Computational analysis of water-based silver, copper, and alumina hybrid nanoparticles over a stretchable sheet embedded in a porous medium with thermophoretic particle deposition effects

Nomenclature

C 1

concentration

ρ

density ( kg m − 3 )

D B

diffusivity ( m 2 s − 1 )

( x , y )

directions ( m )

μ

dynamic viscosity ( kg m − 1 s − 1 )

q w

heat flux

Q 0

heat source/sink ( W m − 3 K − 1 )

Hs

heat source/sink parameter

h s

heat transfer parameter

ν

kinematic viscosity ( m 2 s − 1 )

J w

mass flux

Δ

Newtonian heating parameter

Nu x

Nusselt number

K ⁎

porosity ( m 2 )

λ

porous parameter

Pr

Prandtl number

T r

reference temperature

Re

Reynolds number

Sc

Schmidt number

τ w

shear stress

Sh x

Sherwood number

Cf x

skin friction

ϕ

solid volume fraction

C p

specific heat ( J kg − 1 K − 1 )

a

stretching rate ( s − 1 )

u w

stretching velocity ( m s − 1 )

T 1

temperature ( K )

k

thermal conductivity ( kg ms − 3 K − 1 )

α

thermal diffusivity ( m 2 s − 1 )

τ 1

thermophoretic parameter

V T

thermophoretic velocity

t

time ( s )

γ 1

unsteadiness parameter

( u 1 , v 1 )

velocity components ( m s − 1 )

Subscript

thnf

ternary hybrid nanofluid

nf

nanofluid

f

fluid

hnf

hybrid nanofluid

w

wall

∞

ambient

Abbreviations

BC

boundary condition

BVP

boundary value problem

CWT

common wall temperature

HSS

heat source/sink

LPCM

Laguerre polynomial collocation method

NH

Newtonian heating

ODE

ordinary differential equation

RKF-45

Runge Kutta Fehlberg’s fourth-fifth-order method

T-HNF

ternary hybrid nanofluid

TPD

thermophoretic particle deposition

1 Introduction

The development of metal particles has led to the creation of a new class of fluid recognized as nanofluids, which are made of tiny materials with a diameter of just a few nanometers. A two-phase mixture is created when very small metallic particles, or nanomaterials, are combined with a saturated liquid. The temperature conductivity of these liquids is far more important than that of the basic liquid. It has been discovered that nanofluids offer greater thermal and physical attributes to basic liquids. Under the conditions of the creation of various kinds of innovation based on nanotechnology, a new liquid for heat removal was produced, which led to technical innovation worldwide. There are hybrid nanofluids with enhanced thermal conductivity. Ternary hybrid nanofluids (T-HNFs) are the most recent finding. Even two forms of nanoparticles tightly incorporated into the base liquid offer better possibilities in terms of heat transmission and total efficacy. Albalawi et al. [1] investigated the thermal distribution of nanofluid and its aggregation impacts over a coaxial cylinder. Kumar et al. [2] explored the ternary nanofluid movement in the presence of magnetic dipole over forced, mixed, and free convection scenarios. Vinutha et al. [3] examined the MHD ternary nanofluid stream among two parallel plates with RSM and sensitivity analysis. With the impact of entropy production, Salawu et al. [4] explored the thermal power application of the nanofluid stream through a vertical channel. The flow of nanoliquid past a stretchy plate with the consequence of thermal radiation was probed by Oke et al. [5]. Madhu et al. [6] looked into the heat transport analysis of the nanofluid stream through a revolving cone. Wang et al. [7] probed the influence of the flow of nanofluid across the stretchable surface with heat and mass transport attributes. The stream of ternary nanofluid past a stretching surface with the effect of an entropy generation was scrutinized by Khan et al. [8]. Karthik et al. [9] delineated the flow of ternary nanofluid past a wedge with a radiation impact. Saadeh et al. [10] conducted a study on statistical analysis for computing two-point fuzzy boundary value issues by replicating the kernel scheme.

