Study of nonlinear thermal convection of ternary nanofluid within Darcy-Brinkman porous structure with time dependent heat source/sink

1.   Introduction

Convective heat transfer is one of the most important means of heat exchange in many types of industrial equipment. Several factors influence the convective heat transfer, including device construction, flow patterns, and working fluid characteristics. Traditional working liquids with poor heat conductivity, such as water, oil, ethylene glycol (EG), and lubricants examined by ^[1,2], significantly limit heat transfer performance. Nanofluids, on the other hand, combine excellent thermal conductivity with lower density to facilitate convective heat transfer in a range of applications. The dynamics of nanofluids owing to thermo-capillary convection generated by distinct five nanoparticle morphologies were given by Jiang et al. ^[3] (sphere, blade, brick, cylinder, and platelet). The nanofluid created with spherical nanoparticles had the maximum intensity of thermo-capillary convection, whereas platelet-shaped nanoparticles had the lowest. Nusselt numbers increased by 22.8 percent in blade nanoparticles and 2.8 percent in blade-shaped nanoparticles. A comparable work on nanoparticle shape was published by Arno et al. ^[4] They observed that changing the structure of nanoparticles enhances heat transfer performance, according to Benos et al. ^[5], Liu et al. ^[6] studied the significance of quartic autocatalysis on the dynamics of water conveying 47 nm alumina and 29 nm cupric nanoparticles. They presented a comparative analysis between the dynamics of water conveying 29 nm CuO and 47 nm Al₂O₃ on an upper horizontal surface of a paraboloid of revolution. Sabu et al. ^[7] examined the significance of nanoparticles' shape and thermo-hydrodynamic slip constraints on MHD alumina-water nanoliquid flows over a rotating heated disk with the passive control approach. Several scholars ^[8,9] have done a great job implementing the MHD paradigm to their study, according to the literature. To examine heat and mass transfer on a stretched sheet, vajravelu ^[10] utilized suction or blowing. Concentration and temperature distribution were determined using the isothermal moving plate. The temperature distribution in a flow across a stretched sheet with homogeneous heat flux was explored by Dutta, Roy and Gupta ^[11]. The temperature at a given place drops as the Prandtl number rises. Kumar et al. ^[12] looked at how MD affected a hybrid nano liquid stream running over an elastic surface. In their study, they concentrated on the influence of the ferromagnetic interaction parameter on flow and temperature sketches. In a 2-dimensional stream of micropolar hybrid nano liquid, Ali et al. ^[13] used MD to evaluate the impacts of varying volume percentages on involved distributions.

The influence of rotations on the thermal deterioration of a flexible Walters' (model B') elastico-viscous fluids in porous media was explored by Rana and Kango ^[14] using linearized stability theory. In a Brinkman porous medium, Kuznetsov and Nield ^[15] examined the thermos dynamic instability of a permeable media surface saturated with a nanofluid. While Sheu ^[16] investigated the linear stability of convection in a viscoelastic nanofluid surface. Nonlinear stretching sheets are often used by researchers to better model a variety of real-world flow concerns, such as lowering the weightiness and acoustic of complicated mechanical microstructures. The researchers were interested in the boundary layer flows that occurred from linear/nonlinear stretching sheets of varying geometrical thicknesses because of their significant industrial applications. Lee ^[17] was the first to offer an effective geometrical boundary layer flow structure, which has since sparked a lot of controversy among scientists. Chen's study focused on mixed convective fluxes across thin bodies ^[18]. Fang and his colleagues are a set of people who collaborate to solve difficulties ^[19]. The impact of velocity drops on three-dimensional stretching/shrinking flow was studied by Jusoh et al. ^[20], Jama'ludin et al. ^[21], and Khashi'ie et al. ^[22] using nanofluid. The Boussinesq estimate expresses that while the distinction in inactivity is little, gravity is adequately amazing to make the particular loads of the two liquids vary altogether. The Boussinesq estimate is utilized to tackle issues including liquid stream and warmth move where the temperature of the liquid changes starting with one area then onto the next. Mass, force, and energy are completely saved in the liquid. Varieties in liquid qualities other than thickness are ignored in the Boussinesq estimation, and thickness arises just when it is increased by gravity's velocity increase. Stability analysis was used in these tests to assess the steady flow solution's reliability, further worthwhile the nonlinear buoyancy-induced (quadratic convection) flow of hybrid nanofluid fields are analyzed comprehensively by investigators ^{[23,24,25,26,27,28,29,30,31,32]} and reported that the hybrid nanoparticles have intermediate kind of conductivity in the flow mechanism.

