows a comparison graph of the results obtained in the current study with the values of Mekheimer et al. [38]. The impact of and on velocity are shown in Fig. 3. It is observed that an increase in increases velocity, but reverse action is observed for . The Brownian motion parameter is inversely proportional to the viscosity of the fluid. Hence, the viscosity of the fluid declines for higher values of ; therefore, it enhances the velocity profile. Whereas the thermophoresis force is a ratio of nanoparticle diffusion to nanofluid momentum diffusion, therefore the opposite behavior can be seen for . From Fig. 4, we can observe that the increase in nanoparticle volume fraction Φ and Jeffrey fluid parameter elevates the velocity. The Jeffrey fluid parameter indicates the ratio of relaxation to retardation time. The velocity profile grows as increases; this is achievable only when the relaxation time increases and the retardation time decreases. Fig. 5 depicts that a rise in Hartmann number (H) declines the velocity field, but reverse action is observed for an inclined angle of magnetic field (α) in the same graph, i.e., the velocity enhances by increasing the inclination angle of the magnetic field (α). The Lorentz force is a resistive force generated by the magnetic field that operates in the opposite direction of the flow, producing flow retardation. The fluid velocity decreases as a result of this. This force tends to slow down blood movement. In the same figure, increasing the inclination angle of the magnetic field (α) improves the velocity profile; this is due to an increase in the buoyancy effect. Fig. 6 depicts that the velocity field declines by increasing thermal Grashoff number (), i.e., the flow is slowed over the tube diameter. Thermal buoyancy decelerates viscous flow since it prevents momentum formation and the tube from propelling. Hence enhances the velocity profile. We can observe that velocity is enhanced by increasing nanoparticle Grashoff number () in the same figure. The fluid viscosity decreases due to an improvement, leading to an increase in velocity. Fig. 7 depicts that the velocity field enhances by increasing the velocity of the inner tube (V), and a similar phenomenon is observed for amplitude ratio (). From Fig. 8, we found that Jeffrey nanofluid will have more incredible speed than the Jeffrey fluid.

Figure 2

Comparison of velocity with Mekheimer et al. [38] and present work when δ0 = 0.1, ε = 0.1, θ = 0, z = 0, ϕ = 0.1, t = 0.5, V = 0.1, .

Figure 3

Impact of parameters on velocity field when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, , , β⁎ = 0.01. Gr = 0.5, Gr = 0.5, Φ = 0.05, H = 0.5, , ϕ = 0.1, λ1 = 0.1, and V = 0.1.

Figure 4

Impact of parameters on velocity field when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, , , β⁎ = 0.01. N = 0.5, N = 0.5, Gr = 0.5, Gr = 0.5, H = 0.5, , ϕ = 0.1, and V = 0.1.

Figure 5

Impact of parameters on velocity field when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, , , β⁎ = 0.01. N = 0.5, N = 0.5, Gr = 0.5, Gr = 0.5, Φ = 0.05, λ1 = 0.1, ϕ = 0.1, and V = 0.1.

Figure 6

Impact of parameters on velocity field when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, , , β⁎ = 0.01. N = 0.5, N = 0.5, Φ = 0.05, H = 0.5, , ϕ = 0.1, λ1 = 0.1, and V = 0.1.

Figure 7

Impact of parameters on velocity field when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, , , β⁎ = 0.01. N = 0.5, N = 0.5, Φ = 0.05, H = 0.5, , λ1 = 0.1, Gr = 0.5, and Gr = 0.5.

Figure 8

Comparison of velocity with Jeffrey fluid and Jeffrey nanofluid.

