Entropy Generation and Thermal Analysis on MHD Second-Grade Fluid with Variable Thermophysical Properties over a Stratified Permeable Surface of Paraboloid Revolution.

Stratification is used in a wide range of energy storage fields, including solar thermal energy systems. This paper investigates entropy optimization and the effects of heat production, magnetic field, and various fluid parameters on the flow of second-grade fluid through unstratified and stably stratified paraboloids of revolution. In the heat transfer equation, stratification, linear thermal radiation, and Joule dissipation have all been explored. The similarity transformation is used to convert the governing PDEs into nonlinear ODEs. The HAM (homotopy analysis method) is used to solve dimensionless nonlinear ODEs. The impact of significant elements on various profiles is exposed and explored. Graphical results are used to examine the influence of the velocity profile, temperature, concentration, and entropy formation rate using tables to indicate the characteristics of skin friction, Nusselt number, and Sherwood number for numerous parameters. It is noticed that the velocity is enhanced by raising the stratification parameter, while the opposite behavior is observed for temperature distribution. The concentration profile declined as the solute stratification parameter was enhanced. For both the unstratified and stratified regions, incremental values of the Brinkman number and magnetic parameter depict augmentation in entropy production, while entropy production drops for a large value of the temperature ratio parameter.

Introduction

The dynamics of today’s world have been altered by two sorts of fluids: Newtonian and non-Newtonian fluids. Researchers first focused on Newtonian fluids, but due to the complexity and variety of numerous fluid-dependent procedures, the importance of non-Newtonian fluids is growing. Non-Newtonian fluids are important for a variety of economic, physiological, physical, industrial, medical, mechanical, and technical applications. Therefore, the narrative of non-Newtonian fluids is necessary for a better understanding of such phenomena. It can be executed by contrasting Newtonian and non-Newtonian flow behavior. One significant distinction between these two fundamental kinds of fluids is that Newtonian fluids have a linear association between shear stress and strain rates, but non-Newtonian fluids do not. Non-Newtonian fluids exhibit complex properties as a result of this relationship and are classified into subdivisions such as shear thickening, shear-thinning, thixotropic, and dilatant fluids. Rheologists have discovered that several fluid models, such as second-grade fluid, which explains the properties of shear thickening, shear-thinning, and Newtonian fluid, behave differently under different conditions. Second-grade fluid is famous and valuable among investigators because of its dynamic characteristics. Vajravelu and Roper[1] conducted thermal and dynamical investigations of second-grade fluids. The perturbation technique was used by Rajeswari and Rathna[2] to explore the dynamics of the stagnant second-grade fluid. Rajeswari and Rathna[2] were followed by Grag and Rajagopal,[3] who found complete consensus with their outcomes. Fetecau and Fetecau[4] used the Hankel transform approach to provide a solution for second-grade fluids caused by using an oscillating circular cylinder. Recently, Abbas et al.[5] scrutinize the stagnation point flow of viscoelastic second-grade fluid across a stretching cylinder in the presence of the thermal slip and MHD effects.

Many scientists have looked into fluid flow on a variety of surfaces, including inclined, horizontal, and vertical surfaces. A revolutionary paraboloid is an object with a changeable thickness on the top half face. Davis and Werle[6] developed a new method for resolving the laminar flow across a revolutionary paraboloid in 1972. According to Makinde and Animasaun et al.,[7] fluid flows over surfaces of different thicknesses or non-uniform thicknesses are described as the topmost horizontal surface of a revolutionary paraboloid. Fluid flows over a revolutionary paraboloid have piqued scientists’ interest due to the numerous applications of flows over a variable thickness surface. Zeeshan et al.[8] considered the flow of a Casson fluid containing nanoparticles across a horizontal revolution surface. Magnetite possessed a three-dimensional structure. Abdeljawad et al.[9] clarified Carreau fluid flow on a horizontal revolutionary paraboloid surface with a thermal radiation effect. The Catteneo-Christov model was used by Kalyani et al.[10] in a micropolar fluid with varying thickness.

