1. Introduction
In recent years, the analysis of thermal expansion and liquid flow across permeable surfaces has gained the attention of researchers because of its extraordinary range of utilization in the engineering and biomedical sectors, such as enhanced oil recovery, solid matrix heat exchanges, cooling of nuclear reactors, chromatography, geothermal and petroleum resources, thermal insulation drying of porous solids, ceramic processing, filtration processes, etc. Darcy’s law is only applicable for low-porosity and low-velocity systems. As velocities increase and porosity becomes non-uniform, its application in industrial systems and engineering becomes inefficient. Therefore, the Darcy–Forchheimer (DF) model, which includes the inertial aspects and boundaries, can be employed to overcome the limits of Darcy’s law. For these reasons, Forchheimer [
1] introduced the squared velocity term into the expression of motion in 1901. Ikram et al. [
2] investigated the Darcy effects in nanofluid (NF) flow in the direction of an infinite rotating and stretchy disc with an exponential heat source and thermal heat source. It has been observed that the exponential space heat source has a greater influence on fluid temperature than the thermal heat source. Alotaibi and Mohamed [
3] discussed the DF nanofluid flow across a convectively heated porous expanding surface. The DF flow of Casson material comprising titanium dioxide and graphene oxide in a permeable medium was inspected by Kumar et al. [
4]. The DF relation with variable porosity and permeability flow over a stretchable rotating disk in a porous region was scrutinized by Hayat et al. [
5]. An extending surface with variable thickness incorporated in a DF space was employed by Gautam et al. [
6] to study the Williamson nanofluid flow. It was observed that the rising wall thickness affects the velocity and energy profile. Mallikarjuna et al. [
7] evaluated a double-phase (dust phase and fluid phase) DF-hybrid nanofluid flow with dissipation and melting effects. The consequences reveal that a high melting variable decreases the thermal effect in both phases. Saeed et al. [
8] studied the flow of a couple of stressed hybrid nanomaterials through DF with variable viscosity, Joule heating, and viscous dissipation. Gowda et al. [
9] investigated the incompressible viscous dusty hybrid nanomaterial across a stretching cylinder, with dissipation and DF phenomena. They reported that increasing the curvature parameters increased the temperature and velocity of both liquids. Rasool et al. [
10] investigated how the DF medium and radiation affect the Maxwell NF flow when it is subjected to a stretched surface. DF fluid flow with variable permeability and porosity was examined by several researchers [
11,
12,
13,
14].
Transportation of mass and heat are employed in a broad spectrum of industrial applications, including cars, energy systems, heating and air conditioning, steam-electric power production, electronic devices, and disease detection, but common liquids are unable to fulfill the demand for heat transmission. Consequently, the utilization of nanoparticles in the common fluid is highly suggestive. NF has received much attention in the recent decade, especially in renewable and sustainable energy systems and heat transfer enhancement. NF is utilized in hydropower rotors, ocean power plants, geothermal heat exchangers thermodynamics, wind turbines, and solar collectors [
15]. In an experimental context, the dynamic viscosity of MWCNT-MgO/EG hybrid nano liquid was examined by Soltani and Akbari [
16]. The results reveal that the dynamic viscosity enhances up to 168% by enhancing the number of nanoparticles from 0.1 to 1%. Waqas et al. [
17] analyzed the features of thermal radiation in Sisko nano liquid flow with the mutual effect of heat and mass transmission characteristics using gold nanoparticles over an extending surface. It has been observed that the energy profile of the nanofluid is elevated for both the volume friction of nanoparticles and thermal radiation. Li et al. [
18] used gold nanofluid as an optical filter for increasing thermal collection and balancing the energy in the photovoltaic/thermal system. The behavior of blood carrying gold nanoparticles over a curved surface is examined by Khan et al. [
19]. Mbambo et al. [
20] explore the effectiveness of raising the thermal conductivity of gold nanoparticles in graphene sheets by using an EG-based nanofluid. The results revealed that by changing the structure of nanoparticles, the rate of thermal energy transmission can be boosted. The Buongiorno model is applied by Mishra and Upreti [
21] to evaluate the heat and mass transmission processes of Fe
3O
4-CoFe
2O
4/water-EG hybrid nanomaterial and Ag-MgO/water hybrid nanofluid under the effects of chemical reaction and dissipation. Najiyah et al. [
22] numerically evaluated the 3D flow of the magnetic nanofluids (Mn-ZnFe
2O
4-water, CoFe
2O
4-water, and Fe
3O
4-water) shrinking surface with radiation phenomena. The influence of the Darcy–Forchheimer and Coriolis forces on nanofluid flow consisting of carbon nanotubes (CNTs) in EG over a revolving frame was explained by Ullah et al. [
23]. Entropy is characterized by a growing trend in the temperature ratio and Brinkman number. Ullah et al. [
24,
25] explored the behavior of melting thermal expansion and activation energy on an unstable Prandtl–Eyring fluid caused by an extended cylinder.
