Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints

Nomenclature

ρ (kg/m³)

density fluid [M L⁻³]

Ω (rad/s)

angular velocity [L¹ T⁻¹]

μ (kg/ms)

viscosity of the fluid [M¹ L⁻¹ T⁻¹]

k (W/m K)

thermal conductivity

N _t

thermophoretic parameter

τ (N/m²)

shear stress

η

dimensionless variable

N _b

Brownian motion parameter

Kc

HOM chemical reaction parameter

ρC _p (J/K)

specific heat capacity [M⁰ L² T⁻² K⁻¹]

u, v, w (m/s)

velocity components

m (J/kg)

melting heat parameter

▽

del operator

ν (m/s)

kinematic viscosity

D _B (m²/s)

Brownian diffusivity coefficient

T (K)

temperature [K]

T _w (K)

wall temperature

C (mol/m³)

concentration

A, B

species

C _f

skin friction

θ

dimensionless temperature

φ

dimensionless concentration

P (N/m²)

pressure

Γ

relaxation time constant

A₁ (m/s)²

first Rivlin–Ericksen tensor

b → (N/m³)

body force

V → (m/s)

vector velocity

Sh

Sherwood number

Pr

Prandtl number

Sc

Schmidt number

Re

Reynold number

Nu

Nusselt number

I

identity tensor

Ks

HET chemical reaction

T _∞ (K)

ambient temperature

C _w (mol/m³)

wall concentration

C _∞ (mol/m³)

ambient concentration

1 Introduction

Non-Newtonian fluid flows gained massive attraction in the development of modern industries and different engineering technologies. In particular, the flow of non-Newtonian fluids has many applications such as polymer processing, slurry transport, and coating technologies. Wilkinson [1] presented the uses of such phenomena in terms of production, polymer extrusion of sheets, sheet emulsion, flow of plasma and blood, etc. Because of their rheological properties, the analysis of such fluids cannot be sufficiently described by modeling using the Navier–Stokes equations. However, the viscosity of non-Newtonian liquids depends on the shear stress and shear rate. These liquids exhibit a variety of complex flow behaviors and can be categorized into different types based on their responses to shear. Various non-Newtonian fluid models have been explored owing to their unique characteristics. Rashidi et al. [2] investigated the nonlinear nature of such materials. Yang et al. [3] illustrated the shear-thinning nature of the non-Newtonian flow of liquids across a porous material. Nazir et al. [4] deliberated the impact of chemical reactions on the flow of non-Newtonian liquids with heat and mass balance. Recently, Algehyne et al. [5] explored magnetic field effects on the transport of Maxwell liquid across a three-dimensional interface with passive boundaries.

The concept of nanofluids has become very attractive in recent advances in heat transmission and cooling processes with the inclusion of nanosized particles in the base liquids. However, nanofluids contain nanosized particles. As a result, with the inclusion of such particles, an increase in the thermal conductivity was noted, with the main purposes. Classically, nanomaterials used in nanoliquids contain metals, oxides, carbon nanotubes, and carbides. Choi and Eastman [6] were the first to introduce this phenomenon. They introduced nanoliquids, i.e., fluids that contained nanosized particles (nanoparticles). Later, Buongiorno [7] established the generalized behavior of Brownian motion and thermophoretic forces for producing a unit temperature difference while introducing the motion of nonlinear materials. Kherbeet et al. [8] experimentally investigated the flow characteristics and heat transfer of nanofluids over microscale bidirectional step geometries. Hassan et al. [9] assessed the thermal impact of nanofluids when the base fluid viscosity was considered to be high. They addressed the concepts of Brownian and thermophoretic effects with a declining volume fraction. Payam et al. [10] explored the steady-state nanofluid motion through two parallel planes. Yasmin [11] conducted an analytical study of convective behavior featuring nonlinear properties in a nanolayered liquid film over a graded deforming interface [12]. A numerical analysis of magnetohydrodynamic water-based AA7072 nanofluid flow over a permeable stretching surface with slip conditions. Gangadhar et al. [13] investigated the simulation of radiative nonlinear heat dynamism on Buongiorno-modeled nanoliquids using porous inclined plates with adjustable chemical responses.