The stacking of bodies of water as a result of temperature and concentration changes is known as stratification. The stratification idea is used to retain cold and warm water in the same tank and separate the two distinct masses of water using a thin layer thermocline in various energy storage applications like thermal storage systems or solar thermal utilization systems. Khan et al.[11] recently discussed the impact of temperature and solutal stratification on the chemically reactive flow of the hyperbolic tangent fluid. In a stratified medium, Dawar et al.[12] explored the bioconvection flow of a binary fluid comprising nanoparticles and gyrotactic bacteria. Khan et al.[13] demonstrated the effect of stratification on the nonlinear radiative flow of nanoparticles. The influence of the stratification parameter and Atwood number on the compressible multi-mode Rayleigh–Taylor uncertainty was investigated by Luo and Wang.[14] The dynamics of the buoyant convective flow of a micropolar fluid under temperature stratification were studied by Koriko et al.[15] The existence of stratification was shown to be associated with a decrease in fluid velocity and temperature. Pattnaik et al.[16] investigated the flow of electrically conducting micropolar nanofluids from a vertical stretching surface adjacent to a porous medium in the presence of a transverse magnetic field. They took into account both thermal and solutal stratifications, as well as chemical reaction effects, viscous dissipation, and heat sources.

Entropy optimization is the quantity of entropy produced by irreversible processes such as fluid flow over thermal radiation, heat flow over a thermal resistance, Joule heating, diffusion, fluid viscosity inside a system, friction between solid surfaces, and so on in a thermodynamic system. During a reversible process, the entire entropy of the system remains unaltered, according to the second rule of thermodynamics. It is commonly understood that entropy creation plays a critical role in reducing the system’s required energy sources. The researchers’ main concern in order to enhance effectiveness and achievement in most industrial and engineering applications is to reduce entropy formation. With this in mind, Bejan[17] attempted to examine entropy formation in a convective heat transmission process at first. Shit et al.[18] explored entropy formation in unstable two-dimensional MHD nanoparticles flowing through an exponentially extending surface in a porous medium under the effect of thermal radiation using a mathematical model. Shit and Mandal[19] expanded on this research. They utilized Buongiorno’s model to look into the creation of entropy in an unstable MHD flow of Casson nanoparticles through a stretching vertical plate under the effect of thermal radiation. Bhatti et al.[20] investigated the production of entropy during the interaction of nanomaterials with a stretching permeable sheet and nonlinear thermal radiation. Kumar et al.[21] investigated the 2D stagnation point hydromagnetic flow of Casson nanoparticles across a stretching surface in a non-Darcy permeable medium to Arrhenius activation energy-stimulated binary chemical reaction. Bhatti et al.[22] go into great detail about the behavior of diamond and silica nanoparticles dissolved in a water-based hybrid nanofluid lying above an exponentially elastic surface. They assumed the porous medium when analyzing the flow behavior. Bhatti et al.[23] discuss the development of a mathematical model of hybrid magnesium oxide and nickel nanofluid MHD stagnation point flow impinging on a permeable elastic stretching surface in a permeable medium. Zhang et al.[24] explored the role of the induced magnetic effect on hybrid nanofluid flow via a stretchy surface. Kristiawan et al.[25] evaluated and deduced that the existence of TiO2/water nanofluids in different volumetric concentrations improved the heat transfer coefficient of microfins. A considerable amount of notable work has recently been accomplished.[26−30]

The main goal of this article is to examine the entropy optimization and the effect of variable fluid features on the flow of the second-grade fluid across a surface that is neither inclined nor vertical but rather has a non-uniform thickness. Several factors, including Joule dissipation, heat radiation, stratification, and variable thermal conductivity, have been used to model the energy equation. The flow problem equations are presented and transformed using a suitable similarity transformation. The arising equations are solved using HAM. A concise graphical explanation is provided in a separate section, and the conclusion of relevant material restrictions is accurately depicted. As far as we know, no one has addressed this issue.

