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Abstract

$ZnO - SAE 50$ nanolubricant is one of the most important nanolubricants having widespread uses in heat exchange systems and automobiles. It minimizes friction between moving parts and prevents them from corrosion and scrape, improves durability and performance of the system and also minimizes the consumption of fuel. This work investigates the rheological behavior of magnetized $ZnO - SAE 50$ nanolubricant over moving/stationary Riga plate with viscous dissipation and nonlinear heat generation. The novel features of the current work are the use of $ZnO - SAE 50$ nanolubricant as a flow fluid across a Riga plate and the application of the Patel model to boost the thermal conductivity of the nanolubricant. The governing system of equations is transformed to nonlinear ODEs and then treated analytically by using HAM. The augmentation in the velocity of nanolubricant $ZnO - SAE 50$ is observed due to increasing values of Grashof number. The higher values of nonlinear thermal radiation and nonlinear heat generation parameters upsurge the temperature profile. The value of skin friction increases by increasing modified Hartman number and velocity slip parameter while reverse trend is observed by enhancing magnetic parameter. The radiation parameter, temperature dependent heat source parameter and solid volume fraction tend to augment Nusselt number at the Riga surface.

Keywords

heat generation thermal radiation heat transfer Riga plate

Introduction

It is essential to comprehend the role that fluid motion plays, as it sharply alienates the many characteristics of how different fluids flow past various surfaces. The era of intelligent technology is upon us. According to the most recent research, viscous fluids’ boundary layer flow has changed to nanofluids due to the latter’s greatly increased thermal conductivity as well as to boost heat transmission capability. In the past quarter century, our esteemed scientists and researchers have applied their skills to meet our most urgent research demands. They have succeeded in making numerous nanofluids with exceptional thermal transport for cooling in a variety of microfluidic, heat transfer and bio-medical applications. Applications of these distinctive nanofluid features can be found in an amazingly wide range of industrial domains.^1–9

Nanolubricants are made up of nanoparticles dispersed in common lubricants. These are designed to improve the functionality of engine parts under high temperatures. As a result, many significant technical components, such as those utilized in military, aerospace, and industrial applications, experience less wear and friction in sliding contact. Under different thermal circumstances, nanoparticles are in fact heat stable and only slightly degrade when exposed to high temperatures. The density, viscosity, and specific heat capacity of the base fluid are indeed impacted by the addition of nanoparticles. Due to significant influence on fluid flow, heat transfer, and pumping power, viscosity is among other qualities that are strongly influenced and of major relevance. Therefore, the influence of nanoparticles on fluid viscosity has been extensively researched.^10–17 Additionally, the creation and effective dispersion of the nanoparticles are liable for the smooth surface shape of the nanolubricant and its low wear rate. The drop in wear rate and friction coefficient is primarily caused by these key facets. It is essential to utilize nanolubricants in metal-forming procedures and vehicles because they drastically cut fuel consumption and consequently environmental damage (deep-drawing, improved tool life). Numerous investigations have been made into how heat transfer qualities are affected by the thermal conductivity, viscosity, viscous dissipation, and viscosity index of nanolubricants. The studies revealed that at high temperatures, the nanolubricants exhibited better thermal effects. Nanolubricants are in high demand as they are used in heat pump compressors and offer advantages like to reduce friction which saves fuel, improving efficiency, low wear of moving parts, better thermal dissipation, enhancing horsepower, improving gas mileage, decreasing operating temperature, increasing lubricant life, and longer changeovers, as well as the breakdown of $SOx / NOx$ gases in burning engines, operative corrosion resistance, and eluding filter clogs. The heat dissipation in nanolubricant evaporators is studied theoretically and experimentally by Choi et al.¹⁸ in their publication. To address the limitations of conventional anti-wear and friction decline, the nanolubricant strategy is used.

In order to make $ZnO - SAE 50$ nanolubricants, zinc oxide $(ZnO)$ nanoparticles are normally well suspended in $SAE 50$ engine oil. The contemporary industrial environment requires $SAE 50$ lubricant due to its widespread use in heat exchange systems and in automobiles to lessen friction between numerous affecting components. Moreover, it refines performance, longevity, and lessens fuel consumption by preventing corrosion and abrasion of moving parts. Even with all of these benefits, $SAE 50$ has low heat conductivity. The development of $ZnO - SAE 50$ nanolubricant, which improves the rate of heat transfer of nano-lubricant $(ZnO - SAE 50)$ , was primarily motivated by the need to meet various industrial objectives, such as the thermal requirements in thermal systems that require a higher heat transfer rate.^19,20 It is obvious that using the anti-wear nanolubricant $ZnO - SAE 50$ in engine parts such that shaft, cylinder, gaskets, valve mechanisms, and gear camshaft may lessen the wear effect of nanomaterials. The convection-conduction heat transfer of a fin wetted with $ZnO - SAE 50$ nanolubricant is studied by Gamaoun et al.²¹ Nayak et al.^22–24 used flow behavior on various surfaces to study the heat transmission in $ZnO - SAE 50$ nanolubricant.

