Solution of Water and Sodium Alginate-Based Casson Type Hybrid Nanofluid with Slip and Sinusoidal Heat Conditions: A Prabhakar Fractional Derivative Approach

Abstract

This paper aims to investigate free convection heat transmission in hybrid nanofluids across an inclined pours plate, which characterizes an asymmetrical hybrid nanofluid flow and heat transfer behavior. With an angled magnetic field applied, sliding on the border of walls is also considered with sinusoidal heat transfer boundary conditions. The non-dimensional leading equations are converted into a fractional model using an effective mathematical fractional approach known as the Prabhakar time fractional derivative. Silver (Ag) and titanium dioxide (TiO₂) are both considered nanoparticles, with water (H₂O) and sodium alginate (C₆H₉NaO₇) serving as the base fluids. The solution of the momentum, concentration, and energy equation is found by utilizing the Laplace scheme, and different numerical algorithms are considered for the inverse of Laplace, i.e., Stehfest and Tzou’s. The graphical analysis investigates the impact and symmetry of significant physical and fractional parameters. Consequently, we surmise that water-based hybrid nanofluid has a somewhat higher velocity than sodium alginate-based hybrid nanofluid. Furthermore, the Casson parameter has a dual effect on the momentum profile. Furthermore, the memory effect reduces as fractional restriction increases for both the velocity and temperature layers. The results demonstrate that increasing the heat transmission in the solid nanoparticle volume fractions enhanced the heat transmission. In addition, the numerical assessment examined the increase in mass and heat transmission, while shear stress was increased with an increase in the Prabhakar fractional parameter α.

1. Introduction

During previous years, various attempts have been made to attain the actual thermal effectiveness of diverse systems. This is part of these attempts to enhance the thermal transfer rate by adding different nanoparticles and mixed convection flow [1,2]. The effect of a mixed convection MHD flow along with diverse Grashof numbers, Reynolds numbers, and Hartmann numbers are examined by Al-Salem et al. [3]. They observed that heat convection and flow speed are affected by the path of fluid motion and permit the magnetic field, which sources a deprived transfer of heat. Furthermore, Lorentz’s force in the reverse flow direction is produced using a magnetic force. Another form of nanofluid (NF) that has newly received consideration is known as the hybrid nanofluid (HNF). Simultaneous mixtures of a metallic nanoparticle and a non-metallic form increase the thermal conduction and solidity of the NF [4]. Using this, the characteristics of two or three nanoparticles may be utilized. As yet, several investigations have been made on HNFs. The effects of Al₂O₃-Cu-H₂O and Al₂O₃-H₂O are differentiated through Moghadassi et al. [5]. They determined that the transfer of convective heat is far advanced in the case of HNF. Suresh et al. [6] discussed the viscosity and thermal conductivity of hybrid (Al₂O₃-Cu-H₂O) nano-suspension in a cylinder. They exposed that transfer of heat is elevated when the HNF is utilized. Singh et al. [7] used the quasi-linearization method with an implicit finite difference technique to study the mixed convection flow along a vertical plate. Govindaraj et al. [8] discussed the MHD NF flow over an accelerated vertical plate with different viscosity values and Prandtl numbers. Gnanaprasanna et al. [9] numerically examined a mathematical flow model of Casson NF over a flat plate.

Further, the laminar flow through a hot cylinder filled with HNF (Al₂O₃-Cu-H₂O) was investigated empirically by Suresh et al. [10] and exhibited that the Nusselt number is enlarged in an HNF in contrast with pure water. The HNF (Ag-MgO-H₂O) through a square cavity was premeditated by Ghalambaz et al. [11]. The impacts of disparity of the major constraints, such as the nanoparticle volume fraction, were considered. The impact on the entropy construction and MHD convection of HNF Al₂O₃-Cu in a permeable square addition was deliberated by a numerical scheme in [12]. They noted that heat transmission in convection mode increases by increasing Rayleigh number while it decreases with the increase in Hartmann number. Non-Newtonian fluids are fluids that do not have a linear relationship between rates of deformation and stress. Non-Newtonian fluids are used in a wide range of scientific and technology sectors, such as crude oil extraction and fiber varnishing, and have piqued the interest of many researchers. Because of their flexibility, non-Newtonian fluids do not have all of their characteristics explained in a single equation. Casson fluid, commonly known as a shear-thinning liquid, is a non-Newtonian fluid. Casson-type fluids include honey, human blood, tomato sauce jelly, and others. Casson fluids are studied by engineers, mathematicians, biomedical researchers, and scientists due to their diverse uses. Because of this phenomenon and in nature, several types are established. The Casson model is an important model that is established, and this fluid model explains yield stress. The Casson NF model with the impact of the magnetic field was studied in [13]. A model of non-Newtonian NF because of heat transference in the existence of a porous surface by taking the stagnation point was deliberated by Nadeem et al. [14]. The Casson NF model with the impacts of MHD and heat transfer was inspected by Haq et al. [15]. Alwawi et al. [16] explored the Casson NF model and heat transference produced by the Lorentz force. The water-based boundary layer flow over a stretching surface along with a vertical plate was discussed by employing the Quasilinearization scheme by Govindaraj et al. [17]. The boundary layer flow in a diverging channel along with diverse viscosity was solved numerically with the quasilinearization method [18]. Patil et al. [19] investigated the MHD triple diffusing quadratic and convective Eyring–Powell NF flow over a vertical plate in diffusing liquid oxygen and hydrogen by employing the implicit finite difference estimation. Iyyappan et al. [20] investigated the boundary layer forced convection flow in a diverging channel with viscous dissipation and heat source/sink on momentum and temperature fields numerically. Patil et al. [21] examined a mathematical model of heat and mass transfer in the nonlinear convective Williamson NF with a moving plate. The shape effects of MHD nanoparticles were studied on energy and fluid flow features on a slender cylinder by employing the implicit finite difference system and quasi-linearization technique [22]. The mixed convective HNF flow around a yawed cylinder with one type of nanoparticle was investigated by utilizing the quasilinearization and the finite method [23].

