Entropy and Bejan Number Influence on the Liquid Film Flow of Viscoelastic Hybrid Nanofluids in a Porous Space in Terms of Heat Transfer.

The aim of this study is to determine the influence of the various parameters on the flow of thin film motion on an inclined extending surface. Maxwell fluid is used as a base fluid, and magnesium oxide (MgO) and titanium dioxide (TiO2) are used as nanocomponents. The width of the thin film is considered variable and varied according to the stability of the proposed model. The magnetic field is used in the vertical track to the flow field. The entropy generation and Bejan number are examined under the influence of various embedded parameters. The outputs of the liquid film motion, thermal profile, and concentration field are also shown with the help of their respective graphs based on the collected data. The solution of the model involves key features such as entropy generation, Bejan number, drag force, and heat transfer rate. Brinkman number, magnetic parameter, radiation parameter, thickness parameter β, and unsteadiness parameter S are also deliberated graphically. The percentage improvement for the enhancement of heat transfer has been calculated and compared for both the nanofluid and hybrid nanofluids. The results are validated through comparison with the existing literature.

Introduction

In nature, most of the existing fluids are not Newtonian, and it is not easy to study these fluids through simple Naiver stocks equations without making alterations. Non-Newtonian fluids are needed in our daily lives, including food items, medications, polymers, and so on. These fluids are the combination of shear stresses and normal stresses. The time retardation effect, elasticity, and viscoelastic nature of these fluids make them different from the common fluids. These fluids has the tendency to improve the heat transfer rate efficiently and store energy due to their elastic nature. The researchers have also used these fluids for the cooling and thermal applications. The cooling and heating applications of these fluids are discussed by Hoyt[1] considering friction reduction. The other fruitful applications include pipelines carrying petroleum’s which are made of high polymers in industries. Plastic materials are usually made of elastic fluids, and similarly, chemical and mining industries are dependent on Newtonian fluids. The viscoelastic fluids are more suitable for testing purposes in the form thin layers.

These tiny layers are widely used in the coating industry, including wire and fiber coatings. Thin films are usually used in lubrication to reduce the friction force and improve the device efficiency. Liquid films are more compatible to use for the heat transfer analysis. The basic ideas regarding thin film flow were investigated by Wang.[2] This idea is further improved by the Andersson et al.[3] to add the energy equation. Similarly, new terminologies like the concept of pulse waves were used by researchers[4,5] in the flow regime of the thin film. In the initial studies, a constant thickness of the liquid film was used by the researchers, which was not so effective in maintaining the stability. Maintaining the stability of the thin film is more effective by using variable thickness instead of a constant thickness.[6] Thermal performance produces a dimensionless outcome that are restrained by an appliance that restores energy. The thickness of the thin film is very thin and quite suitable for heat transfer analysis in a short time at the laboratory level. Therefore, the researchers focus on the thin materials like the thin film for the testing purpose in terms of the heat transfer. Akkuş et al.[7] calculated liquid-film vapors considering 2D modeling. The thermal performance of the base liquids is commonly low, and nanoparticles or nanomaterials are usually stably dispersed in the base liquid to improve the thermal performance of the traditional fluid, known as nanofluids. Different mathematical models and various nanomaterials were used by researchers[8−22] to improve the thermal performance of the base liquids. The efficiency in the heat transfer analysis can be seen in the mechanical, industrial, and renewable energy resources, and the proposed model is also a part of the enhancement in heat transfer considering hybrid nanofluids for the applications of renewable energy resources in terms of the theoretical analysis. Tahir et al.[23] have examined the behavior of various parameters using the liquid film flow.

The viscoelastic fluid flow in terms of thin film is commonly used in medication and industries. In fact, blood is a class of Maxwell fluid and some drugs also lie in the viscoelastic class of non-Newtonian fluids. Similarly, elastic materials are another form of the Maxwell fluid usually used in the industries. Maxwell fluid reveals both viscous and elastic performance, which means stress relaxation and elastic recovery after distortion are the main characteristics of the Maxwell fluid to distinguish this fluid from the rest of the non-Newtonian fluids. Nadeem et al.[24] have used the Maxwell fluid as a base fluid for thermal applications by inserting nanoparticles in the base solvent. They focused on the heat transfer rate and calculated all the thermal features related to the Maxwell fluid. Binetti et al.[25] have used the more reliable concepts by treating C6H9NaO7 in direct relation to the viscoelastic behavior of the Maxwell fluid. Sunnapwar and Pawar[26] used the same idea by using the stable thermophysical properties of the solid particles. Later on, the idea of using C6H9NaO7 for thermal performance has also been used.[27,28] These researchers correlated their studies with the agriculture sector.

