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The significance of quadratic thermal radiative scrutinization of a nanofluid flow across a microchannel with thermophoretic particle deposition effects

  • Pullare Nimmy , Rangaswamy Naveen Kumar , Javali Kotresh Madhukesh , Umair Khan EMAIL logo , Anuar Ishak , Kallur Venkat Nagaraja , Raman Kumar , Taseer Muhammad , Laila F. Seddek and Ahmed M. Abed
Published/Copyright: June 15, 2024
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Nanotechnology Reviews
From the journal Nanotechnology Reviews Volume 13 Issue 1

Abstract

The investigation of thermal radiation and thermophoretic impacts on nano-based liquid circulation in a microchannel has a significant impact on the cooling of microscale equipment, microliquid devices, and many more. These miniature systems can benefit from the improved heat transfer efficiency made possible by the use of nanofluids, which are designed to consist of colloidal dispersion of nanoparticles in a carrier liquid. Understanding and precisely modeling the thermophoretic deposition (TPD) of nanoparticles on the channel surfaces is of utmost importance since it can greatly affect the heat transmission properties. This work examines the complex interaction between quadratic thermal radiation, magnetohydrodynamics, and TPD in a permeable microchannel. It aims to solve a significant knowledge gap in microfluidics and thermal and mass transport. The governing equations are simplified by applying suitable similarity restrictions, and computing solutions to the resulting equations is done using the Runge‒Kutta Fehlberg fourth‒fifth-order scheme. The results are shown using graphs, and significant engineering metrics are analyzed. The outcomes show that increased Eckert number, magnetic, and porous factors will improve the thermal distribution. Quadratic thermal radiation shows the greater thermal distribution in the presence of these parameters, while Linear thermal radiation shows the least thermal distribution. The rate of thermal distribution is higher in the linear thermal distribution case and least in the nonlinear thermal radiation case in the presence of radiation and solid fraction factors. The outcomes of the present research are helpful in improving the thermal performance in microscale devices, electronic devices cooling, health care equipment, and other microfluidic applications.

Keywords: nanofluid; microchannel; quadratic thermal radiation; thermophoretic particle deposition

Nomenclature

Symbols

A 1

pressure gradient parameter

Bi i

Biot number

B 0

magnetic field flux density ( A m − 1 )

C

concentration

C 1

fluid concentration

C a

ambient concentration

C p

specific heat ( J kg − 1 K − 1 )

Cf

skin friction

D f

diffusivity ( m 2 s − 1 )

Ec 1

Eckert number

h i

convective heat transfer coefficient ( kg s − 3 K − 1 )

H

distance ( m )

k 1

porous medium permeability ( m 2 )

k

thermal conductivity ( kg ms − 3 K − 1 )

M F

magnetic field parameter

Nu

Nusselt number

P

pressure ( kg m − 1 s − 2 )

Pr

Prandtl number

q r

radiation heat flux

Re 1

Reynolds number

Rd

thermal radiation parameter

S 1

shape factor of the porous media

Sc 1

Schmidt number

Sh

Sherwood number

T r 1

reference temperature ( K )

T

temperature ( K )

T h

fluid temperature ( K )

T a

ambient temperature ( K )

u

axial velocity ( m s − 1 )

V

wall suction/injection velocity ( m s − 1 )

V T

thermophoretic velocity ( m s − 1 )

x , y

coordinates ( m )

Greek letters

θ a

temperature ratio parameter

φ 1

solid volume fraction

τ 1

thermophoretic parameter

k ⁎

mean absorption coefficient ( m − 1 )

σ ⁎

Stefan-Boltzmann constant ( kg s − 3 K − 4 )

σ

electrical conductivity ( Ω − 1 m − 1 )

μ

dynamic viscosity ( kg m − 1 s − 1 )

η

similarity variable

θ

dimensionless temperature

χ

dimensionless concentration

δ

thermophoretic coefficient

υ

kinematic viscosity ( m 2 s − 1 )

ρ

density ( kg m − 3 )

Subscripts

f

fluid

nf

nanofluid

1 Introduction

The rapid growth in engineering technologies demands rapid heat transmission. Ordinary heat-transmitting fluids comprising kerosene, water, engine oil, and thermic fluids exhibit lower thermal conductivity than solid fluids. The augmentation of heat transfer in conventional fluids includes the implementation of fins, inserts, porous surfaces, jet impingement, surface vibration, and fluid vibration. Apart from all of these, heat transfer can be upsurged by improving the thermophysical properties of conventional fluids. Nanometer-sized particles are dispersed into the base fluids owing to their higher heat transfer rate. Into consideration in 1995, Choi and Eastman [1] synthesized nanoparticles in carrier liquid, and later the suspension came to be known as “nanofluid.” Nanofluids exhibit improved thermal conductivity and have more efficient convective transmission of heat if equated with the conventional fluid. For improving thermal performance, nanofluids are applied in the field of cooling electronic and electrical devices, food processing, heat exchangers, transportation industry, and nuclear reactors, and medical domains. Recent advances in nanofluid research have focused on preparation methods and their many present and future uses in mechanical engineering, energy, and biomedical domains were reviewed by Bacha et al. [2]. Over a spinning sphere, Nimmy et al. [3] examined the correlation between endo/exothermic reactions on the magnetohydrodynamic (MHD) nanofluid flow. Regarding solar thermal energy storage, the substantial features of nanoparticles and their influence on molten salts were reviewed by Abir et al. [4]. Ahmad et al. [5] analysed the entropy generation for the natural convection of Casson nanofluid. Babu et al. [6] discussed the entropy generation investigation on convective nanofluid circulation on a 3D extending region because of the wide variety of uses for nanofluids. The Maxwell nanofluid movement over a disk subjected to varying uniform magnetic fields was swotted by Srilatha et al. [7]. Pattanaik et al. [8] explored the movement of a two-phase nanofluid through an elongated surface submerged in a permeable media. Karthik et al. [9] explored the mass and thermal distribution of nanofluid movement over a cylinder.

