Finite difference simulation of natural convection of two-phase hybrid nanofluid along a vertical heated wavy surface

Abstract

This study employs finite difference modelling to investigate the natural convection of a two-phase hybrid nanofluid consisting of Al${}_2$O${}_3$ and Cu nanoparticles dispersed in water along a vertically heated wavy surface. The hybrid nanofluid has unique features that influence convective heat transfer. Numerical solutions analyze fluid flow patterns and temperature distributions, providing insights into heat exchange and thermal management. The research highlights the importance of hybrid nanofluids, notably those containing Al${}_2$O${}_3$ and Cu in a water base, in improving overall heat transfer efficiency. The mathematical model accounts for laminar and incompressible fluid flow with a Prandtl number of Pr = 6.2, Lewis number Le = 10, and maximum $10\mathrm{\%}$ concentration of hybrid nanoparticles. After transforming the governing equations into a non-dimensional form, the implicit finite difference method is used to solve them. The solution employs the in-house FORTRAN 90 code and compares its outcomes with the benchmark results. Various parameters, such as the Schmidt number (Sc = 1 to 10), volume fraction of nanoparticles ($\varphi =0.0$ to 0.1), wavy amplitude (A = 0.0 to 0.3), and $N_{\mathrm{BT}}$ = (0.05 to 0.2), are investigated regarding temperature, velocity, local skin friction coefficient $(C_f)$, local Nusselt number (Nu), streamlines, and isotherms. Elevated volume fraction generally decreases skin friction, increases temperature, and may reduce velocity. Meanwhile, higher skin friction, temperature, velocity, and local Nusselt numbers often correlate with elevated Schmidt numbers. Changes in amplitude and $N_{\mathrm{BT}}$ affect skin friction, temperature, velocity, and local Nusselt numbers, providing information on the dynamics of fluid systems. For example, when the volume fraction (ϕ) increases from 0 to 0.1 at Y = 2, the temperature rises by $75\mathrm{\%}$, while the velocity gradually decreases by $11.11\mathrm{\%}$. This is in contrast to findings according to when the $N_{\mathrm{BT}}$ rises from 0.05 to 0.2 at Y = 1. In this case, the temperature drops by $18.75\mathrm{\%}$, followed by a $9.68\mathrm{\%}$ fall in fluid velocity. Understanding these interactions enhances comprehension of heat transfer and fluid dynamics. By investigating these interactions, we not only improve our understanding of heat transfer and fluid dynamics, but also open the door to new strategies for improving thermal systems and manufacturing procedures.

Keywords:

• Natural convection
• two-phase hybrid nanofluid
• vertical wavy surface
• skin friction
• heat transfer
• implicit finite difference method

Nomenclature

English Symbols

A	=	Amplitude
b	=	Influences Thermal Conductivities
C	=	Concentration of the fluid (kgm${}^3$)
$C_f$	=	Skin-friction Coefficient
$C_{\rho }$	=	Heat Capacity at Constant Density
$C_{\mathrm{\infty }}$	=	Concentration at Infinity
$D_B$	=	Diffusion Coefficient (m${}^2$s${}^{-1}$)
$D_T$	=	Thermal Diffusivity (m${}^2$s${}^{-1}$)
f	=	Dimensionless Stream Function
Gr	=	Grashof Number
g	=	Gravitational Acceleration (ms${}^{-2}$)
k	=	Thermal Conductivity (J m${}^{-1}$ s${}^{-1}$ K${}^{-1}$)
L	=	Width and Height of the Cavity m
$L_e$	=	Equivalent Diameter for Heat Transfer
Nu	=	Local Nusselt Number
$N_{\mathrm{BT}}$	=	Ratio of Brownian to Thermophoretic Diffusivity
Pr	=	Prandtl Number
r	=	Local Radius of the Sphere (m)
Sc	=	Schmidt Number
Sh	=	Sherwood Number
$T_w$	=	Temperature at the Surface (K)
$T_{\mathrm{\infty }}$	=	Temperature at Infinity
T	=	Fluid's Temperature in the Boundary Layer (K)
U	=	Dimensional fluid Velocity Component in the X direction
$\mathrm{\nabla }T$	=	Gradient of the Scalar Field (Km${}^{-1}$)
$v_f$	=	Velocity of Fluid (ms${}^{-1}$)
$\overline{u}$,$\overline{v}$	=	Dimensional Components of Velocity in both Vertical and Horizontal Directions (ms${}^{-1}$)
u, v	=	Components of Velocity without Dimensions
$\overline{x},\overline{y}$	=	Cartesian Coordinates with Dimensions (m)
X, Y	=	Dimensionless Coordinates in Cartesian Space

Greek Letters

α	=	Thermal Diffusivity (m${}^2$s${}^{-1}$)
β	=	Thermal Expansion Coefficient (K${}^{-1}$)
ϵ	=	Viscosity Variation Parameter
μ	=	Dynamic Viscosity (kg m${}^{-1}$s${}^{-1}$)
${\nu }_0$	=	Kinematic Viscosity (m${}^2$ s${}^{-1}$)
ρ	=	Fluid Density (kg m${}^{-3}$)
θ	=	Dimensionless Temperature Function
ϕ	=	Volume Fraction
Φ	=	Non-dimensional Concentration of the Fluid

1. Introduction

Metal nanoparticles added to heat transfer fluids can improve heat transmission in various engineering systems. Combining nanoparticles of different materials with the base fluid can lead to further improvement using hybrid nanofluids [1]. The current study focuses on the computational analysis of thermal gravitational convection in a porous chamber with a slanted periodic magnetic field. When mixed with base fluid, hybrid nanofluids exhibit Brownian motion and the thermophoresis effect, which play crucial roles in heat transfer phenomena. Studying natural convection on complex surfaces requires advanced computational models and numerical techniques for engineering systems. Garoosi et al. [2] investigated simulating the natural and mixed convection phenomena of nanofluid in a square cavity, employing the Buongiorno model. The study discovered an optimal nanoparticle volume fraction for maximizing heat transfer at various Rayleigh and Richardson numbers. It also noted that nanoparticle distribution varies with these parameters: non-uniform at low Rayleigh and high Richardson numbers, and more uniform at high Rayleigh and low Richardson numbers for free and mixed convection.

Natural convection is a critical heat transfer phenomenon in fluids, driven by buoyancy forces resulting from temperature gradients, and it plays a crucial role in both natural and industrial processes [3]. Hemmat et al. [4] studied a simulation involving two phases conducted to examine the natural convection properties of nanofluid within a disturbed enclosure containing a porous medium. They found that higher nanoparticle concentration boosts the Nusselt number, especially in the perturbed region. Isotherms become more disturbed with higher Rayleigh numbers. Maximum horizontal velocity increases with higher Rayleigh numbers. Gebhart [5] conducted Numerical modelling in two stages for hybrid nanofluid heat transfer in minichannel heat sinks with experimental confirmation. The study explored using nanofluid-cooled mini-micro channel heat sinks for better heat transfer in electronics. They tested DI water-based Al${}_2$O${}_3$ nanofluid and Al${}_2$O${}_3$–MWCNT hybrid nanofluid, finding that the hybrid improved heat transfer by $15.6\mathrm{\%}$ with minimal pressure drop increase. Islam et al. [6] investigated the implicit finite difference method's analysis of the MHD ferrofluid's heat transfer characteristics at temperature-dependent viscosity along a vertical thin cylinder. They discovered that local skin friction reduces as the volume fraction of magnetic nanoparticles increases because of a decrease in surface shear stress and an increase in viscosity, which reduces particle mobility. Molla et al. [7] delved into a vertically wavy surface's heat transfer characteristics and fluid flow behaviour. Their study contributed valuable information to understanding the impact of surface geometry on natural convection. However, potential limitations involve the assumption of steady-state conditions, neglecting transient effects that could be present in specific applications. Rahman et al. [8] conducted a numerical study on natural convection over a vertical wavy surface. Their work aimed to enhance knowledge of heat transfer techniques in engineering. While their findings provided valuable insights, potential limitations include the reliance on numerical methods that may have associated discretization errors.