We see that a numerical model involving an integer-order derivative, with the non-linear model, cannot work suitably in many cases. Fractional calculus has numerous implementations in electromagnetics, viscoplasticity, fluid mechanics, fluid dynamics, processing of signals, as well as optics. It is exploited to explain the model’s physics and design forms formulated through fractional approaches. The Caputo fractional derivative (CTFD) with the utilization of time-fractional distribution by employing a fast method for variable order was discussed by Fang et al. [24]. Ali et al. [25] investigated HNFs with continuous reasonable CTFD due to a pressure gradient. A mathematical model demonstrating the human liver with Caputo–Fabrizio derivatives (CF) was studied by Baleanu et al. [26]. Atangana–Baleanu derivatives (AB) are a novel utilization for designing an AB-fractional mask image dispensation communicated in [27]. Saqib et al. [28] examined the heat transfer rate of CNTs-based nanofluid moving on an inclined plate without singular and local kernels definitions of fractional derivatives. In [29], authors investigate the MHD channel flow of BTF containing hybrid nanoparticles with the help of nonlocal definitions of recent fractional derivatives. Moreover, the recent work completed on different steady and unsteady flows can be seen in [30,31,32,33,34]. The heat transfer fractional study based on AB and CF derivatives for MHD mixed convection flow with nanoparticles (copper oxide and silver) through an inclined moving surface using the Laplace method was completed by Bafakeeh et al. [35]. Sadiq et al. [36] used the Laplace approach to investigate the natural convection heat transfer NF fractional model with CF derivative inside a channel with ramped wall conditions under the impacts of radiation, chemical reactions, and the Soret effect.

In 1971, the Prabhakar work was proposed by an Indian mathematician, Professor Tilak Raj Prabhakar, who anticipated a generality of the Mittag–Leffler function involving three parameters. Using the Prabhakar derivative along with precise fractional coefficients might be a valuable path for choosing suitable numerical models that are recognized as a good arrangement between trial and hypothetical outcomes [37,38]. Due to massive uses in fluid mechanics, researchers studied fractional models based on Caputo, CF, AB, and Prabhakar’s time-fractional derivatives to study the memory effects of different Newtonian and non-Newtonian models in [39,40,41,42,43,44]. The carbon nanotube NF, along with Prabhakar-type thermal transport and free convection flow, was deliberated by Elnaqeeb et al. [45]. Shah et al. [46] discussed a Prabhakar fractional of Maxwell fluid with thermal transport and free convection flow model. Raza et al. [47] used the Laplace method to examine viscous natural convection fractionalized fluid flowing based on Prabhakar fractional operator with slip effects, constant mass diffusion, and Newtonian heating over an oscillating inclined plate. Asjad et al. [48] used the Prabhakar operator to obtain a fractional problem of Jeffrey fluid along a moving vertical plate and the equations for energy, and the Laplace approach solved momentum. Recent definitions of fractional derivatives can be seen in [40,49,50,51].

In the literature, researchers dismissed the notion of fractional, particularly the Prabhakar form fractional, which has an extended Mittag–Leffler function as its kernel and can regulate the momentum and thermal boundary layers. This research investigates the mixed convection heat transfer in HNFs across an inclined vertical plate. With an angled magnetic field applied, sliding on the border of walls is also considered. The non-dimensional controlling equations are converted into a fractional model using an effective mathematical fractional method known as the Prabhakar time-fractional derivative. Silver (Ag) and titanium dioxide (TiO₂) are nanoparticles with water and sodium alginate as base fluids. The Laplace scheme is used to solve the momentum, concentration, and energy equation, and several numerical approaches are studied for the inverse of Laplace. The graphical depiction also discusses the effects of physical and flow characteristics.

2. Mathematical Formulation

Assume an unsteady free convection nanofluid mixed with silver (Ag) and titanium dioxide (TiO₂) as nanoparticles. Furthermore, water (H₂O) and sodium alginate (C₆H₉NaO₇) are considered as a base fluid of flowing hybrid nanofluid on an inclined plate with ambient temperature $T_{\infty }$. Moreover, a magnetic field of intensity $B_o^2$ provided to the poured plate at an angle of inclination ${\theta }_1$. Because of the low Reynolds number values, the induced magnetic field is neglected. At the start of time $t=0$, the system is at rest, and the fluid is also motionless with a constant temperature. After some time $t>0^{+}$, the system begins to oscillate with some constant velocity $H\left(t\right)Cos\left(\omega t\right)$, and the hybrid nanofluid begins to flow across the plate owing to oscillations of the inclined plane, as shown in Figure 1.

As a result of the assumptions mentioned above and the Boussinesq’s approximation [52], the significant leading governing equations of free convection-flowing fluid may be described as [53].

