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Abstract

Hybrid nanofluids outperform mono nanofluids in terms of heat transmission. They can be found in heat exchangers, the automobile industry, transformer cooling, and electronic cooling, in addition to solar collectors and military equipment. The primary goal of this study is to scrutinize the magnetohydrodynamic hybrids nanofluid (Copper-oxides and Titanium dioxide/water) $(CuO + Ti O_{2})$ and $(Ti O_{2} / H_{2} O)$ nanofluid flow in two parallel plates using walls suction/injection under the influence of a magnetic field and thermal radiation. Implementing the appropriate transformation, the governing partial differential equations are converted into equivalent ordinary differential equations. In MATLAB software, the built-in numerical technique bvp4c is used to evaluate the final system to generate the graphical results against velocity and temperature gradient for various parameters. The increasing evaluation of Reynolds number and magnetic parameter influenced velocity concentration. The response surface method is used to generate the sensitivity and contour graphs and tables. Skin friction and the Nusselt number have an outcome on flow characteristics as well. This type of research is critical in the coolant process, aviation engineering, and industrial cleaning processes, among other fields. In several cases, comparing current and historical findings reveals good agreement.

Keywords

Squeezing flow hybrid nanofluid variable magnetic field MATLAB thermal radiation

Introduction

Nanofluids are utilized to evolve the thermal conduction and heat transport rate of the main fluid. Base fluids include tri ethylene’s glycols, water, coolants, ethylene’s, lubricating oils and oil, thermoplastic remedies, and biological-fluids. Nanofluids have distinct properties such as homogeneity, highly thermal conduction at low nanoparticle accordion, long-term stability, and minimum obstructing flow paths. As a result, above mention liquids are found in a vast variety of electrical devices, like as micro-electro-mechanical circuits, microchip coolants, micro reactors, and electronic fluid flow projection systems. Nanofluid technologies include structure heating and cooling, heat transfer, conveyance, monitoring, microfluidics, lubrication systems, medicinal procedures, nano cryosurgery, electronic device freezing, cancer therapies, drug delivery, cryopreservation, and imaging. Choi and Eastman¹ postulated that we may produce new category of thermal transport liquid by floating metalized nanoparticles in standard heat transfer fluids. Alghamdi et al.² inspected the impact of MHD and thermal source or sink on the movement of hybrid fluid. Bilal et al.³ are researching the use of thermal radiation to squeeze and dilate permeable walls, allowing a hybrid magnetohydrodynamic nanofluid (Carbon nanotubes and ferrous oxide/water) to flow into a vertical channel. Gul et al.⁴ studied the movement of a hybrids nanofluids made up of copper & aluminum oxide molecules. Waini et al.⁵ described the hybrid nanofluid, turbulent movement and heat transferal past a stretching or shrinking sheet. Huminic and Huminic⁶ investigate the effects of nanofluids and hybrid nanofluids on the creation of volatility in various energy processes with valid boundary constraints and physical settings. Salman et al.⁷ address recent breakthroughs in microscale-faced producing heat transfer enhancement in this study. In Nadeem et al.,⁸ we investigated the hybrid nanofluid flow via a permeable exponentially stretching tube. The goal of this study, according to Yashkun et al.,⁹ was to analyze the heat transport parameters of a magnetohydrodynamic hybrid nanofluid across a longitudinal stretching and shrinking surface in the absence of suction and radiation heat impacts. Jalili et al.,^10-13 Talarposhti et al.,¹⁴ Sheikholeslami et al.,¹⁵ and Jalili et al.¹⁶ list other studies in this field. The study focuses on the boundary layer movement of a hybrid nanofluid across a revolving cylinder in the appearance of oblique magnetism, according to Abbas et al.¹⁷ and Aladdin et al.¹⁸ described the properties of suction and magnetization on a revolving plate of hybrid nanofluid flow consisting of water as the main fluid, Aluminum Oxide as nanomaterials, and Copper (Cu) as nanoparticles. Alsaedi et al.¹⁹ quantitatively estimated the flow of a hybrid nanofluid via two coaxially constructed cylinders. According to Acharya,²⁰ fins of various forms are employed in a range of industrial and technological applications, including semiconductor devices, steam turbines, electrical converters, vehicle radiators, heat transfer, and hydrogen fuel cells, air-cooled automobiles, and so on. Heat generation in high-temperature technological processes under the effect of radiation is a significant event in heat transferable. This phenomenon is momentous in heat transmission and thermal system design. Ashwinkumar et al.²¹ investigated the upshot of nonlinear heat radiation on the two-dimensional magnetohydrodynamic flow of a hybrid nanofluid in two different configurations. The linear form is determined to be incorrect for the high-temperature disparity. In the modeled Rosseland estimate, a non-dimensional variable is utilized, and has no effect on the Prandtl number, according to Magyari and Pantokratoras.²² Recently, nonlinear thermally radiative has been examined, and numerous hybrid nanofluid movements with varied geometries have been reported.^23–27 The magneto-hydrodynamic flow with fluctuating fluid properties has piqued the interest of the scientific community. Technology, physical science, and chemical engineering all benefit from research on magnetization consequences. Contact between an electrical conductor fluid and a magnetization has an impact on a variety of technical systems, including magnetohydrodynamics (MHD) turbines, motors, exchanges, and flow separation processes. A great quantity of research on the unique fluid characteristics across numerous geometries with varying flow conditions has been published due to several applications in geophysics, magnetohydrodynamic power production, and so on. Safdar et al.²⁸ investigated theoretical and computational models for stable MHD Maxwell nanofluid flow along a porosity expanded sheet including gyrotactic organisms. Hussain et al.²⁹ detect entropy in a hybrid nanofluid flow governed by magnetohydrodynamics, variable viscosity, and coupled convection. The radiative and MHD micro-polar fluid movement on stretching or shrinking sheets of $CuO$ and $Ti O_{2}$ nanomaterials is defined by Waini et al.³⁰ Thanesh Kumar et al.³¹ reported on the consequence of titanium oxide nanoparticle morphologies on the magnetohydrodynamic steady movement of Maxwell fluid after passing through a wedge with nonlinear thermal radiation influences. Yahaya et al.³² investigated heat transport on a stretched plate using a $Cu - A l_{2} O_{3} / H_{2} O$ hybrid nanofluid. According to Waini et al.,³³ this study investigates the compressed dual nanofluids movement on a porous sensing area using magnetohydrodynamic (MHD) and radiative influence. The sensitivity and contour graphs and tables are created using the response surface method. In the presence of quadratic velocity, Zainal et al.³⁴ studied the MHD flow of a hybrid $A l_{2} O_{3} - Cu / H_{2} O$ through a permeability stretching or shrinking sheet. Refs.^35–46 offer other works on this topic.