Entropy generation is the quantity of entropy produced by irreversible processes such as diffusion, heat flow via Joule heating, thermal resistance, fluid flow via flow resistance, friction between solid surfaces, the viscosity of a fluid within the system, and so on in a dynamical system [
26]. In a reversible reaction, the net entropy of the system remains constant. The entire entropy of a system increases when a nanofluid moves across a substrate by different irreversible mechanisms such as flow resistance, diffusion, thermal resistance, viscosity-induced friction between fluid layers, Joule heating, and so on. Entropy production in a system plays a significant role in decaying the system’s needed energy sources. Thus, researchers focus on improving the performance and effectiveness of various sectors to decay the production of entropy [
27,
28]. Ikram et al. [
29] discuss the entropy considerations associated with the Darcy–Forchheimer flow of hybrid nanomaterial. Their findings reveal that an increase in temperature caused an increment in the Eckert number and the heat source parameter for both nano liquids and hybrid nano liquids. Fares et al. [
30] applied a finite element approach to characterize the progressive feature of created entropy in a permeable square enclosure. Aziz et al. [
31] discussed entropy optimization in the radiative flow of a nanofluid in a rotating frame with activation energy. Alsallami et al. [
32] have quantitatively inspected the NF flow over a heated spinning disk under the effects of nonlinear radiation, Brownian motion, and thermophoresis. The entropy rate and Bejan number are thought to improve as the chemical process, temperature differential factor, and Schmidt number improve. Ullah et al. [
33] analyzed the numerical investigation of the entropy characteristics of the hybrid nanoparticles with slip effect.
The goal of the current work is to enhance the energy transference rate for industrial and biomedical applications. It has been observed that none of the previous efforts on the Darcy–Forchheimer hybrid NF flow consisting of gold and cobalt ferrite nanoparticles over an extending curved porous space with variable permeability were given in the literature. Additionally, radiation, Joule heating, and the traditional and exponential heat sources are considered to compute the entropy optimization. To fill the gap, the proposed model has been formulated in the form of a system of PDEs, which is converted into the system of ODEs through transformations. The obtained set of nonlinear differential equations is further processed numerically by the NDSolve technique to investigate the variations in entropy, velocity, temperature, Nusselt number, and skin friction against interesting physical variables. The present model is very useful in distinct areas of applied science, biomedicine, and industries like hybrid power generators, the cooling of nuclear reactors, electronic devices, and heat transportation in milk-tissue pasteurization. The following are major considerations for conducting this study:
To investigate how hybrid nanofluid is more efficient than nanofluid and base fluid.
To examine the thermal effects of hybrid nanomaterials with thermal radiation, EHS, and viscous dissipation.
What are the effects of skin friction and Nusselt number in regard to relevant physical constraints?
How does the addition of a Darcy–Forchheimer term with variable permeability and porosity features to the momentum equation impact the hybrid nanofluid flow?
How does the inclusion of cobalt ferrite and gold nano particulates enhance the thermal efficiency of ethylene glycol?
2. Mathematical Modeling
Consider an incompressible two-dimensional hybrid NF flow over curved porous stretching surface with radius
R. The Darcy–Forchheimer law is utilized for present flow analysis with variable permeability. For the improvement of the thermal field, Joule heating is added to energy expression. The surface is extended in the
s-direction with velocity
, where
and
direction is Normal to
direction (see
Figure 1).