Hybrid nanofluids, composed of a base fluid and various nanoparticles, exhibit enhanced thermal and transport properties, offering potential applications across biomedical engineering, fluid mechanics, electronics cooling, and heat transfer [9]. A two-phase hybrid nanofluid, combining metallic and non-metallic nanoparticles to improve thermophysical properties and stability, with potential applications in electronics cooling, nuclear systems, and solar thermal processes by Ahmad and Ibrahim [10, 11]. Huminic et al. [12] provides a comprehensive review of the research conducted on entropy generation in nanofluid and hybrid nanofluid flows in thermal systems. They discovered entropy formation in nanofluids and hybrid nanofluids, showing their potential as alternative thermal systems. Khan et al. [13] have studied to develop a model based on previous ones to predict the thermal conductivity of hybrid nanofluids accurately. Researchers found that modified conventional models and hybrid nanofluids outperform single-particle nanofluids in thermal conductivity data analysis, with less than $3\mathrm{\%}$ prediction errors. Sarkar et al. [14] found that hybrid nanofluids, a blend of base fluid and nanoparticles, enhance thermal and transport properties, offering potential applications in heat transfer, fluid mechanics, electronics cooling, energy production, and biomedical engineering. Two-phase hybrid nanofluids that enhance thermal conductivity and heat transfer studied by Alsabery et al. [15]. They found that regulating heat transport and the concentration distribution of composite nanoparticles requires careful positioning of a solid block and surface undulation. Selimefendigil [16] investigated the performance enhancement of a thermo-electric module in energy systems through the study of surface corrugation and hybrid nanofluids in channel flow. The study found that combining surface corrugation with hybrid nanofluid use considerably improves the power output of a thermoelectric generator module in energy systems, with crucial aspects including Reynolds number, corrugation height, and solid volume percentage of nanoparticles.

Complex wavy surfaces, characterized by undulating peaks, generate low-pressure areas, increase heat transfer, and enhance fluid mixing [17]. Yao et al. [18] demonstrated the significant improvement in heat transfer rates achieved by these intricate surfaces in various engineering applications, such as solar collectors and heat exchangers. Cho et al. [19] delved into natural convection on complex vertical surfaces. It investigated how various characteristics, including the Debye-Huckel parameter and the flow behaviour index, affected the flow behaviour in intricately wavy, rough microchannels. Siddiqa et al. [20] investigated radiative heat transfer in non-Newtonian dusty Casson fluid flow along a complex wavy surface. They found that adding radiation and mass concentration factors to the flow of Casson particle suspension across a complex wavy surface significantly increased the heat transfer rate. Natural convection heat transfer in a cavity with complex-wavy surfaces using non-Newtonian Power-Law fluids studied by Cho et al. [21]. They discovered in non-Newtonian fluid convection, decreasing the flow behaviour index increases the mean Nusselt number; optimization is achieved by modifying the wavy surface. Yao et al. [22] studied natural convection on a vertically wavy surface. The study reveals a transformation approach for analyzing natural convection along irregular vertical surfaces, resulting in a twofold local heat transfer rate frequency and a decrease in the oscillating local Nusselt number amplitude.

Brownian motion, characterized by the random and erratic movement of particles in fluids, holds significant importance in understanding various scientific disciplines such as physics, chemistry, and biology and is often influenced by natural phenomena like pollen grain movement [23]. Klimontovich et al. [24] analyzed Brownian motion theory, focusing on nonlinear Langevin and Fokker-Planck equations. They discovered that their analysis of nonlinear Brownian motion theory included many topics, including friction, Langevin equations, and specific instances such as self-oscillating and chemically reacting systems. Kac et al. [25] investigated the Monte Carlo method, utilizing the number-diffusion process to compute potentials through Brownian motion. They explored probabilistic solutions for Poisson's equation and compared the efficiencies to matrix manipulation. Bevensee et al. [26] studied the application of probabilistic potential theory to electrical engineering challenges was examined. Their findings suggested that probabilistic computations at sparse space-time points on a grid of many points in two or three spatial dimensions may require less time and storage than matrix solutions.

Thermophoresis is a technology that induces directed particle motion in micro and nanomachines [27]. Haddad et al. [28] conducted Brownian and thermophoresis effects increase nanoparticle heat transfer. They discovered that CuO-Water nanofluids have improved heat transmission due to Brownian motion and thermophoresis effects, particularly at low nanoparticle volume fractions. Thermophoresis, observed since 1870, is a particle's motion and force due to temperature differences between hot and cold surfaces [29]. Following Saghir [30], Abdulmajeed et al. [31] examined the Brownian motion and thermophoretic effects in heated channel settings with nanofluids, including Al${}_2$O${}_3$ nanoparticles. It found that increased nanoparticle diameter improves heat extraction efficiency. Thermophoresis phenomena within boundary layer fluid dynamics conducted by Chomiak et al. [32]. The study found that in boundary layer flows, aerosol particle rotation affects their behaviour in thermal gradients. Specifically, fast-rotating particles resist deposition by thermophoresis due to uniform surface temperatures.

Through finite difference simulations, Pathak et al. [33] have delved into the behaviour of two-phase hybrid nanofluids. Their investigations not only shed light on the potential of these nanofluids for improving thermal management but also offer insights into enhancing energy efficiency across various engineering applications. Heris et al. [34] indicated that incorporating nanoparticles into base fluids positively enhances the heat transfer rate. They observed a potential limitation in the performance degradation associated with large nanoparticle volumes. Kashyap et al. [35] investigated how boundary conditions affect heat transmission and entropy creation in a two-phase mixed convection nanofluid flow. The study discovered that altering boundary conditions and Richardson numbers affect heat transfer and entropy formation in nanofluid flow. Selimefendigil [36] studied nanofluid flow over a corrugated bottom wall with obstacles. This study discovered that, among all obstacles, the diamond-shaped obstacle produced the most substantial local heat transfer improvement at the step when compared to circular or square obstacles at high Reynolds numbers.

In this research, we thoroughly investigate natural convection dynamics inside a two-phase hybrid nanofluid framework flowing across a vertically heated wavy surface. Our research focuses on a unique hybrid nanofluid that contains Al${}_2$O${}_3$ and Cu nanoparticles suspended in water, which differs from traditional single-component nanofluids. The synergistic effects of the combination of aluminum oxide (Al${}_2$O${}_3$) and copper (Cu) nanoparticles provide unique chances for improving the nanofluid's thermophysical properties, potentially leading to extraordinary heat and mass transport characteristics. This unique composition distinguishes our research from earlier investigations, which frequently focus primarily on particular nanoparticle kinds or distinct combinations. By investigating the behaviour of this specific two-phase hybrid nanofluid, we provide a new viewpoint on the complicated interaction of thermosolutal convection events across wavy surfaces under the influence of natural convection. However, while simulation approaches bring valuable insights, they have inherent limits. Simplifications in shape and organization may only partially reflect the intricacies of real-world applications, and assumptions about the nanofluid's constant physical characteristics may fail to depict temperature-dependent fluctuations or nanoparticle concentration effects appropriately. To assure the reliability of our findings, we will undertake thorough validation against Yao [22], confirming the accuracy and resilience of our simulation platform. This multidisciplinary study will significantly add to our knowledge of heat transfer dynamics in complicated nanofluid systems, with implications for various engineering applications.

2. Theoretical formulation

2.1. Problem statement

The boundary layer analysis presented below accommodates arbitrary $\overline{\sigma }(\overline{x})$; however, in our comprehensive numerical analysis, we will suppose the surface to have sinusoidal deformations. The following description applies to a portion of the wavy surface [3, 7]: (1) $\begin{array}{r}\overline{y}=\overline{\sigma }\left(\overline{x}\right)=\frac{\alpha }{L}\sin \left(\frac{2\pi \overline{x}}{L}\right)\end{array}$(1) Where L denotes the characteristic length of the wavy surface and $A=\frac{\alpha }{L}$ represents the amplitude of the wavelength, Figure 1 illustrates the geometric layout of the wavy surface within a two-dimensional Cartesian coordinate system. The thermal and physical properties of the base fluid (H₂O), Cu and Al₂O₃ are given in Table 1.

Figure 1. Schematic diagram and coordinate system.

Table 1. The thermal and physical properties of the base fluid (H${}_2$O), Cu and Al${}_2$O${}_3$ [37–40].