The peculiar structure of the Riga Plate produces surface-parallel Lorentz force which stabilizes movement by decelerating growth by altering pressure-gradient driven boundary layer structure. The behavior of nanofluid over Riga surface is investigated by Ahmad et al.²⁵ The role of buoyancy in nanofluid flow by utilizing a Riga surface under the effect of Lorentz force is addressed by Ahmad et al.²⁶ They found that solutal buoyant and thermal forces accelerated fluid movement at the expense of increased wall shear stress. The effects of HHRs on $NaCl - CNP$ nanofluid flowing over Riga surface have been studied by Nayak et al.²⁷ Giri et al.²⁸ investigated the Darcy-Forchheimer flow of nanofluid over Riga plate with chemical reaction. Abbas et al.²⁹ reported the magnetized micropolar fluid transportation over Riga surface. In studies, Shankaralingappa et al.^30,31 discussed the fluid flows over stretching sheet.

In engineering and manufacturing procedures like polymer technology, hot rolling, wire drawing, food fabrication, and the studies,^32–36 it is ingenuous to assume that concern magnetic field will significantly contribute to the creation of a controlled cooling system that provides final product qualities. The investigations on MHD heat transportation in a variety of physical systems has achieved prominent attention due to its substantial industrial and medicinal applications.^37–43 Some of the most important studies discussing the effects of MHD on flow of nanofluids over different geometries can be found in Giri et al.^44–47 and Das et al.⁴⁸

For the situations, such as polymer array, a few suspensions, different emulsion types and froths (liquid behaves as a particle) where the partial slip is the most appropriate BC, it is true that no slip BC is not quite sufficient. Numerous researchers are convinced to conduct research into a few fields related to the partial slip condition because of the significant opportunities in the polymer and electrochemical industries. The HTR grows larger while the skin grating decreases as a result of slip restriction. Aman et al.⁴⁹ reported this. According to Shaw et al.,⁵⁰ the slip parameters cause the domain of occurrence related to dual solution to grow. Using magnetization, Turkyilmazoglu⁵¹ treated viscoelastic slip flow. He looked into how the magnetic field cuts off HTR related to the first branch for fixed non zero slip while significantly enhancing it for the second branch. Recent research on entropy analysis and the effects of thermal slip on radioactive spinning nanofluid was conducted by Rehman et al.⁵² The use of thermal radiation at greater temperatures appears to have always been a feature of devices made to achieve excellent thermal performance. Beyond its original significance, ongoing research and competition between other researchers^6,53–55 have shown its numerous distinct behaviors as a result of its impact on the BLF of different nanofluids. The rate of heat transmission associated to the thermal boundary layer is significantly impacted by the heat generation that occurs simultaneously. Thermal boundary layer structure varies because the heat production altering the rate of heat transport. Sharma et al.⁵⁶ examined how an augmentation in the heat source/sink parameter causes a rise in fluid temperature. Numerous high-tech engineering processes include the combustion of fossil fuels, spacecraft re-entry, solar power technologies, astrophysical flows, and many others.^57–61

The literature survey reveals that only few investigations have been made on the flow behavior of $ZnO - SAE 50$ nanolubricant. None of the study has been conducted yet to discuss the flow behavior and heat transmission features by considering the flow of magnetized $ZnO - SAE 50$ nanolubricant over Riga plate with viscous dissipation and nonlinear heat generation. We aimed to investigate the impacts of viscous dissipation, nonlinear heat source, and thermal radiation on mixed convection Falkner-Skan flow of magnetized $ZnO - SAE 50$ nanolubricant over a moving or fixed Riga plate by taking inspiration from the aforementioned research. The flow takes place under the effects of thermal as well as velocity slips. A whole new perspective that no one had ever considered has been added to the current investigation. Investigation is done into the flow and HT characteristics of $ZnO - SAE 50$ , a brand-new form of nano-lubricant, as they relate to several physical parameters of engineering importance. The novel features of the current work are the use of the $ZnO - SAE 50$ nanolubricant as a flow fluid across a Riga plate and the application of micro-nano-convection model (Patel model) to boost the thermal conductivity of the nanolubricant.

Problem’s formulation

The Falkner-Skan slip flow of a magnetized $ZnO - SAE 50$ nanolubricant via moving/stationary Riga-plate at a constant velocity $U_{w} (x)$ is examined in the present investigation. The investigation is executed by including the viscous heating and nonlinear heat production factors. The electromagnetic Lorentz force that the Riga Plate induces is aligned to the plate surface. With respect to the distance normal to the plate, this force exhibits an exponential decline (Figure 1). Table 1 shows the thermophysical fatures of base fluid and nanoparticles.

Figure 1.

(a) Coordinate system; (b) Flow configuration.

Table 1.