After, thorough literature collection in present study, considered the nonlinear convective flow in suspended Darcy porous layers and ternary nanoparticles. With appropriate transformation, the resulting ruling partial differential system is changed into a governing ordinary differential system, and a solution with two mixture compositions is supplied (Type-I and Type-II). Type-I mixture of Al₂O₃ (Platelet), GNT (Cylindrical), and CNTs (Spherical), Type-II mixture of silver (Platelet), copper (Cylindrical), and copper oxide (Spherical). The combination of liquids is of great importance in various systems such as medical treatment, manufacturing, experimental instrument design, aerosol particle handling and naval academies, etc. Roseland's quadratic and linear approximation of three-dimensional flow characteristics with the existence of Boussinesq quadratic buoyancy and thermal variation. In addition, we combine tertiary solid nanoparticles with different shapes and densities. In many practical applications such as the plastics manufacturing and polymer industry, the temperature difference is remarkably large, causing the density of the working fluid to vary non-linearly with temperature. Therefore, the nonlinear Boussinesq (NBA) approximation cannot be ignored, since it greatly affects the flow and heat transport characteristics of the working fluid. Here, the flow of non-Newtonian elastomers is controlled by the tension of an elastic sheet subjected to NBA and the quadratic form of the Roseland thermal radiation is studied.

2.   Mathematical modelling

The summary and plot are embedded within a Darcy-Brinkman porous structure, where permeability $K\left(x\right)$ is a nonlinear function of x.as shown schematically in Figure 1.

A continuous two-dimensional mixed convection of a radiative Casson Hybrid nanofluid is generated via a wedge shaped thin sheet of varying thickness. Indeed, the Viscoelastic Hybrid nanofluid is chemically created by dissolving a volume fraction of metallic/metallic oxide nanoparticles in a volumetric amount of sodium alginate (SA), which rheological follows the Casson fluid model. In a two-dimensional Cartesian frame $(x, y)$, the present Hybrid nanofluid flow arrangement may be visualised geometrically as a thin sheet stretching upwards in a homogeneous gravitational field of strength g, with a variable velocity ${u}_{h}\left(x\right)$. The varying temperature of the walls Tw warms the stretched sheet unevenly ${T}_{h}\left(x\right)$.

The temperature in the free-stream zone is also maintained at ${T}_{\infty }$ meaning that the Hybrid nanofluid is at rest. Using boundary layer simplifications, the governing equations for the current flow issue are presented ^[23] i.e., $v < < u, \partial /\partial x < < \partial /\partial y, {\partial }^{2}/\partial {x}^{2} < < {\partial }^{2}/\partial {y}^{2}$ and the Oberbeck-Boussinesq approximation in its linearized version ^[24] and Alghamdi et al. ^[31]:

With the following dimensions Boundary Conditions:

The radiative heat flow vector ${q}_{r} = \left({q}_{rx}, {q}_{ry}\right)$ within optically thick medium may be written as follows ^[24,29] using the Rosseland approximation ^[25]:

Given the previous considerations, the powered temperature ${T}^{4}$ is linearized near the free-stream temperature ${T}_{\infty }$ as follows ^[28]:

The following formulas may be used to assess the Hybrid nanofluid thermophysical characteristics provided in the preceding formulation:

We study the viscosity and thermal conductivity of ternary hybrid nanoparticles. To assess the form of the nanoparticles, researchers employed a variety of viscosity and thermal conductivity models for platelet, cylindrical, and spherical nanoparticles. The form of nanoparticles influences their density, therefore this was critical for a correct model. Higher-density nanoparticles contain a nanofluid with changing thermal conductivity and viscosity. The ternary hybrid nanofluid density model is used by Raju et al. ^[24]:

The heat capacity of a ternary hybrid nanofluid may be calculated using the formula given by Raju et al. ^[24]:

For spherical nanoparticles, thermal conductivity models and viscosity are applied.

For cylindrical nanoparticles, the viscosity and thermal conductivity models are applied (Raju et al. ^[24]).

Platelet nanoparticles have thermal conductivity models and viscosity.

The volume of spherical nanoparticles is ${\phi }_{1}$, cylindrical nanoparticles are ${\phi }_{2}$, and platelet nanoparticles are ${\phi }_{3}$. The amount of total volume is expressed as a percentage of the whole volume.

It is ideal for ternary hybrid nanofluids with $\phi$ > 0.02 value, according to models with varied thermal characteristics (Raju et al. ^[24]).

the values of the thermophysical properties of nanoparticles are presented in Table 1.

