Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel

Seelam Ravikumar; Muhammad Ijaz Khan; Salman A. AlQahtani; Sayed M. Eldin

doi:10.1515/phys-2022-0258

Article Open Access

Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel

Seelam Ravikumar , Muhammad Ijaz Khan , Salman A. AlQahtani and Sayed M. Eldin

Published/Copyright: July 3, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

Using mathematical modeling and computational analysis, this study aims to examine the peristaltic blood flow of a non-Newtonian material in a tapered channel with radiative heat flux and response mechanisms. By utilizing a long-wavelength approximation, ignoring the wave number, and performing under conditions of low Reynolds number, closed form solutions for the velocity, temperature, and concentration fields are achieved. Several governing parameters and their effects on the system were analyzed, and relevant diagrams were provided. Increasing the Biot number, Jeffrey material, and thermal radiation parameter of the heat and mass transfer mechanism increases the velocity profile. When the heat source/sink parameter and the heat transfer Biot number increase, the temperature profile improves. The resultant concentration distributions are enhanced when mass transfer Biot number, heat radiation, and chemical processes are all raised. We observe that the pressure rate decreases in all three pumping zones when the heat transfer Grashof number and heat transfer Biot number rise. This is because the pressure rate is affected by the Grashof number and Biot number of heat transmission. The increase in thermal radiation parameter and heat transfer Biot number results in a slower rate of heat transfer than when Prandtl number and heat source/sink parameter increases. When the Soret number, Schmidt number, Biot number, and heat source/sink parameter are all raised, the mass transfer coefficient also rises. This rate, however, decreases as the heat radiation and chemical reaction parameters rise. The findings presented in this study have interesting implications for other aspects of human physiology. The preponderance of organs are permeable. Furthermore, fluids render the location of natural boundaries uncertain. The presented mathematical model can be used to derive predictions about the behavior of various systems. For the study of cancer treatment in biological systems, a mathematical model that includes nanoparticles, viscosity dissipation, and rotation holds much promise. Model development incorporated Soret–Dufour effects and thermal analysis of the digestive system.

Keywords: rate of chemical reaction; tapered channel; heat transfer; Biot number of heat transmission; thermal radiation effect; Biot number of mass transfer; mass transfer

Nomenclature

b: channel width (m)
Bh: heat transfer Biot number
Bm: mass transfer Biot number
c: wave speed (m/s)
C p: specific heat at constant pressure
C ¯: concentration of the fluid
D m: coefficient of mass diffusivity (m²/s)
d: wave amplitude (m)
Ec: Eckert number
Gc: solute Grashof number
Gr: Grashof number
K T: thermal-diffusion ratio
K ₂: reaction rate constant
p: pressure
Pr: Prandtl number
Q ₀: heat generation coefficient
Re: Reynolds number
Rn: thermal radiation parameter (W/m² K)
Sr: Soret number
Sc: Schmidt number
S: chemical reaction parameter
T ¯ m: mean temperature (K)
T ¯: temperature of the fluid
u ¯: velocity along x ¯ direction
ν: heat source/sink parameter
v ¯: velocity along y ¯ direction
λ: wavelength
μ: viscosity of the fluid (kg/m/s)
σ: electrical conductivity of the fluid
ρ: density of the fluid (kg/m³)
ε: amplitude ratio
δ: wave number
λ 1: Jeffery fluid parameter
θ: temperature distribution
ϕ: concentration distribution

1 Introduction

Over the last several decades, a wide range of researchers has studied blood flow via diverse channels. Biomathematics literature has produced a vast array of applications in pharmaceuticals and biology. The medicinal and physiological uses of peristalsis are immense. In the field of medicine known as physiology, this process refers to the passage of food through the body, from the initial propulsion of the food bolus through the digestive tract to the final passage of nutrients through the bowel. Latham [1] has explored several theoretical and experimental approaches to examine the peristaltic action in numerous situations. Pandey and Chaube [2] worked on peristaltic transport in flow of visco-elastic material with non-uniform cross section in a tube as shown in Figure 1.

Figure 1

The gastrointestinal system [3].