A homogenous (HOM) model refers to a system or model in which all the components or elements are identical or similar or consist of uniform entities that have the same characteristics and behavior, or a chemical reaction in which the reactants and products are in the same phase. HOM models are often used in various fields such as Physics, Chemistry, and Computer Science. Chaudhary and Merkin [14] were the first to introduce the phenomena of HOM and HET chemical reactions in the strongest boundary layer flow of the stagnation point. Rajesh et al. [15] studied the effects of HOM/HET reactions on oscillating magnetic boundary layer flow through a curved surface. Suleman et al. [16] analyzed the computational simulation of HOM/HET reactions and Newtonian heat transfer in silver–water nanofluid flowing past a nonlinearly stretched cylinder. A heterogeneous (HET) model refers to a system or model in which the components and elements are different or diverse, or a chemical reaction in which the reactants and products are in different phases. HET models are often used to study complex systems or situations in which the entities involved have different roles, capabilities, or characteristics. Hayat et al. [17] explored the Darcy–Forchheimer motion of carbon nanotube suspensions generated by a rotating disk subjected to convective heating, incorporating uniform and non-uniform chemical reactions. Doh et al. [18] analyzed how uniform and non-uniform reactions influence the nanofluid behavior induced by a rotating disk with varying thicknesses by applying the homotopy analysis technique.

The idea of melting heat is also called the heat of fusion, i.e., the amount of heat energy needed to convert an element from its solid state to its liquid state at a fixed temperature and pressure. This is a particular type of energy phase transition. In fluid dynamics, the concept of melting heat is not directly applicable since fluids (liquids and gases) do not possess a solid state with a well-defined melting point. Instead, they undergo phase transitions such as vaporization and condensation, which involve different thermodynamic properties like the heat of vaporization or heat of condensation. Roberts [19] was the first to initiate the melting concept by considering a hot air stream with the inclusion of an ice slab. Furthermore, Tien and Yen [20] examined the heat balance forced convective concept and observed the effect of melting between the surrounding fluid and the melting body. Yacob et al. [21] approximated the characteristics of the melting phenomenon during the approximation of velocity, temperature, and other related forces opposing the flow of liquid. Sadique et al. [22] explored the behavior of a thermally layered Powell–Eyring fluid, including the melting heat phenomenon. An analytical series solution was obtained using the homotopy analysis procedure. Shuguang et al. [23] and Mamatha et al. [24] employed this method owing to its accuracy and quick convergence.

A rotating disk in a fluid refers to a condition in which a solid disk is placed in a liquid medium and set to introduce rotational motion. This scenario has many applications and can be found in fields such as fluid dynamics, engineering, and physics, particularly in spin coating in thin film manufacturing, biomedical applications, and biomechanics. Karman and Uber [25] analyzed the incompressible viscous flow over an endless flat disk spinning at a constant angular speed. They pioneered the introduction of the concept of disk rotation in the presence of a viscous liquid with a constant density. Theoretical and experimental examinations of fluid motion due to disk rotation were first performed by Stewartson [26]. Watson and Wang [27] studied the unsteady flow over a rotating disk. Turkyilmazoglu [28] investigated the boundary layer development caused by a spinning disk using multiple generalized Newtonian fluid frameworks. Kelson and Desseaux [29] and Bachok et al. [30] explored nanofluid motion and heat transfer across a rotating porous disk. Recently, Ali et al. [31] made effective contributions in this direction. Aldhafeeri and Yasmin [32] conducted a computational study of hybrid nanofluid rotational flow over a unidirectionally stretching surface under velocity and thermal slip effects. Jawarneh et al. [33] explored the use of neural networks to examine magnetohydrodynamic nanofluid flow over a spinning disk, considering the effects of heat production and absorption.

2 Problem description and mathematical model

Consider the three-dimensional, steady, symmetric, and incompressible motion of a Williamson fluid around a spinning cylindrical disk, accounting for both heat and mass transport. The disk rotates clockwise along the z-axis with an angular velocity Ω . T w and C w are the surface temperature and concentration, respectively, and T ∞ and C ∞ are the ambient temperature and concentration, respectively, of the fluid (Figure 1).

Figure 1

Schematic view of the proposed model.