The flow passages whose hydraulic diameters ranging from 10 to 200 micrometers are termed as microchannels. In the field of microelectronics, microchannels which are small and compact in size constitute the primary structural component. In correspondence with the thermal systems employed by the microchannel, their role is bookmarked in various fields of micro heat exchangers, fuel cells, microelectronic cooling, microsensors, bioengineering, space systems, and manufacturing technologies. For proper functioning, microchannels take into account viscous dissipation, compressibility, dilution, surface-to-volume ratio, intermolecular flow, and slip flow. Owing to the wide range of applications exhibited by microchannels, a plethora of researchers studied the nanofluid flow through microchannels. A theoretical and experimental review of nanofluid flow through microchannels was elucidated by Sharma and Sharma [10]. Kambli and Dey [11] reviewed the functioning of microchannels with numerous working fluids and explored their applications in the domain of thermal management. The motion and temperature characteristics of a nanoliquid stream along a microchannel were deliberated by Bouzennada et al. [12]. In a double-layered microchannel, the hydrothermal performance of a hybrid nanofluid was examined by Sarvar-Ardeh et al. [13]. Mahmoodi et al. [14] investigated the laminar circulation of a hybrid nano liquid via a microchannel. The heat transmission and flow of a nanofluid in a 2D microchannel was explained by Li et al. [15]. In an oblique microchannel, thermodynamic evaluation and entropy production in a non-Newtonian liquid were investigated by Anitha and Gireesha [16].

When the temperature of an object exceeds zero, the emission of electromagnetic waves results in thermal radiation. When the temperature differential between the surface and the surrounding liquid increases, the impact of the thermal radiation process becomes more pronounced. Eventually, this alteration also affects the pace of heat transfer. Radiation heat transfer flow plays a vital role in the production industry for designing equipment, gas turbines, nuclear power plants, and numerous other propulsion systems for missiles, space vehicles, airplanes, and satellites. Also, the thermal radiation effect plays an essential part in regulating the heat transfer process in the polymer manufacturing sector. For investigating the consequence of radiative flow at a surface, researchers concentrated on linear thermal radiation (LTR) by implementing linear Rosseland approximation. By considering thermal radiation effects, the MHD flow of nanofluids across a vertical plate was reviewed by ShanthaSheela et al. [17]. Waqas et al. [18] studied temperature transmission in nanofluid through a porous medium by adding thermal radiation phenomena. In the presence of thermal radiation, the influence of mass and heat transfer in Jeffery fluid was investigated by Satya Narayana and Harish Babu [19]. Accompanied by the influence of the Cattaneo Christov heat flux model and with the impact of thermal radiation, Punith Gowda et al. [20] probed the 3D flow of a magnetic nanofluid. Across the surface of solar collectors, the effects of thermal radiation on Powell Eyring nanofluid were explored by Waqas et al. [21]. Over a cylinder, Hussain et al. [22] inspected the influence of radiation on hybrid nanofluid flow. Babu et al. [23] examined the steady flow of a Jeffery nanofluid along a stretching surface with the existence of thermal radiation effects. To detect the entropy production of a Casson nano liquid, Waqas et al. [24] utilized the thermal radiation effects. Together with heat radiation and second-order slip effects, Naidu et al. [25] analysed and formulated a new mathematical model of hybrid nanofluid flow through an elongated surface. The convection-radiation heat transfer of a dovetail fin was investigated by Nimmy et al. [26]. Using a stanching medium and a hybrid nanofluid, Waqas et al. [27] assessed the importance of radiation impacts from temperature. The nanofluid flow through a Riga plate subjected to radiation was studied by Asogwa et al. [28]. Waqas et al. [29] investigated the effect of thermal radiation parameters on the circulation of nanofluid along the stenosed artery. Jagadeesha et al. [30] studied the thermal distribution of an annular fin subjected to radiation effects.

The temperature difference within a flow system was assumed to be a linear function of T and without considering the higher power term of ( T − T a ) : T 4 as determined through the development of the Taylor series about T a . However, this is only applicable to lower temperature differences. Furthermore, it is worth noting that linear radiation is insufficient for constructing thermal equipment with large temperature differences; therefore, non-linear radiation provided by the Rosseland approximation is frequently employed for an array of industrial operations. In nonlinear thermal radiation (NLTR), the temperature function is not modified to encompass the ambient temperature. Several studies looked into the fluid flow with NLTR effects in various configurations. The impact of NLTR on the mass and heat conveyance of a hybrid nanofluid was probed by Sharma et al. [31]. Alamirew et al. [32] analysed the influence of NLTR on Williamson nanofluid over a stretching sheet. The effect of NLTR on MHD flow through a wedge on Jeffrey fluid was studied by Dharmaiah et al. [33]. By incorporating NLTR effects, Qayyum et al. [34] simulated and modelled the hybrid blood nanofluid flow along arteries.

