Mixed Convection Nanofluid Flow with Heat Source and Chemical Reaction over an Inclined Irregular Surface.

Two-dimensional mixed convection radiative nanofluid flow along with the non-Darcy permeable medium across a wavy inclined surface are observed in the present analysis. The transformation of the plane surface from the wavy irregular surface is executed via coordinate alteration. The fluid flow has been evaluated under the outcomes of heat source, thermal radiation, and chemical reaction rate. The nonlinear system of partial differential equations is simplified into a class of dimensionless set of ordinary differential equations (ODEs) through a similarity framework, where the obtained set of ODEs are further determined by employing the computational technique parametric continuation method (PCM) via MATLAB software. The comparative assessment of the current outcomes with the earlier existing literature studies confirmed that the present findings are quite reliable, and the PCM technique is satisfactory. The effect of appropriate dimensionless flow constraints is studied versus energy, mass, and velocity profiles and listed in the form of tables and figures. It is perceived that the inclination angle and wavy surface assist to improve the flow velocity by lowering the concentration and temperature. The velocity profile enhances with the variation of the inclination angle of the wavy surface, non-Darcian term, and wavy surface term. Furthermore, the rising value of Brownian motion and thermophoresis effect diminishes the heat-transfer rate.

Introduction

Surfaces that are roughened or uneven transmit more thermal energy than level interfaces. In many technical and industrial operations, the exterior layer of the items is purposefully nonuniform or uneven to increase the heat conduction rate through it. The reason for this is that rough surfaces speed up the mixing of fluid particles, resulting in a higher heat-transfer rate.[1] Thermal transfer is a field of heat engineering that entails the assembly, use, transformation, and exchange of heat energy in various contexts. Heat transmission is divided into many types, and energy is transferred through stage transitions.[2,3] The heat transfer qualities of fuel cell technology, nuclear plants, microsystems, and transportation are all important in a variety of technical and commercial operations.[4] The analysis of fluid flow induced by a continually turning expanding or contracting sheet is critical in practice. These sorts of streams can be found in a wide range of professional and scientific operations, including electrospinning, film sketching, fiber manufacture, paper production, crystal development, and the condensation process for the production of liquid films.[5,6] Zhao et al.[7] experimented on the influence of varied configurations of wavy and striped fins in a multiple plate-fin compressed heat exchanger on precipitation heat conduction in the heat exchanger. Bilal et al.[8] examined how a wavy oscillating whirling disc transfers energy through the nanofluid flow. In comparison to a uniform interface, the irregular rotational surface enhances energy transmission by up to 15%. Singh and Dewan[9] examined the energy transmission through wavy wall-bounded jets. The heat-transfer augmentation is affected by the offset ratio’s placement and the intensity of the wavy material. Alsabery et al.[10] used a two-phase water base fluid in a 3D wavy permeable medium with top and bottom wavy edges induced with a constant temperature and isothermal circumstances to investigate the Brownian diffusion as convective energy transference enhancement methods. The findings revealed that the nanofluid flow in the wavy channel improved heat transmission, particularly when the Reynolds number and the nanoparticle volume fraction increased. Some recent studies related to the heat transfer through wavy surfaces have been heavily reported by refs (11–14).

