Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel

Nomenclature

u ¯

velocity along x ¯ direction

v ¯

velocity along y ¯ direction

p

pressure gradient

d

channel width (m)

a

wave amplitude (m)

λ

wavelength

μ

viscosity of the fluid (kg/m/s)

σ

electrical conductivity of the fluid

c

wave speed (m/s)

C p

specific heat at constant pressure

ρ

density of the fluid (kg/m³)

T

temperature of the fluid

k

thermal conductivity

k 1

porous medium permeability

β

heat source/sink

Re

Reynolds number

M

Hartmann number

Da

porosity parameter

m

Hall current parameter

ϕ

amplitude ratio

δ

wave number

Ω

rotation parameter

Br

Brinkman number

Rn

thermal radiation parameter (W m²/K)

Pr

Prandtl number

θ

temperature distribution

Sr

Soret number

Sc

Schmidt number

S

chemical reaction parameter

η 1 , η 2

gravitational parameters

1 Introduction

Couple stress fluids are a subclass of non-Newtonian fluids characterized by their consideration of the size of fluid particles. Since fluids displaying couple stress have so many practical and theoretical consequences, their study has attracted a lot of attention from scholars. The couple stress fluid concept may also be applied to understand the dynamics of blood flow inside arterial arteries. Numerous scholars have conducted investigations on the behavior of pair stress fluids in the framework of peristaltic mechanism difficulties under various conditions. Eldabe et al. [1] investigated the role of wall features on the physiological behavior of a couple stress fluid. Abou-zeid and Mohamed [2] did research using the perturbation method to examine the physiological fluid mechanism of a couple stress fluid during magnetohydrodynamic (MHD) circumstances. Alsaedi et al. [3] explored the physiological transport across a homogeneously porous substrate. Researchers [4–14] have used computational techniques to examine couple stress fluid flow with peristalsis in a wide range of contexts.

The peristaltic process has gained popularity in recent decades. The fields of medicine and physiology have found several uses for peristalsis. Some examples of gastrointestinal motility include sperm traveling through the coronary arteries, lymphatic fluid traveling through lymph veins, blood cells migrating through the vasculature, and organs being shifted about in the abdomen. Theoretical and experimental discussions of the peristalsis of viscous fluids may be found in the works of Latham [15] and Shapiro et al. [16]. Numerous studies [17–26] looked at peristaltic flow under the low Reynolds number assumptions of the long-wavelength approximation. Furthermore, in recent years, researchers focusing on geo-substantial solution dynamics have shown a marked uptick in interest in exploring fluid flow models within porous materials. The study carried out by Ramesh Babu et al. [27] looked at the phenomenon of the physiological mechanism of a viscous fluid across a porous substance using the lubrication concept technique. Aman et al. [28] studied how Maxwell fractional fluid through porous material affects heat transmission. Heat and mass transfer are major concerns for MHD fluxes. According to Hayat et al. [29], the convective heat transmission capabilities of Carreau fluid are due to the Joule heating caused by the MHD peristalsis of the fluid. Non-Newtonian fluids’ mixed convective peristaltic transport was investigated by Abbasi et al. [30], who looked at how Ohmic heating affected it. According to Hayat et al. [31], a radiation-heated magneto nanofluid flowing through a porous material exhibits a peristaltic motion. The peristaltic mechanism was studied for its role in heat and mass transfer by Prasad et al. [32]. The study of heat and mass transfer in fluid systems, in conjunction with chemical processes, has gained significance in the fields of metallurgy and chemical engineering. Bestman [33] was the first to investigate chemical reactions in interface-layer streams. Reddy et al. [34] explored the impact of radiation on the expansion of a linearly expanding sheet carrying a chemically reacting Maxwell fluid via a MHD field. Numerous researchers have recently looked at how chemical reactions influence the fluid flow patterns seen on different surfaces [35 –39].

The peristaltic flows of non-Newtonian fluids are the focus of all the aforementioned works. Several of these works also discussed the impacts of reaction mechanism, Joule heating, radiation, and couple stress. The major focus of this work is on studying the peristaltic couple stress fluid under temperature and concentration slip conditions along the rotating effects. Our study may be helpful in directing future research in this area since, to the best of our knowledge, this kind of analytical model has never been explored in any published work previously. Heat and mass transport with Ohmic heating and chemical reaction in an asymmetric channel may be analyzed using a couple stress fluid with a peristaltic mechanism ([9] and [40]) and fluid flow and different techniques. Physical interpretation of penetration energy absorption energy of 2A12 aluminum alloy, relaxation of Epoxy resin on dielectric loss of medium-frequency transformer, and characteristics of centrifugal pump are highlighted in previously published studies [41–43]. Numerous works [44–46] depicted the applications of porous nanosheets for highly efficient photocatalytic degradation of refractory contaminants, coupled thermo-hydro-mechanical mechanism, and colloidal suspension transport in porous media. Previous studies [47–50] highlighted some recent developments in fluid flow versus various flow assumptions. Sun et al. [51] and Xiang et al. [52] recently worked on shear-thickening fluids and micro-fluidic chip structured with micro-wedge array.