Additionally, the thermal difference in applications like solar collectors, power generations, chemical appliances, and energy storage systems are high and the implication of linear Rosseland approximation was found to be not at all acceptable. Therefore, researchers considered the quadratic term as well, rather than ending the series at the linear term. This later came to be described as the “Rosseland quadratic thermal radiation.” When there is a significant temperature change in density, quadratic thermal radiation (QTR) can be used to describe the thermal conditions. Due to the high-temperature difference, the impact of QTR has become a crucial constituent in flow systems and is beneficial for simulating thermal instances that include solar panels, where considerable variation in heat concentration happens. Numerous studies have been conducted on analysing the QTR effects upon various configurations. The melting effect on the Carreau fluid flow across a cylinder with a focus on the effects of QTR was inspected by Jiann et al. [35]. The mass and thermal transfer behaviour of the nanofluid flow produced by the simultaneous QTR was investigated by Bijiga and Gamachu [36]. Over a rotating disk, the effect of QTR along with the convective boundary conditions on a ternary hybrid nanofluid was inspected by Singh et al. [37].

When a mixture of distinct kinds of movable particles are subjected to a temperature gradient, the particles react in a different manner which leads to the phenomenon namely thermophoresis or thermophoretic particle deposition (TPD). Thermophoresis allows the particles to settle down when the surface is cold and the particles will move away from the surface when it is hot. The gas molecules will move rapidly by gaining kinematic energy and move slowly when they are near the cold surface. In the mechanism of fluid flow, TPD signifies its role in thermal engineering and industrial applications involving air purifiers, ventilation systems, heat exchangers, and coal burners. The impact of thermal radiation and TPD on the movement of a ternary hybrid nanofluid over a wedge was deliberated by Karthik et al. [38]. Yasir et al. [39] demonstrated the effect of TPD in the flow of an Oldroyd-B fluid across a stretching cylinder. By considering thermophoretic effects and other parameters, Prasannakumara et al. [40] explored the nanofluid movement through a porous stretching surface. The time-dependent Maxwell nanofluid stream along with TPD was studied by Srilatha et al. [41].

This work is unique since it investigates unexplored microfluid mechanics and thermal transport areas. Previous studies have explored different nanofluid circulation components and heat transfer improvements. However, there needs to be more understanding regarding the impact of three specific ways of thermal radiation (linear, nonlinear, and quadratic) on these complicated phenomena in the context of MHD nanofluid flow through a permeable microchannel. In addition, incorporating the thermophoretic effect, a topic that is gaining greater attention, includes an extra level of intricacy to the investigation. The utilization of nanofluid circulation in a microchannel, where particles are deposited due to thermophoretic processes, has diverse practical applications such as in micro-electromechanical structures, microfluidic appliances, and heat transfer systems. Answering the following questions is the primary goal of the current inquiry.

  1. What is the impact of magnetic field and permeable medium on velocity profile?

  2. How does the thermal behaviour change with linear, nonlinear, and QTR?

  3. What is the impact of thermophoretic particle deposition on mass transfer rate?

2 Mathematical formulation

Consider the movement of a nanofluid through a microchannel which is enclosed in a porous medium and is constrained by two horizontal parallel plates. The y-axis was taken to be perpendicular to the x-axis, which is considered to be along with the direction of the plates. The length and the distance at which the plates are being separated are taken as d 1 and H respectively. The plate receives a uniform magnetic field B 0 which is employed normally and the upper part of the plate is positioned at y = H and the corresponding lower plate is situated at y = 0 . The hot temperature of the fluid is given by T h , and the ambient temperature is taken as T a . Similarly, C a is the ambient concentration, and C 1 is the fluid concentration. Owing to the significant temperature variance, the QTR is considered as of three distinct cases namely linear, nonlinear, and QTR in the temperature equation, and the TPD is considered in the concentration equation. The diagrammatic depiction of the corresponding problem is given in Figure 1.

Figure 1 Schematic view of the problem.
Figure 1

Schematic view of the problem.

The appropriate governing equations are given below based on the aforesaid assumptions [42,43,44].

(1) ρ nf V ∂ u ∂ y = − ∂ P ∂ x + μ nf ∂ 2 u ∂ y 2 − μ nf k 1 u − σ nf B 0 2 u ,

(2) ( ρ C p ) nf V ∂ T ∂ y = k nf ∂ 2 T ∂ y 2 + μ nf k 1 u 2 + σ nf B 0 2 u 2 − ∂ q r ∂ y ,

(3) V ∂ C ∂ y = D f ∂ 2 C ∂ y 2 − ∂ ∂ y ( V T ( C − C a ) ) .

The corresponding boundary conditions are given as follows: [42]

(4) u = 0 , k nf ∂ T ∂ y = ( T − T h ) h 1 , C = C 1 at y = 0 ,

(5) u = 0 , k nf ∂ T ∂ y = − ( T − T a ) h 2 , C = C a at y = H ,

where V represents the wall suction/injection velocity, x , y are the coordinates, u denotes the axial velocity, P is the pressure, k nf is the thermal conductivity of nanofluid, ρ nf is the density of nanofluid, σ nf is the electrical conductivity of nanofluid, k 1 is the permeability of the porous media, C p is the specific heat, μ nf is the dynamic viscosity of nanofluid, T is the temperature of the nanofluid, T h is the hot liquid temperature, T a is the ambient temperature, B 0 is the Magnetic field flux density, H is the distance, D f signifies the diffusivity, C defines the concentration, C 1 is the hot fluid concentration, and C a is the ambient concentration.

The thermophoretic velocity is expressed as [44]

(6) V T = − δ υ nf T r 1 ∂ T ∂ y ,

where δ represents the thermophoretic coefficient and T r 1 is the reference temperature.

The heat flux by radiation is as follows [45]:

(7) q r = − 4 σ ⁎ 3 k ⁎ ∂ T 4 ∂ y ,

where k ⁎ designates the mean absorption coefficient and σ ⁎ specifies the Stefan‒Boltzmann constant.

