Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow

Abdulrahman B. M. Alzahrani

doi:10.1515/phys-2023-0176

Article Open Access

Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow

Abdulrahman B. M. Alzahrani

Published/Copyright: December 31, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

This work investigates the behavior of a nanofluid in a horizontal channel under advection boundary conditions within the domain of magnetohydrodynamic radiative Couette-Poiseuille flow. We utilize the Haar wavelet collocation method (HWCM) to investigate the effects of energy activation. This research relies on the mathematical model introduced by Buongiorno, which effectively captures the flow dynamics and incorporates the influence of chemical processes. To streamline the governing flow equations, we employ boundary layer approximations. The HWCM is employed to numerically solve the non-linear coupled partial differential equations that regulate momentum, heat transport, and mass transfer processes. We examine the impact of several dimensionless convergence parameters on the velocity, temperature, and concentration profiles and give visual representations of these results. It is crucial to highlight that the activation energy of the specific chemical reaction is directly linked to the concentration of nanoparticles. The effect of Brownian motion on nanoparticle concentration varies from that of the thermophoresis parameter.

Keywords: Couette-Poiseuille flow; Haar wavelet collocation method; energy and convective boundary conditions

Nomenclature

A: reaction rate
Bi: Biot number
Br: Brinkman number
C f: skin fraction
C s: heat capacity (J kg^–1 K )
E: activation energy
N b: Brownian motion parameter
N _j: convection diffusion parameter
N r: buoyancy ratio
Nu: Nusselt number
Rd: radiation parameter
Re: Reynolds number
S c: Schmidt number
Sh: Sherwood number

Greek symbols

μ: dynamic viscosity ( N s m − 2 )
( ρ C p ) f: heat capacity of the base fluid ( J K − 1 m − 3 )
ε: momentum accommodation coefficient
α: thermal diffusivity k ( ρ C ) f ( m 2 s − 1 )
β: volumetric volume expansion coefficient ( K − 1 )
τ: parameter defined by ( ρ C p ) s ( ρ C n )
v: kinematic viscosity ( m 2 s − 1 )
ϑ: dimensionless temperature
ϕ: nanoparticle concentration
σ *: Stefan–Boltzmann constant
σ: electrical conductivity ( S m − 1 )
Y: temperature

1 Introduction

Nanofluids, which involve the suspension of nanoscale particles within a convective heat transfer fluid, represent a fascinating scientific frontier where the intricate interplay of thermodynamic forces and fluid dynamics is pivotal in enhancing thermal conductivity. Numerous strategies have been developed to augment the thermal conductivity of these fluids through the dispersion of nano- and micro-scale elements within the liquid matrix. Extensive research has established that including trace amounts of metallic oxide nanoparticles significantly enhances thermal conductivity [1,2,3]. Additionally, the study of natural convection in nanofluids has garnered attention in recent years [4,5], with investigations of natural convective boundary-layer flows across vertical plates, as studied by Kuznetsov and Nield [6] using the Buongiorno model [7]. Boundary flows in porous media with convection dominance, as explored by Wang et al. [8]. Detailed insights into nanofluid flow properties and convective heat transfer are provided in previous literature [9,10,11,12,13].

Moreover, the scientific community has shown significant interest in studying convective heat transfer and magnetohydrodynamic (MHD) motion at the nanofluid interface, focusing on potential applications across various sectors. The peristaltic transport of micro-polar fluids in channels featuring convective boundary conditions and heat source/sink effects has been explored, yielding exact solutions [14,15]. Studies investigating the combined impact of Hall and convective conditions on the peristaltic flow of couple-stress fluids in asymmetric, slanted channels have been conducted, advocating the utilization of peristaltic fluid flow in engineering contexts, incorporating Hall current, thermal deposition, and convective conditions [16,17]. Hayat et al. analyzed mixed convection and varying thermal conductivity in viscoelastic fluids [18]. Further studies have investigated the effects of two-fold stratification and mixed convection in magneto-Maxwell nanofluids [19,20] and the MHD slip flow of micro-polar liquids due to vertically contracting surfaces. The convective flow of Maxwell liquids with thermal radiation was explored by Hayat et al. [21], shedding light on the dynamic alterations experienced by electric fluids moving through magnetic fields created by external sources. Rana et al. [22] researched natural convection flows of carbon nanotube nanofluids featuring Prabhakar-like thermal transfer characteristics. Additionally, Lou et al. [23] investigated the effects of Coriolis force on the dynamics of spinning micropolar dusty fluids under considerable Lorentz force.