Physical properties	Base-fluid (H${}_2$O)	Cu	Al${}_2$O${}_3$
$c_p$ [J kg${}^{-1}$ K${}^{-1}$]	4179	385	765
ρ [kg m${}^{-3}$]	997.1	8933	3970
k [W m${}^{-1}$ k${}^{-1}$]	0.613	401	40
β [K${}^{-1}$]	$21\times {10}^{-5}$	$1.67\times {10}^{-5}$	$0.85\times {10}^{-5}$

The equation below provides the effective density of the hybrid nanofluid [41]: (2) $\begin{array}{rl}{\rho }_{\mathrm{hnf}}& =\varphi +(1-\varphi ){\rho }_f\end{array}$(2) (3) $\begin{array}{rl}{\rho }_{\mathrm{hnf}}& ={\varphi }_{{\mathrm{Al}}_2{O}_3}{\rho }_{{\mathrm{Al}}_2{O}_3}+{\varphi }_{\mathrm{Cu}}{\rho }_{\mathrm{Cu}}+(1-\varphi ){\rho }_f\end{array}$(3) The heat capacitance coefficient of nanofluid is derived using the equation below: (4) $\begin{array}{rl}(\rho C_p{)}_{\mathrm{hnf}}& =(\rho C_p{)}_f(1-\varphi )+\varphi (\rho C_p{)}_s\end{array}$(4) (5) $\begin{array}{rl}(\rho C_p{)}_{\mathrm{hnf}}& ={\varphi }_{{\mathrm{Al}}_2{O}_3}(\rho C_p{)}_{{\mathrm{Al}}_2{O}_3}+{\varphi }_{\mathrm{Cu}}(\rho C_p{)}_{\mathrm{Cu}}\\ & +(1-\varphi )(\rho C_p{)}_f\end{array}$(5) In this case, the hybrid nanofluid's heat capacity is $(\rho C_p{)}_{\mathrm{hnf}}$, while the nanofluid's heat capacity is $(\rho C_p{)}_{\mathrm{hnf}}$. The volume percentage of Al${}_2$O${}_3$ nanoparticles is shown by ${\varphi }_{{\mathrm{Al}}_2{O}_3}$, while that of cu nanoparticles is indicated by ${\varphi }_{\mathrm{cu}}$. (6) $\begin{array}{r}\varphi ={\varphi }_{{\mathrm{Al}}_2{O}_3}+{\varphi }_{\mathrm{cu}}\end{array}$(6) Here, the variable ϕ represents the volume fraction of the hybrid nanoparticles. The Brinkman model [42] formulates the dynamic viscosity for hybrid nanofluids with a Non-Newtonian power-law behaviour as follows: (7) $\begin{array}{r}{\overline{\mu }}_{\mathrm{hnf}}=\frac{{\mu }_f}{(1-\varphi {)}^{2.5}}\end{array}$(7) The parameter ${\mu }_f$ in the power-law model represents its consistency.

Following Maxwell [43], the thermal conductivity $(k_{\mathrm{hnf}})$ for hybrid nanofluid are proposed by Das [44].

The thermal conductivity for hybrid nanofluid is expressed: (8) $\begin{array}{rl}\frac{k_{\mathrm{hnf}}}{k_{\mathrm{bf}}}& =\frac{k_{s_1}+2k_{\mathrm{bf}}-2{\varphi }_1(k_{\mathrm{bf}}-k_{s_1})}{k_{s_1}+2k_{\mathrm{bf}}+{\varphi }_1(k_{\mathrm{bf}}-k_{s_1})}\end{array}$(8) (9) $\begin{array}{rl}\frac{k_{\mathrm{bf}}}{k_f}& =\frac{k_{s_2}+2k_f-2{\varphi }_2(k_f-k_{s_2})}{k_{s_2}+2k_f+{\varphi }_2(k_f-k_{s_2})}\end{array}$(9) where ${\varphi }_1$ and ${\varphi }_2$ represent the volume proportion of Al${}_2$O${}_3$ and Cu, respectively. The solid nanoparticles of copper are represented by $s_2$, and the solid nanoparticles of alumina by $s_1$. The symbols hnf and f stand for hybrid nanofluid and fluid, respectively.

2.2. Governing dimensional equations and boundary conditions

Under Boussinesq's approximation, the dimensional governing equations for the boundary layer issue are the continuity (10(10) $\begin{array}{r}\frac{\mathrm{\partial }\overline{u}}{\mathrm{\partial }\overline{x}}+\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{y}}=0\end{array}$(10) ), momentum (11(11) $\begin{array}{rl}& \overline{u}\frac{\mathrm{\partial }\overline{u}}{\mathrm{\partial }\overline{x}}+\overline{v}\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{y}}=-\frac{1}{{\rho }_{\mathrm{hnf}}}\frac{\mathrm{\partial }\overline{p}}{\mathrm{\partial }\overline{x}}\\ & +\frac{{\mu }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}(\frac{{\mathrm{\partial }}^2\overline{u}}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\overline{y}}^2})\\ & +\frac{g(\mathrm{\rho \beta }{)}_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}(T-T_{\mathrm{\infty }})\end{array}$(11) )–(12(12) $\begin{array}{r}\overline{u}\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{x}}+\overline{v}\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{y}}=-\frac{1}{{\rho }_{\mathrm{hnf}}}\frac{\mathrm{\partial }\overline{p}}{\mathrm{\partial }\overline{y}}+\frac{{\mu }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}(\frac{{\mathrm{\partial }}^2\overline{v}}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\overline{y}}^2})\end{array}$(12) ), energy (13(13) $\begin{array}{rl}& (\overline{u}\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{x}}+\overline{v}\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{y}})={\alpha }_{\mathrm{hnf}}(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{\overline{y}}^2})\\ & +\frac{(\rho C_p{)}_s}{(\rho C_p{)}_{\mathrm{nf}}}[D_B(\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{x}}\frac{\mathrm{\partial }C}{\mathrm{\partial }\overline{x}}+\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{y}}\frac{\mathrm{\partial }C}{\mathrm{\partial }\overline{y}})\\ & +\frac{D_T}{T_{\mathrm{\infty }}}\{{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{x}}\right)}^2+{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{y}}\right)}^2\}]\end{array}$(13) ), and the nanoparticle volume fraction Equation (14(14) $\begin{array}{r}\overline{u}\frac{\mathrm{\partial }\overline{C}}{\mathrm{\partial }\overline{x}}+\overline{v}\frac{\mathrm{\partial }\overline{C}}{\mathrm{\partial }\overline{y}}=D_B(\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{\overline{y}}^2})+\frac{D_T}{T_{\mathrm{\infty }}}(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{\overline{y}}^2})\end{array}$(14) ) which are as follows Buongiorno model for the two-phase nanofluid fluid [45]:

Continuity equation: (10) $\begin{array}{r}\frac{\mathrm{\partial }\overline{u}}{\mathrm{\partial }\overline{x}}+\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{y}}=0\end{array}$(10) u momentum: (11) $\begin{array}{rl}& \overline{u}\frac{\mathrm{\partial }\overline{u}}{\mathrm{\partial }\overline{x}}+\overline{v}\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{y}}=-\frac{1}{{\rho }_{\mathrm{hnf}}}\frac{\mathrm{\partial }\overline{p}}{\mathrm{\partial }\overline{x}}\\ & +\frac{{\mu }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}(\frac{{\mathrm{\partial }}^2\overline{u}}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{\overline{y}}^2})\\ & +\frac{g(\mathrm{\rho \beta }{)}_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}(T-T_{\mathrm{\infty }})\end{array}$(11) v momentum: (12) $\begin{array}{r}\overline{u}\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{x}}+\overline{v}\frac{\mathrm{\partial }\overline{v}}{\mathrm{\partial }\overline{y}}=-\frac{1}{{\rho }_{\mathrm{hnf}}}\frac{\mathrm{\partial }\overline{p}}{\mathrm{\partial }\overline{y}}+\frac{{\mu }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}(\frac{{\mathrm{\partial }}^2\overline{v}}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{\overline{y}}^2})\end{array}$(12) Energy equation: (13) $\begin{array}{rl}& (\overline{u}\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{x}}+\overline{v}\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{y}})={\alpha }_{\mathrm{hnf}}(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{\overline{y}}^2})\\ & +\frac{(\rho C_p{)}_s}{(\rho C_p{)}_{\mathrm{nf}}}[D_B(\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{x}}\frac{\mathrm{\partial }C}{\mathrm{\partial }\overline{x}}+\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{y}}\frac{\mathrm{\partial }C}{\mathrm{\partial }\overline{y}})\\ & +\frac{D_T}{T_{\mathrm{\infty }}}\{{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{x}}\right)}^2+{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }\overline{y}}\right)}^2\}]\end{array}$(13) The nanoparticle volume fraction equation: (14) $\begin{array}{r}\overline{u}\frac{\mathrm{\partial }\overline{C}}{\mathrm{\partial }\overline{x}}+\overline{v}\frac{\mathrm{\partial }\overline{C}}{\mathrm{\partial }\overline{y}}=D_B(\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{\overline{y}}^2})+\frac{D_T}{T_{\mathrm{\infty }}}(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{\overline{x}}^2}+\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{\overline{y}}^2})\end{array}$(14) Where u and v represent the velocity components in the x and r directions, respectively, and T denotes the temperature of the ferrofluid, $T_{\mathrm{\infty }}$ is the ambient temperature, and g indicates the acceleration due to gravity.

