Fractional Analysis of Dissipative Viscous Fluid Flow with Mixed Convection and Variable Viscosity

Time-dependent viscosity and thermal conductivity have been studied in relation to ﬂ ow and thermal energy propagation along a vertical turning cone. Except the density variance, the other ﬂ uid ’ s properties are assumed to be constant. Using the similarities procedure, the nonlinear system of di ﬀ erential equations is simpli ﬁ ed to dimensionless ODEs. The resultant nonlinear ODEs system is computed while using fractional code FDE12, and the ﬁ ndings are quantitatively examined using the bvp4c approach for accuracy and consistency. In form of ﬁ gures and tables, the behavior of momentum, energy, and mass is interpreted versus the physical constraints. The axial and radial velocity, both declines with the variation of unsteadiness parameter S . The axial velocity of the ﬂ uid is considerably increased when the mixed convection parameter is elevated, but the radial velocity is reduced. Similarly, when the variable viscosity increases, the velocity pro ﬁ le develops.


Introduction
The results of this study show that disk-cone appliances are used in a variety of technical applications, including determining the viscosity of a fluid (viscosimetry), convective diffusion, medical devices, and biomedicine for oxygen measurement [1]. The heat transfer through a spinning cone is also addressed in this research. Many academics are drawn to this sort of study to investigate its properties, behavior, and applications. Turkilmazoglu [2] investigated a steady Newtonian viscous fluid over a rotating cone using the homotopy analysis technique. The mixed convective simulation with momentum and heat distribution across a perforated upward spinning cone in an ambient liquid was numerically calculated by Chamkha and Al-Mudhaf [3]. They hypothesized that when cone angular velocity varies, axial, and tangential velocity increases considerably. Garrett et al. [4] investigated the fluid flow across a turning cone (half angle) in axial direction. The MHD (magnetohydrody-namics) nanoliquid flow with Brownian motion and thermophoresis influence over a revolving cone was deled by Nadeem and Saleem [5] and Towers and Garrett [6]. It is worth noting that surface temperature and Mach number destabilise the system, whereas suction stabilises it. Hayat et al. [7] used a shooting method to emphasize the MHD chemical reactivity of an unsteady viscous fluid over a turning cone. Chamkha et al. [8] evaluated the influence of a rotating cone on a 3D CNT hybrid nanofluid in a trapezoid permeable cavity, considering MHD interactions.
The upshot of unsteady viscosity on viscous fluid characteristics causes some variation. The viscosity of liquids, for example, decreases as temperature goes up, but the viscosity of gases improves. The rises in thermal energy cause friction in oily fluids, which affects fluid viscosity, and the viscosity no longer holds consistent. In light of this deficiency, a growing number of researchers are focusing their efforts on demonstrating the impact of changing viscosity phenomena under various situations. The effects of fluctuating viscosity of third-grade dispersed fluid flow via a conduit have been explored by Christie and Massoudi [9]. The numerical outputs were discovered using the finite difference method. Seddeek [10] investigated unstable free convection MHD flow across an infinite plate under the influence of a magnetic field and changing viscosity. The computational solution of the simulated equations is done using the finite difference method. MHD boundary layer flow over an extending heated surface with variable viscosity is reported by Pantokratoras [11]. Their analysis also included a graphical representation of viscosity variations. Mukhopadhyay and Layek [12] looked studied heat exchange across a stretched vertical porous surface with changing fluid viscosity. They discovered that increasing the viscosity, enhances the velocity while decreasing the energy field. Using a discrete variant of HAM known as spectral homotopy evaluation over a stretched surface, Dada and Onwubuoya [13] addressed mass and energy distribution through fluid flow with changing viscosity and activation energy. Hazarika et al. [14] evaluated the upshots of changing viscosity, thermal radiation, and MHD on fluid flow through a vertical cone.
The thermal transition rates are affected by viscous dissipation, which acts as an energy source. The relevance of viscous dissipation is determined by whether the cone is freezing or warmed. Reddy et al. [15] estimated MHD flow and energy propagation across a stretched substrate as a function of heat source and viscous dissipation. Mabood et al. [16] evaluated the MHD flow, energy transport, and chemical reaction of a nanofluid containing copper Cu and aluminium oxide particulates in a porous media under the viscous dissipation influence. Deebani et al. [17] assessed the role of viscous dissipation and MHD across a revolving cone. Gayatri et al. [18] studied viscous dissipation in 2D fluid flow with varying thickness and slip coefficients across a stretched surface. They discovered that the slip parameters increase friction while lowering fluid velocity. Using the Atangana-Baleanu fractional method, Saqib et al. [19] investigated electro-osmotic nanofluid flow.
Fractional calculus, which is an extension of regular calculus, has a 300-year history. This field has exploded in popularity in recent years. Almost all activities in applied sciences are described by signal processing, fluid flow in permeable substances, wave transference in mechanical properties, finance theory, and biological system electric conductance [20,21]. Many definitions exist for the fractional derivative; however, the Caputo and Riemann Liouville fractional derivatives are particularly important in terms of applicability [22]. We know that the kernel was single in both fractional definitions. To overcome this difficulty, Caputo and Fabrizio presented a new point of view of noninteger order derivative with nonsingular kernel [23] in 2016, which is highly beneficial for a variety of physical problems. Manzoor et al. [24] analyzed the uniqueness and existence of solutions of fractional order differential equations with Caputo derivatives. They came up with a set of requirements to assure solution validity while maintaining Hyers-Ulam stability. The fractional assessments for Darcy hybrid nanoliquid flow over a perforated spinning disc were elaborated by Li et al. [25]. The proposed model has been put up using Matlab fractional code Fde12 to produce the fractional solution. The outputs are compared to the fast-approaching numerical Matlab scheme boundary value solver for correctness and validity of the resultant framework.
We generalized the approach of [26] based on the aforementioned literature and its application in the actual world. The goal of this study is to assess the upshots of timedependent viscosity on flow, thermal energy, and mass transfer in a vertical rotating cone. The spinning phenomenon is arranged in the format of a system of PDEs for this reason. Which are solved using the fractional code FDE12, and the results are checked for validity and correctness using the Matlab numerical software boundary value solver (bvp4c). The findings are depicted graphically and shortly reviewed.