The momentum field:

$$\begin{gathered} {\rho }_{nf}\frac{\partial w_{\left(y,t\right)}}{\partial t}=\left(1+\frac{1}{{\mathsf{\Lambda }}_0}\right)\frac{{\partial }^2{w}_{\left(y,t\right)}}{\partial y^2}-\left({\sigma }_{nf}{B}_o^{2}Sin\left({\theta }_1\right)+\left(1+\frac{1}{{\mathsf{\Lambda }}_0}\right)\frac{{\mu }_{nf}}{k}\right)w_{\left(y,t\right)} \\ +g{\left(\rho {\beta }_T\right)}_{nf}\left(T_{\left(y,t\right)}-T_{\infty }\right)Cos\left({\theta }_2\right) \\ +g{\left(\rho {\beta }_C\right)}_{nf}\left(C_{\left(y,t\right)}-T_{\infty }\right)Cos\left({\theta }_2\right) \end{gathered}$$

(1)

The energy equation:

$${\left(\rho C_p\right)}_{nf}\frac{\partial T_{\left(y,t\right)}}{\partial t}=-\frac{\partial {\delta }_{\left(y,t\right)}}{\partial y}, \mathrm{where} {\delta }_{\left(y,t\right)}=-k_{nf}\frac{\partial T_{\left(y,t\right)}}{\partial y}$$

(2)

Concentration field:

$$\frac{\partial C_{\left(y,t\right)}}{\partial t}=-\frac{\partial {\mathfrak{M}}_{\left(y,t\right)}}{\partial y}, \mathrm{where} {\mathfrak{M}}_{\left(y,t\right)}=-D\frac{\partial C_{\left(y,t\right)}}{\partial y}.$$

(3)

With equivalent conditions:

$$\begin{array}{cc}{w}_{\left(0,t\right)}-h{\left.\frac{\partial w}{\partial y}\right|}_{y=0}& =U_{o}H\left(t\right)Cos\left(\omega t\right), T_{\left(y,t\right)}\hfill \\ & =\left\{\begin{array}{c}{T}_{\infty }+\left(T_w-T_{\infty }\right)\frac{t}{t_o}, 0<t\le t_o\\ \\ T_w, t>t_o\end{array}, C_{\left(0,t\right)}=C_w\right.\hfill \\ & w_{\left(y,t\right)}\to 0, T_{\left(y,t\right)}\to T_{\infty }, C_{\left(y,t\right)}\to C_{\infty } ; y\to \infty , t>0 \hfill \end{array}$$

Introducing the consequent non-dimensional variables:

$$\begin{gathered} w^{\ast }=\frac{w}{U_o}, y^{\ast }=\frac{U_o}{{\upsilon }_f}y, t^{\ast }=\frac{t}{t_o}, t_o=\frac{{\upsilon }_f}{U_o^2}, T^{\ast }=\frac{T_{\left(y,t\right)}-T_{\infty }}{T_w-T_{\infty }} \\ {C}^{\ast }=\frac{C_{\left(y,t\right)}-C_{\infty }}{C_w-C_{\infty }}, q^{\ast }=\frac{q}{q_o} \end{gathered}$$

By discarding the static notation in Equations (1)–(3) and comparable conditions, the non-dimensional form of the preceding equations will emerge as

$${\lambda }_o\frac{\partial w_{\left(y,t\right)}}{\partial t}=\frac{{\lambda }_1}{{\mathsf{\Lambda }}_1}\frac{{\partial }^2{w}_{\left(y,t\right)}}{\partial y^2}-\left({\lambda }_{2}MSin\left({\theta }_1\right)+\frac{{\lambda }_1}{k}\right)w_{\left(y,t\right)}+{\lambda }_{3}Gr{T}_{\left(y,t\right)}Cos\left({\theta }_2\right)+{\lambda }_{4}Gr{C}_{\left(y,t\right)}Cos\left({\theta }_2\right)$$

(4)

$${\lambda }_{5}Pr\frac{\partial T_{\left(y,t\right)}}{\partial t}=-\frac{\partial {\delta }_{\left(y,t\right)}}{\partial y}, {\delta }_{\left(y,t\right)}=-\frac{\partial T_{\left(y,t\right)}}{\partial y}$$

(5)

$${\lambda }_{6}Sc\frac{\partial C_{\left(y,t\right)}}{\partial t}=-\frac{\partial {\mathfrak{M}}_{\left(y,t\right)}}{\partial y}, {\mathfrak{M}}_{\left(y,t\right)}=-\frac{\partial C_{\left(y,t\right)}}{\partial y}$$

(6)

With the following transformed conditions:

$$w_{\left(y,0\right)}=0, T_{\left(y,0\right)}=0, C_{\left(y,0\right)}=0; \forall y\ge 0$$

(7)

$$\begin{array}{cc}{w}_{\left(0,t\right)}-\mathrm{h}{\left.\frac{\partial w}{\partial y}\right|}_{y=0}& =H\left(t\right)Cos\left(\omega t\right), \hfill \\ & T_{\left(y,t\right)}=\left\{\begin{array}{c}t, 0<t\le 1\\ \\ 1, t>1\end{array}, C_{\left(0,t\right)}=1\right.\hfill \end{array}$$

(8)

$$w_{\left(y,t\right)}\to 0, T_{\left(y,t\right)}\to 0, C_{\left(y,t\right)}\to 0 ; y\to \infty , t>0$$

(9)

Table 1. The thermal characteristics of nanoparticles and base fluid.