Mathematical Formulation

Consider a steady, MHD, incompressible second-grade fluid flow along the horizontal surface of a paraboloid of revolution, with varying fluid characteristics and stratifications. The domain where is supposed to be occupied by the second-grade fluid flow across the surface with non-uniform thickness. is the velocity index, and and are arbitrary constants related to the thickness of the surface. is less than in this study. The fluid layers on an upper horizontal revolution surface are stretched at , where is the reference velocity. It is assumed that the horizontal surface of the revolutionary paraboloid is porous. The , on the other hand, represents the stretchy surface’s orientation, while the is perpendicular to it.

Under the impact of variable fluid characteristics and stratifications, the equations regulating the dynamics of the second-grade fluid over the upper horizontal surface of the paraboloid of revolution are written as

Subjected to boundary conditions

Batchelor’s[31] experimental results were used to develop a temperature-dependent viscosity mathematical model, along with the temperature-dependent thermal conductivity mathematical model of Charraudeau[30] as

As we describe the thermal stratification at the wall and solutal stratification at the wall , as well as the free stream temperature and concentration , it is critical to show how stratification is included into the energy and concentration equations.where represents the reference temperature; it is worthy to note that stratification happens for all places of on the wall at , as well as for all points of at the ambient temperature as .

The similarity variable and the stream function can be expressed as

The governing partial differential eqs –4, as well as the boundary constraints 5 are changed to a system of nonlinear differential equations represented as a stream function that satisfies the continuity equation via similarity transformation.

It is worth noting that ; the slightest value of that corresponds to the slightest value of the similarity variable is found at the surface

Therefore, the boundary constraints become

The boundary conditions are dependent on , while the modified governing eqs –13 are dependent on . It is imperative to change the domain from to which is carried out by defining and . The non-dimensional governing eqs –13 are rewritten as

The boundary constraints becomewhere

The following are the definitions of the skin friction, the local Nusselt number, and the Sherwood numberwhere and are the surface shear stress and heat flux, respectively. These are defined as

The dimensionless physical quantities are

Entropy Optimization

Entropy optimization is the quantity of entropy produced by irreversible processes such as heat flow over a thermal resistance and Joule heating in thermodynamic system. Such as

Using transformations 9 and 10, the non-dimensional entropy formation is obtained as

Equation is dependent on . It is imperative to change the domain from to which is done by defining , and .Where represents the total entropy formation, represents the temperature ratio parameter, and represents the local Reynolds number.

Bejan Number

The Bejan number (Be) is a number that demonstrates the value of thermal irreversibility in comparison to absolute irreversibility.

Results and Discussion

This section explains how to interpret the behavior of the various parameters on the profiles when they are set to fixed values. The plots in this study took into account the situations of (unstratified surface) and (stably stratified surface). Figures and 2a,b show the outcome of changing the velocity index on the velocity field when and . It can be seen in these images that there is no considerable rise in velocity fields as is altered, although the profiles do slightly increase for bigger values of . Figure c,d depicts the stimulus of the second-grade fluid parameter on the velocity field for and . The velocity field for both cases and is strengthened with increasing values of the second-grade fluid parameter, as seen in figures. This rise in the velocity field occurs because an increase in causes a drop in fluid viscosity, which causes the velocity field in the flow zone to upsurge. In addition, as the second-grade fluid parameter grows, the boundary layer thickness of the momentum also rises. Figure e,f shows the outcome of the magnetic parameter on the velocity field for and . The figures show that at rising levels of the magnetic parameter in the flow zone, the velocity is eventually suppressed. This reduction in velocity field is primarily due to the fact that an effect of magnetic parameter causes an uptick in Lorentz forces involved with the magnetic field, and these magnified Lorentz forces provide further resistance to fluid flow, which causes the velocity field in the boundary layer region to reduce. Figure g,h shows the outcome of the porous parameter on the velocity field for the instances of and , respectively. The velocity field is visibly lowered with higher values of the porosity parameter in the flow zone, as seen in the figures. This drop is mostly because of the fact that increasing the values of the porosity parameter causes more and more opposition to fluid flow in the flow zone, which causes the velocity field to decline. The effects of viscosity dependent on the temperature parameter on the velocity profile for the cases and are shown in Figure i,j. A rise in the viscosity parameter results in a significant change in the velocity profile for both cases, as shown in figures. Since the viscosity parameter is inversely related to dimensionless viscosity, as a result, enhancing the viscosity parameter diminishes the dimensionless viscosity and hence lowers the viscous forces that oppose fluid motion. Therefore, inertia forces outnumber viscous forces, and the liquid accelerates. Figure k depicts the stimulus of the stratification parameter on the velocity profile. It is noticed that the velocity is enhanced by raising the stratification parameter .