By examining the laminar and steady flow of a hybrid nanofluid between two parallel permeable plates with a fluctuating magnetization, the continuing research closes a gap in the investigation. The subject at hand will be investigated for the initial period and has never been studied before. The most current study is a first for the field, to the best of the researcher’s knowledge. The impacts of suction or injection through the channel, as well as the effects of other significant elements on the velocity distribution, temperature distribution, Nusselt number, and skin friction, were also evaluated by the researcher.

Mathematical modeling

Hybrid nanofluid flow is considered through two parallel horizontally plates diverged by a distance of $H$ , where $H$ denotes the space across plates at a time $t = 0$ . $H$ defines as $H (t) = H \sqrt{1 - α t}$ . For $t = \frac{1}{α}$ and $α > 0$ , the plates are crammed together till these collide to each other but both plates are disjoint as $α < 0$ , as drawn in Figure 1. Moreover, the higher plate with velocity $v = dH / dt$ is rapidly approaching the bottom plate. The inspired magnetic field is created using a persuaded magnetic field $(\frac{\partial b_{1}}{\partial x}, 0, \frac{\partial b_{2}}{\partial z})$ . To make a hybrid nanofluid $(CuO + Ti O_{2} / H_{2} O)$ , single and multiwall copper oxides are mixed with major fluid water. The x-axis is parallel to the horizontal axis, while the z-axis is orthogonal to the plate in the geometrical coordinates system. $T_{0}$ and $T_{H}$ are the thermal effect of the bottom and top plate, correspondingly. Because the flow parameters of the hybrid nanofluid $(CuO + Ti O_{2} / water)$ under study are not time-dependent, the fluctuating magnetic field has a greater consequence.

Flow is considered through two horizontal parallel plates.

The effects of thermal radiation and suction/injection are taken under consideration.

Figure 1.

Physical model.

Governing equations

The following are the continuity, momentum, magnetic field, and energy conservation calculations for the abovementioned hybrid nanofluids.^47–49

u_{x} + v_{x} = 0,

(1)

\begin{array}{l} u u_{x} + v u_{y} = - \frac{1}{ρ_{h n f}} p_{x} + \frac{μ_{h n f}}{ρ_{h n f}} (u_{x x} + u_{y y}) \\ - \frac{b_{2} σ_{h n f}}{ρ_{h n f}} ({(b_{1})}_{y} - {(b_{2})}_{x}) \end{array}

(2)

\begin{matrix} u v_{x} + v v_{y} = - \frac{1}{ρ_{hnf}} p_{y} + \frac{μ_{hnf}}{ρ_{hnf}} (v_{xx} + v_{yy}) \\ - \frac{b_{1} σ_{hnf}}{ρ_{hnf}} ({(b_{1})}_{y} - {(b_{2})}_{x}) \end{matrix}

(3)

Maxwell equations.⁴⁹

u {(b_{2})}_{y} + b_{2} u_{y} - v {(b_{1})}_{y} - b_{1} v_{y} + \frac{1}{σ_{hnf} μ_{e}} ({(b_{1})}_{xx} + {(b_{1})}_{yy}) = 0

(4)

v {(b_{1})}_{x} + b_{1} v_{x} - u {(b_{2})}_{x} - b_{2} u_{x} + \frac{1}{σ_{hnf} μ_{e}} ({(b_{2})}_{xx} + {(b_{2})}_{yy}) = 0

(5)

Where $b_{1} & b_{2}$ are the parameters of the magnetic field, the pressure of the fluid is $p$ , the temperature is $T$ and $σ_{hnf}$ is the electrical conductivity of hybrid nanofluid.