Material	$\mathbf{Water} \left({\mathsf{H}}_{\mathsf{2}}\mathsf{O}\right)$	$\mathbf{Sodium} \mathbf{Alginate} \left({\mathsf{C}}_{\mathsf{6}}{\mathsf{H}}_{\mathsf{9}}\mathsf{N}\mathsf{a}{\mathsf{O}}_{\mathsf{7}}\right)$	$\mathbf{Silver} \left(\mathsf{A}\mathsf{g}\right)$	$\mathbf{Titanium} \mathbf{Dioxide} \left(\mathsf{T}\mathsf{i}{\mathsf{O}}_{\mathsf{2}}\right)$
$\mathit{\rho }\left(\mathit{M}/{\mathit{L}}^{\mathsf{3}}\right)$	997.1	898	10,500	425
${\mathit{C}}_{\mathit{p}}\left(\mathit{J}/\mathit{M} \mathit{K}\right)$	4179	4175	235	6862
$\mathit{k}\left(\mathit{W}/\mathit{L} \mathit{K}\right)$	0.613	0.6367	429	8.9538
${\mathit{\beta }}_{\mathit{T}}\left({\mathit{K}}^{-\mathsf{1}}\right)$	21	23	1.89	0.9
$\mathit{\sigma }$	0.05	0.07	$\mathsf{3.6}\mathsf{\times }{\mathsf{10}}^{\mathsf{7}}$	$\mathsf{1}\mathsf{\times }{\mathsf{10}}^{\mathsf{-}\mathsf{7}}$

and

Table 2. Model for thermophysical characteristics of NFs quantities.

Thermal Features	Regular Nanofluid	Hybrid Nanofluid
Density	${\rho }_f=\frac{{\rho }_{nf}}{\left(1-\phi \right)+\phi \frac{{\rho }_s}{{\rho }_s}}$	${\rho }_f=\frac{{\rho }_{hnf}}{\left(\left(1-{\phi }_2\right)\left(\left(1-{\phi }_1\right)+{\phi }_1\frac{{\rho }_{s1}}{{\rho }_f}\right)+{\phi }_2{\rho }_{s2}\right)}$
Dynamic Viscosity	${\mu }_f={\mu }_{nf}{\left(1-\phi \right)}^{2.5}$	${\mu }_f={\mu }_{hnf}{\left(1-{\phi }_1\right)}^{2.5}{\left(1-{\phi }_2\right)}^{2.5}$
Electrical Conductivity	${\sigma }_f=\frac{{\sigma }_{nf}}{\left(1+\frac{3\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\phi }{\left(\frac{{\sigma }_s}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\phi }\right)}$	${\sigma }_{bf}=\frac{{\sigma }_{hnf}}{\left(1+\frac{3\phi \left({\phi }_1{\sigma }_1+{\phi }_2{\sigma }_2-{\sigma }_{bf}\left({\phi }_1+{\phi }_2\right)\right)}{\left({\phi }_1{\sigma }_1+{\phi }_2{\sigma }_2+2\phi {\sigma }_{bf}-\phi {\sigma }_{bf}\left({\phi }_1{\sigma }_1+{\phi }_2{\sigma }_2-{\sigma }_{bf}\left({\phi }_1+{\phi }_2\right)\right)\right)}\right)}$
Thermal Conductivity	$k_f=\frac{k_{nf}}{\left(\frac{k_s+\left(n-1\right)k_f-\left(n-1\right)\left(k_f-k_s\right)\phi }{k_s+\left(n-1\right)k_f+\left(k_f-k_s\right)\phi }\right)}$	$k_{bf}=\frac{k_{hnf}}{\left(\frac{k_{s2}+\left(n-1\right)k_{bf}-\left(n-1\right)\left(k_{bf}-k_{s2}\right){\phi }_2}{k_{s2}+\left(n-1\right)k_{bf}+\left(k_{bf}-k_{s2}\right){\phi }_2}\right)}$$\mathrm{and} k_f=\frac{k_{bf}}{\left(\frac{k_{s1}+\left(n-1\right)k_f-\left(n-1\right)\left(k_f-k_{s1}\right){\phi }_1}{k_{s1}+\left(n-1\right)k_f+\left(k_f-k_{s1}\right){\phi }_1}\right)}$
Heat Capacitance	${\left(\rho C_p\right)}_f=\frac{{\left(\rho C_p\right)}_{nf}}{\left(1-\phi \right)+\phi \frac{{\left(\rho C_p\right)}_s}{{\left(\rho C_p\right)}_f}}$	${\left(\rho C_p\right)}_s=\frac{{\left(\rho C_p\right)}_{hnf}}{\left(1-{\phi }_2\right)\left(\left(1-{\phi }_1\right)+{\phi }_1\frac{{\left(\rho C_p\right)}_{s1}}{{\left(\rho C_p\right)}_f}\right)+{\phi }_2{\left(\rho C_p\right)}_{s2}}$
Thermal Expansion Coefficient	${\left(\rho \beta \right)}_f=\frac{{\left(\rho \beta \right)}_{nf}}{\left(1-\phi \right)+\phi \frac{{\left(\rho \beta \right)}_s}{{\left(\rho \beta \right)}_f}}$	${\left(\rho \beta \right)}_f=\frac{{\left(\rho \beta \right)}_{hnf}}{\left(1-{\phi }_2\right)\left(\left(1-{\phi }_1\right)+{\phi }_1\frac{{\left(\rho \beta \right)}_{s1}}{{\left(\rho \beta \right)}_{f1}}\right)+{\phi }_2{\left(\rho \beta \right)}_{s2}}$

shows the thermal properties and properties of under-conversation nanoparticles and fluids.