The Arrhenius activation energy and mass transfer analysis procedure have received significant attention owing to their relevance in chemical engineering, nuclear reactor cooling, geothermal reservoirs, and thermal oil recovery. This approach allows for an in-depth examination of the intricate relationships between chemical reactions and mass movement, considering various reactant species and production rates. Several studies [24,25,26,27,28,29,30,31] have contributed to a better understanding of nanomaterials enhancing thermal conductivity and thermal performance. Bestman [32] employed activation energy kinetics analysis to investigate convection-driven mass transport in a vertical conduit filled with porous material, employing a perturbation approach to find an analytical solution. Maleque investigated natural convection in a binary chemical reaction system [33]. Mustafa et al. [34] discussed how buoyancy influences a magneto-nanofluid’s chemical reactions and activation energy as it flows across a vertical surface.

Researchers have converged on several key findings: the effects of the thermophoresis parameter on nanoparticle concentration are contrary to those of Brownian motion, and nanoparticle concentration is directly proportional to activation energy and the rate of chemical reaction. Mohyud-Din et al. [35] examined the outcomes of MHD flow in both convergent and divergent channels, concluding that nanoparticle distribution has a diminishing influence as the Reynolds number increases in a divergent channel. Still, it exerts an increasing effect in a converging channel. Previous studies [36,37] considered the flow through a rotating frame with an elastic surface and a binary chemical reaction, expanding our understanding of this complex field. It is of great scientific interest to investigate and assess the activation of energy and thermal dynamics in the context of manufacturing extrusion, considering the impact of radiation phenomena because of the work of some of the above researchers and its applicability in a wide range of scientific and technological disciplines. The Buongiorno fluid model introduces innovative features by considering solid fraction, thermophoresis, Brownian motion, effective thermal conductivity, changeable properties, and non-uniform heat sources. These combined characteristics improve its precision and usefulness in simulating the behavior of nanofluids and heat transfer, making it a valuable tool in multiple scientific and technical fields. This study presents a new method that utilizes similarity transformations to reorganize the partial differential equations efficiently. The transformed equations are then effectively handled using the Haar Wavelet Collocation method (HWCM) [38,39,40]. This method effectively deals with the intricate network of interconnected differential equations that govern the system, a unique aspect of our research. A schematic flow chart of the proposed approach is given in Scheme 1.

Scheme 1

A schematic flow chart of the proposed approach.

An overview of this work is as follows. Section 2 begins by providing the flow geometry in our current study. The methodology and the non-linear coupled partial differential equations describing momentum, heat transfer, and mass transfer phenomena numerically solved using the HWCM are provided in Sections 3 and 4, along with results and discussion. Finally, Section 5 contains the conclusion of our study.

2 Flow geometry

Two indefinitely long parallel plates are separated by distance ( 2 d ) , creating a confined zone through which an incompressible fluid flows continuously. Assume the upper plate moves at a constant velocity ( U ¯ ) , with the bottom plate at temperature ( t 1 ) and the higher plate at temperature ( t 2 ) . Assuming an orthogonal coordinate system ( x , y ) , with the plates at y = − d and ( y = d ) , the y-axis would represent the orthogonal direction concerning the plates. At the same time, the x-axis would be the direction of fluid flow, while the y-axis runs counter to the direction of the gravitational acceleration vector ( g ) and is perpendicular to the channel. The channel wall’s bottom and top must satisfy the convective boundary criteria (Figure 1).