The novel aspects of this study make it exceptional. Due to its significance in public health applications, thermal radiation has gained substantial importance in the area of medical research. Several skin conditions have been successfully treated using infrared radiation techniques in dermatology. These techniques may also promote better blood flow to certain bones by acting as a kind of thermotherapy. Previous studies [53–57] highlighted some recent advancements in the field of fluid dynamics subject to different geometries.

The following questions are at the heart of the study that inspired this article:

• How does the couple stress component affect the fluid velocity and the skin friction coefficient?

• How do the thermal slip parameter, thermal radiation, and couple stress parameter affect heat transfer?

• How do reaction constant, concentration slip parameter, and Soret number influence mass transfer?

• How are the Nusselt number (Nu) and Sherwood number (Sh) at the wall y = h influenced by different parameters?

2 Formulation of the model

Consider the physiological mechanism of a couple stress incompressible, viscous fluid over a uniform thickness in the occurrence of an external magnetic field. Furthermore, the consideration of heat and mass transfer under slip circumstances is included. The fluid is stimulated by sinusoidal wave trains moving along the channel walls at a constant speed c. Figure 1 depicts the physical model of the problem and the flow coordinate system [9].

where a , t , d , and λ are the wave amplitude, time, half-width of the channel, and wavelength.

Figure 1

Physical model illustration.

In a wave reference system, the governing equations are given by [58]:

For thermal radiation, using the Rosseland approximation [59,60], we have

The transformation allows us to extend the fixed (X,Y) frame with a wave frame (x,y) that travels away from it at c:

Non-dimensional measures [58]:

where ε, k ₁, δ, Re, M, Da, α, η ₁, η ₂, Sr, S, ScPr, β are the amplitude ratio, non-uniform parameter, wave number, Reynolds number, Hartman number, porosity parameter, couple stress parameter, gravitational parameters, Soret number, chemical reaction parameter, Schmidt number, Prandtl number, heat source/sink parameter, and thermal radiation.

3 Solution of the problem

Dropping the bars from Eqs. (2)–(6) yields the following non-dimensional form when Eqs. (8) and (9) are implemented:

Eqs. (10)–(14) can be reduced to the following forms if we use the long-wavelength approximation and disregard the wave number, while taking into account low Reynolds scenarios [61]:

Dimensionless boundary conditions

where h = 1 + ϵ sin{2π(x − t)}, and β ₁ and β ₂ are the thermal and concentration slip parameters, respectively.

Using the provided boundary conditions (19) and (20), Eq. (15) becomes

where

By applying boundary conditions (21) and (22) to the outcomes of Eqs. (17) and (18), we acquire

where

f 5 = − Pr β 1 + RnPr , f 6 = M 2 Br 1 + RnPr , f 7 = f 6 f 3 2 16 r 1 2 , f 8 = f 6 f 4 2 16 r 2 2 , f 9 = f 6 f 3 f 4 2 ( r 1 + r 2 ) 2 , f 10 = f 6 f 3 f 4 2 ( r 1 − r 2 ) 2 , f 11 = f 2 f 3 r 1 2 , f 12 = f 2 f 3 r 2 2 , f 13 = f 3 2 2 + f 4 2 2 + f 2 2 , f 14 = ( f 5 − f 13 ) , f 15 = 4 f 7 r 1 , f 16 = 4 f 8 r 2 , f 17 = 2 f 9 ( r 1 + r 2 ) , f 18 = 2 f 10 ( r 1 − r 2 ) , f 19 = 2 f 11 r 1 , f 20 = 2 f 12 r 2 , f 21 = ( f 5 − f 13 ) h 2 2 − 2 f 7 cosh [ 2 r 1 h ] − 2 f 8 cosh [ 2 r 2 h ] − 2 f 9 cosh [ ( r 1 + r 2 ) h ]   − 2 f 10 cosh [ ( r 1 − r 2 ) h ] − 2 f 11 cosh [ r 1 h ] + 2 f 12 cosh [ r 2 h ] , f 22 = f 14 h − f 15 sinh [ 2 r 1 h ] − f 16 sinh [ 2 r 2 h ] − f 17 sinh [ ( r 1 + r 2 ) h ] − f 18 sinh [ ( r 1 − r 2 ) h ] − f 19 sinh [ r 1 h ] + f 20 sinh [ r 2 h ] , f 23 = − f 21 − β 1 f 22 .