Where:

$${\lambda }_o=\left(1-\phi \right)+\phi \frac{{\rho }_s}{{\rho }_f}, {\lambda }_1=\frac{1}{{\left(1-\phi \right)}^{2.5}}, {\lambda }_2=1+\frac{3\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\phi }{\left(\frac{{\sigma }_s}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_s}{{\sigma }_f}-1\right)\phi }$$

$${\lambda }_3=\left(1-\phi \right)+\phi \frac{{\left(\rho {\beta }_T\right)}_s}{{\left(\rho {\beta }_T\right)}_f}, {\lambda }_4=\left(1-\phi \right)+\phi \frac{{\left(\rho {\beta }_C\right)}_s}{{\left(\rho {\beta }_T\right)}_f} {\lambda }_5=\frac{k_{nf}}{k_f}, {\mathsf{\Lambda }}_1=\frac{{\mathsf{\Lambda }}_0{\upsilon }_f}{U_o^2}$$

$$Pr={\left(\frac{\mu C_p}{\kappa }\right)}_f, Gr=\frac{g{\left(\rho {\beta }_T\right)}_f\left(T_w-T_{\infty }\right)}{U_o^3}, Sc=\frac{\nu }{D}, Gm=\frac{g{\left(\rho {\beta }_C\right)}_f\left(C_w-C_{\infty }\right)}{U_o^3}$$

As a result, the equivalent fractional model for the Fourier law of heat conductivity and Fick’s law in terms of Prabhakar time-fractional derivatives is as follows

$${\delta }_{\left(y,t\right)}=-{}_{ }{}^C\mathfrak{D}{}_{\alpha ,\beta ,\alpha }^{\gamma }\frac{\partial T_{\left(y,t\right)}}{\partial y}$$

(10)

$${\mathfrak{M}}_{\left(y,t\right)}=-{}_{ }{}^C\mathfrak{D}{}_{\alpha ,\beta ,\alpha }^{\gamma }\frac{\partial C_{\left(y,t\right)}}{\partial y}$$

(11)

where ${}_{ }{}^C\mathfrak{D}{}_{\alpha ,\beta ,\alpha }^{\gamma }$ is the regularised Prabhakar fractional operator with the necessary preliminaries stated below.

Definition 1. 

The mathematical form of Prabhakar fractional kernel [54]

$$e_{\alpha ,\beta }^{\gamma }\left(\alpha ;t\right)=t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\alpha t^{\alpha }\right), Re\left(\alpha \right)>0$$

Definition 2. 

The mathematical representation of Prabhakar integral

$$E_{\alpha ,\beta ,\alpha }^{\gamma }h\left(t\right)=h\left(t\right)\ast e_{\alpha ,\beta }^{\gamma }\left(\alpha ;t\right)=\underset{0}{\overset{t}{{\displaystyle \int }}}h\left(\tau \right){\left(t-\tau \right)}^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\alpha {\left(t-\tau \right)}^{\alpha }\right)d\tau$$

with its Laplace transform

$$\mathcal{L}\left\{E_{\alpha ,\beta ,\alpha }^{\gamma }h\left(t\right)\right\}\left(q\right)=\mathcal{L}\left\{h\left(t\right)\right\}\left(\frac{q^{\alpha \gamma -\beta }}{{\left(q^{\alpha }-\alpha \right)}^{\gamma }}\right)$$

Definition 3. 

The regularized Prabhakar derivative is distinct as [55]

$${}_{ }{}^C\mathfrak{D}{}_{\alpha ,\beta ,\alpha }^{\gamma }h\left(t\right)=\underset{0}{\overset{t}{{\displaystyle \int }}}{h}^n\left(\tau \right){\left(t-\tau \right)}^{n-\beta -1}{E}_{\alpha ,n-\beta }^{-\gamma }\left(\alpha {\left(t-\tau \right)}^{\alpha }\right)d\tau$$

with its Laplace transform

$$\mathcal{L}\left\{{}_{ }{}^C\mathfrak{D}{}_{\alpha ,\beta ,\alpha }^{\gamma }h\left(t\right)\right\}=q^{\beta -n}{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }\mathcal{L}\left\{h^n\left(t\right)\right\}$$

3. Solution of the Problem

The LT approach is used to solve this issue for both leading equations.

3.1. Energy Profile

By applying the LT to Equations (5) and (10) and inserting Equation (10) into Equation (5), we obtain the ordinary differential equation as follows for the solution of the energy equation.

$${\lambda }_{5}Pr q {\overline{T}}_{\left(y,q\right)}=\left(q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }\right)\frac{d^2\overline{T}}{d{y}^2}$$

After simplifying this ordinary differential equation, we yield the general solution with transformed boundary conditions as follows:

$${\overline{T}}_{\left(y,q\right)}=A{e}^{y\sqrt{ \frac{{\lambda }_{5}Pr q^{1-\beta }}{{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}}}-B{e}^{-y\sqrt{\frac{{\lambda }_{5}Pr q^{1-\beta }}{{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}}}$$

(12)

$${\overline{T}}_{\left(0,q\right)}=\frac{1-e^{-q}}{q^2}, {\overline{T}}_{\left(\infty ,q\right)}=0$$

Using these conditions, the answer to the thermal equation will be as follows:

$${\overline{T}}_{\left(y,s\right)}=\frac{1-e^{-q}}{q^2}{e}^{-y\sqrt{ \frac{{\lambda }_{5}Pr q^{1-\beta }}{{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}}}$$

(13)

The Laplace inverse of Equation (13) will be derived numerically in

Table 3. Numerical analysis of governed profiles by both numerical algorithms.