Figure 1

Flow diagram.

The fluid is y-electrically transmitted through a non-uniform magnetic field ( b ) . The electric field and Hall current are considered insignificant at low Reynolds numbers. In this scenario, it is assumed that the nanofluid’s physical properties are always the same. The fundamental equations for an incompressible fluid, including momentum, mass, and energy conservation, are as follows [41]:

(1) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2) ρ f . ( u . ∇ ) u = − ∇ P + μ ∇ 2 u + g ( C ρ s + ρ f [ ( 1 − β ) ( T − T ∘ ) ( 1 − C ) ] ) + j × b ,

(3) ρ f C ( u . ∇ ) T = ( ∇ 2 T ) k + ρ s C D T T ∘ ( ∇ T ) 2 + D b ( ∇ C . [ ∇ T ] ) + Θ − ∇ . Q r + 1 σ ( j × j ) ,

(4) ρ s . ( u × ∇ ) C = − ∇ × j s − ρ s K 2 T T ∘ m ( C − C ∘ ) e F α κ T .

Here mass flux, magnetic field, current density, and gravitational acceleration are defined as [42],

(5) j s = − ρ s D b ∇ C + D T T ∘ ∇ T , b = ( 0 , b ∘ , 0 ) , j = σ ( u × b ) , g = ( 0 , g ∘ , 0 ) .

The exponential term in Eq. (4) represents the activation energy in a chemical process [43], where Boltzmann and fitted rate constants are defined by ( m ) and ( κ ) . Thus, the fields of Eqs. (1)–(4) can be broken down into their constituent parts along with boundary conditions [44]:

(6) ∂ u 1 ∂ x + ∂ u 2 ∂ y = 0 ,

(7) ∂ P ∂ x = μ ∂ 2 u 1 ∂ y 2 − σ b ∘ 2 u 1 + g ( C − C ∘ ) ( ρ s − ρ f ) + ρ f β ( 1 − C ∘ ) ( T − T ∘ ) ,

(8) u 1 ∂ T ∂ x = ∂ 2 T ∂ y 2 + τ D T T ∘ ∂ T ∂ y 2 + D b ∂ C ∂ y ∂ T ∂ y + μ ( ρ C p ) f ∂ u ∂ y 2 + σ b ∘ 2 u 1 2 ( ρ C p ) f − 1 ( ρ C p ) f ∂ q r ∂ y ,

(9) − D b ∂ 2 C ∂ y 2 = D T T ∘ ∂ T ∂ y 2 − K r 2 T T ∘ m ( C − C ∘ ) e F a κ T ,

(10) u 1 = U ¯ , u 2 = 0 , [ t 2 − T ] h f = − k ∂ T ∂ y , [ c 2 − C ] h s = − D b ∂ C ∂ y , y = d . u 1 = 0 , u 2 = 0 , [ t 1 − T ] h f = − k ∂ T ∂ y , [ c 1 − C ] h s = − D b ∂ C ∂ y , y = − d .

By defining dimensionless parameters [45], one can transform the above system into a non-dimension form

(11) ξ = x a , F = u 1 a , ϑ = T − T ∘ t 1 − T ∘ , p = a 2 P μ U m 1 , n 1 = t 2 − T ∘ t 1 − T ∘ , ζ = y a , G = u 2 a , ϕ = C − C ∘ c 1 − C ∘ , U = U ¯ U m 1 , m 1 = c 2 − C ∘ c 1 − C ∘ .

Since the stream’s path lies along the x-axis, the y-axis velocity component must be null, i.e., the steady pressure difference drives the fluid movement. The maximum speed ( U m 1 ) was assumed to have happened between the two walls. In this situation, we ignore the temperature and concentration scales ( m 1 ) entirely. For this case, the Rosseland approximation describes the radiative heat flux q r = 16 T ∘ 3 σ 3 k ∘ ∂ T ∂ y .