Energy equation.⁴⁶

\begin{matrix} u T_{x} + v T_{y} = \frac{k_{hnf}}{{(ρ C_{p})}_{hnf}} (T_{xx} + T_{yy}) + \frac{μ_{hnf}}{{(ρ C_{p})}_{hnf}} \\ (4 {(v_{y})}^{2} + {(u_{y})}^{2} + {(v_{x})}^{2} + 2 u_{y} v_{x}) - \frac{1}{{(ρ C_{p})}_{hnf}} \frac{\partial q_{r}}{\partial y} \end{matrix}

(6)

The radiative heat flux $q_{r}$ is defined as

q_{r} = - \frac{16 σ_{*} T^{3}}{3 k^{*}} T_{y}

Thermophysical properties

The hybrid nanofluid heat capacity is represented by ${(ρ C_{p})}_{hnf}$ and the kinematic viscosity of hybrid nanofluid is given as $μ_{hnf}$ .^46–51

\begin{matrix} v_{hnf} = \frac{μ_{hnf}}{ρ_{hnf}}, \frac{ρ_{hnf}}{ρ_{f}} = (1 - Φ_{2}) ((1 - Φ_{1} + \frac{Φ_{1} ρ_{1}}{ρ_{bf}}) + \frac{Φ_{2} ρ_{2}}{ρ_{bf}}), \\ \frac{k_{hnf}}{k_{bf}} = \frac{k_{2} + k_{bf} (m - 1) + Φ_{2} (k_{bf} - k_{2})}{k_{2} + (m - 1) k_{bf} + Φ_{2} (m - 1) (k_{bf} - k_{2})} \\ \frac{k_{f}}{k_{f}} = \frac{k_{1} + (m - 1) k_{bf} + Φ_{1} (k_{f} - k_{1})}{k_{1} + (m - 1) k_{f} + Φ_{1} (m - 1) (k_{f} - k_{1})}, \frac{μ_{hnf}}{μ_{f}} \\ = \frac{1}{{(1 - Φ_{1})}^{2.5} {(1 - Φ_{2})}^{2.5}}, \frac{{(ρ C_{p})}_{hnf}}{{(ρ C_{p})}_{f}} \\ = (1 - Φ_{2}) ((1 - Φ_{1} + \frac{Φ_{1} {(ρ C_{p})}_{1}}{{(ρ C_{p})}_{f}}) + \frac{Φ_{2} {(ρ C_{p})}_{2}}{{(ρ C_{p})}_{f}}), \frac{σ_{hnf}}{σ_{f}} \\ = \frac{σ_{2} + 2 σ_{2} - 2 Φ_{2} (σ_{bf} - σ_{2})}{σ_{2} + 2 σ_{bf} + Φ_{2} (σ_{bf} - σ_{2})}, \frac{σ_{bf}}{σ_{f}} \\ = \frac{σ_{1} + 2 σ_{f} - 2 Φ_{1} (σ_{f} - σ_{1})}{σ_{1} + 2 σ_{f} + Φ_{2} (σ_{f} - σ_{1})}, \end{matrix}

(7)

Thermos-physicals properties of base fluid and nano particles are given in Table 1. and $ϕ_{1}, ϕ_{2}$ is the volume fraction of $CuO, Ti O_{2}$ respectively.

Table 1.

Illustrates the thermos-physicals features of nano-particles (titanium dioxide and copper oxide) and base fluid (water).^52,53

Physical properties	$ρ (kg / m^{3})$	$C_{p} (J / kgK)$	$k (W / mK)$	$β \times 10^{5} (K^{- 1})$	$σ (S m^{- 1})$
$H_{2} O$	997.1	4179	0.613	21	0.05
$CuO$	6320	531.8	76.5	1.80	$5.96 \times 10^{7}$
$Ti O_{2}$	4250	686.2	8.9538	0.9	$2.38 \times 10^{6}$

Boundary conditions

The boundary constrains are delineated as follows ⁴⁸:

\begin{matrix} u = ax, v = v_{0}, w = 0, - k_{hnf} [\frac{\partial T}{\partial y}] = h_{f} (T_{f} - T), at y = 0, \\ u = bx, v = v_{L}, w = 0, T = T_{L} at y = L . \end{matrix}

(8)

The transformations of similarity listed below⁴⁶

\begin{matrix} u = ax f' (ζ), v = - aLf (ζ), w = axg (ζ), \\ ζ = \frac{y}{L}, θ (ζ) = \frac{T - T_{L}}{T_{0} - T_{L}} . \end{matrix}

(9)

Transformed set of equation

The resulting group of ordinary differential equations after the operation of factors of resemblance (9) into (2–6)

\begin{matrix} f ″' = \frac{ρ_{hnf}}{ρ_{f}} \frac{μ_{f}}{μ_{hnf}} (S (f' f'' - f f ″')) + M^{2} S {Re}_{m} {(\frac{σ_{hnf}}{μ_{f}})}^{2} \frac{μ_{f}}{μ_{hnf}} (f K^{2}) \\ - M^{2} S^{2} {Re}_{m}^{2} {(\frac{σ_{hnf}}{μ_{f}})}^{3} \frac{μ_{f}}{μ_{hnf}} (f f' K^{2} - f^{2} K K'), \end{matrix}

(10)