Initially, working fluids with millimeter to micrometer sizes were used in renewable energy devices and the performance was not very encouraging. The use of working fluid containing an amalgamation of black liquid and micron-sized particles leads to erosion, sedimentation, and pipe blockage. Nanofluids are basically an amalgamation of nanosized nanofluid (<100 nanometers) metallic particles and the conventional liquid with extended heat transfer capabilities described by Bahiraei et al.[29]

The direct absorption of the nanofluids in the solar collector improved the thermal properties and is more effective in terms of the radiation characteristics as displayed in Al-Rashed et al.[30] The solution of the model is obtained by the researchers using the Keller box scheme.[31−35] The thermal performance of the nanofluids in the form of the direct absorption via solar collector using various nanoparticles, including silver and so on, was analyzed. The experiment depicted that a volume concentration of 3% with an ∼10 mm collector height improves the efficiency by 90%. An analytical solution for the thermal radiation and the Joule heating impact for a thixotropic nanofluid flow is attained elsewhere.[36] A comparatively weaker boundary layer and low velocity of the fluid are witnessed for a strong magnetic field. Similar studies highlighting thermal radiation impacts on solar collectors are found elsewhere.[37,38] Free convection and thermic reaction convection were first introduce by Cess[39] and Arpaci.[40] Gul et al.[41,42] analyzed nanofluid flow using various models for the applications of heat transfer. Takabi & Salehi[51] analyzed hybrid nanofluids using the mathematical model of the sinusoidal enclosures for thermal applications. Hybrid nanofluid is the combination of the stable dispersion of different nanoparticles having different thermophysical and chemical properties in the same base fluid. These are widely used in the heat exchangers, solar systems, and drug deliveries.[43−50]

The Maxwell fluid thin film flow over an extending surface has been examined by Takabi and Salehi.[51] The idea regarding heat and mass transfer including the variable thickness of the liquid film was discussed by Qasim et al.[52] The concept of TiO2 and Ag materials for the drug delivery was analyzed by Charegh and Dinarvand[53] using blood as the base fluid. Gul and Pervez[54] have used the thin film of Maxwell hybrid nanofluids for various thermal applications. They observed the thermophoresis’ and Brownian motion parameters’ impact on the liquid film flow. The viscoelastic fluids were used by researchers[55,56] for heat transfer enhancement applications. The thermal effect is mostly targeted by researchers to improve energy resource efficiency.

Originality/Value

The entropy generation in the liquid film using Maxwell fluid is a new addition.

The Bejan number impact under the influence of various parameters improves the novelty of the suggested model.

MgO and TiO2 nanomaterial dispersion in the Maxwell fluid to perform hybrid nanofluids for the enhancement of heat transformation is a very rare approach in the form of thin film.

The thin film Maxwell hybrid nanofluids in terms of the MgO and TiO2 nanomaterials are a new extension for the heat and mass transfer analysis.

A porous medium, thermal radiation, and magnetic parameters further improve the novelty of the thin film fluid flow on an inclined plane.

Methodology

The proposed flow model was developed through a system of PDEs, which later is transformed into non-linear ODEs. The homotopy analytical method (HAM) in MATHEMATICA has been used for the analytical solution of the proposed model.

Formation

In the proposed flow demonstration, we took into account the thin-layer flow of the Maxwell hybrid nanofluid on an inclined extending surface, and the Maxwell fluid has been used as a base fluid. Two MgO and TiO2 nanomaterials are used in hybrid nanofluid preparation.

As the fluid lies on the extending sheet as force is applied to the sheets, they start moving. h(t) represents the thickness of the film.