Now,

where

f 24 = Sc Sr , f 25 = ( 2 f 15 r 1 ) , f 26 = ( 2 f 16 r 2 ) , f 27 = f 17 ( r 1 + r 2 ) , f 28 = f 18 ( r 1 − r 2 ) , f 29 = f 19 r 1 , f 30 = f 20 r 2 , f 31 = f 14 f 24 b , f 32 = f 24 f 25 4 r 1 2 − b , f 33 = f 24 f 26 4 r 2 2 − b , f 34 = f 24 f 27 ( r 1 + r 2 ) 2 − b , f 35 = f 24 f 28 ( r 1 − r 2 ) 2 − b , f 36 = f 24 f 29 r 1 2 − b , f 37 = f 24 f 30 r 2 2 − b , r 3 = S Sc , b = S Sc , f 38 = − f 31 − f 32 cosh [ 2 r 1 h ] − f 33 cosh [ 2 r 2 h ] − f 34 cosh [ ( r 1 + r 2 ) h ] − f 35 cosh [ ( r 1 − r 2 ) h ] − f 36 cosh [ r 1 h ] + f 37 cosh [ r 2 h ]

f 39 = − 2 r 1 f 32 cosh [ 2 r 1 h ] − 2 r 2 f 33 cosh [ 2 r 2 h ] − ( r 1 + r 2 ) f 34 cosh [ ( r 1 + r 2 ) h ] − ( r 1 − r 2 ) f 35 cosh [ ( r 1 − r 2 ) h ] − r 1 f 36 cosh [ r 1 h ] + r 2 f 37 cosh [ r 2 h ] .

The rate of volumetric flow is referred to as:

where

From Eq. (26), the pressure gradient d p d x is expressed as:

The instantaneous flux Q (x, t) equals

The average volume flow rate is often indicated as:

The pressure gradient can be mathematically represented using Eqs. (27) and (29):

According to reference [62], the skin friction coefficient ( C f ) , the Nusselt number (Nu), and the Sherwood number (Sh) are used to define shear stress, heat transfer rate, and mass transfer rate, respectively, at the wall:

4 Discussion of the problem

In this section, we examine the effects of various emergent components on the distributions of velocity, skin friction, temperature, and concentration. These are the default settings for the parameters used in calculations: ε = 0.2 , ϕ = π 3 , x = 0.5 , t = 0.2 , d = 0.1 , ρ = 0.1 , μ = 0.1 , η 1 = 0.5 , p = − 0.1 , α = 10 , γ = π 2 , Da = 0.2 , M = 2.5 , Br = 0.5 , Rn = 0.5 , Pr = 2.5 , β = 0.5 , Ω = 10 , β 1 = 0.5 , β 2 = 0.3 , Sr = 1.5 , Sc = 0.1 , and S = 0.2 . For numerical analysis, we resort to the Mathematica software.

4.1 Velocity distribution

The fluid velocity as a function of y is illustrated in Figures 2–6. Figure 2 conveys the consequence of the Hartmann number (M) on u. The velocity profile is seen to decrease with increasing M. This is brought on by Lorentz forces’ retarding effect on the velocity of the fluid. The relationship between Da and fluid velocity is seen in Figure 3. Increasing the porosity parameter results in a higher fluid velocity. Physically, fluid flow is enabled by a porous medium with high permeability. Figure 4 displays the change in fluid velocity as a function of Ω. The absolute velocity of the fluid drops as Ω increases. Figures 5 and 6 show that the velocity of the fluid increases when the gravitational and couple stress parameters are both increased.

Figure 2

Significance of Hartmann on velocity (u).

Figure 3

Significance of porosity parameter on velocity (u).

Figure 4

Significance of rotation on velocity (u).

Figure 5

Significance of gravitational parameter on velocity (u).

Figure 6

Significance of couple stress parameter on velocity (u).