K'' = {Re}_{m} S \frac{σ_{hnf}}{μ_{f}} \frac{μ_{f}}{μ_{hnf}} (f' K - f K'),

(11)

\begin{matrix} θ'' = - \frac{1}{(1 + \frac{4}{3} Rd {(1 + (θ_{w} - 1) θ)}^{3}) θ'} \\ [\begin{matrix} S . \Pr \frac{{(ρ C_{p})}_{hnf}}{{(ρ C_{p})}_{f}} \frac{k_{f}}{k_{hnf}} (θ' f) + \frac{μ_{hnf}}{μ_{f}} \frac{k_{f}}{k_{hnf}} \\ PrEc (4 δ f^{' 2} K + f^{'' 2}) \end{matrix}], \end{matrix}

(12)

As well as the evaluate boundary conditions

\begin{matrix} f (0) = α, f' (0) = 1, g (0) = 0, \frac{k_{hnf}}{k_{f}} θ' (0) = - Bi (1 - θ (0)), \\ f (1) = β, f' (1) = λ, g (1) = 0, θ (1) = 0 \end{matrix})

(13)

Where the magnetic variable is signified by $M^{2} (= \frac{H^{2} a σ_{f}}{ρ_{f} v_{f}})$ , $α (= - \frac{v_{0}}{aL})$ , $β (= - \frac{v_{l}}{aL})$ are suction/injection, $λ (= \frac{b}{a})$ is stretching parameter, $S (= \frac{H^{2} a}{2 v_{f}})$ squeeze number, ${Re}_{m} (= σ_{f} v_{f} μ_{e})$ Reynolds’s Magnetized factor, the temperature ratio parameter is given as $θ_{w} = \frac{T_{\infty}}{T_{w}}$ , $\Pr (= \frac{v_{f} {(ρ C_{p})}_{f}}{k_{f}})$ Prandtl number, the thermal radiation is defined as $Rd = \frac{4 σ_{*} T_{\infty}^{3}}{k_{s} k_{f}},$ the Eckert number is denoted by $Ec (= \frac{1}{{(C_{p})}_{f} T_{H}} {(ax)}^{2})$ and $δ (= \frac{H^{2}}{x^{2}})$ .

Physical components

The Nusselt numbers and skin frictions factors are described below,

\begin{matrix} C_{f} = \frac{μ_{nf}}{ρ_{nf} {(\frac{- α l}{2 \sqrt{1 - α t}})}^{2}} {(u_{z} + u_{r})}_{z = h (t)}, N_{u} = \frac{- k_{n} f {(T_{z})}_{z = h (t)}}{k_{f} (T_{0} - T_{u})} \\ (k_{hnf} + (1 + \frac{4}{3} Rd {(1 + (θ_{w} - 1) θ)}^{3})) {(T_{z})}_{z = h (t)}, \end{matrix}

(14)

\begin{matrix} H^{2} {(1 - Φ_{1})}^{2.5} {(1 - Φ_{2})}^{2.5} \\ ((1 - Φ_{2}) ((1 - Φ_{1} + \frac{Φ_{1} ρ_{1}}{ρ_{f}}) + \frac{Φ_{2} ρ_{2}}{ρ_{f}})) Re C_{f} = f'' (1), \\ - θ' (0) = \frac{H \sqrt{1 - α t}}{\frac{k_{2} + (m - 1) k_{bf} + Φ_{2} (k_{bf} - k_{2})}{k_{2} + (m - 1) k_{bf} + Φ_{2} (m - 1) (k_{bf} - k_{2})}} Nu . \end{matrix}

(15)

Numerical scheme

The arrangement of ordinary differential equations is not acutely sensitive to proper clarification to accomplish an appropriate explanation. The MATLAB computational software bvp4c function and shooting method are used to code the above-mentioned combined ODEs (10–12) and the corresponding boundary constraints (13) Bvp4c is essentially a Labotto-III collocation formula that is used to generate numerical results. Higher-order derivatives of the controlling equations have deviated into a first-order set of ODEs

\begin{matrix} f = h_{1}, f' = h_{2}, f'' = h_{3}, f ″' = h_{3}', \\ K = h_{4}, K' = h_{5}, K'' = h_{5}', θ = h_{6}, θ' = h_{7}, θ'' = h_{7}' \end{matrix}},

(16)

Applying these translations into equations (10)–(12), we get

\begin{matrix} h_{3}' = \frac{ρ_{hnf}}{ρ_{f}} \frac{μ_{f}}{μ_{hnf}} (S (h_{2} h_{3} - h_{1} h_{3}')) + M^{2} S {Re}_{m} {(\frac{σ_{hnf}}{μ_{f}})}^{2} \\ \frac{μ_{f}}{μ_{hnf}} (h_{1} h_{4}^{2}) \\ - M^{2} S^{2} {Re}_{m}^{2} {(\frac{σ_{hnf}}{μ_{f}})}^{3} \frac{μ_{f}}{μ_{hnf}} (h_{1} h_{2} h_{4}^{2} - h_{1}^{2} h_{4} h_{5}), \end{matrix}

(17)

h_{5}' = {Re}_{m} S \frac{σ_{hnf}}{μ_{f}} \frac{μ_{f}}{μ_{hnf}} (h_{2} h_{4} - h_{1} h_{5}),

