Carbon nanotubes (CNTs) exhibit cylindrical structures composed of carbon atoms arranged in nature-like configurations, forming thin and elongated pure carbon structures. Their diameters typically range from 1 to 50 nm. When incorporated into certain polymer compounds, CNTs enhance their damping properties and mechanical characteristics. Broadly categorized as single-wall and multi-wall CNTs, they serve both structural and non-structural purposes. Structural applications encompass thermal stability, inter-laminar shear strength, fracture toughness, stiffness, energy absorption, strength, and damping, whereas non-structural applications involve improved thermal conductivity, energy storage, strain sensing, electromagnetic interference mitigation, and so forth. Wang et al. [1] experimentally examined the pressure drop and heat transmission of nanofluid flow containing CNTs in a circular tube lying in a horizontal position. Their findings revealed a notable increase in heat transfer efficiency for high values of Reynolds number. A kerosene-aqueous-based nanofluid flow with CNTs as nanoparticles over an extended surface influenced by an induced magnetic field in the presence of gyrotactic microorganisms is examined by Iqbal et al. [2]. An interesting result disclosed that nanoparticles’ volume fraction improves the strength of the assumed magnetic field. Alharbi et al. [3] performed a comparative analysis of simple nanofluid and hybrid nanofluid flows with CNTs (both single-wall and multi-wall) and ethylene glycol as a base fluid over a two-directional surface with prescribed heat flux and prescribed surface temperature amalgamated with modified Fourier heat flux. It is revealed here that prescribed heat flux and prescribed surface temperature are the main sources for the rise in the temperature of the fluid. The heat transfer analysis of the nanolayer over an aqueous-based nanofluid flow comprising CNTs over a movable wedge is examined numerically by Gul et al. [4]. The model is supported by the convective and the slip conditions at the surface of the wedge. It is witnessed here that fluid velocity is on the decline for slip conditions. Bashir et al. [5] performed a comparative study of the flow of the nanofluid flows comprising CNTs of both types with two base liquids considering the Darcy-Forchheimer and homogeneous-heterogeneous reactions amalgamated with second-order slip at the surface of an extended sheet. The problem is addressed numerically, and outcomes are presented in the forms of graphs and tables. It is comprehended that the effect of the aqueous-based single-wall CNTs fluid performed better than multi-wall CNTs fluid as far as fluid concentration is concerned.

In the study of oceanic sciences and atmospheric, fluid stratification is a phenomenon of liquid layering owing to differences in densities, temperature, or composition. This phenomenon is common in chemical industrial processes, environmental science, meteorology, atmosphere, oceanography, and bodies of water. The dynamics of stratification are also important for solar engineering because stratification may predict the chance of attaining higher energy capability. A strong propensity of the researchers/scientists toward the stratification phenomenon has been observed in the recent past. Hamid et al. [6] discussed the doubly stratified effect of Williamson nanofluids with mixed convection and thermal radiation. They employed the Runge-Kutta Fehlberg method to solve the model. They also reported that for declining values of mass transfer rate, the Brownian motion parameter increases. The double stratification with MHD flow of nanoliquid and slip conditions is investigated by Hayat et al. [7]. They employed HAM to attain the series solution of the system. They also discussed that the concentration profile increases and temperature distribution reduces for higher estimations of the thermophoretic and thermal stratification parameters respectively. Ramzan et al. [8] discovered the influences of Cattaneo-Christov heat flux with stratified media in Darcy Forchheimer nanofluid flow comprising carbon nanotubes and homogeneous-heterogeneous reactions. The effect of nanoliquid flow containing CNTs with an upper permeable wall in a rotating channel by using ethylene-glycol as a base fluid is discussed by Ramzan et al. [9]. It is witnessed that fluid temperature is on the decline for the thermal stratification parameter.

The aforecited literature demonstrates that a lot of effort has been observed by the researchers to discuss the effect of single-wall CNTs on nanofluid flow. However, less attention is paid to the effect of Cattaneo-Christov heat flux and thermal radiation with single-wall CNTs immersed in nanofluid flow. Nonetheless, no work has been done so far on single-wall CNTs with heat transfer and MHD nanoliquid flow with mixed convection over an inclined stretching sheet. The partial slip at the boundary wall is also taken into consideration. The effect of heat generation absorption and stratification boundary conditions is also considered. Irreversibility analysis is also a part of the present study. Utilizing the bvp4c MATLAB software function, the envisaged model is numerically addressed. The outcomes of the mathematical model are reported through diagrams and tables.

Let us assume a two-dimensional, mixed convection, electrically conducting nanoliquid flow along the permeable stretching surface with a velocity, leaning at an angle of (α = 45˚) with the horizontal axis (Figure 1). The nanofluid is comprised of single-wall carbon nanotubes and ethylene glycol mixture. The heat transfer behavior has been examined by considering the uniform electric and magnetic fields with the stratified medium.

The boundary layer governing equations are represented as:

Mathematically, the pattern of heat transfer can be described as follows:

The Cattaneo-Christov heat flux expression is given as under:

For, Equation (4) changes the Fourier’s law. Equation (4) for incompressibility conditions yields:

Removing q from Equations (3) and (5), we obtain:

The boundary constraints are:

To transmute the envisioned model to the system of ODEs, we assume the subsequent transformations:

The above transformation converts the Equations (2), (6), and (7) as given as:

where

The drag force coefficient is presented as:

With

Dimensionless form of drag force coefficient is given as:

where is the local Reynolds number.