Figure 1

Physical sketch of the flow.

Figure 2

(a) Variation in velocity profile for different values of velocity index when . (b) Variation in velocity profile for different values of velocity index when . (c) Variation in velocity profile for different fluid parameter when . (d) Variation in velocity profile for different values of fluid parameter when . (e) Variation in velocity profile for different values of magnetic parameter when . (f) Variation in velocity profile for different values of magnetic parameter when . (g) Variation in velocity profile for different values of porosity parameter when . (h) Variation in velocity profile for different values of porosity parameter when . (i) Variation in velocity profile for different values of variable viscosity parameter when . (j) Variation in velocity profile for different values of variable viscosity parameter when . (k) Variation in velocity profile for different values of thermal stratification parameter .

Effects of variation of on the temperature profile are seen in Figure a,b for and . From both figures, it is clearly seen that the temperature profile declines by enhancing . Figure c,d depicts the temperature profile fluctuations observed for various values of magnetic parameter . The temperature field is amplified for rising values of , as can be seen in both diagrams. An increase in causes the elastic stress variable to drop, resulting in an enlarged temperature field in the region of the flow configuration. Furthermore, because of the existence of the Joule dissipation effect, more heat energy will be released into the working fluid, causing the temperature profile in the boundary layer region to improve for and . Figure e,f portrays temperature field variations for different values of porosity parameter . Because of an escalation in fluid viscosity or a diminution in the permeability of the porous media near the edge of the boundary layer, the non-dimensional temperature is increased in both cases and . Figure g,h shows that the influence of a variable viscosity parameter on the temperature profile for cases and , respectively, and a rise in the value of tends to increase the thickness of the thermal boundary layer. As a result, the values of temperature rise, and the temperature of fluid rises as the rises for unstratified and stratified regions. Figure i,j shows the result of the heat generation parameter on the temperature profile when and . The temperature profile appears to be increasing as the heat generation parameter is increased. Positive changes in the heat generation parameter generate heat in the flow regime, resulting in an increase in fluid temperature for both stratified and unstratified cases. Figure k shows the consequence of the thermal stratification parameter on temperature distribution. Positive values indicate a decrease in temperature because higher values indicate a decrease in convective potential between the ambient temperature and paraboloid of revolution.

Figure 3

(a) Variation in temperature profile for different values of velocity index when . (b) Variation in temperature profile for different values of velocity index when . (c) Variation in temperature profile for different values of magnetic parameter when . (d) Variation in temperature profile for different values of magnetic parameter when . (e) Variation in temperature profile for different values of porosity parameter when . (f) Variation in temperature profile for different values of porosity parameter when . (g) Variation in temperature profile for different values of variable viscosity parameter when . (h) Variation in temperature profile for different values of variable viscosity parameter when . (i) Variation in temperature profile for different values of heat generation parameter when . (j) Variation in temperature profile for different values of heat generation parameter when . (k) Variation in temperature profile for different values of thermal stratification parameter .