$\mathit{y}$	${\mathit{T}}_{\left(\mathit{y},\mathit{t}\right)}\mathbf{by}$ Stehfest	${\mathit{T}}_{\left(\mathit{y},\mathit{t}\right)}\mathbf{by}$ Tzou’s	${\mathit{C}}_{\left(\mathit{y},\mathit{t}\right)}\mathbf{by}$ Stehfest	${\mathit{C}}_{\left(\mathit{y},\mathit{t}\right)}\mathbf{by}$ Tzou’s	${\mathit{w}}_{\left(\mathit{y},\mathit{t}\right)}\mathbf{by}$ Stehfest	${\mathit{w}}_{\left(\mathit{y},\mathit{t}\right)}\mathbf{by}$ Tzou’s
0.1	0.8141	0.8215	0.8342	0.8339	1.1816	1.1923
0.3	0.5591	0.5637	0.5802	0.5797	1.2916	1.3035
0.5	0.3840	0.3868	0.4038	0.4029	1.3134	1.3254
0.7	0.2636	0.2653	0.2808	0.2800	1.2777	1.2894
0.9	0.1810	0.1820	0.1593	0.1946	1.2069	1.2108
1.1	0.1243	0.1248	0.1359	0.1352	1.1165	1.1268
1.3	0.0853	0.0856	0.0945	0.0940	1.0169	1.0263
1.5	0.0585	0.0587	0.0658	0.0654	0.9153	0.9237
1.7	0.0402	0.0402	0.0458	0.0454	0.8163	0.8234
1.9	0.0275	0.0275	0.0319	0.0316	0.7223	0.720

.

3.2. Concentration Profile

We obtain the ordinary differential equation for the simulations of the concentration profile by using the LT on Equations (6) and (11)

$${\lambda }_6 Sc q {\overline{C}}_{\left(y,q\right)}=-\frac{\partial \overline{\mathfrak{M}}}{\partial y}$$

(14)

$${\overline{\mathfrak{M}}}_{\left(y,q\right)}=-\left(q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }\right)\frac{\partial {\overline{C}}_{\left(y,q\right)}}{\partial y}$$

(15)

We introduce Equation (15) into Equation (14) and utilize the corresponding conditions we obtain:

$${\overline{C}}_{\left(y,q\right)}=\frac{1}{q}{e}^{-y\sqrt{\frac{{\lambda }_6 Sc q^{1-\beta }}{{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}}}$$

(16)

The Laplace inverse of the concentration as mentioned above field solution will be computed numerically in Table 3 using Stehfest and Tzou’s techniques.

3.3. Momentum Profile

We use the LT on the modified governing Equation (4) and its following conditions to obtain the momentum profile solution.

$$\begin{array}{cc}\frac{{\partial }^2 {\overline{w}}_{\left(y,q\right)}}{\partial y^2}-\frac{{\mathsf{\Lambda }}_1}{{\mathsf{\lambda }}_1}& \left({\lambda }_{2}Msin\left({\theta }_1\right)+\frac{{\lambda }_1}{k}+{\lambda }_{o}q\right) {\overline{w}}_{\left(y,q\right)}\hfill \\ & =-\frac{{\lambda }_{3}Gr{\mathsf{\Lambda }}_{1}Cos\left({\theta }_2\right)}{{\lambda }_1}{\overline{T}}_{\left(y,q\right)}-\frac{{\lambda }_{4}Gr{\mathsf{\Lambda }}_{1}Cos\left({\theta }_2\right)}{{\lambda }_1}{\overline{C}}_{\left(y,q\right)}\end{array}$$

(17)

$${\overline{w}}_{\left(0,q\right)}-h{\left.\frac{\partial \overline{w}}{\partial y}\right|}_{y=0}=\frac{q}{{\omega }^2+q^2}, {\overline{w}}_{\left(\infty ,q\right)}\to 0$$

Using these circumstances, we obtain the momentum profile simulation shown below.

$$\begin{array}{c} {\overline{w}}_{\left(y,q\right)}\hfill \\ =\frac{1}{1+h\sqrt{\frac{{\mathsf{\Lambda }}_1}{{\lambda }_1}\left({\lambda }_{2}Msin\left({\theta }_1\right)+\frac{{\lambda }_1}{k}+{\lambda }_{0}q\right)}}(\frac{{\lambda }_{3}Gr {\mathsf{\Lambda }}_1 Cos\left({\theta }_2\right)\left(1-e^{-q}\right)}{{\lambda }_1{\mathrm{q}}^2} \frac{1+h\sqrt{ \frac{{\lambda }_{5}Pr q}{q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}}}{\left(\frac{{\lambda }_{5}Pr q}{q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}\right)-\frac{{\mathsf{\Lambda }}_1}{{\lambda }_1}\left({\lambda }_{2}Msin\left({\theta }_1\right)+\frac{{\lambda }_1}{k}+{\lambda }_{0}q\right)}\hfill \\ +\frac{{\lambda }_{4}Gm {\mathsf{\Lambda }}_1 Cos\left({\theta }_2\right)}{{\lambda }_1\mathrm{q}} \frac{1+h\sqrt{ \frac{{\lambda }_{6}Sc q}{q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}}}{\left(\frac{{\lambda }_{6}Sc q}{q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}\right)-\frac{{\mathsf{\Lambda }}_1}{{\lambda }_1}\left({\lambda }_{2}Msin\left({\theta }_1\right)+\frac{{\lambda }_1}{k}+{\lambda }_{0}q\right)}+\frac{q}{{\omega }^2+q^2})e^{-y\sqrt{\frac{{\mathsf{\Lambda }}_1}{{\lambda }_1}\left({\lambda }_{2}Msin\left({\theta }_1\right)+\frac{{\lambda }_1}{k}+{\lambda }_{0}q\right)}}\hfill \\ -\frac{{\lambda }_{3}Gr {\mathsf{\Lambda }}_1 Cos\left({\theta }_2\right)\left(1-e^{-q}\right)}{{\lambda }_1{\mathrm{q}}^2} \frac{e^{-y\sqrt{\frac{{\lambda }_{5}Pr q}{q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}}}}{\left(\frac{{\lambda }_{5}Pr q}{q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}\right)-\frac{{\mathsf{\Lambda }}_1}{{\lambda }_1}\left({\lambda }_{2}Msin\left({\theta }_1\right)+\frac{{\lambda }_1}{k}+{\lambda }_{0}q\right)}\hfill \\ -\frac{{\lambda }_{4}Gm {\mathsf{\Lambda }}_1 Cos\left({\theta }_2\right)}{{\lambda }_1\mathrm{q}} \frac{e^{-y\sqrt{ \frac{{\lambda }_{6}Sc q}{q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}}}}{\left( \frac{{\lambda }_{6}Sc q}{q^{\beta }{\left(1-\alpha q^{-\alpha }\right)}^{\gamma }}\right)-\frac{{\mathsf{\Lambda }}_1}{{\lambda }_1}\left({\lambda }_{2}Msin\left({\theta }_1\right)+\frac{{\lambda }_1}{k}+{\lambda }_{0}q\right)}\hfill \end{array}$$