(18)

\begin{matrix} h_{7}' = - \frac{1}{(1 + \frac{4}{3} Rd {(1 + (θ_{w} - 1) h_{6})}^{3}) h_{7}} \\ [\begin{matrix} S \Pr \frac{{(ρ C_{p})}_{hnf}}{{(ρ C_{p})}_{f}} \frac{k_{f}}{k_{hnf}} (h_{7} h_{1}) + \frac{μ_{hnf}}{μ_{f}} \frac{k_{f}}{k_{hnf}} \\ PrEc (4 δ h_{2}^{2} h_{4} + h_{3}^{2}), \end{matrix}] \end{matrix}

(19)

As well as the modified boundary constraints

\begin{matrix} h_{1} (0) = α, h_{2} (0) = 1, h_{4} (0) = 0, h_{7} (0) + Bi (1 - h_{6} (0)) = 0, \\ h_{1} (1) = β, h_{2} (1) = λ, h_{4} (1) = 1, h_{6} (1) = 0, \end{matrix}

(20)

Mesh size and error tolerance are both 0.001 and $10^{- 6}$ , respectively, assuring precise numerical outputs. The velocity and temperature outlines that satisfy the boundary constraints at infinite are determined based on this estimate result. We can assume a simple starting estimate for the first response for the bvp4c procedure will relate to the first solution even with bad predictions.

We produced ODEs in a q-component group by inserting factor q. We employ q-variable in equations (17)–(19) to generate ODEs in a q-variable group, and therefore

\begin{matrix} h_{3}' = \frac{ρ_{hnf}}{ρ_{f}} \frac{μ_{f}}{μ_{hnf}} (S (h_{2} h_{3} - h_{1} (h_{3}' - 1) q) \\ + M^{2} S {Re}_{m} {(\frac{σ_{hnf}}{μ_{f}})}^{2} \frac{μ_{f}}{μ_{hnf}} (h_{1} h_{4}^{2}) \\ - M^{2} S^{2} {Re}_{m}^{2} {(\frac{σ_{hnf}}{μ_{f}})}^{3} \frac{μ_{f}}{μ_{hnf}} (h_{1} h_{2} h_{4}^{2} - h_{1}^{2} h_{4} h_{5}), \end{matrix}

(21)

h_{5}' = {Re}_{m} S \frac{σ_{hnf}}{μ_{f}} \frac{μ_{f}}{μ_{hnf}} (h_{2} h_{4} - h_{1} (h_{5} - 1) q),

(22)

\begin{matrix} h_{7}' = - \frac{1}{(1 + \frac{4}{3} Rd {(1 + (θ_{w} - 1) h_{6})}^{3}) h_{7}} \\ [\begin{matrix} S \Pr \frac{{(ρ C_{p})}_{hnf}}{{(ρ C_{p})}_{f}} \frac{k_{f}}{k_{hnf}} (h_{1} (h_{7} - 1) q) + \\ \frac{μ_{hnf}}{μ_{f}} \frac{k_{f}}{k_{hnf}} PrEc (4 δ h_{2}^{2} h_{4} + h_{3}^{2}), \end{matrix}] \end{matrix}

(23)

Differentiate by q outcomes in the following method in terms of factor-q values.

Differentiate equations (21)–(23) with respect to $q$

d_{1}' = h_{1} d_{1} + e_{1}

(24)

Where coefficient of the matrix is denoted by $h_{1}$ , $e_{1}$ is the reminder, and $d_{1} = \frac{d P_{i}}{d τ}, 1 \leq i \leq 8 .$

Cauchy Problem is

d_{1} = y_{1} + a_{1} v_{1},

(25)

Vector functions are denoted by $y_{1}, v_{1}$ . We can automatically fulfill the ODEs by determining the two Cauchy problems for each characteristic.

e_{1} + h_{1} (a_{1} v_{1} + y_{1}) = (a_{1} v_{1} + y_{1})'

(26)

With the help of numerical solution:

For the solution to the problem, a complete scheme was adopted.

\frac{v_{1}^{i + 1} - v_{1}^{i}}{Δ η} = h_{1} v_{1}^{i + 1}

(27)

\frac{y^{i + 1} - y^{i}}{Δ η} = h_{1} y^{i + 1} + e_{1}

(28)

Taking the appropriate factors,

Demarcated restrictions are commonly implemented for $P_{i}$ , where $1 \leq i \leq 8,$ for the solution of ordinary differential equations, we are instructed to apply $d_{2} = 0,$ this appears to be a matrix as

l_{1} . d_{1} = 0, or l_{1} . (a_{1} v_{1} + y_{1}) = 0

(29)

Where $a_{1} = \frac{- l_{1} . y_{1}}{l_{1} . v_{1}} .$

Response surface method (RSM)

The RSM is used to establish empirical relationships between a variety of input parameters and a variety of outputs. Given that it provides information on the least and most dominant input factors influencing the answers, it is a particularly excellent tool for evaluating multi-response and multi-variable processes. RSM was founded in 1951 by Box and Wilson,⁵⁴ and its main objective is response optimization. A collection of mathematical and statistical tools known as the Response Surface Methodology (RSM) can be used to model and analyze problems when a variety of variables affect the interest response. The goal is to make this answer better. The link between the responses and the independent factor in RSM problems is frequently ambiguous. Finding a good estimation for the best feasible connection between y and the group of independent variables is thus the first stage in RSM. In several regions of the independent factors, a moderate polynomial is widely used. If the answer can be precisely characterized by a linear function of the independent variables, then the first-order model is used as the approximation function. The response surface methodology was used to finally establish an empirical relationship between the convective heat transfer coefficient and optimization. To plot contour, sensitivity outcomes and tables, the response surface method is employed.⁵⁵

Coding for factors is used.