A nanofluid is a fluid that contains nanomaterials, which are nanometer size objects. These fluids have heterogeneous nanoparticulate concentrations in a base fluid.[15] Metals, iron oxide, carbides, and carbon nanofibers are commonly employed as aggregates in nanofluids. Water, propylene, and petroleum are all frequent base fluids.[16−18] Nanofluids have exclusive geographies that could make them ideal in a diversity of thermal governance domains, such as optoelectronics, fuel cells, generic drug operations, hypervisor motors, engine refrigeration, domestic refrigerator, cooler, exchangers, shredding, milling, and burner flue gas thermal minimization.[19,20] When compared to the base fluid, they have higher thermal conductance and lower thermal diffusivity. It has been discovered that understanding the viscoelastic performance of nanofluids is crucial in determining their feasibility for convective energy transmission purposes.[21] Bhatti et al.[22] noted the fluid flow across a permeable substrate saturated with Williamson nanofluid by simultaneously moving round sheets. The upshot of Prandtl number leads to the enhancement of thermal contour; similar phenomena have been observed with greater Reynolds number. Varun Kumar et al.[23] used the RK-4 strategy to quantitatively assess the outcome of a magnetic flux on the nanoliquid flow. Alharbi et al.[24] described the flow of a highly conductive trihybrid nanoliquid with energy propagation in a boundary layer containing nanocomposites along an extendable cylinder with external magnetic impacts. It has been discovered that varying the tr-hybrid NPs increases the thermophysical properties of the base fluid substantially. Ikram[25] devised a theoretical model for magnetized nanocrystal Darcy–Forchheimer flow through an extending disc with negligible mass flux. The temperature of the nanocrystals is improved through exothermic reactions. Waqas et al.[26] documented the magneto-hydrodynamics (MHD) characteristics of nanoliquid flows across a mixed convection stretched sheet. It was discovered that for increasing scales of the buoyancy ratio constraint, the microbe’s concentration profile and nanoparticle concentration enhanced. Shahid et al.[27] reported the nanoliquid flow over a parabolic reflector perforated surface using MHD. The flow structure and thermal alterations of an intermittent radiation nanoliquid flow caused by the spinning disc were presented by Acharya et al.[28] The energy communication increases with the effect of thermal radiation and nanolayer ratio. It was stated that the heat allocation for nanolayers improved by about 84.61 percent. Many scientists have reported substantial research on the nanofluid flow.[29−32]

The thermal radiation effect is crucial for a multitude of applications, such as heat dissipation, electron microscopy, photodetectors, and energy gadgets.[33] Kumar et al.[34] devised a computational model for nanoliquid flow and heat transmission over an infinite diagonal plate. It has been realized that growing the value of the radiation factor enriches the thermal energy and velocity profiles. Jin et al.[35] addressed the heat conduction effect of Al2O3 nanoliquid flow with mixed convection and thermal radiation through a porous stretched surface. The ethylene glycol–Al2O3 nanofluid is shown to be more important in the freezing phase than the water-based ferrofluid. Raza et al.[36] inspected the 2D MHD stream flow across an absorbent substance with the suction/injection and heat radiation upshots. With clay nanomaterials and thermal radiation, Alrabaiah et al.[37] investigated the time-fractional free laminar flow of stagnation point drilling nanofluid. A quantifiable increase of 11.83% in the energy conversion of boring nanofluid versus radiation effect has been described. Bilal et al.[38] designated the effect of radiation on the stretchable sheet with momentum and mass transformation. Many researchers have recently studied the thermal radiation effect on fluid flow.[39−41]

The goal of this study is to present a numerical simulation for the mixed convection 2D non-Darcy model with the heat generation, thermal radiation, and chemical reaction impact on a nanofluid over a slanted wavy surface. The consequences of Brownian, buoyance ratio, and thermophoretic diffusion are also considered. According to the author’s knowledge, no previous studies have been documented on this topic. The flow stream has been described in terms of partial differential equations (PDEs), which are reduced to a set of ordinary differential equations (ODEs) through a similarity framework. To propose a numerical solution, a computational approach parametric continuation method (PCM) is used. The influence of the physical variables on the dimensionless velocity, energy, and mass contours is depicted graphically.

Mathematical Formulation

We supposed steady 2D mixed convection radiative nanoliquid flow, along with the non-Darcy permeable medium across a wavy inclined surface. The schematic sketch of fluid flow over an inclined surface is portrayed in Figure . The following assumptions have been considered

Figure 1

Nanofluid flow over an inclined wavy sheet.

An incompressible flow

Mixed convection

Laminar nanoliquid flow

The formulation is based on the Boussinesq estimation

The non-Darcy permeable medium

The nanofluid flow under the radiation effect.

The fluid flow system is confined by an infinite wavy inclined plate, stated as[42,43]

The wavy side and typical distance of the wavy surface are documented as a and 2l, respectively. The terms ϕ and T represent the concentration of nanoparticles and temperature. They are calculated in between ∞ and w. The subscripts ∞ and w narrate the position away from the surface and at the irregular surface, respectively. The modeled equations are formulated as[42]

Here, u and v are the horizontal and vertical velocity components. DT is the thermophoretic diffusivity, DB is the Brownian diffusivity, K̃ is the non-Darcian inertial effects, Ke and σ are the mean absorption and Stefan–Boltzmann coefficient, is the thermal diffusion, and β is the volumetric thermal expansion. ϵ(ρc)p and (ρc)f illustrate the heat capacity of the nanomaterials and base fluid, whereas, γ is the ratio between ϵ(ρc)p and (ρc)f.