Several scientists investigated non-Newtonian fluids because they have numerous technical and industrial uses. These fluids exhibit a nonlinear connection between strain and, hence, strain rate. The Jeffrey framework is the most intuitive model of the non-Newtonian fluid. It usually signifies that fluids in the body are circulating via tubes and ducts. In addition, it may be considered a blood model. In addition to temperature gradients, concentration gradients may also result in energy flows and mass fluxes can be generated by the well-known thermal-diffusion effect. We analyze the phenomena of diffusion-thermo effects (Dufour effect) within the context of heat and mass transfer issues. Making an effort may reveal a few relevant research studies [4–11].

Srinivas and Kothandapani [12] looked at how heat moves through an asymmetric channel while studying peristaltic flow. Making an effort may reveal a few relevant research studies [13–19] and some research on heat transfer analysis [20–22]. Peristaltic convective conditions across a channel were studied by Abbasi et al. [23,24]. Ahmed and Tamara [25] investigated the role of heat and mass transfer in Jeffrey transport in the body. Addulhadi et al. [26] presented magnetized peristaltic transport of Jeffrey liquid with heat/mass phenomenon in a curved channel. The mechanism of peristalsis subjected to convective conditions has also been exploited [27–29], along with some others, e.g., classical thermodynamics [30], microwave imaging [31], and modeling vapor–liquid equilibrium [32]. Hamed et al. [33] worked on combined effects of slip and convective conditions in flow of peristaltic nanofluid in an asymmetric channel. Yasmin et al. [34–42] have also studied the peristaltic movement of a non-Newtonian fluid through a conduit in recent years.

No attempt has been made by the authors to analyze the current research, which focused on the modeling and computational analysis of a non-Newtonian fluid over a tapered conduit with radiative heat flux and response mechanisms with peristalsis. Using approximations with a low Reynolds number and long wavelength, the governing equations were able to be simplified. There is a discussion on the significance of relevant flow parameters being included in the flow modeling.

This research is important because it is the first step in determining how blood flow is affected by the presence of fatty plaques of cholesterol and/or arterial blockage in a sick condition. Applications of the current research are not limited to the polymer sector, biomedical engineering (such as magnetic resonance imaging and electrocardiography), clinical diagnostics, and surgery. As well as this research addresses the issue of thermal radiation in blood flow, the findings presented here should be useful in understanding and controlling the therapeutic use of hyperthermia. Furthermore, the results of the present study will be of great use in confirming the results of future identical experimental studies and more complex theoretical investigations.

The article is formatted as follows: In Section 2, the problem formulation is presented. In Section 3, the assumption of an extended wavelength and a low Reynolds number resolves the resulting nonlinear problem. In Section 4, the impacts of essential physical parameters are illustrated via graphs. Section 5 concludes the main findings.

2 Formulation for the problem

Here we explore the effect of sinusoidal wave trains traveling down the transport of a viscous incompressible fluid along the boundaries of a two-dimensional duct at a consistent speed c. Deformations of the walls are defined as follows [9]:

(1) Y = H 1 ¯ = − b − m ′ X ̄ − d sin 2 π λ ( X ̄ − c t ¯ ) + ϕ ,

(2) Y = H 2 ¯ = b + m ′ X ̄ + d sin 2 π λ ( X ̄ − c t ¯ ) ,

where b, d, m ′ , Φ, t, and λ are channel width, wave amplitude, non-uniform parameter, phase variance, time, and wavelength (Figure 2).

Figure 2

Geometry of the problem.

The flow field could be described using the following set of equations [43]:

(3) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(4) ρ ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = − ∂ p ∂ x + ∂ ∂ x ( S xx ) + ∂ ∂ y ( S xy ) + ρ g B t ( T − T 0 ) + ρ g B c ( C − C 0 ) ,

(5) ρ ∂ v ∂ t + u ∂ v ∂ x + v ∂ v ∂ y = − ∂ p ∂ y + ∂ ∂ x ( S xy ) + ∂ ∂ y ( S yy ) ,

(6) ρ C p ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = k ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 + Q 0 − ∂ q r ∂ y ,