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We used a numerical tool, the Stehfest algorithm, to examine the solution of heat and momentum fields to investigate LT’s inverse. Mathematically, the Gaver Stehfest method [56,57,58] may be distinguished as

$$w\left(y,t\right)=\frac{ln\left(2\right)}{t}{\displaystyle \sum }_{n=1}^N{v}_n\overline{w}\left(y,n\frac{ln\left(2\right)}{t}\right)$$

where

$$v_n={\left(-1\right)}^{n+\frac{N}{2}}{\displaystyle \sum }_{r=\left. [\frac{q+1}{2}\right]}^{min\left(q,\frac{N}{2}\right)}\frac{r^{\frac{N}{2}}\left(2r\right)!}{\left(\frac{N}{2}-r\right)!r! \left(r-1\right)! \left(q-r\right)! \left(2r-q\right)!}$$

To compare and confirm the findings acquired by the preceding numerical technique, we also used Tzou’s method, which may be mathematically rigorous.

$$w\left(\xi ,t\right)=\frac{e^{4.7}}{t}\left. [\frac{1}{2}\overline{w}\left(r,\frac{4.7}{t}\right)+Re\left\{{\displaystyle \sum }_{j=1}^N{\left(-1\right)}^k\overline{w}\left(r,\frac{4.7+k\pi i}{t}\right)\right\}\right]$$

3.4. Gradients

This paper uses the following three important fundamental engineering quantities of interest: the Nusselt number, the Sherwood number, and the shear stress. These gradients are mathematically expressed as:

$$Nu=-{\left.\frac{\partial T_{\left(y,t\right)}}{\partial y}\right|}_{y=0}=-{\mathcal{L}}^{-1}\left\{\frac{\partial {\overline{T}}_{\left(0,s\right)}}{\partial y}\right\},$$

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$$Sh=-{\left.\frac{\partial C_{\left(y,t\right)}}{\partial y}\right|}_{y=0}=-{\mathcal{L}}^{-1}\left\{\frac{\partial {\overline{C}}_{\left(0,s\right)}}{\partial y}\right\},$$

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$$C_f=-{\left.\frac{\partial w_{\left(y,t\right)}}{\partial y}\right|}_{y=0}=-{\mathcal{L}}^{-1}\left\{\frac{\partial {\overline{w}}_{\left(0,s\right)}}{\partial y}\right\}.$$

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4. Discussion of Results

We study the application of the recently presented Mittage–Leffler kernel called Prabhakar fractional derivative to Casson HNF ($\mathrm{Ag}$-TiO₂-H₂O and $\mathrm{Ag}$-TiO₂-${\mathrm{C}}_6{\mathrm{H}}_9{\mathrm{NaO}}_7$) and mixed convection over a vertical, inclined plate. The slip on the boundary of walls is also considered with an inclined applied magnetic field. This fractional model is solved by employing the Laplace transform scheme sustaining all initial and boundary conditions. The consequences of fractional and different involved flow parameters on energy, concentration, and momentum are consulted in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.

Figure 2 is planned to see the impacts of fractional parameters $\left(\alpha , \beta , \gamma \right)$ and $Pr$ on the temperature profile. By setting other parameters constant and varying fractional parameters and $Pr$, we see that the temperature rate is decayed by enhancing the $\left(\alpha , \beta , \gamma \right)$ and $Pr$. It means that fluid characteristics may be controlled with fractional parameters. It has been discovered that raising the fractional parameter reduces the temperature, concentration, and profiles due to the Prabhakar fractional kernel. This demonstrates the memory effect of the momentum and temperature at a certain period. Figure 2b depicts the physical influence of the Prandtl number $Pr$ on the temperature profile while keeping the other flow parameters constant. We may argue that when the value of $Pr$ increases, the temperature of the fluid and the thickness of the boundary layer decrease. It is primarily due to an increase in fluid viscosity and a decrease in the thermal boundary layer, which resulted in a slower fluid thermal field. Physically, when the estimations of $Pr$ rise, which source is quickly reducing of the thermal boundary layer thickness, due to which declines the energy profile. Moreover, we perceived that the temperature for silver–titanium dioxide–sodium alginate-based HNF has a relatively more significant value than silver–titanium dioxide–water HNF.

Figure 3 is planned to check the impact of $\left(\alpha , \beta , \gamma \right)$ and $Sc$ on concentration. By setting other parameters fixed and vary $\left(\alpha , \beta , \gamma \right)$ and $Sc$. It is detected that concentration cannot be boosted by mean declines for a more significant estimation of $\left(\alpha , \beta , \gamma \right)$ and $Sc$. Figure 3b depicts the effect of $Sc$, the Schmidt number, on the concentration profile. It is clearly demonstrated that an increase in the value of the Schmidt number $Sc$ results in a decrease in the centration profile. Because the rate of molecular diffusion decreases as $Sc$ increases, the thickness of the boundary layer is reduced. Physically, $Sc$ is the ratio of momentum and mass diffusivity. The fluid layers get more viscosity, so concentration declines.