The characteristics of $SAE 50$ (base fluid) and $ZnO$ (nanoparticles).²⁰

S. NO.		$ZnO$	$SAE 50$
$1$	$ρ (Kg / m^{3})$	$5.606$	$0.906$
$2$	$C_{p} (J / KgK)$	$544$	$1900$
$3$	$k (W / mK)$	$19$	$0.15$
$4$	$μ f (Ns . m^{2})$	−	$0.192543$
5	$df / ds$	60	40
6	$σ$	0.01	−

The vector forms of basic flow equations of Navier Stokes and conservation of energy are given as

Navier Stokes’ equation

ρ_{ZnO - SAE 50} (\frac{\partial V}{\partial t} + V . \nabla V) = - \nabla p + \nabla . τ + F,

(1)

F = F_{1} + F_{2} + F_{3} + F_{4}

(2)

here, $F_{1} = U \nabla U,$ $F_{2} = (ρ β)_{ZnO - SAE 50} g (T - T_{\infty}),$ $F_{3} = σ_{ZnO - SAE 50} B_{0}^{2} V$ , and $F_{4} = (F, 0, 0)$ is the electromagnetic force where $F = \frac{π j_{0} M_{0}}{8} e^{- (\frac{π}{p}) V} .$

Energy equation

\begin{matrix} {(ρ Cp)}_{ZnO - SAE 50} (\frac{\partial T}{\partial t} + V . \nabla T) = (k_{ZnO - SAE 50} + \frac{16 σ^{*}}{3 k^{*}}) \\ \nabla^{2} T + μ_{ZnO - SAE 50} {(\nabla V)}^{2} + Q, \end{matrix}

(3)

here $Q$ represents nonlinear heat generation term.

In component form, the governing equations of the present problem can be written as^22,62

\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0,

(4)

\begin{matrix} u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = U \frac{dU}{dx} + \frac{μ_{ZnO O - SAE 50}}{ρ_{ZnO - SAE 50}} \frac{\partial^{2} u}{\partial y^{2}} \\ + \frac{{(ρ β)}_{ZnO - SAE 50}}{ρ_{ZnO O - SAE 50}} g (T - T_{\infty}) + \frac{π j_{0} M_{0}}{8 ρ_{ZnO - SAE 50}} e^{- (\frac{π}{p}) y} \\ - \frac{σ_{ZnO - SAE 50}}{ρ_{ZnO - SAE 50}} B_{0}^{2} u, \end{matrix}

(5)

\begin{matrix} u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k_{ZnO - SAE 50}}{{(ρ Cp)}_{ZnO - SAE 50}} \frac{\partial^{2} T}{\partial y^{2}} + \frac{16 σ^{*}}{3 k^{*} {(ρ Cp)}_{ZnO - SAE 50}} \frac{\partial^{2} T}{\partial y^{2}} \\ + \frac{μ_{ZnO - SAE 50}}{{(ρ Cp)}_{ZnO - SAE 50}} {(\frac{\partial u}{\partial y})}^{2} + \frac{1}{{(ρ Cp)}_{ZnO - SAE 50}} \\ {Q_{0} (T - T_{\infty}) + Q_{1} (T_{w} - T_{\infty}) \exp (- n^{*} y \sqrt{\frac{A (n + 1) x^{n - 1}}{2 υ_{SAE 50}}})} . \end{matrix}

(6)

The appropriate BCs are²²

\begin{matrix} u = U_{w} (x) + L (\frac{\partial u}{\partial y}), v = 0, T = T_{w} + N (\frac{\partial T}{\partial y}) at y = 0 \\ u = U (x), T \to T_{\infty} as y \to \infty \end{matrix}}

(7)

The thermo physical properties of $ZnO - SAE 50$ nano-lubricant²⁰ are

\begin{matrix} \begin{matrix} μ_{ZnO - SAE 50} = μ_{SAE 50} (1.035 + 0.04336 φ - 0.002407 T) \\ μ_{SAES 0} = 2.174 e^{- 0.06062 T} \\ ρ_{Zn O - SAE 50} = (1 - φ) ρ_{SAE 50} + φ ρ_{Zn O} \\ {(ρ C_{p})}_{ZnO - SAE 50} = (1 - φ) {(ρ C_{p})}_{SAE 50} + φ {(ρ C_{p})}_{ZnO} \\ {(ρ β)}_{ZnO - SAE 50} = (1 - φ) {(ρ β)}_{SAE 50} + φ {(ρ β)}_{Zn O} \end{matrix}} \end{matrix}

(8)

and

σ_{ZnO - SAE 50} = σ_{SAE 50} [1 + \frac{3 (\frac{σ_{ZnO}}{σ_{SAE 50}} - 1) φ}{(\frac{σ_{ZnO}}{σ_{SAE 50}} + 2) - (\frac{σ_{ZnO}}{σ_{SAE 50}} - 1) φ}] .

Where $φ$ represents volume fraction $(%)$ and $T$ denotes temperature $(oC)$ .