Temperature profile

The behavior of several factors on the distribution of temperature (Θ) is described in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14. Fig. 9 shows the impact of thermal jump factor () enhances the heat transfer. This is because the inner tube has been assumed to have a constant heat flux. As the temperature jump coefficient rises, similar to an increase in the thermal contact resistance between the solid and fluid phases, a higher temperature is required to maintain the same heat flux through the system. Figure 10, Figure 11, Figure 12, Figure 13 depict that temperature profile enhances by increasing , , Φ and Pr. The increase of the Brownian motion parameter causes random movement of fluid particles that generate more heat, so the system shows the enhancement of the temperature in Fig. 10. In Fig. 11, the temperature profile of the system enhances for the thermophoresis parameter . This is due to the fact that when the fluid particles were transferred from the cold surface to the warm surface. Fig. 12 depicts that the nanoparticle volume fraction Φ enhances the temperature profile. It's important to note that the volume fraction of solid nanoparticles in a fluid is related to the fluid's thermal conductivity. The thermal conductivity of the nanofluid is proportional to the value of the solid nanoparticle fraction. The ratio of momentum diffusivity to thermal conductivity is referred to as Prandtl number Pr. It is true that increasing Prandtl numbers improves the thermal field of the system. As a result, the heat transmission is improved, which is shown in Fig. 13. Compared to Jeffrey fluid, we can see that Jeffrey nanofluid has a higher heat transmission capability, which is depicted in Fig. 14.

Figure 9

Impact of parameters on temperature profile when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.1. N = 0.5, N = 0.5, Φ = 0.01, and .

Figure 10

Impact of parameters on temperature profile when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.1. β⁎ = 0.01, N = 0.5, Φ = 0.01, and .

Figure 11

Impact of parameters on temperature profile when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.1. β⁎ = 0.01, N = 0.5, Φ = 0.01, and .

Figure 12

Impact of parameters on temperature profile when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.1. β⁎ = 0.01, N = 0.5, N = 0.5, and .

Figure 13

Impact of parameters on temperature profile when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.1. β⁎ = 0.01, N = 0.5, N = 0.5, and Φ = 0.01.

Figure 14

Impact of parameters on temperature profile when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.1.

Nanoparticle volume fraction

Figure 15, Figure 16 depict the impacts of the Brownian motion parameter and thermophoresis parameter . Fig. 15 shows that nanoparticle volume fraction decreases with an increase in . Nanoparticles begin to move away from the boundary into the fluid as their random motion increases, resulting in a drop in the concentration of nanoparticles along the wall, as depicted in Fig. 15. But we can observe the opposite action for in Fig. 16. Thermophoresis operates by warming the wall at low Prandtl and Lewis numbers. As a result, we may deduce that enhancing the thermophoresis parameter lowered the amount of heat and mass exchange.

Figure 15

Impact of parameters on nanoparticle concentration profile when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.1, β⁎ = 0.01, Φ = 0.01, and . N = 0.5.

Figure 16

Impact of parameters on nanoparticle concentration profile when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.1, β⁎ = 0.01, Φ = 0.01, and . N = 0.5.

Pressure rise

Figure 17, Figure 18, Figure 19 depict the impacts of the Brownian motion parameter , thermophoresis parameter , Jeffrey fluid parameter , the velocity of the inner tube V, and radius of the inner tube . Fig. 1 depicts that the pressure rise enhances by increasing . Since the pressure is directly proportional to the density of the fluid, therefore pressure enhances for rising values of . In Fig. 18, we have noticed the opposite phenomenon for , i.e., pressure rise declines by increasing . The temperature gradient plays a vital role in thermophoresis, so the nanoparticle travels from the hot region to the cold region. Hence density of the nanofluid declines; therefore, low pressure is required for the flow of the nanofluid. From Figure 17, Figure 18, we have noticed that the pressure rise enhanced by rising in the retrograde (, ) and peristaltic pumping regions (, ) while decreasing in the augmented pumping region (, ). The space between the inner and outer tube declines for rising the radius of the inner cylinder , i.e., the flow area will decrease. Therefore, more pressure is required for the flow of the nanofluid in retrograde and peristaltic pumping regions, but less pressure is required in the augmented pumping region. enhances by increasing velocity of the inner tube V, but it declines by increasing Jeffrey fluid parameter in retrograde as well as in peristaltic pumping regions, and the opposite phenomenon has been noticed in the augmented pumping region, which is depicted in Fig. 19. The Jeffrey fluid parameter indicates the ratio of relaxation to retardation time. When relaxation time increases and retardation time decreases, then we can observe the decreasing function of pressure in retrograde as well as in peristaltic pumping regions, but the opposite process is noticed in augmented pumping regions.