(7) ∂ C ∂ t + u ∂ C ∂ x + v ∂ C ∂ y = D m ∂ 2 C ∂ x 2 + ∂ 2 C ∂ y 2 + D m K T T m ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 − k 2 ( C − C 0 ) ,

where

S xx = 2 μ 1 + λ 1 1 + λ 2 u ∂ ∂ x + v ∂ ∂ y ∂ u ∂ x , S xy = μ 1 + λ 1 1 + λ 2 u ∂ ∂ x + v ∂ ∂ y ∂ u ∂ y + ∂ v ∂ x ,

S yy = 2 μ 1 + λ 1 1 + λ 2 u ∂ ∂ x + v ∂ ∂ y ∂ u ∂ y ,

where u ¯ , v ¯ , p , ρ , μ , B t , B c , T , C , D m , K T , T m , Q 0 , C p , k 2 are the components of velocity along x ¯ -and y ¯ directions, pressure, density of the fluid, viscosity coefficient, electrical, thermal expansion coefficient, concentration expansion coefficient, temperature, concentration, mass diffusivity, thermal-diffusion ratio, mean temperature, constant heat addition/absorption, specific heat at constant pressure, and chemical reaction constant.

Add a wave frame (x, y) that moves away from the fixed frame (X, Y) at a speed of c by using the transformation

(8) x = X − c t , y = Y .

Non-dimensional quantities:

(9) x ¯ = x λ , y ¯ = y b , t ¯ = ct λ , u ¯ = u c , ε = d b , v ¯ = v c δ , h 1 = H 1 b , h 2 = H 2 b , p ¯ = b 2 ρ c λ μ , δ = b λ , Re = ρ cb μ , Pr = μ c p k , Ec = c 2 c p ( T 1 − T 0 ) , ν = Q 0 b 2 μ C p ( T 1 − T 0 ) , θ = T − T 0 T 1 − T 0 , Sc = μ D m ρ , S = K 2 ρ d 2 μ , Sr = D m ρ k T ( T 1 − T 0 ) μ T m ( C 1 − C 0 ) , ϕ = C − C 0 C 1 − C 0 , Rn = 16 σ * T 0 3 b 2 3 k * μ C p , Gr = ρ g B t b 2 ( T 1 − T 0 ) c μ , Gc = ρ g B c b 2 ( C 1 − C 0 ) c μ ,

where Pr, Gc , ν , Re, Gr , Rn , Sc , Sr , Ec , and S are Prandtl number, solute Grashof number, heat source/sink parameter, Reynolds number, Grashof number, thermal radiation factor, Schmidt number, Soret number, Eckert number, and chemical reaction factor.

3 Solution of the problem

Removing the bars from Eqs. (3)–(7) allows them to transform to the next non-dimensional form using non-dimensional quantities.

(10) δ ∂ u ∂ x + ∂ v ∂ y = 0

(11) Re δ ∂ u ∂ t + u ∂ u ∂ x + v ∂ v ∂ y = − ∂ p ∂ x + δ ∂ ∂ x ( S xx ) + ∂ ∂ y ( S xy ) + Gr θ + Gc ϕ ,

(12) Re δ 3 ∂ v ∂ t + u ∂ v ∂ x + v ∂ v ∂ y = − ∂ p ∂ y + δ 2 ∂ ∂ x ( S xy ) + δ ∂ ∂ y ( S yy ) ,

(13) Re δ ∂ θ ∂ t + u ∂ θ ∂ x + v ∂ θ ∂ y = 1 Pr δ 2 ∂ θ 2 ∂ x 2 + ∂ θ 2 ∂ y 2 + ν + Rn ∂ θ 2 ∂ y 2 ,

(14) Re δ ∂ ϕ ∂ t + u ∂ ϕ ∂ x + v ∂ ϕ ∂ y = 1 Sc + δ 2 ∂ ϕ 2 ∂ x 2 + ∂ ϕ 2 ∂ y 2 + Sr δ 2 ∂ θ 2 ∂ x 2 + ∂ θ 2 ∂ y 2 − k 2 ( C − C 0 ) ,

where

S xx = 2 δ 1 + λ 1 1 + λ 2 δ c d u ∂ ∂ x + v ∂ ∂ y ∂ u ∂ x , S xy = 1 1 + λ 1 1 + λ 2 δ c d u ∂ ∂ x + v ∂ ∂ y ∂ u ∂ x + δ 2 ∂ v ∂ x , S yy = 2 1 + λ 1 1 + λ 2 δ c d u ∂ ∂ x + v ∂ ∂ y ∂ v ∂ y

