Comparative Study of Chemically Reacted Nanofluids SWCNT, MWCNT with Modeling of Cattaneo-Christov Heat Fluxes

This study examines two-dimensional, incompressible flow, which is saturated with SWCNT/MWCNT over a continuously stretched sheet when subjected to an angled magnetic field and a Darcy porous media (see Figure 1). The analysis considers mass diffusion and chemical processes.

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Figure 1. Schematic diagram of flow

The controlling two-dimensional boundary layer flows and the basic equations of this model will be shown in two cases of fluid.

Governing equations [15] are:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$            (1)

$u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}={{\alpha }_{nf}}\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}-\frac{{{\sigma }_{nf}}}{{{\rho }_{nf}}}B_{0}^{2}u{{\sin }^{2}}(\alpha )-\frac{{{\nu }_{nf}}}{K}u.$                (2)

${{\left( \rho {{C}_{p}} \right)}_{nf}}\left( u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y} \right)=-\nabla \left( q+{{q}_{r}} \right).$          (3)

Here u, v are horizontal and vertical directions fluid velocities. And n, m, r, q_,T, and c_pare the kinematic viscosity, dynamic viscosity, density, heat flux, temperature and specific heat respectively.

The Cattaneo-Christov [15] model can be used to determine

$q+\lambda \left( \frac{\partial q}{\partial t}+V\nabla q-q\nabla V+\left( \nabla V \right)q \right)=-k\nabla T.$                 (4)

where, the heat flow relaxation time λ, and thermal conductivity k, are given. When λ = 0, the basic Fourier's law of heat transport can be reduced from Eq. (4). When q is eliminated from Eqs. (3) and (4), we have,

$u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}+\lambda\left(u \frac{\partial u}{\partial x} \frac{\partial T}{\partial x}+v \frac{\partial v}{\partial y} \frac{\partial T}{\partial y}+u \frac{\partial v}{\partial x} \frac{\partial T}{\partial x}+2 u v \frac{\partial^2 T}{\partial x \partial y}+u^2 \frac{\partial^2 T}{\partial x^2}+v^2 \frac{\partial^2 T}{\partial y^2}\right)=\alpha_{n f} \frac{\partial^2 T}{\partial y^2}-\frac{1}{\left(\rho C_p\right)_{n f}} \frac{\partial q_r}{\partial y}$            (5)

$u\frac{\partial c}{\partial x}+v\frac{\partial c}{\partial y}={{D}_{m}}\frac{{{\partial }^{2}}c}{\partial {{y}^{2}}}-{{K}_{r}}\left( c-{{c}_{\infty }} \right)$      (6)

where, D_m is mass diffusion, K_r is chemical reaction.

Considered boundary conditions are [15]:

$u(x,0)={{U}_{w}}=cx,v(x,0)=0,u(x,y\to \infty )=0$

$T(x,0)={{T}_{w}},T(x,y\to \infty )={{T}_{\infty }}$

$c\left( x,0 \right)={{c}_{w}},c(x,y\to \infty )={{c}_{\infty }}.$                  (7)

Blasius similarity transformations are [15]:

$u=cx{{f}_{\eta }}(\eta ),v=-\sqrt{c{{\nu }_{f}}}f(\eta ),\theta (\eta )=\frac{T-{{T}_{\infty }}}{{{T}_{w}}-{{T}_{\infty }}},$

$\phi (\eta )=\frac{c-{{c}_{\infty }}}{{{c}_{w}}-{{c}_{\infty }}},\eta =\sqrt{\frac{c}{{{\nu }_{f}}}}y.$                (8)

Eq. (1) is met by the foregoing changes and Eqs. (2)-(9) become connected nonlinear differential equations.

${{e}_{2}}f'''+{{e}_{1}}(f.f''-f{{'}^{2}})-{{e}_{3}}M{{\sin }^{2}}(\alpha )f'-{{e}_{2}}D{{a}^{-1}}f'=0.$                (9)

$\left( \frac{{{K}_{hnf}}}{{{K}_{f}}}+NR \right)\theta ''+{{e}_{4}}\Pr f\theta '={{e}_{4}}\gamma (f'\theta '+{{f}^{2}}\theta '').$              (10)

$(1-\phi )\phi ''=Sc(Rc-f\phi ').$           (11)

Boundary conditions related to above equations are:

$f=0,\,\,f'=1,\,\,\,\theta =1,\,\,\,\phi =1,\,\,\,\,at\,\,\eta =0$

$f'\to 0,\,\,\,\theta \to 0,\,\,\,\phi \to 0\,\,\,as\,\,\,\eta \to \infty $                  (12)

where,

${{e}_{1}}=\frac{{{\rho }_{nf}}}{{{\rho }_{f}}},$ ${{e}_{2}}=\frac{{{\mu }_{nf}}}{{{\mu }_{f}}},$ ${{e}_{3}}=\frac{{{\sigma }_{nf}}}{{{\sigma }_{f}}},$ ${{e}_{4}}=\frac{{{\left( \rho {{C}_{p}} \right)}_{nf}}}{{{\left( \rho {{C}_{p}} \right)}_{f}}},$ ${{e}_{5}}=\frac{{{k}_{nf}}}{{{k}_{f}}}.$          (13)

Mathematically nanofluids constants are defined as by [19]:

${{\upsilon }_{nf}}=\frac{{{\mu }_{nf}}}{{{\rho }_{nf}}},$ ${{\mu }_{nf}}=\frac{{{\mu }_{f}}}{{{(1-\phi )}^{2.5}}},$ \[{{\rho }_{nf}}=(1-\phi ){{\rho }_{f}}+\phi {{\rho }_{CNT}},\]

$\frac{{{K}_{nf}}}{{{K}_{f}}}=\frac{1-\phi +2\phi \left( \frac{{{k}_{CNT}}}{{{k}_{CNT}}-{{k}_{f}}} \right)\ln \left( \frac{{{k}_{CNT}}-{{k}_{f}}}{2{{k}_{f}}} \right)}{1-\phi +2\phi \left( \frac{{{k}_{f}}}{{{k}_{CNT}}-{{k}_{f}}} \right)\ln \left( \frac{{{k}_{CNT}}-{{k}_{f}}}{2{{k}_{f}}} \right)},$

${{\left( \rho {{C}_{p}} \right)}_{nf}}=(1-\phi ){{\left( \rho {{C}_{p}} \right)}_{f}}+\phi {{\left( \rho {{C}_{p}} \right)}_{CNT}}.$               (14)

Here the nondimensional parameters are given by:

$N R=\frac{16 \sigma T^3}{3 k k^*}$Q is the Radiation number.

$D a=\frac{K}{v_f}$ is the Darcy number.

$M=\frac{\sigma_f B_0^2}{\rho_f c}$ is Magnetic parameter.

$\operatorname{Pr}=\frac{v_f}{\alpha_f}$ is Prandtl number.

$\gamma=c \lambda$  is the relaxation time parameter.

$q_r=\frac{-4 \sigma^*}{3 k^*} \frac{\partial T^4}{\partial y}$ is thermal radiation.

$\alpha_f=\frac{k_f}{\left(\rho c_p\right)_f}$ is thermal diffusivity.

$R c=\frac{k_r\left(C-C_{\infty}\right)}{a}$ is the chemical reaction parameter.

The physical quantities are, "Skin- friction" along $x$ axis $C_{f x}$ "local Nusselt number" $N u_x$, "Sherwood number" $S h_x$ given as:

$\begin{aligned} & R e^{\frac{1}{2}} C_{f x}=\frac{1}{(1-\phi)^{2.5}} f^{\prime \prime}(0) \\ & R e^{-\frac{1}{2}} N u_x=-\frac{K_{n f}}{K_f} \theta^{\prime}(0) \\ & R e^{-\frac{1}{2}} S h_x=-\frac{K_{n f}}{K_f} \phi^{\prime}(0)\end{aligned}$               (15)

where, $R e=\frac{U_w x}{v_f}$ is the restricted Reynolds number.

The comparative study of the Newtonian, two-dimensional, incompressible, continuously stretched sheet when exposed to a Darcy porous media and an inclined magnetic field. The study considers both mass diffusion and chemical processes. A typical collection of velocity $f^{\prime}(\eta)$, temperature $\theta(\eta)$, concentration $\theta(\eta)$ distributions for numerous values of the necessary parameters. Non-dimensional parameters ranges are $\begin{aligned}  0.1 \leq \phi \leq 0.4,0.1 \leq \mathrm{Rc} \leq 0.7,0.1 \leq \mathrm{Sc} \leq 0.8,4 \leq \operatorname{Pr} \leq 7,1 \leq \mathrm{NR} \leq 4, -0.02 \leq \gamma \leq 0.01,0 \leq \alpha \leq \pi / 2 \end{aligned}$. By fixing other variables as zero for different values of volume fractions we found that there exists a good agreement with the previous literature [10], as represented in Table 2.