In the above figure, the extending sheet is placed inclined to make an angle ϕ with the horizontal plane. When we apply force on the sheet, it starts moving with a velocity

The operative elasticity of the velocity is toward the x axis, and the parameter “α” stands for the increment of time in a selected range (0 ≤ α < 1) while “b” represents the elasticity. The surface temperature of the extending sheet is signified as “T”, and the temperatures of the slit are defined as T0 and T. The range of these constraints are referred to as 0 ≤ T ≤ T0.

The unsteady magnetic term in a perpendicular track is defined as

The basic flow equations are

The components of velocity are represented by u, v, which are acting along the directions of x and y, respectively. The physical conditions for the thin film flow is defined ash(t) stands for the liquid film width.

The transformed form is displayed as follows:where S, λr, Pr , λ, γ, Gr, Rd, and Gc are the unsteadiness parameter, porosity term, Prandtl number, Maxwell parameter, chemical reaction, thermal Grashof number, radiation parameter, and mass Grashof number, respectively.

The transformed form of the physical conditions are taken as

These terms are transformed and in the simplified form are displayed as

Entropy Rate

Entropy is an essential idea in engineering, mathematical models, and physics. It plays a vital role in continuum physics, thermodynamics, biology, and economics.[57−60] Entropy is actually a phenomenon dependent on the second law of thermodynamics, which states that entropy increases in an isolated system through any activity. While this idea is further extended in quantum mechanics with the inclusion of a density matrix. In the case of the statistical system or in the theory of probability, entropy is used to measure the uncertainty of the variables that are used in the statistical phenomena.

Here, is the rate of entropy generation, is the diffusive parameter, and is the temperature difference parameter.

Bejan Number

The irreversibility due to heat transfer ratio to total irreversibility is called the Bejan number.

Results and Discussion

The viscoelastic influence can significantly worsen the critical Reynolds number of comparable shear flows. The Oldroyd B and Maxwell fluids are the most systematically studied. The critical Reynolds number is more visible in the tiny layers like thin liquid films. The flow of Maxwell hybrids on an inclined and stretching sheet is taken into consideration.

Solid nanoparticles (NPs) from magnesium oxide (MgO) and titanium dioxide (TiO2) are used to produce hybrid nanofluids. These nanoparticles are dispersed in the Maxwell fluid efficiently up to 5% of the total fluid. The mathematical model in the form of nonlinear ODEs has been applied using a well-known seminumerical technique (HAM). The latest version or package BVPh 2.0 of this method is a more reliable version to sustain the stability, and convergence is used here to find out the solution. The key objective of this study is to investigate the heat transfer rate enhancement using the hybrid nanofluid, and it is observed that hybrid nanofluids are more reliable to improve the heat transfer rate. The physical structure of the problem is revealed in Figure .

Figure 1

Physical sketch of the proposed model.

Figures and 3 show the consequence under β. As β upsurges, the velocity field declines, but the temperature field rises due to the increased film thickness.

Figure 2

Consequence of β vs f′(η).

Figure 3

Consequence of β vs Θ(η).

The film width causes the enhancement in the distribution of velocity components in the downward direction. Physically, the increasing thickness of the liquid film progresses the resistance force and consequently the fluid motion decline. It is also noted that hybrid nanofluids (TiO2 + MgO) are more effective than nanofluids (MgO) used for β variation.

The axwell parameter influence is displayed in Figure .

Figure 4

Consequence of λ vs f′(η).

The fluid movement changes in relation to the Maxwell parameter λ. Therefore, the change of λ disrupts the velocity profile. Thus, the increase of λ decreases the movement of the liquid film. The viscoelasticity parameter increment increases the resistive force and as a result the fluid flow decline.

Figures –7 illustrates the impact of ″S″ in the suggested model; it was noted that the parameter ″S″ has a significant impact on f′(η) and Θ(η).

Figure 5

Consequence of S vs f′(η).

Figure 7

Consequence of S vs Φ(η).

Consequence of S vs Θ(η).

The intensification in ″S″ explains a decrease in the velocity field; likewise, an increase in the temperature distribution is caused by an upsurge in ″S″. Liquid film motion was expressively decreased due to the increasing values of the instability parameter.

