Interaction of multi-walled carbon nanotubes in mineral oil based Maxwell nanofluid

The most pressing issue now is to improve the cooling process in an electrical power system. On the other hand, nanofluids are regarded as reliable coolants owing to their exceptional characteristics, which include excellent thermal conductivity, a faster heat transfer rate, and higher critical heat flux. Considering these fascinating properties of nanofluid, this research looks at the flow of mineral oil based Maxwell nanofluid with convective heat. Moreover, introducing heat radiation, viscous dissipation and Newtonian heating add to the novelty of the problem. The coupled partial differential equations supported by the accompanying boundary conditions are numerically solved using an implicit finite difference method. The simulations are carried out using MATLAB software, and the obtained results are illustrated graphically. It is observed that the velocity of fluid increases concernign the relaxation time parameter but decreases against fractional derivative.


List of symbols
www.nature.com/scientificreports/ where T is the well known Cauchy stress tensor, P is the hydro-static pressure, I is the identity tensor and S is the extra stress tensor, defined by the relationship: Here V is the velocity field, µ is the dynamic viscosity, and is the relaxation time parameter. We shall also define the extra stress tensor S Fractional calculus has been successfully applied to the description of complicated dynamics such as relaxation, wave, and viscoelastic behavior in recent decades. Replacing the first derivative in the Maxwell fluid model with a fractional derivative of order α is a simple approach for adding fractional derivatives into models of linear viscoelasticity. Hence the fractional form of constitutive relation (2) can be written as: where ∂ α V ∂t α is Caputo time-fractional derivative with order α such that 0 ≤ α < 1 . Following definition is required in the sequel.
Definition 1 Let n ∈ N and α ∈ C with ℜ(α) > 0 such that n − 1 < α < n . Then for f in C n (R) , the timefractional Caputo derivative of order α is given by 21 : where Ŵ(·) is the gamma function, defined by The flow and heat transfer of an incompressible Maxwell nanofluid are governed by the following equations: where ρ is the density of the fluid.
here C p is the specific heat at constant pressure and k is the thermal conductivity.
Problem description. Assume that the fluid is confined between two sidewalls of an infinite plate in the xz-plane. A pressure gradient is applied along the x-axis, which initiates the mainstream flow. As a consequence, the velocity field takes on the following form: along with the extra stress tensor: It is simple to verify that the velocity field of the aforementioned form meets the incompressibility criterion automatically. The momentum and energy Eqs. (8) and (9), respectively, reduced to (1) T = − PI + S, (3) S =   s xx s xy s xz s yx s yy s yz s zx s zy s zz   . www.nature.com/scientificreports/ Furthermore, using Eqs. (4), (10) and (11), the extra stress tensor can be computed explicitly, yielding the relations: Multiplying Eq. (12) with 1 + α ∂ α ∂t α and utilizing Eq. (14), we arrived at In conjunction with thermal radiation, the energy Eq. (13) yields to where q r represents radiative heat flux and formulated by Rosseland approximation as 22,23 Here σ b is the Stefan-Boltzman coefficient and k b represents the absorption coefficient. Assume that the difference T − T ∞ inside the flow domain is small enough such that T 4 can be expanded about T ∞ using the Taylor series as The higher orders are ignored since the temperature difference in the approximation is small enough, yielding in Invoking the approximation of T 4 in Eq. (17) results in 24 Differentiating Eq. (20) w.r.t y and incorporating the resultant derivative in Eq. (16) yields to Flow and heat transfer modeling of nanofluid. There are several types of heat transfer modeling available today, including dispersion model, particle migration, single-phase and two-phase models 25 . In this research, we considered single-phase model by replacing traditional fluid's thermal and physical properties with the corresponding properties of nanofluid. Consequently, Eqs. (15) and (21) can be modified as The fluid is initially at rest and at ambient temperature. Therefore, applying the following initial conditions is reasonable.
We impose a no-slip velocity and Newtonian heating conditions along the plate and the walls so that: The natural far field conditions are (14) s xy + α ∂ α s xy ∂t α = µ ∂u ∂y , and s xz + α ∂ α s xz ∂t α = µ ∂u ∂z , where H denotes the Heaviside function, which is defined as follows Thermal and physical properties of nanofluid. Let ϕ represents the volume fraction of nanomaterial, and the subscripts f and nf refer to base fluid and nanofluid, then thermal and physical properties of nanofluid are defined as follows Density of nanofluid. The density of an object is defined as the mass divided by the volume of the object. The mathematical expression for density of nanofluid is given by 26,27 Specific heat capacity of nanofluid. The amount of energy required to raise the temperature of 1 g of a substance by 1 • C is known as the specific heat of a substance. The mathematical expression in terms of base fluid