(12) ∂ 2 F ∂ ζ 2 − M 2 F + R a R e p r ( ϑ − N r ϕ ) − p = 0 ,

(13) ( 1 + R d ) ∂ 2 ϑ ∂ ζ 2 + N b ∂ ϑ ∂ ζ ∂ ϕ ∂ ζ + N t ∂ ϑ ∂ ζ 2 + B r ∂ F ∂ ζ 2 + M 2 B r F 2 − λ F ( ζ ) ,

(14) ∂ 2 ϕ ∂ ζ 2 + N t N b ∂ 2 ϑ ∂ ζ 2 − Sc R e A ( 1 + α ϑ ) m e − ϒ 1 + α ϑ ,

with boundary conditions,

(15) F ( 1 ) = U , ϑ ′ ( 1 ) = − B i ( n 1 − ϑ ( 1 ) ) , ϕ ′ ( 1 ) = − N j ( m 1 − ϕ ( 1 ) ) , F ( − 1 ) = 0 , ϑ ′ ( − 1 ) = − B i ( 1 − ϑ ( − 1 ) ) , ϕ ′ ( − 1 ) = − N j ( 1 − ϕ ( − 1 ) ) ,

where

(16) R a = ( 1 − C ∘ ) ( t 1 − T ∘ ) ν α , M 2 = σ b ∘ 2 a 2 μ , R e = a U m ν , ν = μ ρ , pr = ν α , A = K r 2 a U m , B r = μ U m 2 k ( t 1 − T ∘ ) , N b = τ D b ( c 1 − C ∘ ) α , N t = τ D t ( t 1 − T ∘ ) α T ∘ , N r = ( ρ p − ρ f ) ( c 1 − C ∘ ) ρ f β ( 1 − C ∘ ) ( t 1 − T ∘ ) , R d = 16 σ t 1 3 3 k ∘ k , Sc = ν D b , α = ( t 1 − T ∘ ) T ∘ , ϒ = F a κ T ∘ , B i = k h f a , N j = a h s D b .

3 Methodology

Assuming that A and B are constants, then ζ ∈ [ A , B ] , the ith Haar wavelet family is defined as

(17) H i ( ζ ) = 1 for ζ ∈ [ a 1 , b 1 ) − 1 for ζ ∈ [ b 1 , c 1 ) 0 elsewhere ,

with

(18) a 1 = k / m ¯ 1 , b 1 = ( k + 1 ) / 2 / m ¯ 1 , c 1 = k + 1 / m ¯ 1 ,

where m ¯ 1 = 2 j is the highest attainable resolution, we will refer to this quantity as M ¯ = 2 j , each of the 2 M ¯ subintervals of the interval [ A , B ] has a length ζ = ( B , A ) / 2 M ¯ , where M ¯ is the number of subintervals ( 2 M ¯ ) , a translation parameter k = 0 , 1 , … , m ¯ 1 − 1 and a dilatation parameter j = 0 , 1 , … , J . The formula for the wavelet number i is i = m ¯ 1 + k + 1 .

For Haar function and their integrals

(19) p i 1 ( ζ ) = ∫ 0 ζ H i ( ζ ) ,

(20) p i , l + 1 ( x ) = ∫ 0 ζ p i , l ( ζ ) d ζ , l = 1 , 2 , … ,

To calculate these integrals, we use Eq. (17)

(21) p i 1 ( ζ ) = ζ − a 1 c 1 − ζ 0 for ζ ∈ [ a 1 , b 1 ) for ζ ∈ [ b 1 , c 1 ) elsewhere ,

(22) p i 2 ( ζ ) = 1 2 ( ζ − a 1 ) 2 for ζ ∈ [ a 1 , b 1 ) 1 4 m ¯ 1 2 − 1 2 ( c 1 − ζ ) 2 for ζ ∈ [ b 1 , c 1 ) 1 4 m ¯ 1 2 for ζ ∈ [ c 1 , 1 ) 0 elsewhere ,

(23) p i 3 ( ζ ) = 1 6 ( ζ − a 1 ) 3 for ζ ∈ [ a 1 , b 1 ) 1 4 m ¯ 1 2 ( ζ − b 1 ) − 1 6 ( c 1 − ζ ) 3 for ζ ∈ [ b 1 , c 1 ) 1 4 m ¯ 1 2 ( ζ − β ) for ζ ∈ [ c 1 , 1 ) 0 elsewhere ,

We also present the notation shown below.