The impact of nanoparticle volume fraction ($\phi$) on Concentration Profile $\phi(\eta)$ where (0.1≤$\phi$≤0.4), through surface regime with coordinate (h) shows in Figure 3. This parameter is analytical in defining the concentration of nanoparticles in a fluid and has a direct impact on its physical characteristics and behaviour. It has been observed from the graph that the concentration profile depreciates as nanoparticle volume fraction $\phi$) values increases due to enhanced interaction among nanoparticles. Higher values of $\phi$ leads to greater agglomeration and clustering, which reduces the effective diffusivity of the nanoparticles. This causes a decrease in the uniformity of their distribution, resulting in a lower concentration profile.

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Figure 3. Concentration profile of nanoparticle volume fraction $\phi$

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Figure 4. Concentration profile of Rc

Figure 4 elucidates chemical reaction effect of on concentration profile ϕ(η), where on (0.1≤Rc≤0.7). The chemical reaction parameter (Rc) used to describe effect of a chemical reaction on transport properties of a fluid. It usually estimates the rate at which a chemical species interacts within the fluid, which influences the concentration and dispersion of that species. It has been noticed from the graph that concentration profile depreciates as chemical reaction parameter (Rc) elevates physically because a higher (Rc) indicates a more intense chemical reaction rate. This increased reaction rate accelerates the consumption of nanoparticles, reducing their concentration in the fluid.

Figure 5 illustrates influence of Schmidt number (Sc), on concentration profile $\phi(\eta)$ where 0.1≤Sc≤0.8, Schmidt number is essential in characterizing relative thickness of velocity boundary layer to mass transfer boundary layer. The concentration profile $\phi(\eta)$ dropped with increasing Schmidt number (Sc). Physically, because a greater Sc signifies lesser mass diffusivity as compared to momentum diffusivity. This reduced mass diffusivity hampers the dispersion of nanoparticles within the fluid, leading to a steeper concentration gradient and a lower overall concentration profile.

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Figure 5. Concentration profile of Sc

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Figure 6. Temperature profile of Pr

Figure 6 demonstrates Prandtl number (Pr) effect on temperature profile $\theta(\eta)$, where (4≤Pr≤7), Temperature profile is declining with hiking in (Pr), physically because a higher Pr signifies a higher momentum diffusivity relative to thermal diffusivity. This means that the fluid's ability to conduct heat is lower compared to its ability to transfer momentum, leading to a thinner thermal boundary layer and thus a steeper temperature gradient. Consequently, the temperature profile decreases more rapidly. Comparing SWCNT (solid lines) and MWCNT (dashed lines) nanofluids, MWCNTs show a slightly steeper decrease in temperature profiles, suggesting more effective heat transfer. This difference is due to MWCNTs' higher thermal conductivity, which facilitates more efficient heat dissipation compared to SWCNTs.

Figure 7 portrays influence of radiation number (NR), on θ(η), where (1≤NR≤4), through surface regime with coordinate (η). θ(η) is enhance with an increasing radiation number (NR), physically because a higher (NR) signifies a greater contribution of radiative heat transfer. Radiative heat transfer enhances the overall thermal energy within the system, leading to higher temperatures. Comparing SWCNT and MWCNT nanofluids, SWCNTs (solid lines) generally exhibit a lower temperature profile than MWCNTs (dashed lines) for the same (NR) values, indicating that MWCNT nanofluids, with their higher thermal conductivity, respond more significantly to radiative heat transfer, resulting in a more pronounced temperature increase.

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Figure 7. Temperature profile of Radiation number NR

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Figure 8. Temperature profile of time relaxation parameter γ

Figure 8 portrays relaxation time parameter (γ), on temperature, where (-0.02≤$\gamma$≤0.01). It is particularly relevant in models involving non-Fourier heat conduction, such as the Cattaneo-Christov heat flux model, which accounts for finite speed of thermal propagation. $\theta(\eta)$ lessening with growing in the relaxation time parameter (γ), because a higher γ indicates a delayed response of the material to temperature changes, reducing the overall heat conduction efficiency. This delay causes the heat to dissipate more gradually, resulting in a lower temperature profile. Comparing SWCNT and MWCNT nanofluids, SWCNT show a more pronounced decrease in temperature profiles with increasing (γ), suggesting that SWCNT are more sensitive to the effects of thermal relaxation due to their higher thermal conductivity, which enhances impact of delayed thermal response.