Nanoparticles like silver, copper, and alumina have diverse applications in real life. First, silver nanoparticles play a prominent role in the medical field, which includes dressings for injuries, surgical implants, and delivery of drug devices. Several authors investigated the synthesis and applications of silver nanoparticles. According to Marin et al. [11], silver nanoparticles have a constant quantity of flexible qualities, which supports a wider range of uses in biomedical and related domains. Beyene et al. [12] conducted a review of the manufacturing paradigm and uses of silver nanoparticles. Abou El-Nour et al. [13] looked into the synthesis of silver nanoparticles as well as their uses. Second, copper nanoparticles are well acknowledged because of their exceptional ability to conduct heat and electricity. These features render them essential in the electronics sector, where they are employed in interconnects, printed circuit panels, and as fillers with conductivity in sealants and varnishes. Copper nanoparticles are also being investigated for their applications in antibacterial paints, farming, and as additives in lubrication and polymers. Din and Rehan [14] investigated the possibility of fabricating copper nanoparticles with a variety of structural features and beneficial biological effects by using novel environmentally friendly techniques. According to Crisan et al.’s [15] study, one of the elements that are most prevalent and essential to an organism’s regular operation is copper. Kumar et al. [16] investigated a polyethyleneimine–chromium oxide compound sensor with a perforated copper clad as its substrate for sensing. Further, alumina nanoparticles, which are made of aluminum oxide, possess remarkable hardness, exceptional thermal resistance, and chemical insensitivity. These characteristics render them highly beneficial for the manufacturing of stones, pottery, and refractory components utilized in the creation of cutting instruments, polishing axles, and other products requiring high temperatures. Alumina nanoparticles are utilized as catalyst supports in the discipline of catalysis because of their expansive surface area and temperature durability, which enable a wide range of chemical reactions and treatment of environmental-related issues. Ziva et al. [17] assessed recent advancements in the manufacturing of aluminum oxide nanoparticles. There are several industrial uses for alumina. Omodele et al. [18] looked at the usage of aluminum as an early stage for the creation of nanoparticles for water purification. Pourmadadi et al. [19] investigated the production and properties of porous alumina to create feasible nanotechnology for drug delivery applications. Mahesh et al. [20] evaluated the removal of contaminants from groundwater using alumina-based nanoparticles.

It has already become a crucial word of concept in the 22nd century of engineering due to its wide impact on numerous branches of this vast field. Permeable media (a system of holes or pores interconnected in a solid matrix) has aroused great interest within mechanics-framework for liquids. As we look at more complex engineering problems, such as liquid flow in living tissues, groundwater movement, and storage of the oil reserves that may not be visible to others outside this industry but affect our very lives, porous medium research and application are becoming increasingly important. Alhadhrami et al. [21] investigated the chemical reactions’ impact on the stream of fluid past a stretchy surface. Shamshuddin et al. [22] scrutinized the stream of fluid across a permeable medium with the influence of convection. Wang et al. [23] explored the flow of nanoliquid via a stretchy surface in a porous media. The consequence of the magnetic effect on the liquid stream in a porous stretchable surface was estimated by Alhadhrami et al. [24]. Dharmendar Reddy et al. [25] scrutinized the stream of heat transfer liquid through the stretchy sheet immersed in a permeable medium. While analyzing a situation in which there are significant temperature variations between the liquid at the surface and the environmental liquid, the term “heat source/sink (HSS)” is employed in the study of thermal energy. Because there are so many different ways that heat sources and sinks are used in energy preservation methods, scientists must investigate ways to control their characteristics in engineering difficulties to meet contemporary needs. Wang et al. [26] delineated the stream of liquid past the stretchy shallow with the consequence of HSS. The impact of HSS on the Rabinowitsch liquid stream via a circular tube was delineated by Chu et al. [27]. The flow of nanoliquid across an elongated surface in the presence of HSS was inspected by Thumma et al. [28]. The fractional diffusion equation numerical solutions employing a finite-difference approach were studied by Saadeh [29]. The stream of magnetized micropolar liquid via a porous stretched sheet with an HSS was debriefed by Ram et al. [30].

In many engineering processes, such as heat exchangers, air cleaners, building ventilation systems, nuclear reactor safety, and powdered coal burners, thermophoretic particle deposition (TPD) in the flow of liquid is crucial. The thermophoresis phenomenon is the consequence of different particle kinds responding differently to a temperature gradient. Thermophoresis is a process that dramatically raises the depositing motion of minute particles in the way of decreasing temperature but has no effect on huge particles. Small, minute nanoparticles deposited in a nonisothermal gas will develop a velocity in this process. Thermophoresis is the process by which minute particles settle on a cold surface. Wang et al. [31] inspected the significance of TPD on the fluid movement via an inclined surface. Bai et al. [32] conducted a study on the theoretical computation of the environmental and economic advantages of dam flow using open-source field operations and modification. By using the heat flux model, Bashir et al. [33] probed the flow of liquid through a stretchable surface with the TPD effect. The stream of the Oldroyd-B fluid across the stretchy sheet with the upshot of radiation and TPD was elongated by Wang et al. [34]. The flow of liquid via a cylinder with the consequence of TPD was examined by Yasir et al. [35]. Many researchers were interested in the influence of the fluid flow field on stretching sheets, leading to a large amount of study being done. Shaping is one of the most significant industrial operations that has historically been used to increase ductility and high-precision part manufacture. Numerous technical problems can be solved by studying the fluid flow on stretched sheets, including casting, drawing or plastic films, hot rolling, polymer extrusion, and others. Shamshuddin et al. [36] deliberated the stream of liquid via a stretchy sheet with the magnetic impact. Alsaiari et al. [37] conducted computational research on heat transfer and circulation resistance in a solar-powered air conditioner with a ribbed absorption panel. Saadeh et al. [38] conducted a case study on the mathematical simulation and stability analysis of the innovative fraction model in the derivative of the Caputo operator. Shamshuddin et al. [39] investigated the flow of nanoliquid across a stretchable surface using a quasi-linearization approach. Utilizing homotopic simulation, Shamshuddin et al. [40] probed the motion of nanofluid across a stretchy sheet.