Table 1 and Table 2 represent the thermo-physical values of the nanofluid being considered.

In this section, the outcomes are depicted in the form of graphs for velocity and temperature profiles versus discrete values of the arising flow parameters. The

values of these parameters are fixed as, , , , , , , , , unless otherwise stated.

The impact of the magnetic parameter M on the fluid velocity profile is displayed in Figure 2. It is seen that fluid velocity is declined for the high values of M. This is because of the strong Lorentz force that hinders the fluid flow and a drop in the velocity of the liquid is observed. Figure 3 depicts the behavior of the thermal convection parameter () versus the fluid velocity profile. It is noted

that the fluid velocity is augmented for varied estimates of the (). A surge in the Nusselt number means the high heat transfer rate leads to high thermal convection. This only occurs when there is an increase in the fluid velocity is noted. The association of the suction parameter (S) with the fluid velocity is illustrated in Figure 4. A decline in the fluid velocity is noticed here. The fluid momentum is on the decline when there is a strong effect of the suction. Eventually, the fluid velocity decreases. Figure 5 is displayed to describe the correlation between the thermal relaxation parameter () and the fluid temperature profile. Here, high

estimates of () result in a fall in the fluid temperature. This is due to the increased duration required for fluid particles to convey heat to adjacent particles. The impact of the radiation parameter (Rd) on the temperature profile is depicted in Figure 6. A surge in the fluid temperature profile is witnessed for (Rd). Higher values of (Rd) mean more heat is radiated and is being transferred to the fluid leading to a significant addition to fluid temperature. Figure 7 is demonstrated to perceive the influence of the heat absorption/generation parameter (Q) on the fluid temperature profile. The last thermal equilibrium of the system is disrupted owing to a rise in the values of (Q) causing the fluid temperature to adjust with the latest changes leading to an increment in the fluid temperature profile. The impact of the stratification parameter (P) on the thermal profile is given in Figure 8. The thermal profile falls when we strengthen the estimation of (P). The layers of the fluid behave becomes insulating when there is a strong effect of the thermal stratification leading to insulation effect between the layers of the fluid and the surroundings. So, less heat is transmuted to the liquid leading to a drop in the fluid temperature.

The graph for Eckert number Ec and nanoparticle concentration versus Skin friction coefficient is depicted in Figure 9. It is perceived that the skin friction coefficient shows an opposite trend for Ec and. Figure 10 is drawn to see the impact of the thermal convection parameter and nanoparticle concentration on the Skin friction coefficient. Similar behavior is witnessed here for and.

The volumetric entropy generation N_s is presented as:

The in non-dimensional form is:

where

The Bejan number (Be) is stated as:

The Be in non-dimensional form is given by:

Figure 11 and Figure 12 show important features of the thermal convection parameter and Eckert number Ec versus entropy generation profile N_s. In these figures, less entropy generation is observed for both parameters.

In our current investigation, we have examined the flow of single-wall carbon

nanotubes (CNTs) dispersed in ethylene glycol nanofluid on an inclined extended surface under the influence of mixed convection. Our analysis considers heat transfer in the presence of a heat source/sink, as well as thermal stratification effects and Cattaeo-Christov heat flux. The problem is tackled using numerical methods. The principal findings of our proposed model are outlined below:

・ The velocity profile shows an opposing trend for thermal convection and suction parameters.

・ The fluid velocity is on the decline for a strong magnetic field.

・ The fluid temperature is on the decline for the thermal stratification and thermal relaxation parameters.

・ For heat absorption/generation and thermal radiation parameters, the fluid temperature upsurges.

・ The skin friction coefficient depicts conflicting behavior for the Eckert number and nanoparticle volume concentration.

・ The entropy generation profile is on the decline for the Eckert number and thermal convection parameter.

The authors declare no conflicts of interest regarding the publication of this paper.

Ramzan, M. and Gul, H. (2024) Impact of Cattaneo-Christov Heat Flux in the Nanofluid Flow over an Inclined Permeable Surface with Irreversibility Analysis. Journal of Applied Mathematics and Physics, 12, 1582-1595. https://doi.org/10.4236/jamp.2024.124097

Dimensional constant

E Electrical parameter

Thermal convective parameter

Thermal conductivity of nanofluid

S Suction parameter

Pr Prandtl number

P Thermal stratification parameters

Viscosity of nanofluid

T Liquid temperature

Coefficient of thermal expansion

Nanoparticle volumetric

E₀ Electric field parameter

M Magnetic field parameter

Characteristic entropy generation

Dynamic viscosity of a liquid

U_w,V_w Stretching linear velocity

Be Bejan number

Thermal expansion coefficient

Rd Radiation parameter

C_f Drag force

V₀ Uniform suction

Ratio of heat capacity

Ec Eckert number

Re_x Reynolds number

T_w Temperature on wall

Viscous stress tensor

Density of nanofluid

Specific heat capacity of nanofluid

f Base fluid

Components of velocities

Thermal relaxation parameter

Thermal conductivity

Kinematic viscosity

Entropy generation rate

Heat generation parameter

Br Brinkman number