The given similarity adjustments are included into controlling PDEs to get the dimensionless versions of the boundary layer results:

The following dimensionless ODEs of (Alghamdi et al. ^[31]):

where,

The dimensionless Boundary conditions that regulate Eqs (19) and (20) are as follows:

The nomenclature table explicitly defines the included dimensional/dimensionless quantities and variables. The mixed convection phenomenon is controlled by the following variables in this study:

As a result, the linear relationship below should be used to link the temperature and velocity power parameters (m, n) in order to find adequate similarity solutions:

The following transformations are used to make the succeeding computational tasks easier:

As a result, the following simplified differential structure emerges:

The $xy$-shear stress component ${\tau }_{h}$ and the vertical wall heat flux ${q}_{h}$ are the fundamental quantities of concern in the current convective flow at the stretched sheet, and are defined as:

Those engineering quantities are computed locally in the dimensionless forms as:

in which,

The numbers with no dimensions as previously mentioned, Nux and Cfx signify the local Nusselt number and wall frictional factor, respectively, whereas Nur and Cfr denote their reduced equivalents.

3.   Methodology and authentication

The nonlinear differential equation with the boundary constraints (27) are solved numerically using Runge-Kutta integration method. Initially, the set of nonlinear ODEs converted to 1^st order differential equations, by using the following process is given by (Upadhya and Raju ^[32]).

with boundary conditions as:

We guess the values of ${y}_{3}\left(0\right)$ which are not specified at the initial conditions. The Eqs (35)–(37) are integrated by taking the help of Runge-Kutta method with the successive iterative step length as 0.01. For this, used MATLAB bvp5c solver to solve the first order nonlinear coupled differential equations. The correctness of the supposed values is checked by equating the calculated values ${y}_{1}\left(0\right), {y}_{2}\left(0\right), {y}_{4}\left(0\right)$ at$\varsigma = {\varsigma }_{max}$with their given values at $\varsigma = {\varsigma }_{max}$. If any difference between their corresponding values exists, then the process is continued upto the required good values. Alternatively, we are using the Runge-Kutta method to get the accurately found the initial values of ${y}_{1}\left(0\right), {y}_{2}\left(0\right), {y}_{4}\left(0\right)$ and then integrate Eqs (35)–(37). This process is repeated until the settlement between the designed value and the given condition is within the specified degree of accuracy 10^-5. In order to validate the precision of the current solutions, compared our results with Khader and Megahed ^[33] solutions and found earnest agreement with Khader and Megahed ^[33] solutions under the limited case displayed in Table 2.

4.   Interoperation of physical quantities and their discussions

The results indicate how non-dimensional controlling factors affect dimensionless sketches. ${f}'\left(\eta \right), \theta \left(\eta \right),$ $and\eta$ such as radiation parameter $\left(R\right)$, Dimensional positive characteristics $\left(n\right)$, a nonlinear convection parameter or a quadratic convection parameter $\left(Qc\right)$, the ternary hybrid nanofluid's volume fraction $\left(\phi \right)$, temperature ratio parameter $\left({\theta }_{p}\right)$, heat source parameters $\left(A\&B\right)$, mixed convection parameter $\left(\lambda \right)$, Prandtl number $\left({p}_{r}\right)$ (see Figures 2–19). ${p}_{r} = 6.7, A = B = 0.01, \lambda = 0.5, \phi = 0.5, R = 0.5, {\theta }_{p} = 0.1, Qc = 0.1, P = 1, and\beta = 2$ are the default parameter values utilised in the simulation. These values were kept constant throughout the research, with the exception of the varied values shown in the relevant figure.