Sum of square is types III – Partial

The methodology is proposed to be meaningful by the simulation F-value of 3543.59. These values can be seen in Table 2. An F-value this strong might happen owing to interference only 0.01% of the time.

Table 2.

ANOVA for Local Nusselt number.

Sources	Sum of square	df	Mean squares	F-values	p-values
Models	2.68	9	0.2974	3543.59	<0.0001	Significantly
A-phi	0.0020	1	0.0020	24.16	0.0006
B-Rd	0.0515	1	0.0515	614.07	0.0001
C-Bi	2.49	1	2.49	29,645.44	<0.0001
AC	0.0014	1	0.0014	16.11	0.0025
BC	0.0203	1	0.0203	241.39	<0.0001
A²	0.0156	1	0.0156	186.31	<0.0001
B²	0.0027	1	0.0027	32.60	0.0002
C²	0.0848	1	0.0848	1010.90	<0.0001
Residual	0.0008	10	0.0001
Lack of Fit	0.0008	5	0.0002
Purely Errors	0.0000	5	0.0000
Cor Totals	2.68	19

Simulation aspects are considered meaningful when the p-value is lower than 0.0500. A, B, C, AC, BC, A2, B, and C2 are essential structural components in this instance. Regression coefficients are not meaningful if the value is higher than 0.1000. Simulation minimization may enhance your concept if it has a lot of unnecessary phrases (except those needed to maintain hierarchy).

Table 3 is Boxes-Behnken’s Concept against the real and the coded values, represent the combination of the parameter of volume fraction of nano particles $ϕ$ , thermal radiation parameter Rd and the Biot number Bi response of local skin friction and local Nusselt number.

Table 3.

Boxes-Behnken’s concept.

Std	Runs	Real values			Coded values
Std	Runs	$ϕ$	Rd	Bi	A	B	C	Cf	Nu
4	1	0.4	1	0.5	−1	0	1	−0.3035	−0.4445
12	2	0.25	1.3068	1.25	−1	0	1	−0.3289	−0.8619
20	3	0.25	0.55	1.25	0	0	0	−0.3289	−0.9155
6	4	0.4	0.1	2	1	−1	0	−0.3212	−1.4594
15	5	0.25	0.55	1.25	0	0	0	−0.3289	−0.9155
2	6	0.4	0.1	0.5	1	0	1	−0.3035	−0.4735
8	7	0.4	1	2	0	0	0	−0.3289	−1.2277
7	8	0.1	1	2	0	−1	−1	−0.3289	−1.1686
3	9	0.1	1	0.5	0	1	−1	−0.2853	−0.436
11	10	0.25	−0.2068	1.25	1	0	−1	−0.3212	−1.0505
5	11	0.1	0.1	2	0	0	0	−0.3289	−1.3991
1	12	0.1	0.1	0.5	0	1	−1	−0.3035	−0.4666
9	13	−0.0023	0.55	1.25	0	0	0	−0.3212	−1.001
19	14	0.25	0.55	1.25	0	1	1	−0.3289	−0.9155
17	15	0.25	0.55	1.25	1	1	0	−0.3212	−0.9155
16	16	0.25	0.55	1.25	−1	−1	0	−0.3289	−0.9155
10	17	0.5023	0.55	1.25	0	0	0	−0.2775	−1.0198
18	18	0.25	0.55	1.25	−1	1	0	−0.3289	−0.9155
14	19	0.25	0.55	2.5113	0	0	0	−0.3289	−1.4123
13	20	0.25	0.55	−0.0113	0	0	0	−0.3035	0.0119

For the goodness of fit persistence, relay on several indicators is considered. For instance, the first one is the lack of ft which has a p-value $< 0.001$ for all three fitted models. Secondly adjusted $R^{2}$ which presents how much the models explain the variation in response is used. For $Cf$ the adjusted $R^{2}$ is 96.2%, and for $Nu$ adjusted $R^{2}$ is 99.9%. As a result, all models account for a large amount of the variation in the corresponding responses. Thirdly, a typical residual quantile-quantile plot is used to evaluate the goodness of ft. In an appropriate model that successfully demonstrates the functional relationship between the input parameters and response, the relationship between theoretical and observed quantiles is proved to be one-to-one.

For three responses, three response surface equations are considered. For each model parameters A, B, and C are Nanoparticle volume fraction $(ϕ)$ , thermal radiation $Rd$ , and Biot number $Bi$ . For each of these parameters, three levels are low, medium and high levels coded as (−1, 0, 1) are chosen. The input parameters together with respective notations are presented in Table 3.