The appropriate boundary conditions to solve the governing equations are: (15a) $\begin{array}{rl}& \overline{u}=\overline{v}=0,T=T_w{D}_B\mathbf{n}\cdot \mathrm{\nabla }\overline{C}=-D_T\mathbf{n}\cdot \mathrm{\nabla }T\\ & \mathrm{at}y=0\end{array}$(15a) (15b) $\begin{array}{rl}& \overline{u}\to 0,\overline{v}\to 0,T\to T_{\mathrm{\infty }}\overline{C}\to C_{\mathrm{\infty }}\mathrm{as}y\to \mathrm{\infty }\end{array}$(15b) The following non-dimensional variables are utilized in order to non-dimensionalize the equations above. (16) $\begin{array}{rl}x& =\frac{\overline{x}}{L},y=\frac{\overline{y}-\sigma (\overline{x})}{L}G{r}^{1/4},u=\frac{{\rho }_{f}L}{{\mu }_f}G{r}^{-1/2}\overline{u},\\ p& =\frac{L^2}{{\rho }_{\mathrm{nf}}{{\nu }_f}^2}G{r}^{-1}\overline{p}\\ v& =\frac{{\rho }_{f}L}{{\mu }_f}G{r}^{-1/4}(\overline{v}-{\sigma }_x\overline{u}),\theta =\frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}},\\ \mathrm{Gr}& =\frac{g{\beta }_f(T_w-T_{\mathrm{\infty }})L^3}{{\nu }_f^2},\mathrm{Pr}=\frac{{\nu }_f}{{\alpha }_f},\mathrm{Sc}=\frac{{\nu }_f}{D_B}\\ {\sigma }_x& =\frac{d\hat{\sigma }}{d\overline{x}},\theta =\frac{T-T_{\mathrm{\infty }}}{T_{\omega }-T_{\mathrm{\infty }}},\mathrm{\Phi }=\frac{\overline{C}}{C_b},\mathrm{Le}=\frac{{\alpha }_f}{D_B{C}_b},\\ N_{\mathrm{BT}}& =\frac{D_B{C}_b{T}_{\mathrm{\infty }}}{D_T\mathrm{\Delta }T}\end{array}$(16) The following are the non-dimensional governing equations: (17) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0\end{array}$(17) (18) $\begin{array}{rl}& u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=(1+{\sigma }_x^2)\frac{{\mu }_{\mathrm{hnf}}{\rho }_f}{{\mu }_f{\rho }_{\mathrm{hnf}}}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\sigma }_x{\sigma }_{\mathrm{xx}}}{1+{\sigma }_x^2}{u}^2\\ & +\frac{(\mathrm{\rho \beta }{)}_{\mathrm{hnf}}}{{\rho }_{\mathrm{nf}}{\beta }_f}\frac{1}{1+{\sigma }_x^2}\theta \end{array}$(18) (19) $\begin{array}{rl}& u\frac{\mathrm{\partial }\theta }{\mathrm{\partial }x}+v\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}=\frac{1}{\mathrm{Pr}}(1+{\sigma }_x^2)\\ & \times [\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}+\frac{(\rho C_p{)}_f}{(\rho C_p{)}_{\mathrm{hnf}}}\\ & \times (\frac{1}{\mathrm{Le}}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}\frac{\mathrm{\partial \Phi }}{\mathrm{\partial }y}+\frac{1}{\mathrm{Le}}\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2})]\end{array}$(19) (20) $\begin{array}{rl}& u\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }y}=\frac{1}{\mathrm{Sc}}(1+{\sigma }_x^2)[\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{y}^2}+\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}]\end{array}$(20) For the numerical solution, parabolic transformations are introduced into the above-mentioned non-dimensional governing equations as follows: (21) $\begin{array}{rl}& x=X,y=\frac{Y}{(4X{)}^{1/4}},U=\frac{u}{(4X{)}^{1/2}},V=(4X{)}^{1/4}v,\\ & \theta =\theta ,\mathrm{\Phi }=\mathrm{\Phi }\end{array}$(21) Using the above transformations (21(21) $\begin{array}{rl}& x=X,y=\frac{Y}{(4X{)}^{1/4}},U=\frac{u}{(4X{)}^{1/2}},V=(4X{)}^{1/4}v,\\ & \theta =\theta ,\mathrm{\Phi }=\mathrm{\Phi }\end{array}$(21) ) in the Equations (18(18) $\begin{array}{rl}& u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=(1+{\sigma }_x^2)\frac{{\mu }_{\mathrm{hnf}}{\rho }_f}{{\mu }_f{\rho }_{\mathrm{hnf}}}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\sigma }_x{\sigma }_{\mathrm{xx}}}{1+{\sigma }_x^2}{u}^2\\ & +\frac{(\mathrm{\rho \beta }{)}_{\mathrm{hnf}}}{{\rho }_{\mathrm{nf}}{\beta }_f}\frac{1}{1+{\sigma }_x^2}\theta \end{array}$(18) )–(20(20) $\begin{array}{rl}& u\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }y}=\frac{1}{\mathrm{Sc}}(1+{\sigma }_x^2)[\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{y}^2}+\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}]\end{array}$(20) ), the parabolic non-similar boundary layer governing equations become (22) $\begin{array}{rl}& 4X\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+2U-Y\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}+\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}=0\end{array}$(22) (23) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}+(2+\frac{4X{\sigma }_x{\sigma }_{\mathrm{xx}}}{1+{\sigma }_x^2})U^2\\ & =(1+{\sigma }_x^2)\frac{{\mu }_{\mathrm{hnf}}{\rho }_f}{{\mu }_f{\rho }_{\mathrm{hnf}}}\frac{{\mathrm{\partial }}^{2}U}{\mathrm{\partial }{y}^2}+\frac{(\mathrm{\rho \beta }{)}_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}{\beta }_f}\frac{1}{1+{\sigma }_x^2}\theta \end{array}$(23) (24) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }\theta }{\mathrm{\partial }Y}\\ & =\frac{1}{\mathrm{Pr}}(1+{\sigma }_x^2)[\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}+\frac{(\rho C_p{)}_f}{(\rho C_p{)}_{\mathrm{hnf}}}\\ & +(\frac{1}{\mathrm{Le}}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}\frac{\mathrm{\partial \Phi }}{\mathrm{\partial }y}+\frac{1}{\mathrm{Le}}\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2})]\end{array}$(24) (25) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }Y}\\ & =\frac{1}{\mathrm{Sc}}(1+{\sigma }_x^2)[\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{y}^2}+\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}]\end{array}$(25) The non-dimensional boundary conditions are (26a) $\begin{array}{rl}& U=V=0,\theta =1\frac{\mathrm{\partial \Phi }}{\mathrm{\partial }Y}=-\frac{1}{N_{\mathrm{BT}}}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }Y}\mathrm{at}Y=0\end{array}$(26a) (26b) $\begin{array}{rl}& U\to 0,U\to 0,\theta \to 0,\mathrm{\Phi }\to 0\mathrm{as}Y\to \mathrm{\infty }\end{array}$(26b)

3. Physical quantity

The local shearing stress in terms of the local skin friction coefficient $C_f$ and the local heat transfer rate Nu are two examples of physical variables that may be computed from the numerical solutions of the equations above. Their physical relevance depends on these quantities, which may be described using the following dimensionless relations [46]: (27) $\begin{array}{rl}{C}_f{\left(\frac{\mathrm{Gr}}{4x}\right)}^{1/4}& =\sqrt{1+{\sigma }_x^2}(U^{\mathrm{\prime }}{)}_{Y=0}\end{array}$(27) (28) $\begin{array}{rl}\mathrm{Nu}{\left(\frac{\mathrm{Gr}}{4x}\right)}^{-1/4}& =-\frac{k_{\mathrm{hnf}}}{k_f}\sqrt{1+{\sigma }_x^2}({\theta }^{\mathrm{\prime }}{)}_{Y=0}\end{array}$(28) The local rate of mass transfer in therms of the local Sherwood number is defined as follows: (29) $\begin{array}{r}\mathrm{Sh}{\left(\frac{\mathrm{Gr}}{4x}\right)}^{-1/4}=-\sqrt{1+{\sigma }_x^2}({\mathrm{\Phi }}^{\mathrm{\prime }}{)}_{Y=0}\end{array}$(29)