(24) C i 1 = ∫ 0 L p i 1 ( ζ ) d ζ ,

(25) C i 2 = ∫ 0 L H i ( ζ ) d ζ .

Summation of the Haar wavelet function can be written as follows:

(26) F ( ζ ) = ∑ i = 1 ∞ a i H i ( ζ ) .

We may approximate the highest order derivatives of F , ϑ , and ϕ given for the problems (12)–(14) to build a straightforward and precise HWCM.

(27) F ″ ( ζ ) = ∑ i = 1 2 M a i H i ( ζ ) ,

(28) ϑ ″ ( ζ ) = ∑ i = 1 2 M b i H i ( ζ ) ,

(29) ϕ ″ ( ζ ) = ∑ i = 1 2 M c i H i ( ζ ) .

Integrating Eqs (27–29), we obtain the values F ″ ( ζ ) , F ′ ( ζ ) and F ( ζ ) , respectively,

(30) F ″ ( ζ ) = ∑ i = 1 2 M a i p i , 1 ( ζ ) − 1 L ˆ C i , 1 ,

(31) F ′ ( ζ ) = ∑ i = 1 2 M a i p i , 2 ( ζ ) − 1 L ˆ ζ C i , 1 ,

(32) F ( ζ ) = ∑ i = 1 2 M a i p i , 3 ( ζ ) − 1 L ˆ ζ 2 2 C i , 1 .

Assume L ˆ is a large number. A numerical solution for the ODEs is obtained by substituting Eqs (27)–(32) in Eqs (12)–(14) (Table 1).

Table 1

HWCM convergence, when m 1 = n 1 = 0 , U = ϒ = α = A = R d = B r = 1 , m = 0.5 , Sc = 10 , M = 2 , and λ = 2 , N t = N r = N j = B i = 0.5 , R e = 0.3 , N b = 0.5

Order of approximation	Time	‒ F ' ( ζ )	‒ ϑ ' ( ζ )	‒ ϕ ' ( ζ )
1	3.45319	0.3688	1.0003	0.8873
5	25.84384	0.3681	0.9991	0.8865
11	64.2033	0.3673	0.9982	0.8857
18	147.7525	0.3674	0.9934	0.8849
22	214.2009	0.3657	0.9908	0.8846
25	380.2949	0.3652	0.9900	0.8838

In the next section, we can see a visual representation of the results of the proposed solution.

4 Results and discussion

This section used the HWCM to numerically solve the transformed non-linear differential Eqs (12)–(14) associated with the convective boundary conditions (15) from Figures 2–15, we discuss the effects of various evolving parameters on the flow field. The following analysis and outcomes are achieved by selecting suitable values for emerging parameters:

m 1 = n 1 = 0 , U = ϒ = α = A = R d = B r = 1 , m = N b = N t = 0.5 , N r = N j = B i = 0 . 5 , R e = 0 . 3 , Sc = 10 , M = 2 , and λ = 2 .

$Figure 2 Evolution of F ( ζ ) F(\zeta ) for the spatial parameter ( M ) (M) .$

Figure 2

Evolution of F ( ζ ) for the spatial parameter ( M ) .

$Figure 3 Evolution of ϑ ( ζ ) {\vartheta }(\zeta ) for the spatial parameter ( M ) (M) .$

Figure 3

Evolution of ϑ ( ζ ) for the spatial parameter ( M ) .