The variation in skin friction coefficients and the Nusselt number as the Dimensional positive features parameter $\left(n\right)$ rises the Nusselt number and the friction coefficient is shown in Figure 20(a) for both situations 1 and 2. For both scenarios 1 and 2, the variation in skin friction coefficients and the Nusselt number when the quadratic thermal radiation parameter $R$ enhances the Nusselt number while decreasing the friction coefficient is shown in Figure 20(b). For both Cases 1 and 2, Figure 20(c) depicts the fluctuation in skin friction coefficients and Nusselt number when the volume fraction parameter $\left(\phi \right)$ decreases the Nusselt number and raises the friction coefficient. For both Cases 1 and 2, Figure 20(d) depicts the variation in skin friction coefficients and Nusselt number when the mixed convection parameter $\left(\lambda \right)$ reduces the Nusselt number while increasing the friction coefficient. The variation in the skin friction coefficients and the Nusselt number when the temperature ratio parameter $\left({\theta }_{p}\right)$ raises the Nusselt number and decreases the friction coefficient is shown in Figure 20(e) for both Cases 1 and 2. For both Cases 1 and 2, Figure 20(f) depicts the variation in skin friction coefficients and Nusselt number when Prandtl number $\left(Pr\right)$ decreases the Nusselt number while increasing the friction coefficient. Figure 20(g) shows how the skin friction coefficients and the Nusselt number change when the nonlinear convection parameter $\left({Q}_{c}\right)$ lowers the Nusselt number and raises the friction coefficient in both Case 1 and 2. For both Cases 1 and 2, Figure 20(h, i) depicts the variation in skin friction coefficients and the Nusselt number when Heat source characteristics $\left(A\&B\right)$ lower the Nusselt number while increasing the friction coefficient. The influence of the Dimensional positive features parameter $\left(n\right)$on temperature and velocity sketches is seen in Figures 2 and 3. It has been shown that as the Dimensional positive features parameter is increased, both velocity sketches and temperature sketches increases. Figures 4 and 5 demonstrate the effect of a quadratic thermal radiation parameter $\left(R\right)$ on non-dimensionless velocities and temperature. The figures show that increasing the quadratic thermal radiation parameter increases both the fluid's temperature and velocity sketches. The influence of the Volume fraction parameter of the ternary hybrid nanofluid $\left(\phi \right)$ on temperature and velocity sketches is depicted in Figures 6 and 7. The data show that raising the Volume fraction parameter of the ternary hybrid nanofluid lowers the velocity sketches and increases the temperature profiles.

The effect of the mixed convection parameter $(\lambda)$ on temperature and velocity sketches is seen in Figures 8 and 9. As can be seen, raising the mixed convection value enhances the velocity sketches and decreases the temperature profiles. Figures 10–11 show the effect of the temperature ratio parameter $\left(\theta_p\right)$ on temperature and velocity sketches. When the temperature ratio parameter is raised, both the temperature and velocity sketches are visibly increased. Figures 12 and 13 show the impact of the Prandtl number $\left(\theta_p\right)$ on temperature and velocity curves. It has been discovered that when the Prandtl number increases, so the temperature and velocity sketches decrease. Figures 14 and 15 depicts the influence of the nonlinear convection parameter $\left(Q_c\right)$ on temperature and velocity sketches. It is self-evident that increasing the nonlinear convection parameter raises velocity sketches and decreases the temperature profiles. Figures 16 and 17 depicts the influence of the Heat source factor $(A)$ on temperature and velocity sketches. Increases in the Heat source parameter result in higher velocity and temperature sketches. Figures 18 and 19 show the effect of the Heat source factor $(B)$ on temperature graph. It is obvious that as the Heat source parameter is increased, both temperature and velocity sketches rise.

5.   Conclusions

In recent years, fluid mixtures have become more and more important in various systems, including medicinal treatments, industry, experimental instrument design, aerosol particle processing, and naval academies. The Rosseland approximation on 3D flow characteristics is nonlinear, linear, and quadratic due to the presence of Fourier flux and Boussinesq second order thermal variation. Ternary hybrid nanoparticle densities and various shapes are included. The governing system that results is then tweaked, and a solution with two different mixture compositions is shown (Case-I and Case-II). Graphene (Cylindrical), Carbon nanotubes (Spherical), and aluminum oxide (Platelet) were considered in the first mixture, whereas copper (Cylindrical), copper oxide (Spherical), and silver oxide (Platelet) were studied in the second mixture. In comprehensive literature study it is observed that, many researchers claim that conventional fluids such as water, oil, ethylene glycol, glycerol etc. has comparatively lesser thermal properties compared with solid metals. Considering the enormous demand for energy and on other side diminishing world resources and environmental concerns has impelled the researchers towards the development of more efficient heat transfer fluids with upgraded heat transfer rates. In recent days, fascinated by the applications/physical and chemical properties of nanoparticles researchers have initiated in exploring heat and flow properties of the colloidal mixture convectional fluid with three types of nanoparticles, known as ternary hybrid nanofluid. By considering this into view, investigated the slender surface flow characteristics in the presence of ternary nanoparticles. It is using MATLAB using bvp5c solver. The results are as follows:

1) Case-II ternary mixtures have a narrower temperature variation than Case-I ternary mixtures.

2) Compared to Case-1, Case-2 has a low skin friction coefficient.

3) The ternary hybrid nanofluid velocity increases as n, R, ${\theta }_{p}$, ${Q}_{c}$, A, B and $\lambda$ grow.

4) The ternary hybrid nanofluid velocity is decreases as Pr and $\phi$ grow.

5) The temperature of the ternary hybrid nanofluid rises for the values of n, R, $\phi$, ${\theta }_{p}$, A and B are estimated to be greater.