Figure 17

Impact of parameters on pressure rise when ε = 0.1, θ = 0, z = 0.1, t = 0.3, , β⁎ = 0.01, Gr = 0.5, Gr = 0.5, Φ = 0.01, H = 0.5, , ϕ = 0.1. N = 0.5, λ1 = 0.1, and V = 0.1.

Figure 18

Impact of parameters on pressure rise when ε = 0.1, θ = 0, z = 0.1, t = 0.3, , β⁎ = 0.01, Gr = 0.5, Gr = 0.5, Φ = 0.01, H = 0.5, , ϕ = 0.1. N = 0.5, λ1 = 0.1, and V = 0.1.

Figure 19

Impact of parameters on pressure rise when ε = 0.1, θ = 0, z = 0.1, t = 0.3, , β⁎ = 0.01, Gr = 0.5, Gr = 0.5, Φ = 0.01, H = 0.5, , ϕ = 0.1. N = 0.5, N = 0.5, and δ0 = 0.1.

Pressure gradient

Fig. 20 interprets the impact of thermal Grashoff number () and nanoparticle Grashoff number () on the pressure gradient . It can be noticed that the pressure gradient enhances by increasing , but it decreases by increasing . The impact of Hartmann number (H) and flow rate (Q) on pressure gradient has been shown in Fig. 21. From this figure, it can be noticed that the pressure gradient enhances by increasing H but shows reverse action for Q. The impact of the inclination angle of magnetic field (α) and Jeffrey fluid parameter () on pressure gradient has been shown in Fig. 22. It can be noticed that the pressure gradient rises by increasing but reverse action is observed for α.

Figure 20

Impact of parameters on pressure gradient when ε = 0.1, θ = 0, δ0 = 0.1, t = 0.3, , β⁎ = 0.01, N = 0.5, N = 0.5, Φ = 0.01, V = 0.1, ϕ = 0.1. Q = 0.5, H = 0.5, λ1 = 0.1, and .

Figure 21

Impact of parameters on pressure gradient when ε = 0.1, θ = 0, δ0 = 0.1, t = 0.3, , β⁎ = 0.01, N = 0.5, N = 0.5, Φ = 0.01, V = 0.1, ϕ = 0.1. , λ1 = 0.1, Gr = 0.5, and Gr = 0.5.

Figure 22

Impact of parameters on pressure gradient when ε = 0.1, θ = 0, δ0 = 0.1, t = 0.3, , β⁎ = 0.01, N = 0.5, N = 0.5, Φ = 0.01, V = 0.1, ϕ = 0.1. Q = 0.5, H = 0.5, Gr = 0.5, and Gr = 0.5.

Frictional force

Fig. 23 shows that the Frictional force of inner tube F1 increases by raising the value of , but it declines with the rising radius of the inner tube between , again both the parameters exhibit inverse action between . A similar phenomenon has been noticed for a frictional force of outer tube F2, which is represented in Fig. 25. A rise in correlates to an increase in relaxation time, implying that the particle will take significantly longer to return from the perturbed to the equilibrium state. Therefore, the frictional force increases in the negative volume flow rate, but reverse action is found in the positive volume flow rate since the radius of the inner tube plays an important role in the frictional force. This means a smaller tube radius experiences low friction force in the positive volume flow rate, but reverse action is found in the negative volume flow rate, which is in good agreement with the physical situations. Fig. 24 depicts the impact of Hartmann number (H) and eccentricity parameter (ε) on the frictional force of inner tube F1. It reveals that F1 declines by increasing H, and F1 declines by rising ε between , but reverse action is observed for ε between . A similar phenomenon has been noticed in Fig. 26 for Hartmann number (H) and eccentricity parameter (ε) on the frictional force of outer tube F2.

Figure 23

Impact of parameters on frictional forces when θ = 0, t = 0.3, , β⁎ = 0.01, N = 0.5, N = 0.5, Φ = 0.01, V = 0.1, ϕ = 0.1, Gr = 0.5, Gr = 0.5, and . ε = 0.1, H = 0.5.