4. Numerical methods

The finite difference approach discretize Equations (23(23) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}+(2+\frac{4X{\sigma }_x{\sigma }_{\mathrm{xx}}}{1+{\sigma }_x^2})U^2\\ & =(1+{\sigma }_x^2)\frac{{\mu }_{\mathrm{hnf}}{\rho }_f}{{\mu }_f{\rho }_{\mathrm{hnf}}}\frac{{\mathrm{\partial }}^{2}U}{\mathrm{\partial }{y}^2}+\frac{(\mathrm{\rho \beta }{)}_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}{\beta }_f}\frac{1}{1+{\sigma }_x^2}\theta \end{array}$(23) )–(25(25) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }Y}\\ & =\frac{1}{\mathrm{Sc}}(1+{\sigma }_x^2)[\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{y}^2}+\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}]\end{array}$(25) ), where the central difference scheme represents the diffusion and convection terms. For numerical stability, the backward difference formula uses the x-derivative term. After discretising the governing equations, a set of algebraic system of equations is produced as follows [3]: (30) $\begin{array}{r}{A}_{i,j}{f}_{i,j-1}+B_{i,j}{f}_{i,j}+C_{i,j+1}{f}_{i,j+1}=E_{i,j}\end{array}$(30) The matrix coefficients for the momentum Equation (23(23) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}+(2+\frac{4X{\sigma }_x{\sigma }_{\mathrm{xx}}}{1+{\sigma }_x^2})U^2\\ & =(1+{\sigma }_x^2)\frac{{\mu }_{\mathrm{hnf}}{\rho }_f}{{\mu }_f{\rho }_{\mathrm{hnf}}}\frac{{\mathrm{\partial }}^{2}U}{\mathrm{\partial }{y}^2}+\frac{(\mathrm{\rho \beta }{)}_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}{\beta }_f}\frac{1}{1+{\sigma }_x^2}\theta \end{array}$(23) ) are as follows: (31) $\begin{array}{rl}{A}_{i,j}& =P_1\frac{\mathrm{\Delta }y}{2}+P_3,\end{array}$(31) (32) $\begin{array}{rl}{B}_{i,j}& =-P_2{U}_{i,j}\mathrm{\Delta }{y}^2-2P_3-4X{U}_{i,j}\frac{\mathrm{\Delta }{y}^2}{\mathrm{\Delta }x},\end{array}$(32) (33) $\begin{array}{rl}{C}_{i,j}& =-P_1\frac{\mathrm{\Delta }y}{2}+P_3,\end{array}$(33) (34) $\begin{array}{rl}{E}_{i,j}& =-P_4{\theta }_{i,j}\mathrm{\Delta }{y}^2-4X{U}_{i,j}{U}_{i-1,j}\frac{\mathrm{\Delta }{y}^2}{\mathrm{\Delta }x},\end{array}$(34) where, (35) $\begin{array}{rl}{P}_1& =V_{i,j}-\frac{U_{i,j}Y}{4},P_2=(\frac{1}{2}+\frac{X{\sigma }_x{\sigma }_{\mathrm{xx}}}{1+{\sigma }_x^2}),\\ P_3& =(1+{\sigma }_x^2)\frac{{\mu }_{\mathrm{nf}}{\rho }_f}{{\mu }_f{\rho }_{\mathrm{nf}}},P_4=\frac{(\mathrm{\rho \beta }{)}_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}{\beta }_f}\frac{1}{1+{\sigma }_x^2}\end{array}$(35) The matrix coefficients for the energy Equation (24(24) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }\theta }{\mathrm{\partial }Y}\\ & =\frac{1}{\mathrm{Pr}}(1+{\sigma }_x^2)[\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}+\frac{(\rho C_p{)}_f}{(\rho C_p{)}_{\mathrm{hnf}}}\\ & +(\frac{1}{\mathrm{Le}}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}\frac{\mathrm{\partial \Phi }}{\mathrm{\partial }y}+\frac{1}{\mathrm{Le}}\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2})]\end{array}$(24) ) are as follows: (36) $\begin{array}{rl}{A}_{i,j}& =P_1\frac{\mathrm{\Delta }y}{2}+P_2,\end{array}$(36) (37) $\begin{array}{rl}{B}_{i,j}& =-2P_2-4X{U}_{i,j}\frac{\mathrm{\Delta }{y}^2}{\mathrm{\Delta }x},\end{array}$(37) (38) $\begin{array}{rl}{C}_{i,j}& =-P_1\frac{\mathrm{\Delta }y}{2}+P_2\end{array}$(38) (39) $\begin{array}{rl}{E}_{i,j}& =-4X{U}_{i,j}{\theta }_{i-1,j}\frac{\mathrm{\Delta }{y}^2}{\mathrm{\Delta }x}\\ & +P_2{T}_1(\frac{1}{\mathrm{Le}}\frac{\mathrm{\partial }{\theta }_{\mathrm{ij}}}{\mathrm{\partial }y}\frac{\mathrm{\partial }{\varphi }_{\mathrm{ij}}}{\mathrm{\partial }y}+\frac{1}{\mathrm{Le}}\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2{\theta }_{\mathrm{ij}}}{\mathrm{\partial }{y}^2})\mathrm{\Delta }{y}^2\end{array}$(39) where, (40) $\begin{array}{r}{P}_1=V_{i,j}-U_{i,j}Y,P_2=\frac{1}{\mathrm{Pr}(1+{\sigma }_x^2)}\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f},T_1=\frac{(\rho C_p{)}_f}{(\rho C_p{)}_{\mathrm{hnf}}}\end{array}$(40) As with the energy Equation (24(24) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }\theta }{\mathrm{\partial }Y}\\ & =\frac{1}{\mathrm{Pr}}(1+{\sigma }_x^2)[\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}+\frac{(\rho C_p{)}_f}{(\rho C_p{)}_{\mathrm{hnf}}}\\ & +(\frac{1}{\mathrm{Le}}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}\frac{\mathrm{\partial \Phi }}{\mathrm{\partial }y}+\frac{1}{\mathrm{Le}}\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2})]\end{array}$(24) ), the nanoparticles volume fraction Equation (25(25) $\begin{array}{rl}& 4\mathrm{XU}\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }X}+(V-\mathrm{UY})\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }Y}\\ & =\frac{1}{\mathrm{Sc}}(1+{\sigma }_x^2)[\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{y}^2}+\frac{1}{N_{\mathrm{BT}}}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}]\end{array}$(25) ) is discretized and forms a set of algebraic equations.

Here, $f_{i,j}$ is the generic variable for U-velocity, temperature θ, and nanoparticles volume fraction Φ. The Thomson algorithm is used to solve the above algebraic equations. To get the v-velocity component the continuity Equation (22(22) $\begin{array}{rl}& 4X\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+2U-Y\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}+\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}=0\end{array}$(22) ) is discretised as follows: (41) $\begin{array}{rl}{V}_{I,J}& =V_{i,j-1}+\frac{1}{4}(Y_j+Y_{j-1})(U_{i,j}-U_{i,j-1})\\ & -(U_{i,j}+U_{i,j-1})\mathrm{\Delta }y-4X(U_{i,j}-U_{i-1})\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\end{array}$(41) Iteration begins at X = 0.0 and continues to the downstream region. $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ are set to 0.005 and 0.01, respectively. The tolerance for the iteration procedure is ${10}^{-6}$. The simulation uses an in-house FORTRAN 90 code based on the discretization above, as used in [47].

5. Code validation

Code validation is an essential part of any numerical simulation. A comparison of the present numerical results with the results of Yao [22] in terms of the local Nusselt number along a vertical flat and wavy surface is depicted in Figure 2 while the surface waviness ($\alpha =0.1,0.3$) and Pr = 1. This comparison demonstrates a good agreement, implying that this code is appropriate for non-similar boundary layer flow and heat transfer along a wavy surface.

Figure 2. Comparison between present numerical results in terms of the local Nusselt number (Nu) of Yao [22] while Pr = 1 and the surface waviness $A=0.1,0.3$.

6. Results and discussion

This investigation employs implicit finite difference simulation to analyze natural convection in a two-phase hybrid nanofluid along a vertically heated wavy surface. Aluminum oxide (Al${}_2$O${}_3$) and copper (Cu) nanoparticles distributed in water ($H_{2}O$) make up the hybrid nanofluid under investigation. Our computer analysis goes beyond standard bounds, concentrating primarily on natural convection elements and improving our understanding of the system's peculiar heat transport and fluid dynamic properties. Computational findings of streamline, isotherms, local skin friction, local Nusselt number, velocity, and temperature distribution with the influence of different factors such nanoparticle volume fraction, $\varphi (=0,0.03,0.05,0.8,0.1)$, Schmidt number, $\mathrm{Sc}(=1,5,10)$, Amplitude, $A(=0,0.1,0.2,0.3)$, $N_{\mathrm{BT}}(=0.05,0.1,0.2)$ and the Prandtl number at Pr = 6.2 has been presented and analyzed in the following subsections.

6.1. Influence of volume fraction $(\varphi )$

The results present the influence of the volume fraction parameter (ϕ), distinguishing between $\varphi =0.00$ and $\varphi =0.1$. Two primary kinds of nanofluids, namely dense and diluted, distinguish themselves based on volume fraction [48, 49]. The volume fraction is the ratio of nanoparticle volume to the overall volume of the nanofluid [50, 51]. At first, the effect of ϕ on velocity (u), and temperature (θ) profiles will be discussed in Figure 3, and then $C_f$, local Nusselt number will be discussed in Figure 4. After that, the local Sherwood number in Figure 5 and streamline and isotherm in Figure 6 will be analyzed. In all instances, the Sc number has been maintained constant at 10, $N_{\mathrm{BT}}$ is 0.2, and the amplitude is 0.2.

Figure 3. Effect of volume fractions, $\varphi =(0,0.03,0.05,0.08,0.1)$ on (a) velocity profile, and (b) temperature profile at A = 0.2, $S_c=10$, $N_{\mathrm{BT}}$=0.2 and X = 0.5.