The effective thermal conductivity of nano-lubricant by following Patel et al.⁶³ is

\begin{matrix} \begin{matrix} \frac{k_{ZnO - SAE 50}}{k_{SAE 50}} = 1 + \frac{k_{ZnO} A_{ZnO}}{k_{SAE 50} A_{SAE 50}} + c k_{ZnO} P_{e} (\frac{A_{ZnO}}{k_{SAE 50} A_{SAE 50}}), \\ \frac{A_{ZnO}}{A_{SAE 50}} = \frac{d_{SAE 50}}{d_{ZnO}} (\frac{φ}{1 - φ}), \\ Pe = \frac{u_{ZnO} d_{ZnO}}{α_{SAE 50}}, u_{ZnO} = \frac{2 k_{B} T}{π μ_{SAE 50} d_{ZnO}^{2}}, c = 25, 000 . \end{matrix}} \end{matrix}

(9)

The suitable transformations of our model are

\begin{matrix} u = A x^{n} F^{'} (η), \\ v = - \sqrt{\frac{A (n + 1) v_{SAES 0}}{2} x^{\frac{n - 1}{2}}} [F (η) + (\frac{n - 1}{n + 1}) η F^{'} (η)] \\ θ (η) = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, η = y \sqrt{\frac{A (n + 1) x^{n - 1}}{2 v_{SAES 0}}} . \end{matrix}

(10)

By applying the transformations given in equation $(10)$ , our governing problem is modified as

\begin{matrix} (\frac{(1.035 + 0.04336 φ - 0.002407 T)}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) F^{″'} + F F^{″} - β {(F^{'})}^{2} + β λ^{2} \\ + (2 - β) [\begin{matrix} (\frac{Gr}{R e^{2}}) (\frac{(1 - φ) + φ {\frac{{(ρ β)}_{ZnO}}{{(ρ β)}_{SAES 0}}}}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) θ \\ + Ha (\frac{1}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) \exp (- γ η) \end{matrix}] - (2 - β) \\ (\frac{1}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) (\frac{σ_{ZnO - SAE 50}}{σ_{SAE 50}}) MF' \\ = 0, \end{matrix}

(11)

\begin{matrix} (\frac{k_{ZnO - SAE 50}}{k_{SAE 50}} + \frac{4}{3} Rd) (\frac{1}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) θ^{″} \\ + \Pr (\frac{1}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) [Q_{t} θ + Q_{e} \exp (- n^{*} η)] \\ + EcPr (\frac{(1.035 + 0.04336 φ - 0.002407 T)}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) {F^{″}}^{2} + \Pr F θ^{'} = 0 . \end{matrix}

(12)

The modified BCs are

\begin{matrix} F^{'} (0) = 1 + Γ F^{″} (0), F (0) = 0, θ (0) = 1 + α θ^{'} (0) at η = 0 \\ F^{'} = λ, θ \to 0 as η \to \infty \end{matrix}}

(13)

Where

\begin{matrix} Ha = \frac{π j_{0} M_{0}}{8 A^{2} x^{2 n - 1} ρ_{SAE 50}}, γ = \frac{π}{p} \sqrt{\frac{2 x v_{SAE 50}}{(n + 1) U_{w}}}, G_{r} = \frac{g β_{SAE 50} (T_{w} - T_{\infty}) x^{3}}{v_{SAE 50}^{2}}, \\ Re = \frac{x U_{w}}{v_{SAE 50}}, β = \frac{2 n}{n + 1}, \Pr = \frac{v_{SAE 50}}{α_{SAE 50}}, Rd = \frac{4 σ^{*} T_{\infty}^{3}}{k^{*} k_{SAES 0}}, \\ Γ = L x^{\frac{n - 1}{2}} \sqrt{\frac{A (n + 1)}{2 v_{SAE 50}}}, λ = \frac{B}{A}, Ec = \frac{U_{w}^{2}}{{(Cp)}_{SAE 50} (T_{w} - T_{\infty})}, \\ Q_{t} = \frac{Q_{0} x}{{(ρ Cp)}_{SAE 50} U_{w}}, Q_{e} = \frac{Q_{1} x}{{(ρ Cp)}_{SAE 50} U_{w}}, α = N x^{\frac{n - 1}{2}} \sqrt{\frac{A (n + 1)}{2 v_{SAE 50}}} and \\ M = \frac{σ_{SAE 50} B_{0}^{2} x}{ρ_{SAE 50} A} . \end{matrix}

(14)

It is worth mentioning that $β < 0$ and $β > 0$ represent adverse pressure gradient and favorable pressure gradient respectively. Blasius flow is however represented by $β = 0$ . Additionally, $λ < 0$ and $λ > 0$ show the motion of Riga plate in opposite and same directions to the free stream velocity. Though, $λ = 0$ depicts the surface at rest.

The skin friction is locally defined as

{Re}^{\frac{1}{2}} C_{fx} = (1.035 + 0.04336 φ - 0.002407 T) \sqrt{\frac{n + 1}{2}} F^{″} (0) .

(15)

The Nusselt number is given as

R e^{- \frac{1}{2}} N u_{x} = - \frac{k_{ZnO - SAE 50}}{k_{SAE 50}} (1 + \frac{4}{3} Rd) \sqrt{\frac{n + 1}{2}} θ' (0) .