The expansion of the Taylor series is exploited to develop T 4 about T which is given as,

(8) T 4 ≈ T a 4 + 4 ( T − T a ) T a 3 + 6 ( T − T a ) 2 T a 2 + …

For adequately large temperature deviations, the quadratic approximation should be utilized since the term would have a substantial impact on the transfer of heat. Thus, we have

(9) T 4 ≈ 3 T a 4 + 6 T a 2 T 2 − 8 T a 3 T .

In the thermal analysis, the Rosseland thermal radiation mechanism was examined from three perspectives. The equation with LTR case yields [43].

(10) ( ρ C p ) nf V ∂ T ∂ y = k nf ∂ 2 T ∂ y 2 + μ nf k 1 u 2 + σ nf B 0 2 u 2 − ∂ ∂ y − 16 σ ⁎ T a 3 3 k ⁎ ∂ T ∂ y .

The equation with NLTR case yields [43]

(11) ( ρ C p ) nf V ∂ T ∂ y = k nf ∂ 2 T ∂ y 2 + μ nf k 1 u 2 + σ nf B 0 2 u 2 − ∂ ∂ y − 16 σ ⁎ T 3 3 k ⁎ ∂ T ∂ y .

The equation with QTR effect is given as [43]

(12) ( ρ C p ) nf V ∂ T ∂ y = k nf ∂ 2 T ∂ y 2 + μ nf k 1 u 2 + σ nf B 0 2 u 2 − ∂ ∂ y − 4 σ ⁎ 3 k ⁎ 6 T a 2 ∂ T 2 ∂ y − 8 T a 3 ∂ T ∂ y .

The similarity variables utilized here are specified below:

(13) W = u H υ f , υ f = μ f ρ f , X = x H , η = y H , θ = T − T a T h − T a , χ = C − C a C 1 − C a , P ¯ = H 2 ρ f P μ f 2 ,

where υ f is the kinematic viscosity of the fluid, μ f is the dynamic viscosity of the fluid profile, and χ is the dimensionless concentration profile.

The respective nondimensional parameters are quantified as follows: A 1 = − ∂ P ¯ ∂ X represents the pressure gradient parameter, Re 1 = V H υ f denotes the Reynolds number, Pr = μ f C p k f symbolizes the Prandtl number, Ec 1 = υ f 2 C p ( T h − T a ) H 2 is the Eckert number, S 1 = H 2 k 1 designates the shape factor of the porous media, M F = σ f B 0 2 H 2 μ f implies the magnetic field parameter, τ 1 = − δ ( T h − T a ) T r 1 quantifies the thermophoretic parameter, Sc 1 = υ f D f indicates the Schmidt number, Rd = 4 σ ⁎ T a 3 k ⁎ k f is the thermal radiation parameter, Bi i = H h i k f is the Biot number where i = 1 , 2 , h i is the convective heat transfer coefficient, and θ a = T h T a represents the temperature ratio parameter.

By applying equation (13), we get the following simplified form of the controlling equations and boundary conditions:

(14) 1 δ 1 W ″ − δ 2 Re 1 W ′ − 1 δ 1 S 1 W + A 1 − σ nf σ f M F W = 0 .

LTR case:

(15) k nf k f + 4 3 Rd θ ″ − δ 3 Re 1 Pr θ ′ + Pr Ec 1 1 δ 1 S 1 W 2 + σ nf σ f M F W 2 = 0 .

NLTR case:

(16) k nf k f + 4 3 Rd [ ( θ a − 1 ) θ + 1 ] 3 θ ″ − δ 3 Re 1 Pr θ ′ + Pr Ec 1 1 δ 1 S 1 W 2 + σ nf σ f M F W 2 = 0 .

QTR case:

(17) k nf k f + 4 Rd ( θ a − 1 ) θ + 4 3 Rd θ ″ − δ 3 Re 1 Pr θ ′ + 4 Rd ( θ a − 1 ) θ ′ 2 + Pr Ec 1 1 δ 1 S 1 W 2 + σ nf σ f M F W 2 = 0 .

(18) χ ″ − Sc 1 Re 1 χ ′ − 1 δ 1 δ 2 Sc 1 τ 1 ( θ ″ χ + θ ′ χ ′ ) = 0 ,

where,

(19) δ 1 = ( 1 − φ 1 ) 2.5 , δ 2 = 1 − φ 1 + φ 1 ρ s ρ f , and δ 3 = 1 − φ 1 + φ 1 ( ρ C p ) s ( ρ C p ) f ,

where φ 1 represents the solid volume fraction.

The boundary conditions (4) and (5) are reduced to

(20) W = 0 , k nf k f θ ′ − Bi 1 ( θ − 1 ) = 0 , χ = 1 at η = 0 ,

(21) W = 0 , k nf k f θ ′ + Bi 2 θ = 0 , χ = 0 at η = 1 .

Table 1 lists the nanofluid’s thermophysical characteristics [46].

Table 1

Thermophysical properties of nanofluid

Property Expression
Density ρ nf = ρ f 1 ‒ φ 1 + ρ S ρ f φ 1
Heat capacity ( ρ C p ) nf = ( ρ C p ) f 1 ‒ φ 1 + φ 1 ( ρ C p ) S ( ρ C p ) f
Dynamic viscosity μ nf = μ f ( 1 ‒ φ 1 ) 2.5
Thermal conductivity k nf = k f k s + 2 k f ‒ 2 ( k f ‒ k s ) φ 1 k s + 2 k f + ( k f ‒ k s ) φ 1
Electrical conductivity σ nf = σ s + 2 σ f ‒ φ 1 ( σ f ‒ σ s ) 2 σ s + 2 σ f + ( σ f ‒ σ s ) φ 1 σ f

The thermophysical characteristics of nanofluid and the base fluid are mentioned in Table 2 [42].