The governing flow equations for this problem, namely continuity and momentum equations, are [34]

where ( u , v , w ) are the components of the velocity field V, P is the pressure, ρ is the density of the material, and τ is the stress tensor, which is defined as [34]:

By plugging Eqs. (3) and (4) in Eqs. (1) and (2), the leading equations in cylindrical coordinates are as follows [35,36]:

The non-Newtonian Williamson fluid viscosity model (tensor) is μ = μ ∘ 1 − ( Γ 2 γ ⋅ ) 1 2 − 1 , where Γ is the relaxation time constant and γ is given as follows [31]:

The boundary conditions (BCs) are

Using T = T ( r , θ , z ) and C = C ( r , θ , z ) [34], the constitutive vector equations of the energy and concentration are as follows:

where q and J are the heat and mass fluxes, respectively, and are defined as

In the above expression, D refers to the diffusion coefficient, and the other term signifies the concentration gradient. Eqs. (11) and (12) are reduced as

where D A and D T are the diffusion and thermophoretic coefficients, T ∞ is the ambient temperature, k c C a C b 2 are the HOM reaction rate and k s C a C b 2 the HET chemical reaction rate, ( C a , C b ) are the concentrations of the chemical reaction (A, B), respectively, and C p is the specific heat.

The associated BCs are defined as:

All the aforementioned equations are dimensional, and by maintaining the physical significance, we converted the above set of equations into a dimensionless set of PDEs by introducing the following dimensionless similarity variables [37,38]:

Using the above transformations in Eqs. (5)–(9) and Eqs. (15)–(18), then we obtain

where

The dimensionless factors appearing in the above equations are listed in the following table:

The BCs are reduced as

For a smooth and accurate, less time-consuming solution, we need to transform the final PDEs to ODEs for which the following local similar variables are expressed as [37]

Then the final governing ODEs are obtained as

where

and

If ϕ ( η ) + φ ( η ) = 1 , then for δ = 1 , Eqs. (34) and (35) are reduced to the following form:

The system is subjected to the following BCs:

3 Results and discussion

The velocity field (V), temperature (T), and concentration (C) profiles are examined here with respect to the following parameters: the relaxation time constant (Γ), Schmidt number (Sc), Prandtl number (Pr), thermophoretic parameter ( N t ) , Brownian motion parameter ( N b ) , and HOM/HET chemical reaction parameters ( Kc ) and ( Ks . The values of all the parameters are fixed for finding the approximate results, i.e.,

Γ = 0.1 , Pr = 0.7 , N t = 0.3 , N b = 0.7 , Kc = 0.1, Ks = 4.5, Sc = 1, and m = 0.5.