These outputs depict the nature of the unstable parameter resulting in a drop in the liquid film thickness β. The distinction in the liquid film thickness bears the stability of the fluid motion, and also the convergence is mainly based on the thickness of the liquid film. Also, the concentration profile decreases with the higher values of S. Figure illustrates the behavior of the porosity parameter ″ λr″. It is seen that for increasing the porous parameter, the velocity declines. The declining trend is due to the fact that the increased porous parameter dominates the frictional effects. MgO and TiO2 + MgO nanomaterials are melted in the Maxwell liquid to perform nanofluid and hybrid nanofluids. The resistance force is initiated due to the larger liquid film thickness β, and the same influence is produced with the rising value of the porosity parameter ″λr″. Therefore, the resistance force is improved with the increasing value of the porosity parameter.

Figure 8

Consequence of λr vs f′(η).

Figures and 10 display the influence of the Gc (Mass Grashof number) over the liquid film motion. The mass Grashof number is acting along the flow direction, and therefore the augmentation of Gc enhances the fluid motion. The increasing effect is further improved in terms of the hybrid nanofluids..

Figure 9

Consequence of Gc vs f′(η).

Figure 10

Consequence of Gr vs f′(η).

The increasing values of the thermal Grashof number in terms of linear temperature difference enhance the fluid motion and this effect is more effective by using hybrid nanofluids (MgO and TiO2).

Figures and 12 show the nanoparticle volume fraction (ϕ1, ϕ2) for velocity and thermal profiles. The increasing values of the nanoparticle volume fraction in a particular range (ϕ1, ϕ2 = 0.01,0.02,0.03) reduce the velocity field and boost the temperature distribution.

Figure 11

Consequence of ϕ1, ϕ2vs f′(η).

Figure 12

Consequence of ϕ1, ϕ2vs Θ(η).

The obtained outputs show that those hybrid nanofluids containing MgO + TiO2 nanomaterial progress the thermal characteristics of the conventional fluids. The target of the proposed model is to progress the thermal transport of the conventional fluids, and the addition of the radiation term shown in Figure improves the temperature distribution. Again, the increasing amount of Rd boosts the temperature profile.

Figure 13

Consequence of Rd vs Θ(η).

Increasing values of chemical reaction γ reduce the concentration field as shown in Figure . The cohesive forces between the molecules increase due to the increasing values of the chemical reactions, which reduces the concentration field. The augmentation in the Schmidt number (Sc) also reduces the concentration profile, and this happens due to the reduction in the molecular diffusion as shown in Figure .

Figure 14

Consequence of γ vs Φ(η).

Figure 15

Consequence of Sc vs Φ(η).

The entropy regime enhances with the cumulative values of the porosity parameter while the opposite trend is achieved in the case of the Bejan number as shown in Figures and 17. Similarly, the thermal growth upsurges with the accumulation in the radiation parameter, and thus the entropy of the fluid upsurges while the Bejan number decreases as shown in Figures and 19.

Figure 16

Consequence of λr vs NG.

Figure 17

Consequence of λr vs Be.

Figure 18

Consequence of Rd vs NG.

Figure 19

Consequence of Rd vs Be.

A plot of the Bejan number versus Brinkman number is displayed in Figure . The Bejan number declines with the augmentation in the Brinkman number. In fact, fluid and entropy upsurge with an increase in the Brinkman number, and the Bejan number plays a reverse role of entropy.

Figure 20

Consequence of Br = EcPr vs Be.

The statistical analysis was also done for the imperative parameters like skin friction (drag force), Nusselt number (heat transfer rate), and Sherwood number. The comparative analysis for the nanofluid and hybrid nanofluid has been performed in terms of the above physical parameters. The increment in the mass Grashof number Gc improves the liquid film flow, as displayed in Figure . Physically, the ways of the flow and mass Grashof number are in the same direction and the increasing value of the mass Grashof number improves the fluid velocity, and therefore the skin friction declines.

Figure 21

Consequence of Gc vs skin friction.