Using a long-wavelength approximation while disregarding the wave number and operating under conditions of low Reynolds number for Eqs. (10)–(14), we have

(15) ∂ 2 u ∂ y 2 = ( 1 + λ 1 ) ∂ p ∂ x − ( 1 + λ 1 ) Gr θ − Gc ϕ ,

(16) ∂ p ∂ y = 0 ,

(17) ( 1 + Pr Rn ) ∂ 2 θ ∂ y 2 + Pr ν = 0 ,

(18) ∂ 2 ϕ ∂ y 2 + ScSr ∂ 2 θ ∂ y 2 − S Sc ϕ = 0 .

The dimensionless boundary conditions

(19) u = − 1 at y = h 1 = − 1 − k 1 x − ε sin [ 2 π ( x − t ) + ϕ ] ,

(20) u = − 1 at y = h 2 = 1 + k 1 x + ε sin [ 2 π ( x − t ) ] ,

(21) ∂ θ ∂ y − B h θ = − B h , ∂ ϕ ∂ y − B m θ = − B m at y = h 1 = − 1 − k 1 x − ε sin [ 2 π ( x − t ) + ϕ ] ,

(22) θ = 0 , ϕ = 0 at y = h 2 = 1 + k 1 x + ε sin [ 2 π ( x − t ) ] ,

where ε = d b is the amplitude ratio.

Solving Eqs. (15)–(18) using the boundary conditions (19)–(22), we get

(23) θ = G + Fy + m 1 y 2 2 ,

(24) ϕ = K sin h [ α y ] + J cosh [ α y ] + m 2 ,

(25) u = R + Qy + m 11 p y 2 − ( m 12 + m 17 ) y 2 − m 13 y 3 − m 14 y 4 − m 15 sinh [ α y ] − m 16 cosh [ α y ] ,

where

α = S Sc , m 1 = − Pr ν 1 + Pr Rn , m 2 = Sr m 1 S , m 3 = α cosh [ α h 1 ] − B m sinh [ α h 1 ] , m 4 = α sin h [ α h 1 ] − B m cos h [ α h 1 ] , m 5 = ( 1 + λ 1 ) Gr G , m 6 = ( 1 + λ 1 ) Gr F , m 7 = ( 1 + λ 1 ) G r m 1 2 , m 8 = ( 1 + λ 1 ) Gc K , m 9 = ( 1 + λ 1 ) Gc J , m 10 = ( 1 + λ 1 ) Gc m 2 , m 11 = ( 1 + λ 1 ) p 2 , m 12 = m 5 2 , m 13 = m 6 6 , m 14 = m 7 12 , m 15 = m 8 α 2 , m 16 = m 9 α 2 , m 17 = m 10 2 , m 18 = − m 11 ( h 1 + h 2 ) , m 19 = − ( m 12 + m 17 ) ( h 2 2 − h 1 2 ) − m 13 ( h 2 3 − h 1 3 ) − m 14 ( h 2 4 − h 1 4 ) ( h 1 − h 2 ) + − m 15 ( sinh [ α h 2 ] − sinh [ α h 1 ] ) − m 15 ( cosh [ α h 2 ] − cosh [ α h 1 ] ) ( h 1 − h 2 ) , m 20 = ( − 1 − Q h 1 + ( m 12 + m 17 ) h 1 2 + m 13 h 1 3 + m 14 h 1 4 + m 15 sinh [ α h 1 ] ) + ( m 16 cosh [ α h 1 ] ) , G = − F h 2 − m 1 h 2 2 2 F = − B h − m 1 h 1 − B h m 1 2 ( h 2 2 − h 1 2 ) ( 1 − B h h 1 ) + B h h 2 K = − J cosh [ α h 2 ] − m 2 sinh [ α h 2 ] , J = − B m sinh [ α h 2 ] + m 2 ( B m sinh [ α h 2 ] + m 3 ) m 4 sinh [ α h 2 ] − m 3 cosh [ α h 2 ] , Q = m 18 p + m 19 , R = m 20 − m 21 p − Q h 1 .