The effect of Γ on the velocity components F ( η ) , G ( η ) , and H ( η ) and pressure P ( η ) of the fluid is shown in Figure 2(a)–(d). When the values of Γ are increased, then the radial and tangential velocity components and the pressure of the fluid are increased and the axial velocity component of the fluid is reduced. Physically, the fluid takes longer to adjust to changes in flow conditions, leading to more pronounced radial and tangential velocities as the fluid redistributes its momentum and slows down the adjustment of the fluid along the axial direction, potentially leading to reduced axial velocity. Fluid takes longer to equilibrate, resulting in higher radial pressure gradients to achieve balance within the flow. The contribution of the melting heat parameter m to the radial velocity component of the fluid is presented in Figure 3, where the melting heat parameter m enhanced F(η). Physically, the melting heat parameter can influence the radial velocity field in a fluid during motion by affecting the distribution of the temperature gradient, which can alter the viscosity and density of the fluid. This can lead to changes in fluid flow patterns, including variations in radial velocity profiles and the formation of different flow structures. The contribution of the Prandtl number to the radial velocity component and temperature of the fluid is shown in Figure 4(a) and (b). A higher Prandtl number reduced the radial velocity component and temperature of the fluid. The Prandtl number affects the boundary layer development and heat transfer characteristics of the fluid flow. In terms of the radial velocity component, a higher Prandtl number typically results in a more pronounced boundary layer and slower radial velocity near the walls, as thermal diffusion dominates over momentum diffusion. The Prandtl number influences the reduction in the thermal boundary layer thickness. In heat transfer scenarios, the Pr governs the comparative growth of the momentum and thermal boundary layers. When Pr increased, the thermal diffusivity of the fluid decreased, causing the thermal boundary layer to become thinner. The effects of ( N t ) and ( N b ) on the temperature and concentration fields of the fluid are shown in Figure 5(a)–(d). The thermophoretic parameter enhanced the temperature and concentration fields, whereas the Brownian motion parameter enhanced the temperature and reduced the concentration of the fluid. Physically, Brownian motion increases the kinetic energy of the particles in the fluid, leading to more collisions and thus more energy transfer between the particles. This increased energy transfer can increase the temperature. In the presence of temperature gradients, particles with different thermophoretic motilities experience forces that drive them toward regions of higher or lower temperatures, depending on their properties. ( N b ) has a significant physical effect on the concentration field in the fluid. In simple terms, N b influences the spread and movement of particles within the fluid, leading to changes in the concentration distribution over time. N b impacts the stochastic movement of particles in the fluid. As the ( N b ) factor enhanced, the particle exhibited more rapid and erratic movement, leading to greater dispersion throughout the fluid. The thermophoretic motion factor influences the concentration field in a fluid by affecting particle movement in response to temperature gradients. When a temperature gradient exists in a fluid, the thermophoretic motion parameter influences the migration of particles, leading to changes in the concentration distribution. Essentially, it affects the movement of particles in response to temperature differences, which in turn affects the concentration field within the fluid. The effects of the HOM/HET reaction parameter and Schmidt number on the concentration field are examined. For a higher HOM reaction parameter, the concentration of the fluid is reduced, and when the HET reaction parameter is varied, the concentration of the fluid is decreased. With an increase in the Schmidt number, the concentration field increased, as shown in Figure 6(a)–(c). It is clear that Kc and Ks decrease the concentration distribution because a large quantity of reactants is involved in the HOM reaction. Because Sc is the ratio of momentum diffusivity to mass diffusivity, higher Sc values indicate greater momentum diffusivity, resulting in an improved concentration profile. The contributions of the relaxation time constant Γ , melting heat parameter m, and the Prandtl number Pr to the radial and tangential skin frictions are shown in Table 2. The relaxation time constant enhanced the radial and tangential skin frictions. The variation in the Prandtl number tends to increase the radial and reduce the tangential skin friction, and a higher Prandtl number usually results in thicker boundary layers, leading to increased radial skin friction. The melting heat parameter typically reduces the radial and increases the tangential skin fraction because a higher melting heat parameter indicates that more heat is required to cause the fluid to transition from a solid to a liquid state. The effects of the thermophoretic, Brownian motion, and HOM/HET chemical reaction parameters on the Nusselt and Sherwood numbers are shown in Table 3. The thermophoretic parameter reduced the Nusselt and Sherwood numbers, and the Brownian motion parameter reduced the Nusselt number and enhanced the Sherwood number. The HOM reaction parameter enhanced and the HET reaction parameter reduced the Nusselt and Sherwood numbers. The thermophoretic parameter reduced the Nusselt and Sherwood numbers because it tends to drive particles away from hotter regions, hindering heat and mass transfer. Conversely, the Brownian motion parameter reduces the Nusselt number because it leads to more chaotic particle movement, making it harder for them to carry heat; however, it enhances the Sherwood number as it increases the chances of particles interacting with the fluid, improving mass transfer. HOM reactions enhance the Nusselt and Sherwood numbers because they increase the contact between the fluid and the reacting species, promoting heat and mass transfer. However, HET reactions reduce these numbers because they create surface barriers that impede the transfer of heat and mass between the fluid and the solid surface where the reaction occurs.

Figure 2

(a)–(d) Impact of Γ on the velocity components and pressure of the fluid.

Figure 3

Influence of m on F(η).

Figure 4

(a) and (b) Influence of Pr on F(η) and θ(η).

Figure 5

(a)–(d) Impact of Brownian and thermophoretic parameters on θ(η) and φ(η) fields.

Figure 6

(a)–(c) Impact of HOM/HET reactions and Schmidt number on φ(η).

4 Concluding remarks and future work

This study was initiated to explore the unique features of the rotary motion of a dilatant fluid, such as the Williamson material. The same materials were effectively analyzed in the presence of generalized Brownian motion and the melting heat concept. The thermal analysis of the materials was explicitly demonstrated by incorporating the influence of the thermophoretic forces. Two types of chemical reactions were tested to explore the significance of the material concentration. Each new impact was then presented through graphs of flow speed, concentration, energy, and pressure. The valid physical characteristics of the material were briefly demonstrated for specific values of the various controlling parameters. This typical work can be further extended by considering the clockwise and anti-clockwise rotations of the stretching/shrinking double disks. The key findings of this study are as follows:

• The melting heat parameter reduced the flow speed away from the disk surface.

• The material factor enhanced the F(η) and G(η) components of the flow speed.

• Both thermophoresis and Brownian motion increase the material temperature.

• The material concentration was enhanced for higher values of the thermophoretic force, and a reverse trend was observed for increasing values of N _b .

• Higher HET and HOM reaction factors reduced the material concentration.

• The resistive forces in the radial direction were reduced, and the opposite conduct was observed in the case of the tangential resistive force when considering a higher melting factor.