Mathematical Formulation
We considered axisymmetric, an incompressible and unsteady fluid flow with an angular velocity Ω across a rotating cone. The u, v, and w are the velocity component has been considered along x, y, z direction. The gravity g impact is downward, and the buoyancy force exist due to the temperature variation. The tangential direction influences temperature T w variability near the cone's edge, while temperature T ∞ away from the cone kept fixed. Fluid flow mechanism over a rotating cone is depicted through Figure 1. Furthermore, the variable thermal conductivity and viscosity are expressed as [26]: the variable viscosity model used here is the Reynold's model. The Taylor series expansion has been used to obtain. Where Here, k 0 and μ 0 is the fluid conductivity and dynamic viscosity.
The governing system of nonlinear PDEs for momentum, mass, and energy may be written as by utilising Boussinesq approximation boundary layer theory and the aforementioned assumption [26]: ∂v ∂t 2 Journal of Nanomaterials Where α * , β, k, ρ , and g are the cone vertical angle, thermal expansion, thermal conductivity, density, and gravity, respectively.
The boundary conditions are: To simplify the PDEs to ODES, we commence the preceding similarity variables [26]: By plunking Equation (9) in Equation (3)- (8), we track down: The reduced conditions are:  Ref. [26] Present work     Journal of Nanomaterials The parameters generated are rebound as: Where Pr, Sc, and Ec are the Prandtl, Schmidth, and Eckert number, respectively. While S is the unsteadiness and λis the mixed convection coefficient.
The skin friction, mass transfer, and Nusselt number are written as:

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Equations (14) and (15) dimensionless set up are: , Where

Preliminaries
Definition 1. For a function g : R + ⟶ R, the fractional integral of order α ≻ 0 is defined as: Definition 2. For function g ∈ C n , the Caputo noninteger order derivative is defined as: Clearly c D α t ðgðtÞÞtends to g ′ ðtÞas α ⟶ 1:  Journal of Nanomaterials

Result and Discussion
The purpose of this section is to quantify and compare the functionality of energy and mass transition rate based on various physical factors. The findings are produced using Matlab fractional package (fde12), and a fast-approaching numerical technique bvp4c, has been applied to ensure the validity and correctness of the outcomes. Table 1 shows the adequacy of the current study when compared to [26]. While the fractional and numerical methods have been compared in Tables 2 and 3, respectively. For mixed convection and Prandtl number Pr, the numerical out-comes of tangential and azimuthal skin friction is shown in Table 2. For mixed convection and Prandtl number, Table 3 shows the quantitative results of Sherwood Nusselt number.
The schematic drawing of a revolving cone is depicted in Figure 1.   Figures 8(a) and 8(b) show a reduction in thermal gradient when the variable viscosity factor is increased. The interaction forces within fluid molecules grow as the amount of variable viscosity rises, which results in the lowering fluid energy profile. It is self-evident that when the Prandtl number Pr increases, the fluid temperature significantly reduces. Because a fluid with a higher Prandtl number has a lower thermal diffusivity (Figures 9(a) and 9(b)).
In Figures 10(a) and 10(b), an increase in energy profile is observed against the Eckert number Ec. The Eckert number describes a fluid's self-rising thermal rate as a result of 10 Journal of Nanomaterials viscous dissipation. The fluid temperature is efficiently increased using a variable thermal conductivity as revealed through Figures 11(a) and 11(b). The result of Schmidt number vs mass transmission rate is shown in Figures 12(a) and 12(b). The fluid's mass transfer rate is reduced as the Schmidt number is increased.

Conclusion
The rotating flow of a viscous fluid with time-dependent viscosity and thermal conductivity across a vertical cone is evaluated in this study. A comparison of the numerical bvp4c method and fractional fde12 package is also emphasized. The study's compelling observations are listed below: (i) With a positive increase in unsteadiness entity S, both the axial and radial velocity declines  Semi vertical angle of cone T: Fluid temperature (K) T ∞ : Temperature away from cone surface ε: Variable thermal conductivity coefficient η: Similarity variable μ 0 : Dynamic viscosity ν 0 : Kinematic viscosity Pr: Prandtl number θ: Dimensionless temperature ρ: Density c p : Specific heat z-axis: Axial or normal to cone C f x : Skin friction Nu x : Nusselt number Bvp4c: Boundary value solver u, v, w: Velocity component α: Fractional order T w : Wall temperature β: Temperature expansion coefficient g: Gravity λ: Mixed convection t * : Dimensionless time k: Thermal conductivity

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Data Availability
All the data exist in the manuscript.

Conflicts of Interest
The authors declare that they have no conflicts of interest.