The influence of variation on the concentration fields is shown in Figure a,b). When the concentration field is observed to be enhanced for larger values of , whereas when the concentration field is observed to be increased from the negative zone for larger values of . Physically, the observed trend is caused by stratification, which creates a transition zone between the temperature and concentration gradients of cold and hot fluid zones. The effects of the chemical reaction parameter variation on the concentration fields are shown in Figure c,d for the unstratified and stratified regions of the paraboloid of revolution. The concentration field is observed to be enhanced for larger values of for both cases. Figure e,f depicts the influence of Schmidt number on the concentration profile for cases and . In Figure e, a slight increase in the concentration field is observed for variation in an unstratified region of the paraboloid of revolution, whereas in Figure f, a significant increase in the concentration field is observed for variation in a stably stratified region of the paraboloid of revolution. Physically, the ratio of momentum diffusivity (viscosity) to mass diffusivity is the Schmidt number, which is a dimensionless quantity. As a result, as the Schmidt number rises, the concentration profile rises as well. The variation in fluid concentration in relation to the solutal stratification parameter is depicted in Figure g. It is observed that the concentration distribution reduces for larger values of the solutal parameter . This is due to a decrease in the potential difference between surface and ambient concentrations.

Figure 4

(a) Variation in concentration profiles for different values of velocity index when . (b) Variation in concentration profiles for different values of velocity index when . (c) Variation in concentration profiles for different values of chemical reaction parameter when . (d) Variation in concentration profiles for different values of chemical reaction parameter when . (e) Variation in concentration profiles for different values of Schmidt number Sc when . (f) Variation in concentration profiles for different values of Schmidt number Sc when . (g) Variation in concentration profiles for different values of solute stratification parameter .

The outcome of the Brinkman number on the entropy formation rate for and is seen in Figure a,b. For both the unstratified and stratified cases, the finding demonstrates that the entropy generation rate within the paraboloid of revolution grows as the Brinkman number escalates. The impact of the Brinkman number on the Bejan number is demonstrated in Figure a,b for both unstratified and stratified cases, which shows that the Bejan number drops down for a higher Brinkman number. The explanation for this is simple: the Brinkman number is a heat source, and heat is produced inside the layers of moving fluid particles. Entropy formation within the paraboloid of revolution flow is encouraged by the heat created and the heat transmitted from the heat wall. As a result, the Brinkman number must be kept under control in order to decrease entropy. The influences of the magnetic parameter on entropy production and the Bejan number are shown in Figures c, 6c and 5d, 6d for and cases, respectively. The figures show that as the magnetic parameter is increased, the extent of local entropy generation increases, while the Bejan number decreases in both unstratified and stratified regions. The magnetic field adds to the fluid’s entropy. At the surface, there is also a lot of entropy creation. Figure e,f depicts the impact of the temperature ratio parameter on entropy formation for unstratified and stratified cases. From the figures, it is observed that for both unstratified and stratified regions, entropy production declines as Ω improves. Figure e,f indicates the influence of the temperature ration parameter on the Bejan number for both unstratified and stratified regions. The Bejan number grows up as improves.

Figure 5

(a) Variation in entropy optimization for different values of Brinkman number Br when . (b) Variation in entropy optimization for different values of Brinkman number Br when . (c) Variation in entropy optimization for different values of magnetic parameter when . (d) Variation in entropy optimization for different values of magnetic parameter when . (e) Variation in entropy optimization for different values of the temperatureratio parameter when . (f) Variation in entropy optimization for different values of the temperature–ratio parameter when .

Figure 6

(a) Variation in Bejan number for different values of Brinkman number Br when . (b) Variation in Bejan number for different values of Brinkman number Br when . (c) Variation in Bejan number for different values of magnetic parameter when . (d) Variation in Bejan number for different values of magnetic parameter when . (e) Variation in Bejan number for different values of temperature ratio parameter when . (f) Variation in Bejan number for different values of the temperature ratio parameter when .

