A New Heat Dissipation Model and Convective Two-Phase Nanofluid in Brittle Medium Flow over a Cone

A time-dependent convectional flowof a two-phase nanofluid over a rotating conewith the impact of heat andmass rates is elaborated in this article. /e instability in the flow field is induced by the cone angular velocity that depends on the time. /e Navier–Stokes selfsimilar solution and the energy equations are obtained numerically. Here, the achieved solution is not only for Navier–Stokes equations but also for the equations of the boundary layers. In this work, the concentration, Brownian motion, and thermal buoyancy effects have important significance. We have assumed viscous dissipation with heat-absorbing fluid. Similarity answers for spinning cones with divider temperature boundary conditions give an arrangement of nonlinear differential conditions that have been handled numerically. /eMATLABmethodology BVP4C is used to resolve the reduced structure of nonlinear differential equations numerically. Observation for skin friction and Nusselt number is also taken into account. Velocity and temperature impact is depicted graphically, while the outward shear stress values and heat allocation rate are included in tables.


Introduction
e design of reliable equipment in manufacturing industries relies heavily on convective flow ended a cone through radiative heat and form transfer. Due to its relevance in modern technology and applications in geothermal engineering, as well as other hydrological and astrophysical biofluid studies, researchers have shown a strong interest in heat and mass transfer in Newtonian flows in recent years. e analysis of fluid flow in a cone encompasses a wide variety of subjects. It is used in plastic processing, elastic sheet cooling, polymer technology, polymer chemistry, and engineering, to name a few. So, because of its enormous applications, the researchers are holding a purpose in this area. Dependence of viscosity on temperature plays a vital role in the realm of fluids flow. e viscosity of water decreases as the temperature increases, while the viscosity of gases rises as the temperature rises. e increase in temperature in lubricating fluids causes internal friction that affects the fluid's viscosity and will no longer remain constant. Because of this inadequacy, many researchers are interested to understand the effects of using different variable viscosity models.
Nanofluids remain a class of heat allocation fluids which caused suspended nanoparticles are distributed in the fluid (1-100 nm). In base fluids, locomotive oil, polymer solutions, bio-fluids, other critical fluids, water, and organic fluids (e.g., ethylene and diethylene) are commonly used. Nanoparticles are generally made of carbon in diverse edifices (e.g., carbon nanotubes, black lead, and diamond), metals (e.g., copper, hoary, and gold), and metal oxides (e.g., titanium and zirconia), besides functionalized nanoparticles. A wide-range of possible applications has been initiated for the use of nanofluids. Choi remained the first to research updraft conductivity development in nanofluids [1][2][3][4][5][6][7][8][9][10][11][12]. Nanofluids, bio and pharmacological nanofluids, remedial nanofluids, environmental nanofluids, and other heat transfer fluids are categorized according to their applications. Many researchers have looked into how extent, concentration, form, and other assets affect the warmth transfer rate of a fluid. For the Prandtl number of airs, Hering and Grosh [13] investigated a steady mixed convection boundary layer flow from a vertical cone in an ambient fluid. For a broad range of Prandtl numbers, Himasekhar et al. [14] solved the similarity solution of the mixed convection boundary layer flow over a vertical rotating cone in an ambient fluid. Body solutions of unstable mixed convection flow from a rotating cone in a rotating fluid were obtained by Anil Kumar and Roy [15] a few years ago. Chamkha and Mudhaf [16] examined heat generation, consumption, and unstable heat and mass transfer from a revolving vertical cone with a magnetic field. Ravindran et al. [17] suggested a new approach for investigating the effects of fluid flow (suction/ injection) on a vertical porous cone's steady natural convection boundary layer flow. e impact of heat-dependent viscosity with viscous heat generation on third-grade fluid flow in a standard pipe was examined by Nadeem and Hussain [18]. Exploitation the finite simple difference method, numerical solutions were obtained. Different fluids like water, ethylene glycol, and oil, due to their poor thermal conductivity, have low heat transfer properties. It is now realized that by retaining nano-sized metal flakes such as Al, titanium, silver, gold, Cu, or their oxides, the thermophysical characteristics of such fluids could be enhanced, ending in what is commonly known as nanofluid [19]. Several researchers have spent the last few years studying the edge layer movement of nanofluid fluids in various geometries and under various conditions. Kameswaran et al. [20] investigated the flow of hydromagnetic nanofluids due to a shrinking surface. Over a stretching field, Kameswaran et al. [21] discovered solutions for the deflation-point flow equations. Fauzi et al. [22] explored the time-independent nanofluid boundary film flowing along a perpendicular cone in a brittle medium. Boutra et al. [23] investigated unrestricted convection induction in a nanofluid-filled framework through round heaters, and Ambethkar and Kumar [24] investigated 2D noncompressible flow solutions with the transfer of heat in a powered square cavity singing stream function-vorticity model. Cheng [25] addressed natural convection flowing through a formatted cone in a brittle medium in the boundary layer. Chamkha et al. [26] examined the issue of mixed convection boundary layer flow in a continuous, turbulent flow over a rigid cone enclosed in a brittle thermal radiation medium. Nadeem and Saleem [27] examined turbulent nanofluid flow in a turning cone subjected to an induced magnetic field. In this paper, we look at a two-phase nanofluid flow along a vertically stretching cone. e work of Heiring and Grosh [28] on natural convection over a multi-isothermal cone is one of the most recent cone-shaped surface boundary layer studies. A theoretical study of forced convection flow in relation to a rotating cone was suggested by Tien and Tsuji [29]. Koh and Price [30] have evaluated the transfer of heat past a pivoting cone. Ellahi et al. [31] have studied the analysis of simplified third-grade slide Couette fluid flow. ere are some related studies about this phenomenon provided in References [32][33][34][35][36][37][38][39][40][41][42][43]. Extrusion processes, plastic product processing, polymers, and silicone slips, wire and copper-coating, glass and fiber optic production, hot spinning manufacture, metal rolling, food preparation, and a variety of other topics are frequently mentioned. e combination of fluids has a broad range of applications, including cooling systems, heating processes, as well as biomedical and automotive science and technologies that control heat and mass transfer rates. e persistence of this article is to inspect the flow of liquid-based two-phase nanofluids (copper oxide and silver added to water) over a rotating cone. Two-phase flows include the flows that transition from pure liquid to vapor as a result of external heating, separated flows, and scattered two-phase flows in which phase is observed in a continuous carrier phase in the form of particles, droplets, or leaks. We have solved governing differential equations with the assistance of the BVP4V scheme under MATLAB. It also describes the possessions of relevant bodily parameters that affect the velocities, surface strain tensors, temperature, and convection rate with the help of graphs and tables.