Figure 9 illustrates influence of magnetic inclination angle(α), on the $f^{\prime}(\eta)$ where (0≤$\alpha$≤π/2), It shows that the enhancement in aligned angle declines the velocity profiles of SWCNTs/H₂O and MWCNTs/H₂O nanofluids. It is frequently owing to the aligned angle, which fortifies magnetic field. Opposing force to the flow owing to the increasing magnetic field, which is called the Lorentz force, this declines thickness. When α=0, magnetic flux has no influence on velocity. However, when α=π/2, the nanofluid particles have the highest resistance.

Figure 10 demonstrates impact of Darcy number (Da), velocity profile $f^{\prime}(\eta)$ where 0.01≤Da≤0.04, $f^{\prime}(\eta)$ elevates significantly as Da values appreciates physically because a higher Da indicates higher permeability of the porous medium, allowing fluid to flow more freely. This reduces resistance to fluid motion, resulting in higher velocity profiles. Comparing SWCNT (solid lines) and MWCNT (dashed lines) nanofluids, SWCNTs exhibit a slightly higher velocity profile than MWCNTs for the same (Da) values, likely due to SWCNTs' higher thermal conductivity and better dispersion properties, enhancing fluid flow and reducing viscous drag within the medium.

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Figure 9. Velocity profile of inclined angle α

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Figure 10. Velocity profile of Darcy number Da

Figure 11 emphasis influence of magnetic inclination angle (α), on temperature profile $\theta(\eta)$, where 0≤α≤π/2. It shows that the enhancement in aligned angle steadily rises the temperature profile $\theta(\eta)$ of SWCNTs/H₂O and MWCNTs/H₂O nanofluids because a higher values of α enhances the influence of magnetic field on fluid flow, increasing resistive force against the fluid motion. This results in more energy being dissipated as heat within the fluid, raising the temperature profile. Comparing SWCNT and MWCNT nanofluids, SWCNTs show a slightly lower temperature profile than MWCNTs for the same α values. This indicates that MWCNT nanofluids, with their higher thermal conductivity, exhibit a more significant temperature increase due to the combined effects of enhanced thermal energy dissipation and better heat transfer capabilities.

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Figure 11. Temperature profile of Inclined angle α

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Figure 12. Velocity Profile of Magnetic parameter M

The influence of Magnetic parameter M, (1≤M≤4) on Velocity profile $f^{\prime}(\eta)$ discussed via Figure 12. The magnetic parameter M hikes physically a stronger magnetic field increases Lorentz forces, which act as a drag on the fluid flow. This magnetic drag resists the motion of the fluid, leading to a reduced velocity. Additionally, the increased magnetic field enhances thermal boundary layer, contributing to a further decline in velocity. Moreover, Figure 12 indicates that the velocity of SWCNT nano fluid upsurges more than the velocity of MWCNT nanofluid.

The influence of Darcy number (Da), on concentration profile $\phi(\eta)$, where (0.01≤Da≤0.04), demonstrated in Figure 13. The concentration profile decreases with increasing Darcy number (Da) more for SWCNT than MWCNT due to the higher permeability in SWCNT-based nanofluids. As the Darcy number rises, the porous media becomes more permeable, enhancing fluid flow while decreasing diffusion residence time. For SWCNTs, which have higher surface area and reactivity, this increased permeability accelerates the transport of the chemical species, leading to a steeper decline in concentration compared to MWCNTs.

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Figure 13. Concentration profile of Darcy number Da

In Table 3, it has been observed that for SWCNT/H₂O, Shx is enhancing for the parameters Rc, Sc, Da, ϕ and diminishes for α. For g, Da, α parameters C_fx is decreasing, Have no effect for remaining parameters. Also, for time relaxation parameter γ, NR shows a substantial decay in (Nu_x).

In Table 4, it was found that for MWCNT/H₂O, Shx is growing for the parameters Rc, Sc, Da, ϕ and drops for α. Also, for time relaxation parameter γ, α, NR shows a substantial decay in (Nu_x).

Future research could extend this study to three-dimensional flows and explore more complex fluid models, such as non-Newtonian and hybrid nanofluids. Experimental validation of the numerical results would enhance reliability. Investigating other nanoparticle combinations and base fluids could identify more effective formulations. Optimization techniques could be employed to maximize heat transfer efficiency. Assessing the environmental impact and stability of SWCNT and MWCNT nanofluids in various applications is essential. Finally, applying the findings to real-world engineering problems like solar collectors and cooling systems could demonstrate practical benefits. Studying the influence of additional external forces, such as electric fields or variable magnetic fields, on the behavior of nanofluids could further enhance our understanding of their control and manipulation in advanced engineering systems.