Figure 4a shows that the velocity curve drops as fractional parameters are improved, and the velocity field shows a dual behavior by varying Casson parameter (${\mathsf{\Lambda }}_1$) as shown in Figure 4b. There are separate peaks for momentum profile layer thickness as well. The velocity near the plate is more incredible, but as the fluid flows away from the plate, its value decreases and becomes zero as y approaches infinity, as depicted in the figures. Figure 5a signified the influence of $Pr$ on the velocity profile and observed that the fluid’s velocity enhancements as $Pr$ declines. The velocity layer increases thickness because of the lesser rate of thermal diffusion, $Pr$ directs the relative viscosity of momentum boundary layers in energy transfer model problems. As predicted, raising the values of $Pr$ lowers thermal conductivity, making the fluid thicker, and therefore reducing the thickness of the thermal boundary layer. Figure 5b displays that the velocity behavior is also seen contrary to the $Sc$ since as the estimations of $Sc$ increase, it drops mass diffusivity by raising the kinematic viscosity. In Figure 6a,b, we see that fluid velocity amplified for more significant estimations of thermal Grashof number ($Gr)$ and mass Grashof number ($Gm),$ respectively. Because $Gr$ creates natural convection owing to buoyancy and viscosity forces acting on the fluid, large values of $Gr$ cause buoyancy forces to increase, forcing the flow to accelerate. Physically, $Gr$ is that it characterizes the ratio of the buoyant forces because of spatial disparity in the fluid density (produced by temperature variation) to the preventive force because of the fluid viscosity. Grashof number signifies how the buoyant force is dominant, which controls the convection because of which velocity is enlarged. A similar trend is seen for $Gm$. Because $Gm$ is the ratio of viscous forces to buoyant forces owing to concentration gradient, raising the value of $Gm$ increases fluid velocity and boundary layer thickness in both circumstances. The influence of inclined magnetic field $M$ considered in Figure 7a shows that the velocity decays by increasing the estimation of $M$ due to Lorentz forces. It is a type of resistant force that sources velocity decay. In Figure 7b it is realized how the volume fraction ($\phi$) impacts the velocity. We realize that fluid velocity is reduced for great values of $\phi$. This is physically suitable as the fluid gains much viscosity with growing $\phi ,$ which sources a reduction in the fluid velocity and eventually displays the fluid motion. It is discovered that increasing the value of $\phi$ reduces the velocity. The velocity is most significant for $“\phi =0”$ (pure water), while it is lowest for $“\phi =0.04”$. As it rises, the viscous forces get more extraordinary, and the velocity drops. Furthermore, sodium alginate-based hybrid nanofluid is denser than pure water-based hybrid nanofluid. Figure 8a,b are planned to compare two diverse numerical approaches, Stehfest and Tzou, for temperature, concentration, and momentum profiles. The consequences from different profile curves have overlapped slightly, indicating this research work’s validity. Figure 9 is designed to see the validity of our achieved results compared to Khalid et al. [59] velocity outcomes. By overlapping both curves, it is appreciated from these graphs that our attained outcomes match those developed by Khalid et al. [59]. The comparison of governed equations with different numerical schemes is analyzed in Table 3 and

Table 4. Numerical analysis of Nusselt number, Sherwood number, and skin friction.

$\mathit{\alpha }$	$\mathit{N}\mathit{u}$	$\mathit{S}\mathit{h}$	${\mathit{C}}_{\mathit{f}}$
0.1	1.4057	1.4801	7.5847
0.2	1.4845	1.5480	7.9900
0.3	1.5811	1.6169	8.3908
0.4	1.6826	1.6808	8.7468
0.5	1.8124	1.7340	9.0116
0.6	1.9319	1.7736	9.1780
0.7	2.0423	1.7989	9.2379
0.8	2.1374	1.8115	9.2163
0.9	2.2135	1.8123	9.1376

with nusselt number, Sherwood number, and skin friction.

5. Conclusions

We investigated a Casson-type sodium alginate and water-based hybrid nanofluid combined with silver (Ag) and titanium dioxide (TiO₂) and flowing on an inclined plate in this study. The effects of an inclined applied magnetic field saturated porous plate and sinusoidal thermal conditions are also studied. The LT approach examines both governing equations’ semi-analytical solutions. The behavior of various parameters is visually and quantitatively evaluated. The key findings of the graphical research are listed in bullet form below.

• With fractional and Prandtl number augmentation, both velocity and temperature fields exhibit opposing behavior.
• The memory effect is reduced for both concentration and temperature profiles by increasing the fractional value restriction.
• Because of its physical properties, the energy field fluctuates with the fluctuation in volume fractional parameters.
• The momentum profile accelerates as the Grashof number increases due to an improvement in the boundary layer of the flowing fluid.
• The permeability parameter and the applied magnetic field retorted the velocity profile for water-based hybrid nanofluid and sodium alginate nanofluid.
• The significant comparison of the momentum profile with the current physical literature adds to the study’s originality.
• When comparing numerical methodologies, the curves of both methods overlap, indicating that our obtained results are legitimate.

Recent advances in the study of fractional order frameworks include the fractional natural decomposition method (FNDM), the fractional Shehu transform, and the modified generalized Taylor fractional series method (MGTFSM). Researchers in the future can correlate their findings to those we found utilizing the Caputo–Fabrizio, Atangana–Baleanu, and Prabhakar fractional methods in our investigation.