Table 2

Thermophysical characteristics of nanofluid and the base fluid

Symbols S.I. unit TiO 2 H 2 O
ρ kg m ‒ 3 4,250 997.1
C p J kg ‒ 1 K ‒ 1 686.2 4,179
k kg ms ‒ 3 K ‒ 1 8.9538 0.6131
σ Ω ‒ 1 m ‒ 1 2.38 × 10‒6 5.5 × 10‒6
Pr — — 6.3

The engineering coefficients, Skin friction Cf , Nusselt number Nu , and Sherwood number Sh , are defined as [46].

(22) C f = H 2 μ nf ρ f υ f 2 ∂ u ∂ y y = H Nu = H q w k f ( T h − T a ) Sh = − H D f ∂ C ∂ y y = H D f ( C 1 − C a ) ,

where the heat flux is defined as

(23) q w = − k nf ∂ T ∂ y + q r .

The reduced form of the engineering coefficients is detailed below:

(24) C f = μ nf μ f W ′ ( 1 ) .

LTR case:

(25) Nu = − θ ′ ( 1 ) k nf k f + 4 3 Rd .

NLTR case:

(26) Nu = − θ ′ ( 1 ) k nf k f + 4 3 Rd [ ( θ a − 1 ) θ ( 1 ) + 1 ] 3 .

QTR case:

(27) Nu = θ ′ ( 1 ) − k nf k f + 4 3 Rd + 4 Rd ( θ a − 1 ) θ ( 1 ) ,

and

(28) Sh = − χ ′ ( 1 ) .

3 Numerical approach

Due to the higher order and linked character of the simplified equations (14)–(18) and boundary conditions (20)–(21), solving them analytically is very challenging. In order to solve this problem numerically, we may begin by converting the provided system of equations into first order. This can be achieved by using the following substitutions.

( W , W ′ , θ , θ ′ , χ , χ ′ ) ≈ ( q 1 , q 2 , q 3 , q 4 , q 5 , q 6 )

(29) W ″ = − − δ 2 Re 1 q 2 − 1 δ 1 S 1 q 1 + A 1 − σ nf σ f M F q 1 δ 1 .

LTR case:

(30) θ ″ = − − δ 3 Re 1 Pr q 4 + Pr Ec 1 q 1 2 1 δ 1 S 1 + σ nf σ f M F k nf k f + 4 3 Rd − 1 .

NLTR case:

(31) θ ″ = − − δ 3 Re 1 Pr q 4 + Pr Ec 1 q 1 2 1 δ 1 S 1 + σ nf σ f M F × k nf k f + 4 3 Rd [ ( θ a − 1 ) q 3 + 1 ] 3 − 1 .

QTR case:

(32) θ ″ = − − δ 3 Re 1 Pr q 4 + 4 Rd ( θ a − 1 ) q 4 2 + Pr Ec 1 q 1 2 1 δ 1 S 1 + σ nf σ f M F × k nf k f + 4 Rd ( θ a − 1 ) q 3 + 4 3 Rd − 1 .

(33) χ ″ = − − Sc 1 Re 1 q 6 − 1 δ 1 δ 2 Sc 1 τ 1 ( [ RHS of specific case ] q 5 + q 4 q 6 ) .

The boundary conditions (4) and (5) are reduced to

(34) q 1 = 0 , k nf k f q 4 − Bi 1 ( q 3 − 1 ) = 0 , q 5 = 1 at η = 0 ,

(35) q 1 = 0 , k nf k f q 4 + Bi 2 q 3 = 0 , q 5 = 0 at η = 1 .

The translated equations are solved numerically using the RKF-45 approach, which is a scheme based on the Runge‒Kutta Fehlberg fourth- to fifth-order method. This is done by assigning values to the parameters, selecting the step size of about 0.01, and the error tolerance of about 10−6. The present numerical scheme’s outcomes have been validated by comparing them to previous studies in the literature. It has been determined that they closely correspond to each other, as shown in Table 3. The flow chart of the present numerical scheme is shown in the figure Figure 2.

Table 3

Validation of the numerical value of W ( η ) for variation in η in the absence of S 1 , M F , δ 1 , δ 2 keeping Re 1 and A 1 to 1

η → 0 0.2 0.4 0.6 0.8 1
Ref. [47] Numerical 0 0.07114875 0.11376948 0.12154600 0.08676372 0
Analytical 0 0.07114875 0.09639032 0.12154600 0.08676372 0
Present work RKF-45 0 0.07114881 0.09939754 0.12154691 0.08676379 0
Figure 2 Flow diagram of the numerical scheme.
Figure 2

Flow diagram of the numerical scheme.

The algorithm of the RKF-45 method is given below:

(36) y i + 1 = y i + 25 216 K 1 + 1 , 408 2 , 565 K 3 + 2 , 197 4 , 104 K 4 − 1 5 K 5 ,

(37) z i + 1 = y i + 16 135 K 1 + 6 , 656 12 , 825 K 3 + 28 , 561 56 , 430 K 4 − 9 50 K 5 + 2 55 K 6 .

The six steps of the method are given below

(38) K 1 = hf ( x i , y i ) ,

(39) K 2 = hf x i + 1 4 h , y i + 1 4 K 1 ,

(40) K 3 = hf x i + 3 8 h , y i + 3 32 K 1 + 9 32 K 2 ,

(41) K 4 = hf x i + 12 13 h , y i + 1 , 932 2 , 147 K 1 − 7 , 200 2 , 147 K 2 + 7 , 296 2 , 147 K 3 ,

(42) K 5 = hf x i + h , y i + 439 216 K 1 − 8 K 2 + 3 , 680 513 K 3 − 845 4 , 104 K 4 ,

(43) K 6 = hf x i + 1 2 h , y i − 8 27 K 1 + 2 K 2 − 3 , 544 2 , 565 K 3 − 1 , 859 4 , 104 K 4 − 11 40 K 5 .