The influences of HSS, TPD, and porous media are examined concerning the time-dependent T-HNF stream across a stretchy surface. Further, Newtonian heating (NH) and common wall temperature (CWT) scenarios are considered for inspecting the heat transfer analysis in the present investigation. Suitable similarity transformations are used to convert the governing equations of the studied model into ordinary differential equations (ODEs). By using Laguerre polynomial collocation method (LPCM), the resulting dimensionless equations are solved and the RKF-45 is employed for the comparison of the achieved numerical results. The influence of various factors on the different profiles is exhibited with the aid of graphical representations.

The following research questions are the focus of the present work’s investigation.

1. What are the changes observed in the velocity profiles for changing values of porous parameters?

2. How does the HSS parameter affect the temperature profile with CWT and NH case?

3. How will the thermophoretic parameter impact the concentration profile?

2 Mathematical background of the problem

Let’s examine an incompressible, time-dependent, two-dimensional T-HNF flow with the consequence of HSS and TPD across a stretched surface. Let ( u 1 , v 1 ) be the velocity in the ( x , y ) direction as depicted in Figure 1. The temperature and concentration are taken as T 1 and C 1 . The ambient temperature and far-field concentration are assumed as T ∞ and C ∞ . Wall concentration is denoted by C w , and wall temperature is represented by T w .

Figure 1

Design of the flow problem.

The stretchable velocity is given as u w = a x ( 1 − β t ) , Q 0 = Q 1 ( 1 − β t ) is the rate of heat generation/absorption, and the porous medium permeability is indicated as K ⁎ = K 1 ( 1 − β t ) − 1 . On the basis of the aforementioned assumptions, the governing equations are given as follows [41–44]:

Boundary conditions (BCs) [44]:

The subsequent similarity variables are presented as follows:

3 Elucidation of LPCM

The following expression is appropriate to develop Laguerre polynomials:

In addition, the aforementioned equation satisfies the recursive correlation as shown by

The first two terms L 0 ( ζ ) and L 1 ( ζ ) are obtained by using equation (15), and the subsequent terms can be determined by applying equations (18) and (19). The initial five terms, which are polynomials of different degrees, are given as follows:

The boundary value problem (BVP) of order n can be approximated by a linear combination of the Laguerre polynomials given in equation (20) as follows:

where a 0 , a 1 , … , a N are constants to be determined. On differentiating the aforementioned equation up to n number of times and approximating the nth order BVP yields:

Like coefficients of equation (22) ( a j , j = 0 , 1 , … , N ) are accumulated after each term in equation (22) has been expanded, leading to:

Given that there is the same number of BCs and order of BVP in this study, with the help of the BCs, l number of equations are produced, l 2 for each lower and higher boundary, respectively. At the collocation points, equation (23) is used to produce the remaining N − l + 1 equations. To proceed this, the collocation points are produced using the following expression:

By using these collocation points, equation (23) can be approximated as follows:

for i = 1 , 2 , … , N − ( l − 1 ) and a j , j = 0 , 1 , … , N .

4 Results and discussion

This section presents the interpretation of arising physical characteristics on three distinct profiles. Temperature and concentration profiles for the NH and CWT examples were shown along with the physical explanation. The approximate LPCM is shown for velocity, temperature, and concentration profiles. In addition, Table 1 displays the thermophysical characteristic values of nanoparticles and the base liquid.

Figure 2 exhibits the role of the porosity parameter λ on velocity. It is obtained that an increase in λ yields a reduction in fluid movement. This is because a greater amount of porous media will suffer drag force and impede fluid travel, which will lower the fluid’s velocity. Figure 3 shows the role of λ on the temperature profile θ ( η ) for NH and CWT cases. Advance in λ will enhance the system’s temperature dispersal. This is due to the thermal boundary layer has grown due to an improvement in the porosity factor. The illustration makes it evident that there is less thermal dispersion in the NH case than in the CWT scenario.

Figure 2

Impact of λ on f ′ .

Figure 3

Impact of λ on θ .