Figure 4. (a) Skin friction ($C_f$), (b) Local Nusselt number (Nu) for different volume fractions, $\varphi =(0,0.03,0.05,0.08,0.1)$ while A = 0.2, Sc = 10, and $N_{\mathrm{BT}}=0.2$.

Figure 5. Local Sherwood number (Sh) for different volume fractions, $\varphi =(0,0.03,0.05,0.08,0.1)$ while A = 0.2, Sc = 10, and $N_{\mathrm{BT}}=0.2$.

Figure 6. (a) Streamlines and (b) Isotherms for $\varphi =0.0$ (solid line) and $\varphi =0.1$ (dashed line) while Sc = 10, A = 0.3 and $N_{\mathrm{BT}}=0.1$.

Each phase of the two-phase hybrid nanofluid system independently follows the rules of mass and momentum conservation, and each phase is represented by its volume fraction. The flow velocity vector field facilitates the quantitative depiction of fluid motion. The flow speed serves as a comprehensive measure of the flow velocity. The vector field that describes fluid velocity at a specific time and location is fluid flow velocity. Figure 3(a) vividly illustrates that larger values of $(\varphi$) result in lower fluid velocity. Examining the temperature profiles of a vertically wavy surface for nanoparticles, ranging from $(\varphi =0.00)$ to $\varphi =0.1$ in Figure 3(b), reveals intriguing dynamics. The graphical representation demonstrates that fluids within the boundary layer exhibit accelerated motion at high surface temperatures. In contrast, the fluids in the boundary layer flow at the slowest possible speed at low surface temperatures. This observation implies that the region with the highest temperature experiences increased shear stress and faster heat transfer. Additionally, it has been found that rising values of $(\varphi$ raise the fluid temperature in the boundary layer, which affects the complex interactions between fluid dynamics and heat transport processes. There is a continuous trend in the velocity behaviour as the volume fraction rises from 0.0 to 0.3, the velocity gradually decreases 11.11$\mathrm{\%}$, and the temperature increases 75$\mathrm{\%}$ when ϕ increases from 0 to 0.1 at Y = 2.

Figure 4(a,b) vividly illustrates the consequential effects of skin friction and local Nusselt number under conditions A = 0.2, Sc = 10, and $N_{\mathrm{BT}}=0.2$. In fluid dynamics, the volume fraction, denoted by $(\varphi )$, signifies the percentage of a specific component or phase within a given mixture volume. Notably, as $(\varphi )$ values ascend, the impact on fluid dynamics becomes evident in Figure 4(a), showcasing a reduction in the fluid's surface shear stress and subsequently diminishing skin friction. The interplay between shear stress and volume fraction reveals an intriguing inverse relationship. As the velocity decreases, the increasing viscosity leads to a higher volume fraction, counteracting the buoyant force and influencing skin friction downward due to surface shear stress. This nuanced relationship unveils the intricate dynamics governing fluid behaviour in the presence of nanoparticles. Turning our attention to Figure 4(b), the local Nusselt number exhibits a declining trend as $(\varphi )$ rises. Elevated volume fractions of nanoparticles introduce modifications to fluid flow patterns, inducing higher drag and altering fluid behaviour. Such alterations may reduce heat transfer efficiency, particularly when the flow becomes turbulent or sluggish. The decrease in fluid velocity due to rising fluid viscosity, associated with an increased volume percentage, contributes to a potential decline in the Nusselt number, reflecting reduced convective heat transfer efficiency.

At various volume fractions ($\varphi =0,0.03,0.05,0.08,0.1$), we examined the local Sherwood number (Sh) while keeping the parameters A = 0.2, Sc = 10, and $N_{\mathrm{BT}}=0.2$ constant in Figure 5. The findings shown in Figure 5 show a distinct trend: the Sherwood number falls with increasing volume proportion. Many characteristics of the behaviour of hybrid nanofluids in natural convection give birth to this phenomenon. First, the fluid's mass transport and thermal characteristics are changed when nanoparticles are added. Higher volume fractions cause the nanoparticles to be more noticeable, which improves convective heat transfer near the heated wavy surface. Simultaneously, the increased volume fraction leads to a denser distribution of nanoparticles, which hinders the solute particles' ability to diffuse and, as a result, lowers the Sherwood number. Mass transport continues to be limited at greater volume fractions due to agglomeration and interactions between nanoparticles. Thus, the volume fraction-Sherwood number inverse connection highlights the complex equilibrium between mass transfer inhibition and convective heat transfer increase in hybrid nanofluids. These results emphasize how important it is to optimize volume fractions in practical applications in order to achieve optimum mass transfer and heat transfer performance.

Figure 6, elucidates a comprehensive depiction of the impact of variations in streamline and isotherm for different volume fractions ($\varphi =0.0,0.1$) under conditions $A=0.3$, Sc = 10, and $N_{\mathrm{BT}}=0.1$. Notably, the figure reveals a pronounced waviness and denser clustering of fluid concentration. In Figure 6(a), the fluid concentration is observed to form denser clusters nearer to the boundary layer, indicating a more significant deceleration of fluid velocity. This deceleration results in a narrowing of streamlines closer to the wavy surface, suggesting slower particle motion within the fluid. Consequently, streamlines exhibit a more pronounced clustering, particularly within the boundary layer adjacent to the surface. Examining the thermal effects in Figure 6(b), we observe an intricate interplay between particles and convective heat transport. Higher volume fractions induce intensified fluid-particle interactions, facilitating enhanced mixing of temperature gradients. This interaction leads to modifications in the configuration and dispersion of isotherms. Specifically, as convective heat transmission diminishes, the temperature gradient within the fluid increases, causing isotherms to converge, particularly within the boundary layer. This convergence signifies a more significant temperature differential in proximity to the surface. These findings offer critical insights into heat transmission dynamics, particle dispersion, and fluid motion in two-phase hybrid nanofluid natural convection along a vertical heated wavy surface.

6.2. Effect of Schmidt number (Sc)

The Schmidt number (Sc) is a dimensionless parameter that characterizes how a fluid's momentum and mass diffusivity interact. A high Sc suggests dominating mass transfer, stressing molecular diffusion, whereas a low Sc emphasizes momentum transport. This number governs the equilibrium between momentum and mass transport, significantly impacting fluid flow dynamics.

Figure 7 clarifies the effects of different Schmidt numbers $(\mathrm{Sc}=1,5,10)$ on the concentration profile and temperature distribution at A = 0.2, $N_{\mathrm{BT}}=0.2$, and $\varphi =4\mathrm{\%}$. Interestingly, the symbol Sc represents the reciprocal ratio of mass (species) diffusivity to momentum diffusivity. In Figure 7(a), a distinct pattern emerges, indicating that velocity increases proportionally with an increase in the Schmidt number. This phenomenon is attributed to species' decreasing chemical molecular diffusivity as Sc values rise, leading to a narrower concentration boundary layer than the viscous (momentum) boundary layer. Figure 7(b) shows the thermal distribution response to differences in Sc, revealing that temperature increases as Sc increases. Interestingly, despite these changes in concentration and temperature, the overall distribution hasn't altered significantly. This observation suggests that the thermal profile remains relatively stable, even with variations in Schmidt numbers, emphasizing the complex and nuanced nature of heat transfer phenomena in two-phase hybrid nanofluid, a vertically heated wavy surface using natural convection.

Figure 7. (a) velocity profile, (b) temperature profile for the effect of different Schmidt numbers, $\mathrm{Sc}=(1,5,10)$ while A = 0.2,$N_{\mathrm{BT}}=0.2$, and $\varphi =0.04$.

Figure 8(a,b) shows how different Schmidt numbers $(\mathrm{Sc}=1,5,10)$ affect skin friction, the average Nusselt number while keeping A = 0.2, $N_{\mathrm{BT}}=0.2$, and $\varphi =4\mathrm{\%}$ constant. In contrast to the overall pattern, Figure 8(a) shows that skin friction rises in proportion to the value of Sc. This surprising result defies the widely held belief that an increasing Schmidt number indicates slower mass diffusion relative to momentum. The results point to the necessity of a sophisticated interpretation of the Sc effect on skin friction in this particular fluid dynamics scenario by pointing to a complicated interplay between mass diffusion and momentum transmission. More knowledge is necessary to fully understand the underlying mechanisms causing this relationship to seem counterintuitive. Figure 8(b) shows that increasing Sc results in a noticeable drop in the local Nusselt number. The local Nusselt number characterizes convective heat transmission in a fluid flow. It is affected by fluid velocity and temperature gradients. A higher Schmidt number causes slower mass diffusion, resulting in less effective species. Heat transport is inefficient due to mixing. As a result of quicker mass, local Nusselt numbers decreased. Diffusion reduces the efficiency of heat transport.