Ordinary differential equations are frequently encountered in engineering and scientific problems, and they are typically solved using the Runge‒Kutta method [9,38,40]. RKF-45 is an embedded method from the Runge‒Kutta family, in which methods with different orders and similar error constants are created by combining identical function evaluations. The procedure combines a fifth-order scheme with a fourth-order Runge‒Kutta scheme that helps in approximating the ODE solutions with a high degree of accuracy when compared with lower-order methods. One of the primary benefits of RKF-45 method is its ability to adaptively adjust the step size during integration which aids in maintaining accuracy and capturing rapid modifications in the corresponding solution. The fifth-order component helps in accuracy improvement, whereas the fourth-order component offers the computational efficiency which makes RKF-45 an efficient choice solving ODEs in numerous kinds of engineering and scientific applications. For modelling the behaviour of physical systems, for analysing the signals in telecommunication processes and for computer simulation, Runge‒Kutta method can be implemented.

4 Results and discussion

The movement of a nanofluid in a microchannel which is enclosed in a porous medium and is constrained by two horizontal parallel plates is explored in the current study. Furthermore, the velocity, concentration, and temperature profiles are analysed with respect to increasing values of various parameters involved in the study. The influence of S 1 on W is explored in Figure 3. Here, the upsurge in values of S 1 declines the W . Because it decreases the permeability of the medium and enhances the opposition to liquid circulation, a rise in the form factor of a permeable media results in a decrement in the velocity field of a nanoliquid. The presence of a porous component will function as a barrier to the fluid flow. The enhancement in the shape factor that displays the drag and frictional force on the liquid has resulted in the restriction of the flow routes that the liquid takes as it moves through the system.

Figure 3 Impact of S 1 {S}_{1} on W W (Blue S 1 = 0.1 {S}_{1}=0.1 , Red S 1 = 0.5 {S}_{1}=0.5 , Green S 1 = 1 {S}_{1}=1 ).
Figure 3

Impact of S 1 on W (Blue S 1 = 0.1 , Red S 1 = 0.5 , Green S 1 = 1 ).

Figure 4 explores the impact of S 1 on θ for three distinct cases, namely, liquid flow with LTR, liquid flow with NLTR, and liquid flow with QTR. In all three scenarios, increasing values of S 1 enhances heat transmission. A rise in the porous medium shape factor might lead to a surface structure that is more specified and sophisticated inside the porous medium. Because of this, there is a potential for an enhanced surface area for contact between the solid matrix and the liquid, which might result in an improvement in heat transmission. Because of the increased permeable media complexity, there is a possibility that the nanoliquid may experience enhanced turbulence or mixing, which might result in an increase in the thermal field. Moreover, the fluid flow with QTR shows increased heat transfer compared to remaining two cases with respect to S 1 .

Figure 4 Impact of S 1 {S}_{1} on θ \theta (Blue S 1 = 0.1 {S}_{1}=0.1 , Red S 1 = 0.5 {S}_{1}=0.5 , Green S 1 = 1 {S}_{1}=1 ).
Figure 4

Impact of S 1 on θ (Blue S 1 = 0.1 , Red S 1 = 0.5 , Green S 1 = 1 ).

Figure 5 represents the influence of M F on W . The escalation of M F decays the W . The liquid experiences the Lorentz force due to the existence of a magnetic field. This force causes a reduction in velocity and operates in a direction that is perpendicular to the motion of the liquid and the magnetic field. In addition, the velocity field declines as the magnetic parameter increases. Figure 6 signifies the impact of M F on θ for three different cases, namely, liquid flow with LTR, liquid flow with NLTR, and liquid flow with QTR. Here, the rise in values of M F enhances the heat transmission for all three instances. Within a nanofluid, an upsurge in the magnetic field parameter may result in an enhancement in the heat transfer. The application of M F to a carrying liquid, like a nanofluid in a microchannel, causes the generation of Lorentz forces that impact its properties of movement. The Lorentz forces modify the velocity profile W and circulation patterns in the microchannel, resulting in improved thermal transmission. Moreover, the fluid flow with QTR shows increased heat transfer compared to the remaining two cases with respect to M F .

Figure 5 Impact of M F {M}_{F} on W W (Blue M F = 1 {M}_{\text{F}}=1 , Red M F = 2 {M}_{\text{F}}=2 , Green M F = 3 {M}_{\text{F}}=3 ).
Figure 5

Impact of M F on W (Blue M F = 1 , Red M F = 2 , Green M F = 3 ).

Figure 6 Impact of M F {M}_{\text{F}} on θ \theta (Blue M F = 1 {M}_{\text{F}}=1 , Red M F = 2 {M}_{\text{F}}=2 , Green M F = 3 {M}_{\text{F}}=3 ).
Figure 6

Impact of M F on θ (Blue M F = 1 , Red M F = 2 , Green M F = 3 ).