and nanoparticle is defined as Later, Xuan and Roetzel amended this correlation by considering thermal equilibrium between nanomaterial and the liquid phase and modified the above equation by taking the density into account [28][29][30] Thermal conductivity of nanofluid. The ability of a material to conduct heat is referred to as thermal conductivity. It is an essential factor for determining the heat transfer capacity in a thermal system. Well-known Maxwell theory for the effective thermal conductivity of a liquid with a dilute suspension of spherical particles is given as 31,32 Later, Hamilton and Crosser extended Maxwell model by taking shape factor of nanoparticle into account. Their proposed model is [33][34][35] where n = 3/̟ is the shape factor with sphericity ̟ . Note that Hamilton Crosser model reduces to Maxwell model for ̟ = 1 . Xue 36 claimed that the existing models are unable to reveal the influence of CNT space distribution on thermal conductivity. In general, CNTs are considered as rotating elliptical nanoparticles with a very large axial ratio, which ensures that current models might not work on CNT-based composites. According to Xue, the thermal conductivity for model CNT-based composites is Dynamic viscosity of nanofluid. Dynamic viscosity is a measure of internal resistance to flow. Einstein's formula for calculating the effective viscosity µ nf of a linearly viscous fluid with viscosity µ f and a dilute suspension of small spherical particles is Einstein's equation was extended by Brinkman 37,38 as  39 performed an experiment to measure the apparent viscosity of the oil-water and water-copper nanofluid at temperatures ranging from 20 to 59 • C. The findings of the experiment show that Brinkman's theory is fairly accurate 28 .
Non-dimensional flow and heat transfer model. To comprehend the physics of the presented problem, non-dimensional representation is required. Therefore, we introduced the following non-dimensional parameters Incorporating the non-dimensional parameters (37) in Eqs. (22), (23), (24), (25) and (26) yield to following (after removing the bars for simplicity) subject to the initial and boundary conditions given that
(41)  Table 1. The trend of velocity for different parameters is depicted in Figs. 1, 2 and 3. The axial velocity pattern for various fractional derivative parameter α is shown in Fig. 1. It is worth noting that when α gets higher, the amplitude of the velocity gets smaller. Moreover, the obtained surface plot resembles a Gaussian distribution or a traditional heat kernel graph with increasing standard deviation when the fractional derivative power has increased. For larger estimates of relaxation time parameter , an increase in fluid velocity is visualized, see Fig. 2. The influence of nanotube volume fraction ϕ is ascertained in Fig. 3. As illustrated in this graph, the velocity of fluid decreases as the value of ϕ increases. Generally, raising the volume concentration of nanomaterials inside a fluid increases its viscosity. As a result, the velocity of the fluid substantially reduces.
The trend of temperature profile for several parameters are illustrated in Figs. 4, 5, 6, 7, 8 and 9. The effects of the thermal radiation parameter Rd on the temperature distribution are outlined in Fig. 4. It is observed that, with higher Rd values, more heat is generated. As a result, a rise in the fluid's temperature is perceived. The influence of dissipation parameter E on the temperature distribution is portrayed in Fig. 5. The parameter E is used to determine the effect of self-heating due to the dissipation properties. At high flow rates, the thermal field in a fluidic framework is swamped by the temperature gradient present in the framework and the effects of dissipation due to internal friction. As seen in Fig. 5, the temperature field is more significant for higher values of E . The temperature profile improves because more heat energy is stored in the fluid due to friction forces as E increases. Figure 6 is provided to visualize the effect of conjugate heat parameter γ on temperature distribution of Maxwell nanofluid. The temperature is increased when γ is increased. Physically, it was expected because more heat is transferred from the heated surface to the cold fluid. As a result, the temperature of the fluid rises.
By fixing the y-coordinate in Figs. 7, 8 and 9, a one-dimensional temperature profile is drawn for MWCNTS/ mineral oil and SWCNTs/mineral oil. The same conclusions as the surface plots can be drawn; however, the temperature of fluid with SWCNTs is more significant than MWCNTs.

Conclusions
The mineral oil base nanofluid flow with MWCNTs accompanied by the radiative heat, viscous dissipation and Newtonian heating is deliberated numerically. A finite difference method is used to solve the formulated mathematical problem, and the graphical results are generated in MATLAB software. The following are the most important findings of this research: www.nature.com/scientificreports/ • Lower velocity is associated with nanomaterial volume concentration.
• The relaxation time parameter corresponds to higher velocity flow.
• Friction forces generated by the viscous dissipation factor increase the temperature.
• The temperature of Maxwell nanofluid significantly raises against the conjugate heat transfer parameter.