Final Equation in coded Units

\begin{matrix} C 8 = N u_{r} = - 0.9156 - 0.0122 * A + 0.0614 * C - 0.4268 * C \\ - 0.000 A * B - 0.0130 * A * C + 0.0503 * B * C - 0.0329 * A^{2} \\ - 0.0138 * B^{2} + 0.00767 * C^{2} \end{matrix}

In terms of model variables, sensitivity is usually calculated from the response function. Sensitivity analysis, as opposed to model vigor estimation, examines the unusual requirements provided by model output as assigned by input variables.

Final Equation in Actual Factor

\begin{matrix} C 8 = N u_{r} = - 0.091596 + 0.0795635 * ϕ + 0.025160 * Rd \\ - 0.963282 * Bi - 0.000741 * ϕ * Rd - 0.115556 * Bi * ϕ \\ + 0.149111 * Rd * Bi - 1.46404 * ϕ^{2} - 0.068041 * R d^{2} \\ + 0.136411 * B i^{2} \end{matrix}

Results and discussions

The influence of precise emerging variables in the governing flow framework on the velocity concentration, temperature panel and magnetic distribution is evaluated through graphically and tabular form. The influence of different included components such as Prandtl number, magnetic Reynolds number ${Re}_{m}$ , magnetic parameter, Eckert number squeezing component and nanomaterials volume fractions $ϕ$ have analyzed the behavior of heat and mass transfer. The bvp4c methodology, which is a significant numerical method, is used to decode the difficulties at hand. The response surface method is used to plot the sensitivity and contour graphs and tables. The ranges of pertinent parameters are given as $0.1 \leq S \leq 1.2, 0.1 \leq M \leq 1.5, 1.0 \leq {Re}_{m} \leq 4.0, 0.1 \leq Ec \leq 1.0,$ $0.1 \leq Rd \leq 1.5,$ and $1.5 \leq θ_{w} \leq 1.8 .$ The consequence of squeezing components $S$ via velocity panel $f$ is demonstrated in Figure 2. As the movement on the surface, the fluids are squeezed into the channel. From this figure, the velocity of fluids at the boundary region is increased due to increasing outcomes of the squeeze variable. Figure 3 depicts the effect of magnetic factor $M$ via flow panel $f$ . The velocity distribution $f$ of fluids is diminished owing to the upward valuation of $M$ . Similarly, the Lorentz force has been detected as an outcome of the valuation of the magnetic factor rising the resistance in the flow zone. The movement of fluid through the channel wall is inhibited as the plates advance, resulting in greater flow in the boundary region. The consequence of the Reynolds number ${Re}_{m}$ on velocity gradient $f$ is drawn in Figure 4. Due to enhancing values of Reynolds number the velocity of given fluids is boosted down. This figure shows that if we increase the outcomes of the Reynolds number the kinematic viscosity of fluids is enhanced. Enhancing the value of kinematic viscosity, the density of fluids is decreased. Figure 5 exemplifies the nature of the magnetic component $M$ on the axial velocity gradient $f'$ . The enhancing outcome of a magnetic factor $M$ the velocity of fluids is diminished. The Lorentz force is formed as a result of the magnetic field. The Lorentz force had an effect on the channel’s boundary area. This downward tendency is noticed because the introduction of a transversal magnetization causes the energy of the resistance nature known as Lorentz force, which tends to hinder fluid flow and so decelerates velocity. Figure 6 has drawn the implication of the Reynolds number variable ${Re}_{m}$ on the axial flow panel $f'$ . By enhancing outcomes of the Reynolds number factor ${Re}_{m}$ the velocity $f'$ of fluids is increased. The influence of the squeezing component $S$ on the axial velocity $f'$ is shown in Figure 7. By increasing the outcomes of the squeezing variable, $S$ the velocity of fluids is raised. The kinematic viscosity of the fluid divided by the top plate’s flow rate is known as the squeezing factor. It’s worth noting that large or small $S$ values denote the sluggish or quick flow. Greater $S$ outcomes suggest that the lower plate is moving away from the top plate, or that the distance between the two plates is growing, whereas diminishing $S$ values show that the top plates is revolving outside of the bottom plate, or that the gap among the two plates is decreasing.

Figure 2.

Upshot on velocity distribution $f$ of squeezing variable $S$ .

Figure 3.

Upshot on velocity distributions $f$ of the magnetism variable $M$ .

Figure 4.

Upshot on velocity distribution $f$ of Reynolds number ${Re}_{m}$ .

Figure 5.

Upshot on velocity distributions $f'$ of Magnetism factor $M$ .

Figure 6.

Upshot on velocity distribution $f'$ of Reynolds number ${Re}_{m}$ .

Figure 7.

Upshot of $S$ on velocity distribution $f'$ .