6) The ternary hybrid nanofluid temperature is a decreasing as of Pr, ${Q}_{c}$, and $\lambda$ grow.

Data Availability

Data available on request from the authors.

Acknowledgements

This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation, (grant number B05F650018).

Conflict of interest

The authors declare no conflicts of interest.

List of Symbols

$n$    Positive dimensional features parameter

$(A, B)$    Parameters of the heat source

$q$    Net heat flux, [W.m^-2]

${C}_{fx}$    Coefficient of skin friction simplified

$g$    The acceleration of gravity, [m.s^-2]

${Q}_{c}$    ${Q}_{c} = \frac{{\beta }_{1}}{{\beta }_{0}}\left({T}_{w}-{T}_{\infty }\right)$convective parameter (quadratic or nonlinear)

$G{r}_{x}$    Local Grashof number

${\rho }_{bf}$    The density of the base fluid (kg/m³)

${\rho }_{hnf}$    The ternary hybrid nanofluid's density (kg/m³)

${T}_{h}$    Temperature of the wall dimensions (K)

${C}_{p}$    Specific heat, [J.kg^-1.K^-1]

$T$    A ternary hybrid nanofluid's dimensional temperature (K)

${\left(\rho {c}_{p}\right)}_{hnf}$    The ternary hybrid nanofluid's heat capacity (kg/m³)

${T}_{\infty }$    Fluid temperature at a distance from the wall (K)

${\mu }_{1}and{\kappa }_{1}$    A spherical nanofluid's viscosity and heat conductivity. (Ns/m² & W/m K)

${\mu }_{2}and{\kappa }_{2}$    A cylindrical nanofluid's viscosity and heat conductivity. (Ns/m² & W/m K)

${\mu }_{3}and{\kappa }_{3}$    A platelet nanofluid's viscosity and thermal conductivity (Ns/m² & W/m K)

$u$    The motion's dimensional velocity in the x-direction (m/s)

${u}_{h}$    The motion along the x-axis is being stretched at a faster rate. (m/s)

$v$    The motion's dimensional velocity in the y-direction (m/s)

${\mu }_{hnf}$    The ternary hybrid nanofluid's dynamic viscosity (Ns/m²)

${\phi }_{1}$    spherical nanoparticles volume/amount (m³)

${\phi }_{2}$    cylindrical nanoparticles volume/amount (m³)

${\phi }_{3}$    platelet nanoparticles volume/amount (m³)

$Nu$    Nusselt number

${\phi }_{hnf}$    The ternary hybrid nanofluid's volume percent (m³)

$qr$    Radiative heat flux, [W. m^-2]

$\theta p = \frac{{T}_{h}}{{T}_{\infty }}$    is the temperature ratio parameter.

$Re$    Reynold number

$R$    quadratic thermal radiation parameter, $R = \frac{4{\sigma }^{*}{T}_{\infty }^{3}}{k{k}^{*}}$

${u}_{h}\left[x\right]$    velocity of stretch, [m. s^-1], ${u}_{h}\left[x\right] = [a+x{]}^{n}b$

$\left[x, y\right]$    Cartesian coordinates

$P$    Porosity parameter $= \frac{{\mu }_{f}}{\left({k}_{0}b{\rho }_{f}\right)}$

$\left[u, v\right]$    Components of velocity, [m/s]

Subscripts

$r$    Thermophysical qualities that are relative

$s$    Nanoparticles in solid form

$\infty$    Condition of free-flow

$f$     a base liquid

$w$    Condition of the walls

Greek symbols

$\beta$    Casson parameter for hybrid nanofluids

${\beta }_{T}$    Coefficient of thermal expansion, [K^-1]

$\phi$    portion of a volume

$\mu$    Dynamic viscosity, [Pa.s]

${\beta }_{R}$    The absorption coefficient of Rossland, [m^-1]

$\sigma$    Electrical conductivity in a hybrid nanofluid, [S.m^-1]

${\sigma }_{e}$    Constant of Stefan-Boltzmann, [W.K^-4.m^-2]

$\lambda$    Mixed convection parameter, $\lambda = \frac{G{r}_{x}}{{\mathit{Re}}_{x}^{2}} = \left[\frac{c\left(\rho {\beta }_{T}\right)g}{{b}^{2}{\rho }_{f}}\right]$

$\rho$    Density, [kg.m^-3]

$\left[\rho {C}_{P}\right]$    Capacitance to heat, [J.m^-3.K^-1]

$\nabla$    Gradient vector, $\nabla = \left(\frac{\partial }{\partial x}, \frac{\partial }{\partial y}\right)$

$\gamma = \mu /\rho$    Kinematic viscosity, [m².s^-1]