$Figure 4 Evolution of ϕ ( ζ ) \phi (\zeta ) for the spatial parameter ( M ) (M) .$

Figure 4

Evolution of ϕ ( ζ ) for the spatial parameter ( M ) .

$Figure 5 Evolution of F ( ζ ) F(\zeta ) for the spatial parameter ( N r ) . ({N}_{\text{r}}).$

Figure 5

Evolution of F ( ζ ) for the spatial parameter ( N r ) .

$Figure 6 Evolution of ϑ ( ζ ) {\vartheta }(\zeta ) for the spatial parameter ( R d ) ({R}_{\text{d}}) .$

Figure 6

Evolution of ϑ ( ζ ) for the spatial parameter ( R d ) .

$Figure 7 Evolution of F ( ζ ) F(\zeta ) concerning the spatial parameter ( A ) (A) .$

Figure 7

Evolution of F ( ζ ) concerning the spatial parameter ( A ) .

$Figure 8 Evolution of ϑ ( ζ ) {\vartheta }(\zeta ) for the spatial parameter ( A ) (A) .$

Figure 8

Evolution of ϑ ( ζ ) for the spatial parameter ( A ) .

$Figure 9 Evolution of ϕ ( ζ ) \phi (\zeta ) to the spatial parameter ( A ) (A) .$

Figure 9

Evolution of ϕ ( ζ ) to the spatial parameter ( A ) .

$Figure 10 Evolution of F ( ζ ) F(\zeta ) concerning the spatial parameter ( B i ) ({B}_{\text{i}}) .$

Figure 10

Evolution of F ( ζ ) concerning the spatial parameter ( B i ) .

$Figure 11 Evolution of ϑ ( ζ ) {\vartheta }(\zeta ) concerning the spatial parameter ( B i ) ({B}_{\text{i}}) .$

Figure 11

Evolution of ϑ ( ζ ) concerning the spatial parameter ( B i ) .

$Figure 12 Evolution of ϕ ( ζ ) \phi (\zeta ) concerning the spatial parameter ( B i ) ({B}_{\text{i}}) .$

Figure 12

Evolution of ϕ ( ζ ) concerning the spatial parameter ( B i ) .

$Figure 13 Evolution of F ( ζ ) F(\zeta ) concerning the spatial parameter ( N j ) ({N}_{\text{j}}) .$

Figure 13

Evolution of F ( ζ ) concerning the spatial parameter ( N j ) .

$Figure 14 Evolution of ϕ ( ζ ) \phi (\zeta ) concerning the spatial parameter ( N t ) ({N}_{\text{t}}) .$

Figure 14

Evolution of ϕ ( ζ ) concerning the spatial parameter ( N t ) .

$Figure 15 Evolution of ϕ ( ζ ) \phi (\zeta ) concerning the spatial parameter ( N b ) ({N}_{\text{b}}) .$

Figure 15

Evolution of ϕ ( ζ ) concerning the spatial parameter ( N b ) .

The combined effect of the magnetic parameter ( M ) on the velocity, temperature, and concentration fields is shown in Figures 2–4. The phenomenon of hydromagnetic flow corresponds to ( M = 0 ) , while that of hydrodynamic flow is led by ( M > 0 ) . Figure 2 depicts a noticeable reduction in nanofluid flow velocity with its increase, attributed to the amplification of opposing Lorentz forces acting on it. This phenomenon illustrates that as the flow of the nanofluid intensifies, the resistance it encounters from the Lorentz forces proportionally augments, resulting in the observed slowdown. Compared to the hydrodynamic flow, a more uniform temperature distribution characterizes hydromagnetic flow. The Lorentz force exerts more effect in a strong magnetic field. Figure 3 displays the enhanced temperature distribution resulting from the amplified Lorentz force, demonstrating a more uniform and effective thermal profile throughout the system. The increased distribution of temperature not only facilitates more efficient dissipation of heat but also reduces the presence of temperature gradients, enhancing the system’s overall performance.