The impact of the squeezing factor $S$ on the thermal gradient $θ$ is drawn in Figure 8. By enhancing the consequences of the squeezing component, the thermal implication of fluids is downward. Practically, the increasing outcome of $S$ the fluid flow causes friction, which induces heat, as well as enhances the fluid’s kinematic energy, which also generates heat. Figure 9 plotted the impact of nanomaterials volume fraction $ϕ$ on the temperature panel $θ$ . By rising the outcomes of nanomaterials volume fraction $ϕ$ the fluid’s temperature is upward. The temperature gradient $θ$ of nanofluids becomes parabolic as the nanomaterials volume fraction $ϕ$ increases, approaching the bottom in the center of the channel. The consequence of the Eckert number $Ec$ on the temperature panel $θ$ is sketched in Figure 10. The thermal impact $θ$ of nano-fluids is enhanced by the increment in consequences of the Eckert number $Ec$ . Flow friction in nanoparticles is created with heightened efficiency as the $Ec$ increases. Kinetic energy is transferred into thermic energy in this physical process, which aids in the increment in a temperature gradient, and for greater results of Eckert number, the temperature panel turn into parabolic and reaches its extreme value in the center of the channel. Furthermore, boosting the volume proportion of nanomaterials causes the density of fluid particles to increase. Fluid flow is valued in this physical process, and the thermic characteristics of the fluid flow bits are lowered. The influence of the radiation component $Rd$ via temperature gradient $θ$ is shown in Figure 11. By enhancing outcomes of radiation parameters $Rd$ the temperature panel $θ$ of fluids is upsurge. Temperature and associated boundary layer thickness decrease as heat radiation increases. Modifications of the value of thermal radiation are due to the existence of conduction over radiation. Figure 12 exemplified the behavior of $θ_{w}$ on the thermal distribution $θ$ . The intensifying values of the temperature ratio factor have a positive effect on the temperature profile as observed from the graph. Figure 13 illustrated the features of the squeezing component $S$ on the $K$ . The magnetic profile $K$ of liquids is decline by increasing impact of the squeezing variable $S$ . It shows that the $K$ is mostly linear at $S = 0.1$ , but becomes parabolic as $S$ rises, approaching the maximal outcomes in the channel’s middle.

Figure 8.

Upshot on the temperature profile $θ$ of $S$ .

Figure 9.

Upshot on the thermal profile $θ$ of nanomaterials volumes fractions $ϕ_{1} = ϕ_{2}$ .

Figure 10.

Upshot on the temperature profile $θ$ of Eckert number $Ec$ .

Figure 11.

Upshot of radiation parameter $Rd$ on the temperature profile $θ$ .

Figure 12.

Upshot on the temperature profile $θ$ of temperature ratio parameter $θ_{w}$ .

Figure 13.

Upshot magnetic field profile $K$ of squeezing component $S$ on.

Figure 14 described the residual plots for squeezing flow. This figure describes the percentage of residual on normal probability, residual via fitted value on versus fits, frequency of residual on the histogram, and residual via observation order on versus order. Figure 15 Discussed the contour plot of response via thermal radiation parameter and volume fraction of nano particles. Figure 16 defines the Contour plot of response volume fraction and Biot number. Figure 17 depicted the contour plot of response via radiation parameter and Biot number. Figure 18 mentioned the 3-D surface plot between thermal radiation parameter and volume fraction parameter. Figure 19 represented the 3-D surface plot between radiation and volume fraction. Figure 20 highlighted the 3-D surface plot between Biot number and volume fraction.

Figure 14.

Residual plots for squeezing flow.

Figure 15.

Contour plot of Nu via thermal radiation and volume fraction.

Figure 16.

Contour plot of Nu against nano particle volume fraction and Biot number.

Figure 17.

Contour plot of Nu via radiation parameter $Rd$ and Biot number.

Figure 18.

Surface plot Nu via $Rd$ and volume fraction.

Figure 19.

Surface plot Nu via radiation parameter and Biot number.

Figure 20.

Surface plot Nu via volume fraction and Biot number.

Conclusion

A hybrid liquid flow between two plates is investigated in the existence of a magnetized factor. Utilizing resemblance modification, the suggested flow model’s controlling equations are turned into highly nonlinear structures of ordinary differential equations. For the flow model’s formulation, the simulation model bvp4c was operated. On the flow panel, thermal profile, and magnetic panel, the effects of several related factors are presented using various graphs and tables. The response surface method is employed to plot the sensitivity and contour graphs and tables. The key values of the current research are given following:

❖ Due to the improving value of nanoparticle volume fraction, the boundary layer’s consistency is diminished, and the velocity of fluids is raised.

❖ Due to rises outcomes of the Eckert number the temperature of fluids is boosted.

❖ The magnetic distribution of fluids is reduced by increment in the squeezing factor.

❖ Residual plots for squeezing flow, third surface plot via different parameters, and contour plots via various factors are also discussed.

❖ The insertion of nanoparticles into the base fluid significantly develops the velocity of the hybrid nanofluid and the rate of energy transfer.

Applications for hybrid nanofluid flow include petroleum engineering, geothermal engineering, automobile manufacturing, nuclear waste storage, thermal extrusion systems, heat exchangers, energy resources, ventilation, grain storage, and other industries. Because of their high thermal conductivity and low viscosity, hybrid nanofluids may have an impact on heat transfer systems.^54,55

Footnotes

Appendix

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hassan Waqas

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Thermal transport analysis of squeezing hybrid nanofluid flow between two parallel plates

Abstract

Keywords

Introduction

Mathematical modeling

Governing equations

Maxwell equations. 49

Energy equation. 46

Thermophysical properties

Boundary conditions

Transformed set of equation

Physical components

Numerical scheme

Response surface method (RSM)

Results and discussions

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

References

Maxwell equations.⁴⁹

Energy equation.⁴⁶