4.2 Skin friction

Figures 7–11 show the results of varying M , Da , Ω , η 1 , and α to demonstrate how they affect the skin friction coefficient C _f. The observed trend in Figures 7 and 8 indicates that an increase in both M and Ω leads to a rise in the skin friction coefficient inside the region x ∈ ( 0 , 0.5 ) , whereas a drop is seen in the other section of the channel, x ∈ ( 0.5 , 1 ) . The C f as affected by Da is seen in Figure 9. The aforementioned graph shows that the C f decreases between x = 0 and x = 0.5 , while it improves between x = 0.5 and x = 1 . In Figure 10, the influence of η 1 on C f is presented. In the area of the channel between x = 0 and x = 0.5 , the skin friction coefficient decreases as η 1 rises. However, the skin friction coefficient measurements are unremarkable along the sides of the channel. From Figure 11, as α is increased, we see that the skin friction coefficient improves in the range of x = 0 to x = 0.5 .

Figure 7

Significance of Hartmann on skin friction ( C f ) .

Figure 8

Significance of rotation on skin friction ( C f ) .

Figure 9

Significance of porosity parameter on skin friction ( C f ) .

Figure 10

Significance of gravitational parameter on skin friction ( C f ) .

Figure 11

Significance of couple stress parameter on skin friction ( C f ) .

4.3 Heat transfer analysis

Figures 12–19 exhibit the temperature pattern ( θ or θ ( y ) ) as a function of y for different values of the parameters. Figure 12 demonstrates that as M increases, so does the temperature of the fluid. The force of Lorentz, which opposes the motion of the fluid, causes an increase in the resistance of the fluid to motion, which in turn causes an increase in the temperature of fluid. Figure 13 shows how Da affects the temperature profile. This graph shows that when Da increases, the fluid temperature drops because the boundary layer becomes thicker. Figure 14 shows the temperature variation as a function of the Prandtl number. As Pr rises, so does the temperature of fluid. A rise in the Prandtl number results in a magnification of the interfacial shear stress between adjacent layers of the fluid. Consequently, an increased Pr value indicates an increased temperature. Figure 15 depicts the influence of Rn on θ ( y ) . In reality, when the thermal radiation parameter rises, the fluid temperature falls. The fluid temperature increases with increasing β 1 , as seen in Figure 16. As the slip parameter ( β 1 ) enhances, the temperature profile rises because more heat is transmitted from the surface to the liquid. The temperature distribution with β is shown in Figure 17. A rise in the heat source/sink parameter results in higher-temperature profiles because the thermal boundary layer is thinner. The impact of Br on θ ( y ) is seen in Figure 18. Since the Brinkman number is proportional to the amount of viscous dissipation, it can be shown that it has a major impact on the flow and that more accurate Brinkman number estimates lead to a more desirable temperature profile. Figure 19 illustrates the fluctuation in temperature profile in response to different values of α . The graph elucidates that the progressive increments in the values of α result in an increase in the temperature profile.

Figure 12

Significance of Hartmann on temperature ( θ ) .

Figure 13

Significance of porosity parameter on temperature ( θ ) .

Figure 14

Significance of Prandtl number on temperature ( θ ) .

Figure 15

Significance of thermal radiation on temperature ( θ ) .

Figure 16

Significance of thermal slip parameter on temperature ( θ ) .

Figure 17

Significance of heat source parameter on temperature ( θ ) .

Figure 18

Significance of Brinkman number on temperature ( θ ) .

Figure 19

Significance of couple stress parameter on temperature ( θ ) .

4.4 Mass transfer analysis

There are provided graphical representation s 20–29 of how diverse factors influence the concentration profile ( Φ or Φ ( y ) ) . Figures 20 and 21 depict the role of Da and Rn on the distribution of concentration. These charts show that a rise in Da and Rn results in a corresponding increase in fluid concentration. As depicted in Figure 22, the fluid concentration improves as the chemical reaction parameter increases. Concentration distribution changes caused by M , Pr , α , Br , β 2 , Sr , and Sc are shown in Figure 23 through 29. The graphs demonstrate that the fluid concentration falls as the values of M , Pr , α , Br , β 2 , Sr , and Sc increase (Figures 24–29).

Figure 20

Significance of porosity parameter on concentration ( Φ ) .

Figure 21

Significance of thermal radiation on concentration ( Φ ) .

Figure 22

Significance of chemical reaction parameter on concentration ( Φ ) .

Figure 23

Significance of Hartmann number on concentration ( Φ ) .

Figure 24

Significance of Prandtl number on concentration ( Φ ).

Figure 25

Significance of couple stress parameter on concentration ( Φ ).

Figure 26

Significance of Brinkman number on concentration ( Φ ).

Figure 27

Significance of concentration slip parameter on concentration ( Φ ).

Figure 28

Significance of Soret number on concentration ( Φ ).

Figure 29

Significance of Schmidt number on concentration ( Φ ).