In the wave model, the volumetric flow rate is represented as follows:

(26) q = ∫ h 2 h 1 udy = ∫ h 2 h 1 ( R + Qy + m 11 py 2 − ( m 12 + m 17 ) y 2 − m 13 y 3 − m 14 y 4 − m 15 sinh [ α y ] − m 16 cosh [ α y ] ) d y = p m 22 + m 23 ,

where

m 22 = ( − m 21 − m 18 h 1 ) ( h 1 − h 2 ) + m 18 2 ( h 1 2 − h 2 2 ) + m 11 3 ( h 1 3 − h 2 3 ) , m 23 = m 20 ( h 1 − h 2 ) − m 19 h 1 ( h 1 − h 2 ) + m 19 2 ( h 1 2 − h 1 2 ) − ( m 12 + m 17 ) 3 ( h 1 3 − h 2 3 ) + − m 13 4 ( h 1 4 − h 2 4 ) − m 14 5 ( h 1 5 − h 2 5 ) − m 15 α ( cosh [ α h 1 ] − cosh [ α h 2 ] ) + − m 16 α ( sinh [ α h 1 ] − sinh [ α h 2 ] ) .

The expression for the pressure gradient derived from Eq. (26) is as follows:

(27) d p d x = q − m 23 m 22 .

In the laboratory frame, the instantaneous flux Q (x, t) is given by

(28) Q = ∫ h 2 h 1 ( u + 1 ) d y = q − h .

The typical volumetric flow rate of peristaltic waves is as follows:

(29) Q ̄ = 1 T ∫ 0 T Q d t = q + 1 + d .

The pressure gradient, which may be derived from Eqs. (27) and (29), is given by

(30) d p d x = ( Q ¯ − 1 − d ) − m 23 m 22 .

4 Discussion of the problem

In this subsection, we look at how different emergent factors affect the patterns of velocity, temperature, pressure rise, and concentration. When calculations are performed, the following settings are used for the default parameters: ε = 0.2 , ϕ = π 6 , x = 0.6 , t = 0.4 , k 1 = 0.1 , Pr = 3 , ν = 0.1 , λ 1 = 0.5 , Q ¯ = 2 , d = 1 , Gr = 0.5 , Sr = 2 , Sc = 0.4 , Gc = 1.5 , Bh = 0.5 , Bm = 0.5 , S = 0.5 , and Rn = 0.25 . Mathematica software is used to determine the numerical outcomes.

4.1 Validation of the model

Table 1 displays a comparison of our study results with those of Ravi Rajesh and Rajasekhara Gowda [44]. The approach of Ravi Rajesh and Rajasekhara Gowda is used as a reference point since it is purely analytical. In Table 1, we can observe a comparison between the calculated velocity and the velocity distribution values calculated using Ravi Rajesh and Rajasekhara Gowda’s solution. The calculated results for the Jeffrey parameter λ₁ = 0.1, 0.5, and 1 are found to be in close agreement with Rajesh and Rajasekhara Gowda’s equivalent result, showing that the solution obtained and the analysis offered are accurate

Table 1

Velocity distribution for various values of λ ₁ (comparison of values from present work with that of Rajesh and Rajasekhara Gowda’s work [43])

y value	λ ₁ = 0.1 (Ravi Rajesh work)	λ ₁ = 0.1 (Present work)	λ ₁ = 0.5 (Ravi Rajesh work)	λ ₁ = 0.5 (Present work)	λ ₁ = 1 (Ravi Rajesh work)	λ ₁ = 1 (Present work)
−1.25	−1	−1	−1	−1	−1	−1
−0.75	0.096123	0.096208	0.821013	0.827013	1.55082	1.55782
−0.25	0.562933	0.563033	1.59505	1.60505	2.639708	2.64708
0.25	0.4947075	0.494875	1.48746	1.49146	2.48104	2.48804
0.75	−0.042956	−0.04356	0.588406	0.594067	1.228169	1.23169
1.25	−1	−1	−1	−1	−1	−1