Here,
indicates the static sheet,
b < 0 demonstrates the shrinking, and
specifies the stretching sheet. In this work, we scrutinize the flow across a stretching sheet; therefore, we only consider
in this study. Cobalt ferrite and gold nanoparticles were added to ethylene glycol and water to form the hybrid nanofluid. A magnetic field with strength
is incorporated in the
r-direction. The temperature at the surface is specified as
Heat source and thermal radiation in the energy expression is added to further investigate the temperature variation. Additionally, the second law of thermodynamics is employed to compute entropy production. On the basis of the above-described presumptions, the leading equations can be stated as [
34,
35,
36,
37]:
where
demonstrate the velocity component, while
k* is the porosity term,
is the surface porosity,
is surface permeability,
shows the constant having dimension of length,
demonstrates the kinematic viscosity,
is the variable permeability and
is the porosity,
is electrical conductivity,
is density,
is the Stefan Boltzmann coefficient,
is thermal conductivity,
is the drag coefficient,
is the magnetic field strength,
Q is the heat generation/absorption,
is the exponential heat source, and
is the volumetric heat capacity [
34,
35].
The relevant boundary conditions are: [
34]
Considering the variables:
Equation (1) is identically satisfied and the system of ODEs form Equations (2)–(4) and are transformed as:
By putting Equation (9) into Equation (8) we have:
where transformed boundary conditions are:
The dimensionless variables are:
In Equation (14), K represent the dimensionless curvature variable, S is the heat source, is the Prandtl number, is the non-uniform inertia factor, is the exponential heat source variables, is the permeability term, M is the magnetic variable, is the Eckert number, Ra is the radiation parameter, and Br depict the Brinkman number.
Skin friction is
and Nusselt number are
specified as:
where wall shear stress
and heat flux
are specified as:
Upon using Equation (7), the above equations become:
where Reynolds’s number
5. Discussion or Outcomes
This section analyzes the variation of velocity, skin friction, and temperature gradient against the variation of numerous physical parameters such as curvature parameter
magnetic field
, variable permeability
, variable porosity
inertia coefficient
permeability parameter
Brinkman number
exponential heart source (EHS) parameter
radiation parameter
heat source parameter
, and Eckert number
for hybrid nanoliquid and nanofluid. For such aims,
Figure 2,
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10,
Figure 11 and
Figure 12 and Tables are declared. In these figures, solid lines manifest the nanofluid, whereas dashed lines show the hybrid nanofluid.
Table 1 and
Table 2 depict the thermophysical relation of nano and hybrid nanomaterials, respectively.
Figure 2 elaborates on the role of volume friction on
The resistance to fluid flow rises as the volume fraction of nanoparticles rises; as a result, the velocity declines. It is also manifest from
Figure 2 that hybrid nanofluid has more contribution to the reduction in velocity when compared with nanofluid.
Curves for velocity against various values of the curvature parameter and magnetic field is depicted in
Figure 3a,b, respectively. An increasing behavior of velocity in
Figure 3a is observed for different values of curvature parameter. A high estimation of
K leads to uplift in the surface radius, which improves the velocity. The features of
M on
is seen in
Figure 3b. Here, velocity of the hybrid nanofluid slows down for a higher estimation of
M. The magnetic effect induces the Lorentz force, which causes a resistive force to the flow field; consequently, the radial velocity of the flow reduces. A zero value of the magnetic parameter, i.e.,
M = 0, corresponds to the hydrodynamic situation. When compared with nanofluid, hybrid nanofluid has a greater contribution toward enhancing the velocity, as shown by
Figure 3a, whereas
Figure 3b demonstrates the contrary tendency.
Figure 4.
The variation in velocity (a) versus variable permeability (b) versus variable porosity .
Figure 4.
The variation in velocity (a) versus variable permeability (b) versus variable porosity .
Figure 4a,b demonstrate the impact of
In this case, velocity
increases as the variable permeability
increases (see
Figure 4a), while
experiences the opposing trend (see
Figure 4b). Moreover,
Figure 4a indicates that hybrid nanofluid contributed more to velocity increase than nanofluid, while the opposite trend is observed in
Figure 4b. The behavior of the inertia coefficient
on
is illustrated in
Figure 5a. An abatement is observed in velocity for a higher value of
because of an increase in internal force.
Figure 5b describes the effects of
on velocity
. By enlarging the value of
the velocity of the nanomaterial is also augmented. When compared with nanofluid, it is clear that hybrid nanofluid has a greater contribution to the decrease in velocity, as shown by
Figure 5a.