The thermal Grashof number influence on the skin friction is displayed in Figure . Again, the greater values of the Gr improve the liquid film flow and decline the skin friction. Figure shows the consequences of ϕ1, ϕ2. The thermal profile and thermal transport rate are more enhanced due to the increasing amount of the nanoparticle volume fraction. The hybrid nanofluid is more effective in comparison with nanofluids and conventional fluids to enhance the heat transfer rate as shown in Figure . The percentage-wise enhancement in the heat transfer provides more evidence to show that hybrid nanofluids have the tendency to improve the heat transfer rate more efficiently. The thermophysical properties of the solid nanoparticles are displayed in Table . The comparisons of the proposed model with the published work are shown in Tables and 3. The Prandtl number values remain fixed in all the existing and current work. As per the experimental approach, the Prandtl number is fixed and does not vary for the regular fluid. Therefore, the common parameter, the unsteadiness S, was compared with the published work,[2,52] and as a result a much closer agreement was attained. As Wang’s[2] study is limited up to heat transfer, the variation in the Schmitt number, which is a common parameter of the concentration profile among the present work and that of Qasim et al.,[52] has been compared for the various values. The comparison shows the authentication of the present results with the existing literature.

Figure 22

Consequence of Gr vs skin friction.

Figure 23

Consequence of ϕ1, ϕ2vs Nusselt number enhancement in %.

Table 1

Thermophysical Features of the Nanoparticles

parameter	APS (average particle size), nm	mMorphology	true density, g cm–3	thermal conductivity, W m–1 k–1	specific heat capacity, J kg–1 K–1
MgO	25–45	nearly spherical	3.57	5.112	0.852
TiO2	18–23	nearly spherical	3.95	5.407	0.835
80 wt % MgO-20 wt % TiO2	10–45	spherical	2.87	4.768	0.842

Table 2

Comparison between the Published Work and Present Work for the Surface and Wall Temperature Gradients Considering Common Factors Using the Regular Fluid Having Pr = 7.56a

	Wang[2]	Wang[2]	Qasim et al.[52]	Qasim et al.[52]	present	present
S	Θ(1)	–Θ′(0)	Θ(β)	–βΘ′(0)	Θ(β)	–βΘ′(0)
0.1	0.34201	0.5302	0.34810	0.53263	0.348320	0.532865
3	0.65321	1.20437	0.67224	1.31026	0.681210	1.34720
6	1.02531	1.72810	1.05168	1.81321	1.062102	1.85210
9	1.73106	2.23017	1.82021	2.37630	1.881312	2.413021

Note that they used small and variables values of the Prandtl number.

Table 3

Comparison between the Published Work and Present Work for the Sherwood Number Considering a Common Factor Using the Regular Fluid Having Pr = 7.56, S = 10

	Qasim et al.[52]	Qasim et al.[52]	present	present
Sc	Φ(β)	–Φ′(0)	Φ(β)	–Φ′(0)
10	0.31264	0.72018	0.31301	0.723510
12	0.21235	0.68382	0.21423	0.69310
14	0.13641	0.52103	0.13910	0.55102

Conclusions

The current study delineates the effect of nanocomposites on the Maxwell hybrid nanofluid. The heat transfer rate for various parameters in the presence of radiation, magnetic field, and porosity is analyzed in the form of physical and computational data. Mass transfer and entropy factors are considered in this model.

The problem is tackled through the homotropy analysis method (HAM) in MATHEMATICA. The proposed flow model is very worthwhile and valuable in various physical processes involving heat and mass phenomena. This work demonstrated its productivity and valuable usage in medical sciences, engineering, and other industries.

The key conclusions are the following:

The growing inputs of the thermal Grashof number increase the fluid motion.

The thermal field grows with the augmentation in the volume fraction of the nanoparticles, and this effect is more effective in terms of the hybrid nanofluids.

Increasing the width of the film would cause the velocity profile to increase.

The large value of the magnetic moment parameter would cause the Lorentz force enhancement, and therefore the deceleration in fluid motion occurs.

The percentage-wise improvement in the heat transfer rate confirms that the hybrid nanofluids are more effective at enhancing heat transfer.

The entropy of the hybrid nanofluids increases with increasing porosity parameter, and for the same variation of the porosity parameter, the Bejan number decreases.

The radiation parameter increases the entropy of the hybrid nanofluids for its larger values and reduces the Bejan number.