Figure 7 portrays the impact of Ec 1 on θ for three different cases as mentioned above. Here, the upsurge in values of Ec 1 improves the θ . Moreover, the fluid flow with NLTR and LTR shows less heat transfer compared QTR case with respect to an increase in Ec 1 values. If the Eckert number goes up, it means that the fluid’s velocity has increased to the point that it has overtaken its thermal energy. Because of the increased kinetic energy, the liquid can be mixed and distributed more effectively, which leads to an increment in the amount of heat that is transmitted. The Eckert number will play an essential role in the electronic and industrial equipment’s cooling. Impact of Bi 1 and Bi 2 on θ is depicted in Figures 8 and 9, respectively for three different cases. Here, the rise in values of Bi 1 improves the heat transport for all three cases but converse trend is depicted for increase in Bi 2 values. Physically, the Biot numbers refer to the ratio of the resistance to heat distribution at the fluid-solid interface to the internal thermal resistance of the fluid. A substantially lesser resistance during the solid and fluid contact phase is indicated by higher Biot number values. Due to the presence of extra nanoparticles in the base liquid, the nanofluid displays a greater degree of heat dispersion in the vicinity of the bottom plate. However, in the upper plate, the nanofluid will prevail because of a variety of characteristics and dynamics of flow that are peculiar to the top plate. The dispersion features of the upper plate also differ from those exhibited in the bottom plate. Moreover, the fluid flow with NLTR and LTR shows less heat transfer compared QTR case with respect to increase in Bi 1 and Bi 2 values.

Figure 7 Impact of Ec 1 {\text{Ec}}_{1} on θ \theta (Blue Ec 1 = 1 {\text{Ec}}_{1}=1 , Red Ec 1 = 2 {\text{Ec}}_{1}=2 , Green Ec 1 = 3 {\text{Ec}}_{1}=3 ).
Figure 7

Impact of Ec 1 on θ (Blue Ec 1 = 1 , Red Ec 1 = 2 , Green Ec 1 = 3 ).

Figure 8 Impact of Bi 1 {\text{Bi}}_{1} on θ \theta (Blue Bi 1 = 0.5 {\text{Bi}}_{1}=0.5 , Red Bi 1 = 0.8 {\text{Bi}}_{1}=0.8 , Green Bi 1 = 1.2 {\text{Bi}}_{1}=1.2 ).
Figure 8

Impact of Bi 1 on θ (Blue Bi 1 = 0.5 , Red Bi 1 = 0.8 , Green Bi 1 = 1.2 ).

Figure 9 Impact of Bi 2 {\text{Bi}}_{2} on θ \theta (Blue Bi 2 = 0.5 {\text{Bi}}_{2}=0.5 , Red Bi 2 = 0.8 {\text{Bi}}_{2}=0.8 , Green Bi 2 = 1.2 {\text{Bi}}_{2}=1.2 ).
Figure 9

Impact of Bi 2 on θ (Blue Bi 2 = 0.5 , Red Bi 2 = 0.8 , Green Bi 2 = 1.2 ).

Figure 10 signifies the influence of Rd on θ for three different cases, namely, liquid flow with LTR, liquid flow with NLTR, and liquid flow with QTR. Here, the rise in values of Rd declines the heat transport for all three cases. When the values of the radiation parameter are high, this might hinder the convection heat transport that occurs in the channel. The temperature differential between the liquid and the walls of the channel is essential to the convection process. Convective heat transport is less efficient when radiative heat transport is dominant. This is because the temperature differential between the liquid and the walls can become smaller when radiative heat transfer is dominant. Moreover, the fluid flow with LTR case shows less heat transfer compared to the remaining two cases. Impact of τ 1 on χ is portrayed in Figure 11. Here, the growth in values of τ 1 improves the χ . The τ 1 is linked to the thermophoretic force, which describes a phenomenon that influences the motion of nanoparticles, in reaction to differences in temperature. Temperature gradients may have an effect on the motion of particles. The thermophoretic force may have some effect on the nanoparticles distribution in a nanoliquid contained inside a channel which results in an enhanced τ 1 .

Figure 10 Impact of Rd \text{Rd} on θ \theta (Blue Rd = 1 \text{Rd}=1 , Red Rd = 2 \text{Rd}=2 , Green Rd = 3 \text{Rd}=3 ).
Figure 10

Impact of Rd on θ (Blue Rd = 1 , Red Rd = 2 , Green Rd = 3 ).

Figure 11 Impact of τ 1 {\tau }_{1} on χ \chi (Blue τ 1 = 0.1 {\tau }_{1}=0.1 , Red τ 1 = 0.5 {\tau }_{1}=0.5 , Green τ 1 = 1 {\tau }_{1}=1 ).
Figure 11

Impact of τ 1 on χ (Blue τ 1 = 0.1 , Red τ 1 = 0.5 , Green τ 1 = 1 ).

The discrepancy of C f for change in φ 1 and M F is displayed in Figure 12. Here, the rise in values of φ 1 and M F improves the C f . The C f goes up because of the Lorentz force, which is generated whenever there is a rise in the values of M F and acts in opposition to the motion of the fluid. Figure 13 portrays the behaviour of Nu for change in φ 1 and Rd for three different cases, namely, liquid flow with LTR, liquid flow with NLTR and liquid flow with QTR. Here, the rise in φ 1 value improves the Nu but conflict trend is seen for increased values of Rd . Here, the nanoliquid flow with QTR shows less heat transfer rate compared to remaining two cases. This trend is due to the addition of φ 1 will improves the surface area to the distribution of the heat. Figure 14 portrays the behaviour of Sh for change in φ 1 and τ 1 . Here, the rise in φ 1 value improves the Sh but conflict trend is seen for increased values of τ 1 . When τ 1 is intensified, nanoparticles encounter increased hindrance to their travel across temperature variations, leading to a reduction in the total Sh . On the other hand, when φ 1 is increased, more particles are added to the framework, increasing the amount of surface area accessible for the transfer of mass. The rise in particle number facilitates heightened occurrences of collisions and contacts among nanoparticles and the adjacent liquid, resulting in an escalation of Sh . Further, Table 4 is drawn to show the heat transfer rate for different values of dimensionless constraints for three different cases. It is found that the QTR case performs well in the distribution of temperature.