(16)

Solution methodology

The solution of our resulting nonlinear problem has been found by using HAM. Liao⁶⁴ hosted HAM approach for solving nonlinear differential equations for the first time. The role of HAM is vital in order to get convergent series solution. The followings are the chosen initial guesses and linear operators:

F_{0} (η) = \frac{e^{- η} (- 1 + λ + e^{η} (1 + (- 1 + η + η Γ) λ))}{1 + Γ},

(17)

θ_{0} (η) = \frac{e^{- η}}{1 + α}, E

(18)

L_{F} (F) = \frac{d^{3} F}{d η^{3}} + \frac{d^{2} F}{d η^{2}}, L_{θ} (θ) = \frac{d^{2} θ}{d η^{2}} + \frac{d θ}{d η},

(19)

where, $L_{F}$ and $L_{θ}$ satisfy the following conditions

L_{F} [C_{1} + C_{2} e^{- η} + C_{3} e^{η}] = 0, L_{θ} [C_{4} e^{- η} + C_{5} e^{η}] = 0 .

(20)

Zeroth order deformation

h_{F} N_{F}^{*} [F (η; q)] = \frac{1 - q}{q} L_{F} [F (η; q) - F_{0} (η)],

(21)

h_{θ} N_{θ}^{*} [F (η; q), θ (η; q)] = \frac{1 - q}{q} L_{θ} [θ (η; q) - θ_{0} (η)],

(22)

B.Cs.

\begin{matrix} {F (η; q) |}_{η = 0} = 0, {\frac{\partial F (η; q)}{\partial η} |}_{η = 0} = 1 + Γ {\frac{\partial^{2} F (η; q)}{\partial η^{2}} |}_{η = 0}, \\ {\frac{\partial F (η; q)}{\partial η} |}_{η \to \infty} = λ, \end{matrix}

(23)

{θ (η; q) |}_{η = 0} = 1 + α {\frac{\partial θ (η; q)}{\partial η} |}_{η = 0}, {θ (η; q) |}_{η \to \infty} = 0 .

(24)

In above equations $q ϵ [0, 1]$ is an embedding parameter, $h_{F}$ , $h_{θ}$ are auxiliary parameters and $N_{F}^{*}$ , $N_{θ}^{*}$ are auxiliary functions. Also, $N_{F}^{*}$ and $N_{θ}^{*}$ are nonlinear operators and are defined as:

\begin{matrix} N_{F}^{*} [F (η; q)] = (\frac{(1.035 + 0.04336 φ - 0.002407 T)}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) \\ \frac{\partial^{3} F}{\partial η^{3}} + F \frac{\partial^{2} F}{\partial η^{2}} - β {(\frac{\partial^{2} F}{\partial η^{2}})}^{2} + β λ^{2} \\ + (2 - β) [\begin{matrix} (\frac{Gr}{R e^{2}}) (\frac{(1 - φ) + φ {\frac{{(ρ β)}_{ZnO}}{{(ρ β)}_{SAES 0}}}}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) θ \\ + Ha (\frac{1}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) \exp (- γ η) \end{matrix}] \\ - (2 - β) (\frac{1}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) (\frac{σ_{ZnO - SAE 50}}{σ_{SAE 50}}) M \frac{\partial F}{\partial η}, \end{matrix}

(25)

\begin{matrix} N_{θ}^{*} [F (η; q), θ (η; q)] = (\frac{k_{ZnO - SAE 50}}{k_{SAE 50}} + \frac{4}{3} Rd) \\ (\frac{1}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) \frac{\partial^{2} θ}{\partial η^{2}} + \Pr F \frac{\partial θ}{\partial η} \\ + EcPr (\frac{(1.035 + 0.04336 φ - 0.002407 T)}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) {(\frac{\partial^{2} F}{\partial η^{2}})}^{2} \\ + \Pr (\frac{1}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) [Q_{t} θ + Q_{e} e^{- n^{*} η}] . \end{matrix}

(26)

Mth order deformation

The Taylor’s series expansions of $F (η; q)$ and $θ (η; q)$ with respect to $q$ are:

F (η; q) = F_{0} (η) + \sum_{m = 1}^{\infty} q^{m} F_{m} (η), F_{m} (η) = {\frac{1}{m!} \frac{\partial^{m} F}{\partial q^{m}} |}_{q = 0},

(27)

θ (η; \hat{p}) = θ_{0} (η) + \sum_{m = 1}^{\infty} q^{m} θ_{m} (η), θ_{m} (η) = {\frac{1}{m!} \frac{\partial^{m} θ}{\partial q^{m}} |}_{q = 0} .