Figure 8. Impact of different Schmidt number, $\mathrm{Sc}=(1,5,10)$ on (a) skin friction ($C_f$), (b) Local Nusselt number,(Nu) at A = 0.2, $N_{\mathrm{BT}}=0.2$, $\varphi =0.04$, and X = 0.5.

The impact of the Schmidt number, (Sc), on fluid dynamics and temperature distribution is examined under specific conditions, with A = 0.2, $N_{\mathrm{BT}}=0.2$, and ϕ = $4\mathrm{\%}$. Shifting Schmidt numbers causes changes in streamline patterns on the left side. The Schmidt number, representing the momentum-to-mass transfer ratio, is critical in establishing the fluid's diffusion characteristics, influencing the streamlines' course and behaviour. Changes in the Schmidt number can affect diffusion rates, resulting in changes in velocity profiles and flow patterns. The left portion of Figure 9(a–e) shows that when Sc increases, momentum diffusion's role declines compared to mass diffusion. As a result, concentration gradients take longer to equilibrate than velocity gradients, producing more diffuse streamlines. The increased importance of mass diffusion caused by the shift in Sc changes the fluid behaviour into a diffusion-dominated phase with more scattered and less well-defined flow patterns.The isotherm patterns on the right side, which indicate constant temperature contours, undergo alterations as the Schmidt number changes. The right side of Figure 9(b–f) shows that the isotherm patterns, which indicate constant temperature contours, undergo alterations as the Schmidt number changes. Mass diffusion becomes more significant as Sc increases, causing slower equilibration of temperature gradients and limiting heat transfer effectiveness. Isotherm profiles reflect the increased relevance of mass diffusion with more significant fluctuations and slower corrections. This complex interplay between Schmidt number fluctuations and their effects on streamlines and isotherms provides essential insights into the dynamic behaviour of fluid systems under the given conditions.

Figure 9. Effect of different Schmidt number, $\mathrm{Sc}=(1,5,10)$ on (a), (c), (e) Streamlines and (b), (d), (f) Isotherms at A = 0.2, $N_{\mathrm{BT}}=0.2$ while $\varphi =0.0$ (solid line) and $\varphi =0.1$ (dashed line).

6.2.1. Effect of amplitude (A)

The amplitude of fluid flow influences its dynamics by influencing mixing, flow patterns, and heat and mass transfer efficiency. It fosters intense convective movement and changes convective dynamics and concentration profiles by accelerating fluid mixing.

Figure 10 illustrates the impact of Amplitude, A (with values of 0.0, 0.1, 0.2, 0.3) on the velocity and temperature for Sc = 10, $N_{\mathrm{BT}}$ = 0.2, and $\varphi =4\mathrm{\%}$. Amplitude plays a crucial role in affecting the velocity and temperature of fluid flow. Its influence on temperature dispersion within the fluid movement is evident. Establishing that ϕ's inclusion and rising values of ϕ impede fluid movement. The analysis of velocity (U) response in Figure 10(a) further substantiates this, revealing a direct correlation between increasing A values and rising viscosity coupled with heightened thermal conductivity. The fluid initially shows mobility from a static state, but beyond a certain point, it experiences a sudden decrease due to increased limitations within the boundary layer. The graphical representation in Figure 10(a,b) indicates that higher A values result in decreased velocity and increased temperature within the boundary layer. Additionally, we observe that the peak U value for A = 0 is slightly larger than the peak values for other non-zero A values. The temperature profile representations in Figure 10(b) depict similar changes, showing an ascent in temperature distribution with increasing amplitude values. As mentioned earlier, for larger Y, the peak U value decreases while the particle temperature distribution level rises. The behaviour of two critical variables has a consistent pattern across the Amplitude spectrum, which spans from 0.0 to 0.3. The fluid's velocity decreases by $55.55\mathrm{\%}$, but the temperature rises from 0.075 to 0.55 as the amplitude(A) increases from 0.0 to 0.3 at Y = 1.5.

Figure 10. Effect of different Amplitude, $A=(0.0,0.1,0.2,0.3)$ on (a) velocity profile (b) temperature profile at Sc = 10, $N_{\mathrm{BT}}=0.2$, and $\varphi =0.04$, and X = 0.5.

In Figure 11, the fluctuation of skin friction ($C_f$) and local Nusselt number (Nu) demonstrates that a rise in fluid density inside the boundary layer induces a reduction in skin friction owing to the increase in volume fraction. Because of both the falling velocity associated with the volume fraction and the increasing velocity related to the amplitude, the mean velocity of fluid flow drops. Nevertheless, the mean velocity and velocity gradients at the border often influence skin friction. If the reduction in velocity gradients is more critical than other factors, the overall mean velocity decreases, resulting in less skin friction. However, although an increase in amplitude and volume fraction alone may result in a lower mean velocity, the natural effect on skin friction is contingent upon how these variables interact with velocity gradients at the boundary. The local Nusselt number steadily drops as the volume percentage and amplitude increase. Heat and mass transmission likewise fall with higher amplitude values as the velocity drops. Reduced fluid velocity causes transportation to be compromised, which lowers the Nusselt number – a measure of the proportion of conductive to convective heat transfer. Although an increase in amplitude and volume fraction alone may result in a lower mean velocity, the natural effect on skin friction depends on how these variables interact with velocity gradients at the boundary. Reduced fluid velocity causes transportation to be compromised, which lowers the Nusselt number, a measure of the proportion of conductive to convective heat transfer.

Figure 11. (a) skin friction ($C_f$),(b) Local Nusselt number (Nu) for the effect of different $A=(0.0,0.1,0.2,0.3)$ at Sc = 10, $N_{\mathrm{BT}}=0.2$, and $\varphi =0.04$.

Figure 12 depicts the change of the local Nusselt number for both single and two-phase situations. The amplitude is carefully adapted to 0.1 and 0.3 for the two cases, with Sc = 5, $N_{\mathrm{BT}}=0.1$, and $\varphi =4\mathrm{\%}$. For Amplitude = 0.1, shown by solid and dotted lines for single and two phases, respectively, the local Nusselt number is higher in the single-phase scenario (0.80) than in the two-phase scenario (0.78). This is a $2.5\mathrm{\%}$ drop at X = 0 while switching from single-phase to two-phase circumstances.Similarly, at Amplitude= 0.3, the local Nusselt number for the two-phase scenario (0.58) is significantly lower than for the single-phase scenario (0.60), resulting in a $3.3\mathrm{\%}$ drop at X = 0. The pattern implies that switching from single-phase to two-phase circumstances lowers the local Nusselt number. This drop might be attributable to the additional complexity caused by nanoparticles in the fluid, which can hamper convective heat transmission. Nonetheless, despite a modest drop in the local Nusselt number, using a two-phase hybrid nanofluid provides advantages such as increased thermal conductivity and heat transmission capabilities.

Figure 12. (a) skin friction ($C_f$),(b) Local Nusselt number (Nu) for the effect of different $A=(0.1,0.3)$ at Sc = 5, $N_{\mathrm{BT}}=0.1$, and $\varphi =0.04$.

The impact of amplitude (A) on fluid dynamics and temperature distribution is explored within the defined conditions of Sc = 10, $N_{\mathrm{BT}}$ = 0.2, and ϕ= $4\mathrm{\%}$. The representation of streamlines in the left panel highlights the crucial role that oscillation amplitude plays in shaping flow patterns. The streamlines, representing the paths of fluid particles, exhibit discernible variations as the amplitude (A) increases. These variations are indicative of changes in the velocity field. Greater amplitudes provide more noticeable variations in flow patterns influenced by turbulence and vorticity. The right side of the isotherm illustration shows how amplitude affects temperature distribution. Changes in temperature gradients and the overall thermal profile can result from elevated amplitudes affecting the heat transfer rate. A thorough understanding of how amplitude influences streamlines and isotherms reveals the system's dynamic behaviour under the given conditions. According to Figure 13(a,c,e,g), the fluid displays increasingly noticeable dispersion when the temperature or velocity amplitude varies. More significant temperature swings and better convective heat transmission result from higher amplitudes, which disturb ordered flow patterns through increased fluid motion and mixing. Higher amplitudes lead to more enormous temperature differentials, isotherm distribution variations, and more noticeable and variable temperature gradients as convective heat transfer becomes more and more dominant, as the depiction in Figure 13(b,d,f,h) further highlights.

Figure 13. Effect of Amplitude, $A=(0.0,0.1,0.2,0.3)$ on (a), (c), (e) and (g) Streamlines, (b), (d), (f) and (h) Isotherms at Sc = 10, $N_{\mathrm{BT}}=0.2$, while $\varphi =0.0$ (solid line), $\varphi =0.1$ (dashed line), and X = 0.5.