4.2 Velocity distribution

The velocity distribution is plotted as a function of y in Figures 3–8. As seen in the Figure 3, the Soret number has a substantial influence on the velocity. As the values of Sr increase, it is seen that the distribution of velocities improves [44]. The influence of Bh in heat transmission on the velocity is seen in Figure 4. This pattern suggests that when Bh increases, so does velocity. From Figure 5, the velocity distribution improves as the Grashof number (Gr) increases. The influence of Bm on the velocity distribution is seen in Figure 6. Based on the plots, we have observed that a rise in Bm improves velocity profiles. This is due to the fact that buoyancy forces induce a greater velocity distribution [44]. Figures 7 and 8 show how the Jeffrey fluid parameter and thermal radiation affect the distribution of velocities. When λ 1 and Rn values rise, the velocity distribution also rises. Therefore, an increase in λ 1 always causes an increase in elastance, which in turn increases the fluid velocity [45]

Figure 3

Significance of Sr on u.

Figure 4

Significance of Bh on u.

Figure 5

Significance of Gr on u.

Figure 6

Significance of Bm on u.

Figure 7

Significance of Rn on u.

$Figure 8 Significance of λ 1 {\lambda }_{1} on u.$

Figure 8

Significance of λ 1 on u.

4.3 Temperature distribution

Figures 9–12 exhibit how the temperature profile (θ) varies for different factors. The Biot number is associated with the surface convective boundary condition. As shown in Figure 9, as the Biot number increases, the temperature gradient near the surface increases, resulting in an increase in the temperature near the surface and the thermal boundary layer thickness [46]. The temperature distribution is exhibited in Figure 10. The findings show that increasing the heat source/sink parameter value improves the temperature dispersion. Figure 11 shows the significance of Pr on the distribution of temperature. The temperature of a fluid increases in proportion to the Prandtl number. Due to an increase in Prandtl number, the thermal conductivity of the material decreases, making it less effective at conducting heat. Additionally, the rate of heat transmission is accelerated [45]. Figure 12 displays how Rn affects the overall temperature pattern. This plot suggests that increasing Rn values reduces the fluid flow temperature. Here we can make the significant observation that thermal boundary layer thickness decreases as thermal radiation increases.

Figure 9

Significance of Bh on θ.

$Figure 10 Significance of ν \nu on θ.$

Figure 10

Significance of ν on θ.

Figure 11

Significance of Pr on θ.

Figure 12

Significance of Rn on θ.

4.4 Concentration distribution

Figures 13–19 exhibit the consequences of various factors on the concentration profile (Φ). Figures 13 depicts the importance of Bm on the concentration profile. As shown in the concentration profile, the results improve as Bm increases. It happens because mass diffusivity decreases [46]. Even more clearly, Figures 14 and 15 show that increasing Rn and S enhanced the fluid’s concentration distribution. As can be seen in Figures 16–19, the concentration distribution is affected by a diversity of factors, including the Pr, ν, Sr, and Sc. These results demonstrate that the concentration profile flattens out as Pr, ν, Sr, and Sc levels are increased. This is because the increased concentrations of Pr, v, Sr, and Sc reduce the mobility of fluid molecules [35].

Figure 13

Significance of Bm on Φ.

Figure 14

Significance of Rn on Φ.

Figure 15

Significance of S on Φ.

Figure 16

Significance of Pr on Φ.

$Figure 17 Significance of ν \nu on Φ.$

Figure 17

Significance of ν on Φ.

Figure 18

Significance of Sr on Φ.

Figure 19

Significance of Sc on Φ.

4.5 Pressure rise

Figures 20–23 illustrate the association between the pressure increase per wavelength and the volumetric flow rate in the vertically tapered channel for various physical parameter values. Peristaltic transport is widely known to have three pumping zones, which are characterized as the pumping region ∆ p > 0 (peristaltic pumping region Q ̅ > 0 and retrograde pumping region Q ̅ < 0 ), the free pumping region ∆ p = 0 , and the co-pumping region when ∆ p < 0 . Figure 20 displays the effect of λ 1 on the pressure rise. As λ 1 increases, the pressure rate drops in the free and retrograde pumping sectors, whereas it increases in the co-pumping sector. Figure 21 shows how ν affects the value of Δp. Retrograde, free pumping, and co-pumping rates may significantly increase with the increase in values of ν . Figures 22 and 23 depict the effect of Gc and Bh on the pressure rate, respectively. We observe that the pressure rate in three pumping zones is reduced as a result of the prominent impact of external heat flow resistance.