Figure 5.
The variation in velocity (a) versus inertia coefficient (b) versus permeability parameter .
Figure 5.
The variation in velocity (a) versus inertia coefficient (b) versus permeability parameter .
Figure 6.
The variation in temperature (a) versus volume friction (b) versus EHS parameter .
Figure 6.
The variation in temperature (a) versus volume friction (b) versus EHS parameter .
Figure 7.
The variation in temperature (a) versus inertia coefficient (b) versus Brinkman number .
Figure 7.
The variation in temperature (a) versus inertia coefficient (b) versus Brinkman number .
An escalating behavior is perceived in
Figure 6a for different values of volume friction on temperature. As the volume fraction of nanoparticles in the fluid raises, more friction between the particles of the fluid arises, by which the temperature of the hybrid nanomaterial is elevated. As depicted in
Figure 6b,
has a significant impact on
. A rise in
causes an enhancement in
Figure 6a,b show that hybrid nanofluid boosted temperature more than nanofluid.
Figure 7a depicts the rising trend of inertia coefficient
versus temperature
The inertia coefficient increases fluid motion resistance, resulting in more heat being created and strengthening the temperature field.
Figure 7b captured that
is an increasing function of
because
has a direct relation with the formation of heat through fluid friction, which leads to greater
Figure 7a,b demonstrate hybrid nanofluid raises temperature more than nanofluid.
Figure 8.
The variation in temperature (a) versus variable permeability (b) versus variable porosity .
Figure 8.
The variation in temperature (a) versus variable permeability (b) versus variable porosity .
The variation in variable permeability
and variable porosity
against temperature
is seen in
Figure 8a,b. An improvement in
is reported for a higher value of
(see
Figure 8a), whereas the reverse tendency is exhibited against a different value of
(see
Figure 8b).
Figure 8a shows that hybrid nanofluid dropped temperature more than nanofluid, while
Figure 8b illustrates the inverse.
Figure 9.
The variation in and temperature (a) versus radiation parameter (b) versus heat source parameter .
Figure 9.
The variation in and temperature (a) versus radiation parameter (b) versus heat source parameter .
Figure 9a is devoted to analyzing the influence of
on temperature. The motion of charged particles in a fluid accelerates as thermal radiation rises, which boosts the temperature. As portrayed in
Figure 9b, that increment in heat source parameter
improves the temperature. Greater heat source parameters result in a thicker thermal boundary layer, which raises the temperature.
Figure 10.
The impact of (EHS) parameter versus .
Figure 10.
The impact of (EHS) parameter versus .
As we observed in previous graphs that with a high value of EHS parameter
, thermal radiation
, magnetic field
, Brickman number
, and the temperature of the fluid increase. Vibrations and internal displacement are two additional phenomena that occur as the fluid temperature rises. Consequently, the fluid entropy improves. As indicated in
Figure 10,
Figure 11a,b, and
Figure 12a,b, a rise in radiation parameter
, magnetic field
Brinkman, and
Eckert numbers leads to development in the production of entropy in the fluid. The entropy optimization is demonstrated to be enhanced significantly more by hybrid nanofluid than by nanofluid in
Figure 10,
Figure 11 and
Figure 12.
Figure 11.
The variation in . (a) versus radiation parameter (b) versus magnetic field .
Figure 11.
The variation in . (a) versus radiation parameter (b) versus magnetic field .
Figure 12.
The variation in (a) versus Brinkman number (b) versus Eckert number .
Figure 12.
The variation in (a) versus Brinkman number (b) versus Eckert number .
Table 3 illustrates the influence of skin friction coefficient
against various variables such as curvature parameter
magnetic parameter
variable permeability
variable porosity
inertia coefficient
, and permeability parameter
. The coefficient of skin friction diminishes for higher values of variable permeability
. The behavior of the Nusselt number
for the different physical parameters is depicted in
Table 4. Nusselt number
enhances because there is more heat owing to thermal radiation
EHS parameter
Brinkman number
variable permeability
and heat source parameter
Table 5 exemplifies the comparison of the current study with the previously published work [
37,
38,
39] for m = 1 and various values of curvature parameters
The analysis reveals a very close match.