The initial and boundary conditions are

The stream function and dimensionless variables are[42]

As a result of eq , the dimensionless system of equations is obtained aswhere Ra is the Rayleigh number, Q is the dimensionless velocity, Pe is the Peclet number, Fc is the non-Darcian term, Nt is the thermophoresis effect, Nb is the Brownian diffusivity, Le is the Lewis number, R is the radiative factor, and Δ is the mixed convective term defined as[43]

Eq is employed to convert the wavy surface impact to controlling formulas from the boundary restrictions

Replacing eq into eq by making Ra → ∞, we get

The transform conditions are

The local heat and nanomaterial mass transfer from the wavy surfaceHere, is a unit perpendicular vector and is a thermal radiation heat flux.

The dimensionless Nusselt and Sherwood numbers are[43]

Numerical Solution

The following are the basic steps while applying the PCM approach, and its future direction is as follows.[44−49]

Step 1: Reduce eqs –15 to first order

By putting eq in eqs –15, we get

The transform conditions are

Step 2: Introducing parameter p

Step 3: Apply Cauchy Principal and discretized eqs –25

After discretization, the obtained set of equations is computed through the Matlab code of PCM.

Result and Discussion

This section describes the physical process behind each figure and table. The following observations have been noted:

Velocity profile (f′(η))

Figure a–c explains the velocity outlines versus the inclination angle of a wavy surface A, non-Darcian term Fc, and the wavy surface term δ, respectively. Throughout the analysis, (Δ = −3) is used for opposing flow and (Δ = 5) for adding the flow, while the arrow represents the (↓) reducing and (↑) elevating behaviors of velocity, energy, and concentration profiles. Figure a displays that the velocity field intensifies with the difference of the inclination angle of wavy surface A. Figure b elaborates that the effect of the non-Darcian term Fc boosts the fluid velocity within the case of (Δ = 5). Physically, the non-Darcian inertial effects improve, while the kinetic viscosity of fluid diminishes with the augmentation of Fc, which results in the diminution of the momentum boundary layer. Similarly, Figure c demonstrates that the velocity outline increases with the variation of wavy surface term δ. Physically, due to the zigzag motion of the fluid particles, the kinetic energy of the fluid increases, which results in the advancement of the velocity field (Δ = 5), while an inverse behavior has been noticed (Δ = −3).

Figure 2

Velocity outlines (f′(η)) vs the (a) inclination angle of a wavy surface A, (b) non-Darcian term Fc, and (c) wavy surface term δ.

Temperature Profile θ(η)

Figure a–e exemplifies the energy distribution (θ(η)) outlines versus the inclination angle of a wavy surface A, non-Darcian term Fc, heat source term hs, radiation term R, and wavy surface term δ, respectively. Figure a,b shows that the energy contour declines with the upshot of an inclination angle of a wavy surface A, while it enhances with the non-Darcian term Fc in case of (Δ = 5). The energy field shows an opposite trend against both parameters in the case of (Δ = −3). Figure c,d reports that the energy profile magnifies with the effects of heat source hs and radiation term R. The consequences of both constraints provide additional heat to the fluid, which raises the energy of the fluid, and as a result, the energy profile accelerates. Figure e displays the effect of the wavy surface term δ on the energy profile. Physically, due to the zigzag motion of the fluid particles, the kinetic energy of the fluid increases, which results in the elevation of the energy field (Δ = 5), while an inverse behavior has been noticed with (Δ = −3).

Figure 3

Outlines of energy distribution (θ(η)) vs the (a) inclination angle of a wavy surface A, (b) non-Darcian term Fc, (c) heat source term hs, (d) radiation term R, and (e) wavy surface term δ.

Concentration Profile ϕ(η)

Figure a–d describes the outlines of mass distribution ϕ(η) versus the inclination angle of a wavy surface A, non-Darcian term Fc chemical reaction term Kr, and wavy surface term δ, respectively. Figure a,b reports that the mass transport rate lessens with the effect of inclination angle, while it boosts with the impact of the non-Darcian term. Physically, the fluid particles need more energy to move over an inclined surface, that is, the increment in the angle of wavy surface retards the mass-transfer rate as elaborated in Figure a. Figure c explains that the influence of chemical reaction constraints boosts the mass propagation rate because its effect improves the kinetic energy of fluid particles, which promotes the mass transition rate as shown in Figure c. Figure d discloses that the mass profile shrinks with the variation of the wavy surface term. Physically, for the fluid particles, it is difficult to move over an irregular surface; that is why the variation of δ declines the mass transmission. Figure reveals the flow chart of the solution methodology.