(28)

The obtained $mth$ order deformation equations are

L_{F} [F_{m} (η) - χ_{m - 1} F_{m - 1} (η)] = h_{F} R_{m}^{F} (η),

(29)

L_{θ} [θ_{m} (η) - χ_{m - 1} θ_{m - 1} (η)] = h_{θ} R_{m}^{θ} (η),

(30)

\begin{matrix} {F_{m} (η; q) |}_{η = 0} = 0, {\frac{\partial F_{m} (η; q)}{\partial η} |}_{η = 0} - Γ {\frac{\partial^{2} F_{m} (η; q)}{\partial η^{2}} |}_{η = 0} = 0, {\frac{\partial F_{m} (η; q)}{\partial η} |}_{η \to \infty} = 0, \end{matrix}

(31)

{θ_{m} (η; q) |}_{η = 0} - α {\frac{\partial θ_{m} (η; q)}{\partial η} |}_{η = 0} = 0, {θ_{m} (η; q) |}_{η \to \infty} = 0,

(32)

where,

\begin{matrix} R_{m}^{F} (η) = (\frac{(1.035 + 0.04336 φ - 0.002407 T)}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) F_{m - 1}^{″'} + \sum_{k = 0}^{m - 1} F_{k} F_{m - 1 - k}^{″} - β \sum_{k = 0}^{m - 1} F_{k}^{'} F_{m - 1 - k}^{'} \\ + β λ^{2} + (2 - β) [(\frac{Gr}{R e^{2}}) (\frac{(1 - φ) + φ {\frac{{(ρ β)}_{ZnO}}{{(ρ β)}_{SAES 0}}}}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) θ + Ha (\frac{1}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) \exp (- γ η)] \\ - (2 - β) (\frac{1}{(1 - φ) + φ (\frac{ρ_{Zn 0}}{ρ_{SAE 50}})}) (\frac{σ_{ZnO - SAE 50}}{σ_{SAE 50}}) {MF}_{m - 1}^{'}, \end{matrix}

(33)

\begin{matrix} R_{m}^{θ} (η) = (\frac{k_{ZnO - SAE 50}}{k_{SAE 50}} + \frac{4}{3} Rd) (\frac{1}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) θ_{m - 1}^{″} (η) \\ + EcPr (\frac{(1.035 + 0.04336 φ - 0.002407 T)}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) \sum_{k = 0}^{m - 1} F_{k}^{'} F_{m - 1 - k}^{'} + \Pr F θ_{m - 1}^{'} \\ + \Pr (\frac{1}{(1 - φ) + φ {\frac{{(ρ C_{p})}_{ZnO}}{{(ρ C_{p})}_{SAES 0}}}}) [Q_{t} θ_{m - 1} + Q_{e} \exp (- n^{*} η)] \end{matrix}

(34)

χ_{m} = {\begin{matrix} 0, & m \leq 1 \\ 1, & m > 1 \end{matrix}

(35)

We solved the above equations by using Mathematica and attained the $mth$ term in the form

F_{m} (η) = F_{m}^{*} (η) + C_{1} + C_{2} e^{η} + C_{3} e^{- η},

(36)

θ_{m} (η) = θ_{m}^{*} (η) + C_{4} e^{η} + C_{5} e^{- η},

(37)

here, $F_{m}^{*} (η)$ and $θ_{m}^{*} (η)$ are particular solutions and $C_{1} - C_{5}$ are constants.

Hence, the analytical series solutions of $F (η)$ and $θ (η)$ are

F (η) = F_{0} (η) + \sum_{m = 1}^{\infty} F_{m} (η),

(38)

θ (η) = θ_{0} (η) + \sum_{m = 1}^{\infty} θ_{m} (η) .

(39)

Convergence

HAM is found to be the best method to find the solution of such type of systems of nonlinear ODEs. The mandatory convergent region has been adjusted by using auxiliary parameters $h_{F}$ and $h_{θ}$ . HAM solution converges in the range of $ξ (0 \leq ξ \leq \infty)$ for $h_{F} = h_{θ} = - 1 .$ The suitable ranges for auxiliary parameters are $- 2.8 \leq h_{F} \leq 0.1$ and $- 1 \leq h_{θ} \leq 0.5$ , which have been exposed in Figure 2(a) and (b).

Figure 2.

(a and b) h-curves for HAM solution.

Results and discussion

The present work analyzes the rheological behavior of magnetized $ZnO - SAE 50$ nanolubricant flow over moving/stationary Riga plate. The flow occurs in the presence of viscous dissipation and nonlinear heat source. The effects of velocity and thermal slips and thermal radiation have also been taken into account. The nonlinear transformed ODEs have been treated analytically by HAM. The motive behind this section is to discuss and analyze the behaviors of velocity and temperature profiles through graphs by varying different emerging parameters of interest.