6.2.2. Effect of $N_{\mathrm{BT}}$

We examine natural convection in a two-phase flow system using finite difference simulation, and our primary focus is on the critical parameter of the $N_B$/$N_T$ ratio, which we refer to as $N_{\mathrm{BT}}$. As a crucial factor in deciphering the intricate dynamics of the system, the $N_{\mathrm{BT}}$ ratio symbolizes the interaction between buoyancy $(N_B)$ and temperature ($N_T$) effects. We obtain critical insights into the relative dominance of buoyancy-induced density fluctuations and thermal impacts by methodically varying the $N_{\mathrm{BT}}$ value. A high $N_{\mathrm{BT}}$ ratio indicates natural solid convection and the predominance of buoyancy forces, whereas a low ratio implies that buoyancy plays a secondary role in other influencing variables.

Figure 14 provides a comprehensive view of the influence of $N_{\mathrm{BT}}$ ($=0.05,0.1,0.2$) on temperature and velocity distributions, with Sc $=10$, A $=0.1$, and $\varphi =4\mathrm{\%}$, particularly emphasizing Figure 14(a,b). The nuanced variations in temperature and velocity distributions within the fluid boundary layer become apparent as $N_{\mathrm{BT}}$ increases. These alterations, induced by rising $N_{\mathrm{BT}}$ values, significantly impact the decreased velocity, temperature, and dispersion of nanoparticles, as discerned from a closer examination of the graphical representation. The data further highlights the inverse relationship between $N_{\mathrm{BT}}$ and temperature/velocity within the system, demonstrating that as $N_{\mathrm{BT}}$ increases, both temperature and velocity experience a notable decline. This analysis advances our knowledge of the relationship between $N_{\mathrm{BT}}$ and velocity/temperature in the context of two-phase hybrid nanofluid natural convection along a vertical heated wavy surface by illuminating the complex interactions between mass diffusion, fluid dynamics, and nanoparticle dispersion. As we delve into the intricate dynamics of temperature and velocity influenced by $N_{\mathrm{BT}}$, these findings contribute to the fundamental understanding of nanofluid behaviour and pave the way for advancements in designing more efficient and tailored thermal systems. With the increase in the $N_{\mathrm{BT}}$ from 0.05 to 0.2 at Y = 1, the temperature shows a significant reduction of $18.75\mathrm{\%}$ and the fluid's velocity gradually decreases by $9.68\mathrm{\%}$.

Figure 14. Impact of $N_{\mathrm{BT}}=(0.05,0.1,0.2)$ on (a) velocity profile, (b) temperature profile at A = 0.1, Sc = 10, $\varphi =0.04$, and X = 0.5.

The investigation of $N_{\mathrm{BT}}=0.05,0.1,$ and 0.2 on skin friction and the local Nusselt number, presented in Figure 15(a,b) for conditions A = 0.1, Sc = 10, and $\varphi =4\mathrm{\%}$, reveals intriguing dynamics. Figure 15(a) demonstrates a decreasing trend in skin friction with increasing $N_{\mathrm{BT}}$ – a consequence of the slower diffusion of mass compared to momentum, leading to a rise in $N_{\mathrm{BT}}$. As $N_{\mathrm{BT}}$ increases, the significance of mass diffusion diminishes relative to momentum transport. Simultaneously, the velocity reduction due to the $N_{\mathrm{BT}}$ effect impacts skin friction downward through surface shear stress. Figure 15(b) further illustrates that elevated $N_{\mathrm{BT}}$ correlates with a reduction in the Nusselt number (Nu), reflecting convective heat transfer. Slower mass diffusion associated with higher $N_{\mathrm{BT}}$ results in less effective heat transfer and reduced species mixing, leading to a decline in Nu. These results contribute to our knowledge of the complex interaction between fluid dynamics, convective heat transfer, and mass diffusion in the setting of two-phase hybrid nanofluid natural convection along a vertical heated wavy surface.

Figure 15. (a) Skin friction ($C_f$), (b) Local Nusselt number (Nu) for the effect of different $N_{\mathrm{BT}}=(0.05,0.1,0.2)$ at A = 0.1, Sc = 10, and $\varphi =0.04$.

In Figure 16, the intricate dynamics of fluid flow and temperature distribution are meticulously examined, with a specific focus on the variable $N_{\mathrm{BT}}$ under defined conditions (A = 0.1, Sc = 10, and $\varphi =4\mathrm{\%}$). The left panel showcases the evolving streamlines, offering insights into the complex alterations in liquid particle paths with changing $N_{\mathrm{BT}}$. Notably, heightened values of $N_{\mathrm{BT}}$ appear to impact flow separation and boundary layer thickness, introducing nuanced variations in various boundary layer dynamics parameters. The isotherm representation in the right panel reveals the subtle impact of $N_{\mathrm{BT}}$ on the temperature distribution. Changes in $N_{\mathrm{BT}}$ are pivotal in shaping temperature gradients and the overall thermal profile, emphasizing the intricate relationship between $N_{\mathrm{BT}}$, streamlines, and isotherms. This study contributes to our comprehension of the complicated relationship between the quantity of boundary and thermal layers, offering critical new perspectives on the intricacies of natural convection of a two-phase hybrid nanofluid along a vertically heated wavy surface.

Figure 16. Effect of different $N_{\mathrm{BT}}=(0.05,0.1,0.2)$ on (a), (c), (e) Streamlines (b), (d), (f) Isotherms at A = 0.1, Sc = 10, $\varphi =0.0$ (solid line) and $\varphi =0.1$ (dashed line).

7. Conclusion

In this comprehensive analysis, we investigated various factors within a two-phase hybrid nanofluid system, including the Schmidt number ($1\le \mathrm{Sc}\le 10$), volume fraction ($0\le \varphi \le 0.1$), Amplitude ($0\le A\le 0.3$), $N_{\mathrm{BT}}$ ($0.05\le N_{\mathrm{BT}}\le 0.2$), and Prandtl number ($\mathrm{Pr}=6.2$). The different volume fractions clarify how the concentration of nanoparticles affects thermophysical characteristics, and the Schmidt numbers clarify the relative significance of mass diffusion and momentum. Furthermore, investigating various amplitudes offers a more profound comprehension of surface waviness's impact on convective heat transmission. The differences in $N_{\mathrm{BT}}$ highlight how effective nanofluids are in increasing convective heat transfer as compared to base fluids, and the Prandtl number clarifies the properties of the thermal boundary layer. The model was verified by comparing the reported findings to well-cited literature. We built the computational code using FORTRAN. The analysis's findings demonstrate the importance of these elements in establishing the distributions of skin friction, local Nusselt, streamline, temperature, velocity, and isotherm. This work provides insights into the complex behaviour of thermosolutal convection along a vertically wavy surface in hybrid nanofluids. The primary conclusions of the study are listed below:

• A higher volume fraction results in a reduction in velocity and an increase in temperature.

• As the volume fraction increases, skin friction lowers, and the local Nusselt number falls.

• As the fluid speed decreases, the streamlines near the wavy surface become smaller, indicating slower movement of particles. When the volume fraction of the fluid increases, the shape of the isotherm alters by improving the interaction between the fluid and particles. The closer isotherms signify a more noticeable temperature difference near the surface due to reduced heat transfer.

• An increase in the Schmidt number, which results in a temperature rise, correlates with an increase in velocity.

• When the Schmidt number rises, we observe a significant increase in skin friction, correlating with a drop in the local Nusselt numbers.

• The impact of the Schmidt number on streamlines and isotherms is critical. Greater Schmidt numbers encourage mass diffusion, changing isotherm patterns and resulting in more diffuse streamlines. This complex interaction provides essential insights into the dynamics of fluid systems, particularly in phases when diffusion is dominant.

• A discernible pattern appears as the amplitude increases, in which the velocity decreases in tandem with an increase in temperature.

• With increasing amplitude, a pattern arises in which skin friction reduces, and the local Nusselt numbers drop synchronously.

• Changes in the amplitude of a fluid flow can affect the streamlines, revealing differences in the flow patterns and causing turbulence. Isotherms demonstrate how amplitude affects gradients and convective heat transfer in the temperature distribution. Higher amplitudes result in better dispersion, more significant temperature fluctuations, and improved heat transfer.

• A pattern is apparent when the $N_{\mathrm{BT}}$ rises, showing a simultaneous drop in temperature and velocity.

• When the $N_{\mathrm{BT}}$ is upward, there is a pattern where skin friction reduces with an increase in the local Nusselt values.

• Variations in $N_{\mathrm{BT}}$ can modify temperature gradients and the overall thermal profile by affecting flow separation, boundary layer dynamics, and temperature distribution. Understanding this relationship is crucial for comprehending how $N_{\mathrm{BT}}$ affect fluid behaviour and heat transfer processes.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

This study is not associate with any third part data.

Additional information

Funding

The corresponding author acknowledges gratefully to the (i) North South University, Grant No.: CTRG-23-SEPS-11, and (ii) the Ministry of Science and Technology, Government of Bangladesh, Grant No.: EAS/SRG-232406 for the financial support.