Figure 25

Impact of parameters on frictional forces when θ = 0, t = 0.3, , β⁎ = 0.01, N = 0.5, N = 0.5, Φ = 0.01, V = 0.1, ϕ = 0.1, Gr = 0.5, Gr = 0.5, and . ε = 0.1, H = 0.5.

Figure 24

Impact of parameters on frictional forces when θ = 0, t = 0.3, , β⁎ = 0.01, N = 0.5, N = 0.5, Φ = 0.01, V = 0.1, ϕ = 0.1, Gr = 0.5, Gr = 0.5, and . δ0 = 0.1, λ1 = 0.1.

Figure 26

Impact of parameters on frictional forces when θ = 0, t = 0.3, , β⁎ = 0.01, N = 0.5, N = 0.5, Φ = 0.01, V = 0.1, ϕ = 0.1, Gr = 0.5, Gr = 0.5, and . δ0 = 0.1, λ1 = 0.1.

Trapping

In peristaltic transport, trapping is an essential phenomenon to be aware of. Trapping is the process of closed streamlines forming an internally circulating bolus of fluid, which is pushed forward by the peristaltic wave. Thrombus development in the blood and food bolus movement in the gastrointestinal system may be caused by physical events. From Figs. 27(a-b), we have observed that the bolus's size is enhanced by the magnetic field's rising inclination angle α. Figs. 28(a-b) depict that the size of the bolus shrinks, and the number of boluses increases for higher values of H.

Figure 27

(a and b): Streamlines for different values of α.

Figure 28

(a and b): Streamlines for different values of H.

Validation of results

Ebaid [4], [7], [23], [33], [34], [43] assessed that the Homotopy Perturbation technique has to be revised appropriately when applied to peristaltic problems; otherwise, a forged physical sight is required. Additional terms must be added to the series solution in order to get a more accurate answer. As a result of the aforementioned findings, the current research has been thoroughly scrutinized. The series solutions for temperature and nanoparticle concentration are obtained using the symbolic software Mathematica 11.0 for more iterations. The comparison of the present exact solution and analytical solution (HPM) of temperature and nanoparticle concentration is depicted in Figure 29, Figure 30, respectively. We can conclude from these figures that the analytical solution obtained by HPM approaches the exact solution.

Figure 29

Comparison of the exact solution and HPM for temperature when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.2, Φ = 0.01, β⁎ = 0, N = 0.5, N = 0.5 and .

Figure 30

Comparison of the exact solution and HPM for nanoparticle concentration when δ0 = 0.1, ε = 0.1, θ = 0, z = 0.3, t = 0.3, ϕ = 0.2, Φ = 0.01, β⁎ = 0, N = 0.5, N = 0.5 and .

Conclusion

Analysis of the peristaltic flow of Jeffrey fluid with Ag nanoparticles in the gap between two eccentric tubes is reported in this study. Long-wavelength and low Reynolds number assumptions are used to evaluate the issue. The current study's findings are compared to those published in the scientific literature. The following are the most critical findings from the study mentioned above.

It is observed that the velocity profile falls by enhancing H; this is due to the existence of Lorentz force, but inverse action is observed for α.

It is noticed that temperature is enhanced by increasing thermal jump , and Prandtl number Pr.

It is noticed that in peristaltic flow, the pressure rise enhances for , V but it declines for and , also observed that pressure rise enhances by rising in the retrograde pumping region and peristaltic pumping region while decreasing in the augmented pumping region.

The Jeffrey nanofluid achieves higher velocity and temperatures than the Jeffrey fluid.

This analysis can better evaluate the syringe's injection speed and fluid flow features during cancer treatment, artery blockage removal, and reduced bleeding throughout the surgery.

Declarations

Author contribution statement

Asha S. Kotnurkar, Vijaylaxmi T. Talawar: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Wrote the paper.

Funding statement

Vijaylaxmi Talawar was supported by the Government of India Ministry of Tribal Affairs under the NFST scheme [201920-NFST-KAR-02350].

Data availability statement

No data was used for the research described in the article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.