Mathematical Formulations
We deliberate the flow of a compressible viscous nanofluid along an erect turning cone enclosed in a brittle medium as a two-dimensional time-dependent boundary layer. Figure 1 displays the scheme of coordinates and the corporal model. We have used a rectangular coordinate system in which the x− axis is determined along a meridian, the y− axis is determined along a round section, and the z− axis is determined on the cone's surface. Let u, v, and w be velocity gears, with x (tangential), y (azimuthal), and z (horizontal) orders (normal). e equations can be written as follows: e boundaries conditions are set, subject to initial terms and conditions as follows: Reference [15] provides the momentum, temperature, initial conditions, and boundary conditions for this issue. e following transformation is defined where A is the Deborah number and c 1 and c 2 are the buoyancy parameters. e things of physical importance use in the two-phase model are as follows: α nf is the thermal diffusivity, ] nf is the kinematic viscosity, μ nf is the effective dynamic viscosity, ρ nf is the density, (ρC P )nf is the heat capacity, κ nf is the nanofluid thermal conductivity, and (ρβ) nf is the nanofluid thermal expansion coefficient: e transformation's equations (7) and (6) are replaced into (1)-(3). After that equation (1) will be fulfilled automatically, and equations (2)− (4) diminish to the form as follows: Now, the boundary conditions are e skin friction C fx across the x-axis, C fy along y-axis, and Nusselt sum Nu x are physical quantities of our distinct interest where or in the form of dimensionless In dimensionless form, the heat transfer coefficient is given as Now, the Reynolds number is (13)