$Figure 20 Significance of λ 1 {\lambda }_{1} on Δ p . \Delta p.$

Figure 20

Significance of λ 1 on Δ p .

$Figure 21 Significance of ν \nu on Δ p \Delta p .$

Figure 21

Significance of ν on Δ p .

$Figure 22 Significance of Gc on Δ p \Delta p .$

Figure 22

Significance of Gc on Δ p .

$Figure 23 Significance of Bh on Δ p \Delta p .$

Figure 23

Significance of Bh on Δ p .

Tables 2 and 3 show the heat/mass transfer coefficient rate at the wall y = h ₁ for dissimilar values of the embedded parameters: ε = 0.2 , ϕ = π 6 , x = 0.6 , t = 0.4 , k 1 = 0.1 , Pr = 3 , ν = 0.1 , Rn = 0.25 , Sr = 2 , S = 0.5 , Sc = 0.4 , Bh = 1 , and Bm = 0.1 . The findings reveal that the heat transfer coefficient decreases as Pr and v increase, whereas it increases with Rn and Bh. Generally, the mass transfer coefficient decreases with the increase in Pr, Sr, Sc, and Bm but increases with the increase in Rn and S.

Table 2

Numerical values of heat/mass transfer rate at the wall h ₁

Pr	ν	S	− θ ′ ( h 1 )	− ϕ ′ ( h 1 )
3	0.1	0.5	0.131717	0.10854
5			0.086229	0.11561
7			0.057282	0.12010
3	0.1		0.131717	0.10854
	0.2		−0.021803	0.13238
	0.3		−0.175325	0.15623
	0.1		0.131717	0.10854
			0.177774	0.10139
			0.202573	0.09753
		0.5		0.10854
		1		0.10546
		1.5		0.10359

Table 3

Numerical values of heat/mass transfer rate at the wall h ₁

Sr	Sc	Bm	Rn	Bh	− θ ′ ( h 1 )	− ϕ ′ ( h 1 )
2	0.4	0.1	0.25	1		0.10854
3						0.12046
4						0.13238
2	0.4					0.10854
	0.6					0.11710
	0.8					0.12362
	0.4	0.1				0.10854
		0.5				0.33668
		1				0.45665
		0.1	0.25		0.131717	0.10854
			0.5		0.177774	0.10139
			0.75		0.202573	0.09753
			0.25	0.1	0.036925
				0.5	0.102485
				1	0.131717

5 Conclusion

This research aims to simulate and analyze computationally the peristaltic blood flow of a non-Newtonian material via a tapered channel subjected to a radiative heat flux. While few research works [47–54] highlight recent developments in the fluid flow in the presence of different flow assumptions, others [55–57] indicate important numerical solving techniques. The most important findings are shown below:

The velocity increases with the increase in λ₁, Rn, Bh, and Bm.
When Bh and Pr are increased, the fluid temperature increases.
An increase in Pr, ν , Sr, and Sc causes a decrease in fluid concentration.
Heat transfer coefficient decreases as Pr and ν increase, whereas it increases with Rn and Bh.
Mass transfer coefficient decreases with the increase in Pr, Sr, Sc, and Bm but increases with the increase in Rn and S.

Acknowledgments

This Research is funded by Research Supporting Project Number (RSPD2023R585, King Saud University, Riyadh, Saudi Arabia.

Funding information: This Research is funded by Research Supporting Project Number (RSPD2023R585), King Saud University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2023-02-23

Revised: 2023-05-10

Accepted: 2023-05-26

Published Online: 2023-07-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2022-0258

Keywords for this article

rate of chemical reaction; tapered channel; heat transfer; Biot number of heat transmission; thermal radiation effect; Biot number of mass transfer; mass transfer

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