Figure 4 demonstrates the significance of Hs on θ ( η ) for CWT and NH cases. The heat circulation inside the system will intensify as Hs values rise. The heat sink will physically act as an exchanger, bringing heat from the outside into the nanofluid. As a significance, with heat sink, the thermal circulation is minimum, and in the case of the heat source, the surface produces the temperature. In this case, the heat source shows improved thermal dispersion than the heat sink. In comparison to the CWT scenario, the NH case exhibits a greater amount of heat dispersion. The influence of τ 1 on χ ( η ) is exposed in Figure 5. The upsurge in τ 1 values will decline χ ( η ) . This is because an elevation in the temperature gradient causes particle mobility. Consequently, as τ 1 values rise, the concentration decreases. In comparison to CWT cases, NH cases have greater concentration.

Figure 4

Impact of Hs on θ .

Figure 5

Impact of τ 1 on χ .

Figure 6 displays the impact of the porosity parameter and ϕ 3 on skin friction. Upsurge in ϕ 3 reduces the surface drag force. This causes due to increase in solid volume percentage raise the momentum boundary layer’s thickness and increasing the porosity parameter progressively alter the liquid’s flow, which decreases the surface drag force. Figure 7 illustrates the consequence of Nu x on Hs for numerous values of ϕ 3 . As the values of ϕ 3 rise, the rate of heat transmission increases. Furthermore, the figure illustrates how the rate of thermal dispersion falls when the Hs values increase from −0.1 to 0.1. Specifically, in the CWT scenario, the thermal distribution rate decreases from sink to source, but in the NH situation, the opposite tendency is seen. Also, improvement of ϕ 3 rises the thermal boundary layer leading to enhancement in Nu x with the upsurge of Hs from sink to source. Compared to the NH instance, the CWT case has a higher heat transmission rate. Figure 8 depicts the difference of Sh x on τ 1 for numerous values of ϕ 3 . The mass transport rate drops with the increase in values of ϕ 3 and τ 1 . The reason for this is that temperature differences in τ 1 cause molecules to move more quickly, which lowers the mass transport rate. The addition of ϕ 3 will reduce the mass transport rate due to vary in τ 1 values. Further, Table 2 is presented to exhibit the results of numerical solutions achieved using LPCM, which are compared against RKF-45 results. The compared outcomes of − Cf x , Nu x , and Sh x reveal appealing convergence among them. Table 3 is the validation table for the current work with the existing literature by limiting the values A 1 = A 2 = γ 1 = 0 . The outcomes will best match each other.

Figure 6

Impact of λ and ϕ 3 on Cf x .

Figure 7

Impact of Hs and ϕ 3 on Nu x .

Figure 8

Impact of τ 1 and ϕ 3 on Sh x .

Table 4 shows the computational values of Cf x % , Nu x % (CWT and NH cases) [52], and Sh x % for γ 1 = 0.1 , ϕ 1 = ϕ 2 = ϕ 3 = 0.01 for ternary nanofluid and ϕ 3 = 0.01 for nanofluid. It has been determined that the rate of thermal distribution is enhanced by approximately 2.73% in the CWT case, 4.73% in the NH case, and 2.99% in the rate of mass transfer when transitioning from a ternary hybrid nanofluid to a (normal) nanofluid, for different values of parameters. Hence, the ternary nanofluid shows significant improvement than nanofluid in engineering factors.

5 Conclusions

This study investigates the effects of HSS, TPD, and porous media on the time-dependent T-HNF stream over a stretchable surface under NH and CWT conditions. The researched model’s governing equations are converted into ODEs with the use of appropriate similarity transformations. The LPCM is employed to solve the resulting dimensionless equations. The key elements identified in the current study are as follows:

• As the porosity parameter increases, the velocity profile decreases.

• The thermal profile will intensify in the occurrence of permeable media and a HSS parameter.

• With an improvement in the thermophoretic parameter, the concentration falls.

• The surface drag force is high in the presence of porosity factor and solid volume fraction.

• Compared to the NH scenario, the CWT case has a higher rate of heat dispersion.

• By comparing the LPCM outcomes with the RKF-45 findings, the suggested approach’s efficiency and accuracy were confirmed. Also, the computational approach proved trustworthy for the under considered polynomials and collocation points.

• The solution resulting from LPCM highlights the effectiveness of this approach in dealing with the developed nonlinear problem. Furthermore, it has been demonstrated to obtain appropriate approximations for this problem by employing a minimal number of Laguerre polynomials expansion terms.

The present study is limited to examine the heat and mass transfer of HSS, TPD, and porous media impacts on the time-dependent T-HNF stream over a stretchable surface with NH and CWT conditions. The present work can be extended to examine various nanoparticle combinations, different fluid models, and aggregation impacts with different physical conditions.