Shifting our focus to Figure 4, the concentration curve is depicted in correlation with different magnetic field parameter values. The observed concentration profile demonstrates a clear positive correlation, indicating that a proportional increase in the concentration of the targeted compounds within the system accompanies an increase in the magnetic field parameter. This occurrence underscores the significant association between the intensity of the magnetic field ( M ) and the concentration of the desired components, indicating the possibility of meticulous regulation and enhancement of these processes.

Higher values of ( N r ) seen in Figure 5 also influence the fluid motion. The drop in velocity profile directly results from the increase in opposite bouncing due to the ambient nanoparticle volume percentage. The effect of the radiation parameter ( R d ) on the temperature distribution is seen in Figure 6. As the radiation parameter increases, a more significant impact is seen on the nanofluid’s temperature distribution. The physical mechanism underpinning this remarkable improvement is increased radiative energy transfer within the nanofluidic channel.

The velocity, temperature, and concentration profiles for various values of the chemical reaction parameter ( A ) are depicted in Figures 7–9, respectively. A steeper velocity profile may be observed as the chemical reaction parameter is raised. The chemical reaction causes a decrease in physical velocity, yet the velocity at the surface increases to its maximum value compared to the surface boundary. The velocity of the fluid decreases when the chemical reaction parameter, ( A ) , is more significant than zero. The nanofluid’s temperature and the nanoparticle volume concentration decrease with the increase in ( A ) . This behavior indicates the diminutive impact of buoyant force owing to the concentration gradient on the concentration profile.

As shown in Figure 10, the peak velocity is modest regardless of the Biot number ( B i = 0 ) . Thermal resistance of the plate decreases as the convection Biot number rises. As a result, both the peak velocity and surrounding velocities increase substantially. In Figure 11, we can see how the temperature changes as a function of the Biot number ( B i ) (the convection conduction parameter) since ( B i ) is linked to convective cooling at the surface. Improving the thermal boundary layer due to a rise in ( B i ) temperature leads to a rise. The concentration field as a function of the Biot number ( B i ) is shown in Figure 12. For a higher thermal Biot number ( B i ) , both the concentration field and the thickness of the corresponding boundary layer are magnified.

Figure 13 shows F ( ζ ) for both cases, with and without ( N j ) . The plot shows that the flow is smooth everywhere in the geometry, but it speeds up significantly as the parameter increases. We see the effects on the concentration profile in Figures 14 and 15. Concentration boundary layer thickness decreases with the increase in Brownian motion parameter values; however, the thermophoresis parameter’s effect on nanoparticle concentration follows the opposite trend.

The numerical results for many physical parameters are shown in Tables 2–4. These include the strength of the magnetic field, buoyancy ratio, Rayleigh number, radiation properties, Prandtl number, dimensionless reaction rate, non-dimensional activation energy, fitted rate constant, temperature gradient, thermophoresis effects, Schmidt number, Biot number, and convection-diffusion parameters. These factors directly impact the Sherwood number, Nusselt number, and skin friction coefficient. Notably, the skin friction and Nusselt numbers positively correlate with the magnetic field and buoyancy ratio parameters, while the activation energy and Sherwood numbers show an inverse relationship.

Table 2

Skin friction coefficient for different values of M , N r , Ra a n d Pr .

( M )	( N r )	( Ra )	( P r )	‒ F ' ( ‒ 1 )	‒ F ' ( 1 )
0.0	0.5	2	7	‒ 0.004573	0.004745
2.0				−0.004044	0.004606
4.0				−0.003453	0.004187
6.0				−0.002801	0.003489
	0.5			−0.004044	0.004606
	1.0			−0.004251	0.005911
	1.5			−0.004457	0.007216
	2.0			−0.004664	0.008522
		1		−0.005388	0.006848
		2		−0.004044	0.004606
		3		−0.002700	0.002364
		4		−0.001356	0.000122
			7	−0.004044	0.004606
			8	−0.004380	0.005166
			9	−0.004642	0.005602
			10	−0.004851	0.005951

Table 3

Nusselt number for different values of M , N t , R d , N b and B r .