Numerical Solution
e numerical technique performs the solutions of the joined linear sinusoidal differential equations. e following initial estimates and nonlinear operators are f 0 , g 0 , and θ 0 , respectively, for velocity components and temperature fields.
New variables are defined simplifying differential equations of high order in the form of first-order equation, i.e., Now, the new equations are Along with limitation, 4 Mathematical Problems in Engineering e secondary velocity g is also seen to have the greater magnitude for α 1 . A vertically spinning or expanding cone was analyzed on the unstable frontier layer flow of both water-based nanofluids. e flow was contrary to viscous debauchery, the cohort of excess heat, and a natural process. e numerical technique is used to resolve the equations. We studied the belongings on the nanofluid velocity (f&g) and temperature (θ) profiles and even the skin friction (C fx &C fy ) coefficient, energy, and mass exchange coefficients of the nanoparticle volume segment, buoyancy parameter c 1 , heat production, and chemical reaction. We considered nanoparticles of copper oxide (Cuo) and silver (Ag) with water as the basis fluid. In Figures 2(a) Figure 7(a) shows that C fy values increase in Cuo but Figure 7(b) shows decline values of C fy . Physically, we can conclude that the surface temperature is higher than the fluid temperature close to the cone boundaries; therefore, larger c 1 gives the greater values of skin friction. It is examined that the coefficient of tangential skin friction (C fx &C fy ) decreases as α 1 increases (see Figures 7(a) and 7(b)). Figures 8(a) and 8(b) and Figures 9(a) and 9(b) show that the primary skin friction coefficients increase or decrease with the rise in α 1 , also the same behavior for c 1 . In Figures 8(a) and 9(a), C fy values decline for Cuo, and we see that in Figures 8(b) and 9(b), the values of C fx are enhanced when the values of α 1 and c 1 are increased. Since the impact of Pr in the primary − f ′ and secondary g directions on the velocity profiles is relatively small, therefore, the profiles are ignored. In Figures 10(a) Table 1 and Table 2 represent the variations of α 1 , c 1 , s, and Pr on the coefficient of skin friction (C fx & C fy ) and local Nusselt number (Nu x ). Table 3 shows the physical properties of copper oxide and silver.

Concluding Remarks
\In this numerical study, we have considered an unstable twophase nanofluid flow and heat transfer attitudes over a cone littered with two diverse metal types specifically Ag and Cuo. Unsteady mixed convection flow has been investigated in a moving viscous fluid on a turning cone. Numerical solution of ordinary differential equations BVP4C has been implemented successfully. e afresh determined outcomes are recognized to be accessible in the literature in traditional agreement with the findings previously reported. Viscous dissipation was found to have the consequence of swelling the nanofluid temperature within the gravity effects area when the heat transfer rate from the layer decreases with an increase in a viscous heat generation. e analysis summary is as follows: (1) For increasing c 1 and α 1 , the tangential velocity field − f ′ declines. However, s near to the boundary often causes − f ′ to decrease and increases it far away from the boundary for increasing c 1 and α 1 . (2) On elevating α 1 , the azimuthal velocity field g decreases and decreases with increase in s. (3) For higher values of Pr, the temperature profile θ(η) increases for Ag but decreases for Cuo. (4) Growing the importance of shape effects has raised the temperature profile and also the local Nusselt number. (5) e two-phase nanoparticle model has always a bigger impact than nanoparticles on the temperature profile.

List of Symbols
Pr: Prandtl number T: Temperature C fx : In the x-direction, there is local skin friction C fy : Skin friction in y-direction f, g: e velocity of a dimensionless stream function component in x− and y− direction, respectively K, L: ermal conductivity and area of the indentation, respectively, Km − 1 K − 1 μ: Dynamic viscosity Nms − 2 Nu x : Local Nusselt number Re x : Reynold number based on x Re L : Reynold number based on L (ρC p ) nf : Heat capacity of nanofluid jk − 1 (u, v, w): Velocity components ms − 1 t, t * : Dimensional and dimensionless times, respectively (x, y, z): e distance measured along the meridian of a circular segment parallel to the cone's superficial α: Semiupright angle of the cone η: Similarity variable θ: Dimensionless temperature c 1 : Buoyancy parameter due to temperature ]: Kinematic viscosity m 2 s − 1 ρ: Density kgm − 3 ρ nf : Nanofluid density kgm − 3 μ f : e viscosity of fluid Nms − 2 μ nf : Nanofluid viscosity Nms − 2 α nf : Nanofluid thermal diffusivity m 2 s − 1 .

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.

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