Figure 4

Outlines of mass distribution ϕ(η) vs the (a) inclination angle of a wavy surface A, (b) non-Darcian term Fc, (c) chemical reaction term Kr, and (d) wavy surface term δ.

Figure 5

Flow chart of solution methodology.

Table reveals the relative assessment of the present results with the available work to ensure the validity of the proposed method. It can be perceived that the present results are correct and reliable. Table illustrates the numerical outcomes of the Nusselt number (θ(η)) toward and opposing the flow field. It can be observed that the rising value of the Brownian motion thermophoresis effect lessens the heat-transport rate. Table elaborates the numerical outcomes of the Sherwood number (θ(η)) toward and opposing (Δ = −3), the flow field.

Table 1

Statistical Evaluation of the Present Results with the Published Studies for Nusselt Number −θ′(0)

	–θ′(0)	–θ′(0)	–θ′(0)		–θ′(0)	–θ′(0)	–θ′(0)
Δ	ref (50)	ref (51)	present work	Δ	ref (50)	ref (51)	present work
0.0	0.5641	0.56415775	0.56415974	–0.2	0.5269	0.526911	0.528814
0.5	0.6473	0.6473651	0.64737520	–0.4	0.4865	0.486533	0.486732
1.0	0.7205	0.72055401	0.72055503	–0.6	0.4420	0.442021	0.442323
3.0	0.9574	0.95744512	0.95744610	–0.8	0.3916	0.391663	0.391732
10	1.5160	1.51623967	1.51623864	–1.0	0.3320	0.332021	0.332324
20	2.0660	2.0660	2.06701200	0.0

Table 2

Numerical Outcomes of Nusselt Number (θ(η)) toward and Opposing to the Flow Field

R	Nb	Nt
0.3	0.1	0.2	0.5653657	0.447610
0.5			0.6098267	0.482763
0.7			0.6519716	0.514105
	0.3		0.6838035	0.538975
	0.5		0.6519725	0.514106
		0.4	0.6347640	0.497362
		0.6	0.6182457	0.487163

Table 3

Numerical Outcomes of Sherwood Number ϕ(η) toward and Opposing to the Flow Field

Nb	Nt	Le
0.1	0.1	2.0	0.5653687	0.448712
0.3			0.6098267	0.481864
0.5			0.6519716	0.513206
	0.3		0.6838034	0.537872
	0.5		0.6519716	0.513206
	0.1	4.0	0.6347631	0.499463
		6.0	0.6182457	0.486262

Conclusions

The aim of this work is to study the two-dimensional mixed convection radiative nanofluid flow, along with the non-Darcy permeable medium across a wavy inclined surface. The nonlinear system of PDEs is simplified into a class of dimensionless sets of ODEs through a similarity framework, where the obtained set of ODEs is further determined by employing the computational technique PCM via MATLAB software. The relative evaluation of the current outcomes with the earlier existing literature studies confirmed that the present findings are quite reliable and the PCM technique is satisfactory. The core findings from the above examination have been offered below:

The velocity profile enhances with the variation of the inclination angle of the wavy surface A, non-Darcian term Fc, and wavy surface term δ.

The energy profile declines with the upshot of the inclination angle of a wavy surface A, while it enhances with the effect of the wavy surface term δ, non-Darcian term Fc, heat source hs, and thermal radiation parameter in the case of (Δ = 5).

The mass-transfer rate reduces with the effect of the wavy surface term and the inclination angle of the wavy surface, while it boosts with the impact of the non-Darcian term and chemical reaction.

The rising value of Nt and Nb effect diminishes the heat-transfer rate.

The inclination angle and wavy surface assist to improve flow by lowering the concentration and temperature.

The present mathematical may be extended to other types of fluids. It may be modified with Das and Tiwari’s model and can be solved using other numerical techniques, analytical methods, and fractional derivatives.

The present mathematical model may be extended to other types of fluids (i.e., non-Newtonian) by considering different sorts of physical effects over different geometries and may be solved through fractional and analytical techniques.