Tables , 2, and 3 depict the variation of skin friction coefficients, Nusselt number, and Sherwood number for various parameters, respectively. Table indicates the variant of skin friction coefficients for incremental values of the second-grade fluid parameter, magnetic parameter, porosity parameter, and variable viscosity parameter for unstratified and stratified cases. Incremental values of the fluid parameter declined the skin friction coefficient, whereas growing values of the magnetic parameter, porosity parameter, and viscosity parameter upsurge skin friction coefficients for both unstratified and stratified regions. Table indicates the variant of Nusselt number for numerous values of the magnetic parameter, heat generation parameter, radiation parameter, and Eckert number for and . From the table, it is observed that for both unstratified and stratified cases, the Nusselt number upsurges for rising values of , , , and . Tables and 4 indicate the variant of Sherwood number for incremental values of the Schmidt number and chemical reaction parameter for the cases and . The finding demonstrates that the Sherwood number declined for escalations in the chemical reaction parameter and Schmidt number for both unstratified and stratified regions.

Table 1

Variation of Skin Friction for Numerous Parameters

0.0				–0.79999	–0.74799
0.5				–1.54439	–1.40753
1.0				–2.22858	–2.01121
	0.1			–2.032141	–1.88437
	0.3			–1.86643	–1.72229
	0.5			–1.70385	–1.56335
		0.1		–1.84897	–1.64243
		0.3		–1.561946	–1.48505
		0.5		–1.28781	–1.33079
			0.1	–1.72802	–1.58771
			0.3	–1.67888	–1.53819
			0.5	–1.62656	–1.48549

Table 2

Variation of the Nusselt Number for Various Parameters

0.1				1.366413	–0.173012
0.3				1.381303	–0.1509507
0.5				1.394915	–0.1302655
	0.1			1.966641	–0.17356005
	0.3			2.111575	–0.020312
	0.5			2.256512	0.132937
		0.1		1.013456	–0.0995668
		0.3		1.263412	–0.1198638
		0.5		1.530895	–0.140836
			0.1	1.373282	–0.1521856
			0.3	1.416549	–0.1083453
			0.5	1.459818	–0.0645039

Table 3

Variation of the Sherwood Number for Different Parameters

0.1		1.1295706	–0.0045168
0.3		1.1042614	–0.01355805
0.5		1.0789438	–0.0226091
	0.1	1.1212254	–0.0135447
	0.3	1.1099196	–0.0135536
	0.5	1.0985996	–0.0135625

Table 4

Comparison of the Obtained Result of With That of Kalyani et al[10]

m1	present paper	Kalyani et al.[10]
0.1	–0.86727272	–0.86710093
0.2	–0.85333334	–0.8624051
0.3	–0.84153846	–0.8584863
0.4	–0.83142857	–0.8481544

Conclusions

Second-grade fluid flow over a horizontal surface of a paraboloid of revolution has been studied, with emphasis on the unstratified and stably stratified regions. The dimensionless equations have been numerically solved, and the major key outcomes of the study are shortened as follows:

Velocity profiles increase when the velocity index and variable viscosity parameter upsurge, whereas conflicting behavior is shown for upsurge values of the fluid parameter, magnetic parameter, and porosity parameter for both unstratified and stratified cases.

Temperature distribution is an increasing function of the magnetic parameter, porosity parameter, and variable viscosity parameter, whereas it is a declining function of velocity index and heat generation parameter for both unstratified and stratified cases.

For unstratified and stratified cases, there is enhancement in the concentration profile for large values of the velocity index parameter, chemical reaction parameter, and Schmidt number.

Velocity and temperature profiles declined for large values of the thermal stratification parameter.

For large values of the solute stratification parameter, concentration profiles fall down.

For both unstratified and stratified regions, incremental values of the Brinkman number and magnetic parameter depict augmentation in the entropy production, while entropy production drops for a large value of the temperature ratio parameter.

Skin friction coefficients rise up for incremental values of the magnetic parameter, porosity parameter, and variable viscosity parameter while decline for incremental values of the fluid parameter for both unstratified and stratified regions.

The Nusselt number declined for growing values of the magnetic parameter, heat generation parameter, radiation parameter, and Eckert parameter for both unstratified and stratified regions.

For large values of the chemical reaction parameter and Schmidt number, the Sherwood number for unstratified and stratified cases declined.