( M )	( N t )	( R d )	( N b )	( B r )	ϑ ' ( ‒ 1 )	ϑ ' ( 1 )
0.0	0.5	11	0.5	1	0.492929	0.504004
2.0					0.493246	0.504333
4.0					0.493493	0.504745
6.0					0.493810	0.505321
	0.5				0.493246	0.504333
	1.0				0.494116	0.506943
	1.5				0.494990	0.509565
	2.0				0.495869	0.512199
		1			0.493246	0.504333
		2			0.493262	0.504369
		3			0.493278	0.504406
		4			0.493293	0.504442
			0.5		0.493246	0.504333
			1.0		0.493848	0.506208
			1.5		0.494451	0.508085
			2.0		0.495055	0.509964
				0	0.492212	0.501538
				1	0.493246	0.504333
				5	0.494280	0.507129
				7	0.495314	0.509924

Table 4

Sherwood number for different values of M , N t , R d , N b a n d B r .

( A )	( E )	( n 1 )	( ϒ )	( Sc )	( R d )	‒ ϕ ' ( ‒ 1 )	‒ ϕ ' ( 1 )
0.0	2.0			10.0	1.0	‒0.744148	2.330771
1.0						−0.870893	2.552822
2.0						−1.007160	2.795660
5.0						−1.153320	3.060250
	0.0					−0.124560	1.082631
	1.0					−0.241022	1.606220
	2.0					−0.870893	2.552820
	3.0					−1.842190	4.057940
		−1.0				−3.411570	6.359712
		0.0				−1.512590	3.509851
		0.5				−0.870893	2.552823
		1.0				−0.397690	1.851932
	1.0		0.0		0.5	0.052703	0.356023
	1.0		1.0			0.409739	0.715164
	1.0		2.0			2.652001	2.867631
	1.0		3.0			9.993070	9.019430
	1.0			1.0		5.312660	2.974801
	1.0			3.0		6.700660	4.696381
	1.0			5.0		8.257800	6.704832
	1.0			10.0		9.993070	9.019430

5 Conclusion

We have extensively analyzed the convective boundary conditions in a horizontal channel for Couette-Poiseuille MHD radiative heat and mass transfer. Our analysis incorporates the Buongiorno model, activation energy, chemical processes, and concentration fields. We do a thorough analysis that considers Joule heating, viscosity dissipation, radiation, and energy distribution. By utilizing an appropriate similarity transformation, we have converted the flow issue into a form that lacks dimensions and efficiently employed the Haar Wavelet Collocation approach to solve it. The graphical depictions of diverse, dynamic variables for nanofluid have been clarified. The main discoveries from our investigation demonstrate several essential observations.

The magnetic field parameter substantially influences the distribution of heat and mass, causing a corresponding impact on the flow field. An increase in the magnetic parameter leads to a decrease in the velocity of the nanofluid. Heat and mass distribution behavior during a chemical process shows unexpected patterns. The shear effects on both walls produce contrasting effects on the buoyancy ratio and magnetic parameter, and the heat transfer effects on both walls intensify with an unspecified factor. These discoveries enhance our comprehension of the intricate interconnections within this system and provide useful perspectives for future research in this subject.

Acknowledgments

The author would like to extend his sincere appreciation to the Researchers Supporting Project number (RSPD2023R920), King Saud University, Saudi Arabia.

Funding information: The Researchers Supporting Project number (RSPD2023R920), King Saud University, Saudi Arabia.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.

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Received: 2023-08-17

Revised: 2023-12-16

Accepted: 2023-12-18

Published Online: 2023-12-31

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

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0176

Keywords for this article

Couette-Poiseuille flow; Haar wavelet collocation method; energy and convective boundary conditions

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