Tables 1 and 2 show the Nusselt number and Sherwood number at the y = h wall for an assortment of fixed values of ε = 0.2 , ϕ = π 3 , x = 0.5 , t = 0.2 , and η 1 = 0.5 respectively. Nusselt and Sherwood values demonstrate the efficacy of surface heat and mass convection. Sherwood number for the concentration boundary layer is comparable to the Nusselt number for the thermal boundary layer. According to Table 1, an increase in M , Br , Pr , β , Ω , and α causes the Nusselt number to increase, whereas an increase in Da and Rn causes the Nusselt number to lessen. Sherwood number results decrease with increasing M , Br , Pr , β , Ω , Sc , and Sr , but increase with increasing Da , Rn ,  and S , as shown in Table 2.

Table 1

Nusselt number numerical values at the wall y = h

M	Da	Br	Rn	Pr	β	Ω	α	Nusselt number (Nu)
0.5	0.2	0.5	0.5	2.5	0.5	10	10	0.371857
1.5								0.432244
2.5								0.5128
2.5	0.2							0.5128
	0.3							0.505677
	0.4							0.500363
	0.2	0.1						0.452018
		0.5						0.5128
		1						0.588778
		0.5	0.5					0.5128
			1					0.393962
			1.5					0.33767
			0.1	2.5				0.5128
				3.5				0.536334
				4.5				0.552627
				2.5	0.5			0.5128
					0.7			0.615508
					0.8			0.718216
					0.5	10		0.5128
						15		0.515326
						20		0.516794
						10	10	0.5128
							20	0.541496
							30	0.543337

Table 2

Sherwood number numerical values at the wall y = h

M	Da	Br	Rn	Pr	β	Ω	S		Sc	Sr	Sherwood number (Sh)
0.5	0.2	0.5	0.5	2.5	0.5	10	0.2		0.1	1.5	−0.0534638
1.5											−0.0673244
2.5											−0.0821678
2.5	0.2										−0.0821678
	0.3										−0.0814932
	0.4										−0.0809408
	0.2	0.1									−0.070517
		0.5									−0.0821678
		1									−0.0967312
		0.5	0.5								−0.0821678
			1								−0.0634352
			1.5								−0.0545619
			0.1	2.5							−0.0821678
				3.5							−0.0850308
				4.5							−0.0870129
				2.5	0.5						−0.0821678
					0.7						−0.0973229
					0.8						−0.112478
						10					−0.0821678
						15					−0.0941432
						20					−0.120531
						10	0.2				−0.0821678
							0.5				−0.0800682
							0.8				−0.0780732
							0.2		0.1		−0.0821678
									0.2		−0.161512
									0.3		−0.238176
									0.1	1.5	−0.0821678
										2	−0.109557
										2.5	−0.136946

5 Conclusions

Examining the peristalsis process in the flow of a pair stress fluid through an inclined channel with a porous material present is the focus of this study. The study takes into account not only the role of reaction mechanism and Ohmic heating, but also slip boundary conditions. The most significant results are referenced as follows:

1. The fluid flow of velocity enhances with a rise in Da , η 1 , and α , whereas it decelerates with an increase in Ω and M .

2. The skin friction coefficient rises in the channel region x = 0 to x = 0.5 , whereas it decelerates in the other part of the portion x = 0.5 t o x = 1 with an increase in M and Ω , while the trend is reverse in case of Da and η 1 .

3. The fluid temperature reduces with a rise in Da and Rn , while it improves with a rise in M , Pr , β 1 , β , Br ,  and  α .

4. The fluid concentration decelerates as the values of M , Pr , α , Br , β 2 , Sr , and Sc increase.

5. An increase in M , Br , Pr , β , Ω , and α causes the Nusselt number to improve, whereas an increase in D a and R n causes the Nusselt number to lessen.

6. Sherwood number results decrease with increasing M , Br , Pr , β , Ω Sc ,  and Sr but increase with increasing Da ,  Rn ,  and S .

Several aspects of human physiology may be profoundly affected by the findings of this study. Porosity is frequently observed in a substantial proportion of human organs. Consequently, the mathematical model presented in this study can be used to predict the effectiveness of various systems. Incorporating nanoparticles, nanoparticle hybrids, and viscous dissipation into a comprehensive mathematical framework has the potential to advance research into cancer treatment strategies in physiological systems. Incorporating Dufour effects and convective boundary conditions is one possible methodology for developing a thermal analysis model of the digestive system. The aforementioned research demonstrates the diversity of possible applications for peristalsis in a variety of disciplines of study.