Flow enactment of nanolubricant (ZnO – SAE50)

The effects of Grashof number $Gr$ , modified Hartman number $Ha$ , velocity slip parameter $Γ$ , pressure gradient parameter $β$ and solid volume fraction $φ$ on non-dimensional velocity $F' (η)$ of $ZnO - SAE 50$ nanolubricant for moving as well as stationary Riga plate have been depicted in Figures 3 to 7. The impact of thermal Grashof number $Gr$ on velocity profile $F' (η)$ for moving/static Riga plate is depicted in Figure 3. The larger values of thermal Grashof number $Gr$ support the motion of nanolubricant $ZnO - SAE 50$ and hence the velocity profile $F' (η)$ enhances. Figure 4 clarifies the effect of modified Hartman number $Ha$ on velocity profile $F' (η)$ for moving/stationary Riga plate. The fluid motion accelerates obviously by rising the values of modified Hartman number. For all three situations of the Riga plate, response of velocity profile is found to be the same (accelerated). This accelerated response occurs because the flow along the plate is supported by the Lorentz force which is parallel to the surface of the Riga plate. An asymptotic behavior exists for $F' (η)$ toward ambient fluid. The behavior of velocity profile $F' (η)$ with varied velocity slip parameter $Γ$ is shown is Figure 5. An augmentation in $F' (η)$ is observed by enhancing the values of $Γ$ . For $λ > 0$ (motion of Riga plate is parallel to free stream velocity), this augmented trend is however dominant and distinct. Overshoots occur in the slip flow regime, near the plate at the same time. This happens because of the influence of pressure gradient that supports the upright flow beside the Riga plate where the fluid velocity overcomes the free stream velocity. Additionally, for all three states of the Riga plate an asymptotic convergence of $F' (η)$ is determined by ambient fluid. The nature of velocity profile $F' (η)$ in correspondence to different Riga plate positions, in reaction to evocative values of pressure gradient parameter $β$ is revealed in Figure 6. It is obvious to see that for all three cases $(λ = 0, λ > 0 and λ < 0)$ , enhancement in pressure gradient parameter $β$ deeply influences the velocity profile $F' (η)$ in the decreasing sense. The dominance of pressure gradient force over inertial and viscous forces is responsible of this decreasing trend in $F' (η)$ . The effect of solid volume fraction $φ$ on the velocity profile $F' (η)$ is presented in Figure 7. The flow is hampered by the rise in liquid viscosity brought on by increasing volume fraction of nanoparticles and hence the velocity profile $F' (η)$ tends to decease.

Figure 3.

Effect of $Gr$ on $F' (η)$ .

Figure 4.

Effect of $Ha$ on $F' (η)$ .

Figure 5.

Effect of $Γ$ on $F' (η)$ .

Figure 6.

Effect of $β$ on $F' (η)$ .

Figure 7.

Effect of $φ$ on $F' (η)$ .

Thermal enactment of nanolubricant (ZnO – SAE50)

The thermal behavior of $ZnO - SAE 50$ nanolubricant corresponding to varied values of solid volume fraction $φ$ , Prandtl number $\Pr$ , thermal radiation parameter $Rd$ , thermal slip parameter $α$ , Eckert number $Ec$ , temperature dependent heat source parameter $Q_{t}$ and exponential dependent heat source parameter $Q_{e}$ over moving or stationary Riga plate is brought into real focus in Figures 8 –14. Figure 8 elaborates the impact of volume fraction parameter $φ$ on the temperature profile $θ (η)$ . The heat absorption capacity and viscosity of the nanolubricant $ZnO - SAE 50$ enhance with the addition of more quantity of nanoparticles ( $ZnO$ ), as a result temperature of the fluid rises. The effect of Prandtl number $\Pr$ on non-dimensional temperature $θ (η)$ of nanolubricant $ZnO - SAE 50$ is manifested in Figure 9. The enhancement in $\Pr$ causes a tremendous decline in the temperature profile $θ (η)$ . The thermal diffusivity gets weakened by increasing the value of $\Pr$ and is responsible of this decline in the temperature profile $θ (η)$ . Figure 10 exposes the effects of thermal radiation parameter $Rd$ on the temperature profile $θ (η)$ . Irrespective of the Riga plate situations (moving or stationary), the temperature profile $θ (η)$ shows increasing response toward higher values of $Rd$ . The purpose of applying thermal radiation in the current study is to enhance temperature of the nanolubricant $ZnO - SAE 50$ . The thermal systems require higher heat transmission and by applying thermal radiation we can achieve the desired temperature. Physically, more heat enters into the fluid by strengthening $Rd$ and boundary layer thickness gets enhanced which causes the fluid temperature to intensify. The representation of temperature profile $θ (η)$ with increasing values of thermal slip parameter $α$ is made in Figure 11. The decline in the temperature profile $θ (η)$ is observed by enhancing the values of $α$ . The higher values of $α$ shrinks the thermal boundary layer which results a decline in the temperature profile $θ (η)$ of the nanolubricant. Figure 12 brings out the impact of Eckert number $Ec$ on temperature profile $θ (η)$ . The kinetic energy of the fluid overcomes the enthalpy by increasing $Ec$ and hence in the entire flow domain temperature of the nanolubricant goes up. The effects of temperature dependent heat source parameter $Q_{t}$ and exponential dependent heat source parameter $Q_{e}$ on the temperature profile $θ (η)$ are presented in Figures 13 and 14. The thermal boundary layer significantly enhances by enhancing the value of temperature dependent heat source parameter $Q_{t}$ . As a result, an augmentation in the temperature profile $θ (η)$ happens and similar response is reported by increasing the value of exponential dependent heat source parameter $Q_{e}$ .