Figure 12 Variation of C f {C}_{\text{f}} for change in φ 1 {\varphi }_{1} and M F {M}_{\text{F}} (Blue φ 1 = 0.01 {\varphi }_{1}=0.01 , Red φ 1 = 0.03 {\varphi }_{1}=0.03 , Green φ 1 = 0.05 {\varphi }_{1}=0.05 ).
Figure 12

Variation of C f for change in φ 1 and M F (Blue φ 1 = 0.01 , Red φ 1 = 0.03 , Green φ 1 = 0.05 ).

Figure 13 Variation of Nu \text{Nu} for change in φ 1 {\varphi }_{1} and Rd \text{Rd} (Blue φ 1 = 0.01 {\varphi }_{1}=0.01 , Red φ 1 = 0.03 {\varphi }_{1}=0.03 , Green φ 1 = 0.05 {\varphi }_{1}=0.05 ).
Figure 13

Variation of Nu for change in φ 1 and Rd (Blue φ 1 = 0.01 , Red φ 1 = 0.03 , Green φ 1 = 0.05 ).

Figure 14 Variation of Sh \text{Sh} for change in φ 1 {\varphi }_{1} and τ 1 {\tau }_{1} (Blue φ 1 = 0.01 {\varphi }_{1}=0.01 , Red φ 1 = 0.03 {\varphi }_{1}=0.03 , Green φ 1 = 0.05 {\varphi }_{1}=0.05 ).
Figure 14

Variation of Sh for change in φ 1 and τ 1 (Blue φ 1 = 0.01 , Red φ 1 = 0.03 , Green φ 1 = 0.05 ).

Table 4

Numerical estimation of Nu for different dimensionless constraints keeping A 1 = Re 1 = 1 , φ 1 = 0.01 & θ a = 0.9 ( for NLTR and QTR cases )

Parameters Nu
S 1 M F Rd Bi 1 Bi 2 Ec 1 LTR NTLR QTR
0.1 0.1 1 0.5 0.5 1
0.1 0.93090 0.83959 1.104920
0.5 0.93657 0.84412 1.112782
1.0 0.94254 0.84888 1.121080
1 0.94223 0.84863 1.120648
2 0.95114 0.85574 1.133071
3 0.95738 0.86071 1.141802
1 0.93090 0.83959 1.104920
3 1.57084 1.41590 1.926169
5 2.11835 1.90680 2.533648
0.5 0.93090 0.83959 1.104920
0.8 0.95013 0.84950 1.121576
1.2 0.96117 0.85506 1.130945
0.5 0.93090 0.83959 1.104920
0.8 1.33443 1.23621 1.585268
1.2 1.75772 1.67344 2.081818
1 0.93090 0.83959 1.104920
2 0.93403 0.84209 1.109258
3 0.93716 0.84459 1.113600

5 Conclusion

This work examines the complex interaction between QTR, MHD, and TPD in a permeable microchannel. The results of the current analysis are enumerated as follows:

  • The increase in porosity and magnetism will decrease velocity while enhancing thermal dispersion.

  • The existence of magnetic, porous, and Eckert numbers leads to a more pronounced thermal distribution, especially in QTR case.

  • The improvement in the thermophoretic constraint will improve the concentration.

  • The bottom plate exhibits enhanced heat dispersion, whereas the top plate demonstrates the opposite behavior in the presence of Biot number.

  • Surface drag force and mass transfer rate improve with the rise in the volume percentage corresponding to magnetic and thermophoretic constraints.

  • The rate of thermal distribution is higher in the case of linear thermal distribution and lower in the case of NLTR, considering the influence of radiation and solid fraction factors.

The results of this research have applications in various fields, including the construction of pipelines, wind power generation, turbo machinery, improving the thermal performance in microscale devices, electronic devices cooling, health care equipments and the marine industry. This information may be used by engineers to optimize the building and maintenance of a wide variety of systems, with the goals of reducing energy consumption, enhancing efficacy, and making cost savings, all while supporting sustainable practices. The present work is carried out to investigate QTR, MHD, and the TPD of particles in a permeable microchannel. The current work can be extended to examine the heat and mass transfer over different kinds of non-Newtonian nano, hybrid, and ternary nanofluids under various physical aspects.

Acknowledgments

This work has been funded by the Universiti Kebangsaan Malaysia project number “DIP-2023-005.” Also, the authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University, Abha, Saudi Arabia for funding this work through a Large Research Project under grant number RGP.2/2/45. In addition, this study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1445).

  1. Funding information: This work has been funded by the Universiti Kebangsaan Malaysia project number “DIP-2023-005.” Also, the authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University, Abha, Saudi Arabia for funding this work through a Large Research Project under grant number RGP.2/2/45. In addition, this study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1445).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets used and/or analysed during the current study are available from the corresponding author upon reasonable request.

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Received: 2024-01-24
Revised: 2024-05-18
Accepted: 2024-05-27
Published Online: 2024-06-15

© 2024 the author(s), published by De Gruyter

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

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  124. Future prospects of gold nanoclusters in hydrogen storage systems and sustainable environmental treatment applications
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https://doi.org/10.1515/ntrev-2024-0045
Keywords for this article
nanofluid; microchannel; quadratic thermal radiation; thermophoretic particle deposition
Creative Commons
BY 4.0

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  137. Special Issue on Advanced Nanomaterials and Composites for Energy Conversion and Storage - Part II
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