Figure 8.

Effect of $φ$ on $θ (η)$ .

Figure 9.

Effect of $\Pr$ on $θ (η)$ .

Figure 10.

Effect of $Rd$ on $θ (η)$ .

Figure 11.

Effect of $α$ on $θ (η)$ .

Figure 12.

Effect of $Ec$ on $θ (η)$ .

Figure 13.

Effect of $Q_{t}$ on $θ (η)$ .

Figure 14.

Effect of $Q_{e}$ on $θ (η)$ .

Validation of results

Table 2 represents the comparative values Nayak et al.²² by using the various values of radiative parameter Rd. Both the results have nice match with each other in limiting approach.

Table 2.

Comparison of present results with Nayak et al.²² when $M = Ec = Q_{t} = Q_{e} = 0$ .

$Rd$	Heat flux $ϕ = 0.0006250$		$ϕ = 0.0021875$		$ϕ = 0.003750$
	Nayak et al.²²	Present study	Nayak et al.²²	Present study	Nayak et al.²²	Present study
$0.5$	$16.7675$	$16.767510$	$18.7204$	$18.720410$	$20.6536$	$20.653602$
$1.0$	$22.5990$	$22.599003$	$25.2682$	$25.268211$	$27.9149$	$27.914920$
$2.0$	$33.4653$	$33.465311$	$37.4793$	$37.479304$	$41.4687$	$41.468710$

Concluding remarks

$ZnO - SAE 50$ nanolubricant is of one of the most important nanolubricants having widespread uses in heat exchange systems and automobiles. It minimizes friction between moving parts and prevents them from corrosion and scrape, improves durability and performance of the system and also minimizes the consumption of fuel. The present work explains the rheological behavior of nonlinear Falkner- Skan flow of magnetized nanolubricant $ZnO - SAE 50$ induced by the movement of Riga plate. The magnetic field plays vital role in controlling and adjusting the flow behavior. With the help of applied magnetic field, we can control the fluid speed and can direct the flow toward desired direction. The flow takes place in the presence of thermal and velocity slips along with mixed convection, heat radiation, and nonlinear heat generation. The nonlinear heat generation provides more heat to the fluid and works as a key factor to enhance temperature of the nanolubricant and to achieve the required cooling rates in thermal systems. The analytical solution of transformed nonlinear ODEs has been determined by using HAM. The major achievements of this study are as follows:

The velocity profile gets enhanced for higher values of thermal Grashof number in case of static as well as moving Riga plate conditions.

The larger values of modified Hartman number accelerate the motion of nanolubricant $ZnO - SAE 50$ irrespective of the Riga plate situations (moving or static).

The augmentation in the velocity profile of nanolubricant $ZnO - SAE 50$ is observed by giving rise to velocity slip parameter.

For all three cases (moving/static Riga plate), the increasing values of pressure gradient parameter decline the velocity profile.

The enhancing values of radiation parameter and Eckert number improved temperature profile.

The decline in temperature is revealed by the growing thermal slip factor and Prandtl number.

The higher values of nonlinear heat generation parameters upsurge the temperature profile.

The value of skin friction increases by increasing modified Harman number and velocity slip parameter while reverse trend is observed by enhancing magnetic parameter (see Table 3).

The radiation parameter, temperature dependent heat source parameter and solid volume fraction tend to augment the value of Nusselt number at the Riga surface (see Table 4).

The presence of nonlinear heat radiation and nonlinear heat generation vividly improve the rate of heat transfer of the nanolubricant $Zno - SAE 50$ , which is the significant finding of our study. This idea can be used in industries and automobiles to improve heat transmission rate, which is the dire need of the present age.

The computations reveal that our results are in excellent agreement with the existing study.

Table 3.

Effects of different parameters on skin friction $(C_{fx})$ .

$M$	$α$	$β$	$Γ$	$Ha$	$Gr$	$C_{fx}$
$0.1$	$0.5$	$0.4$	$0.7$	$0.8$	$0.9$	$2.90160$
$0.2$						$2.52220$
$0.3$						$2.14280$
	$0.1$					$2.94230$
$0.2$	$0.2$	$0.4$	$0.7$	$0.8$	$0.9$	$2.94280$
	$0.3$					$2.94300$
		$0.1$				$2.44432$
$0.6$	$0.5$	$0.3$	$0.7$	$0.8$	$0.9$	$2.14881$
		$0.5$				$1.84421$
			$0.1$			$1.56220$
$0.6$	$0.5$	$0.4$	$0.2$	$0.8$	$0.9$	$2.16280$
			$0.3$			$2.99320$
				$0.1$		$2.67350$
$0.6$	$0.5$	$0.4$	$0.7$	$0.2$	$0.9$	$2.78320$
				$0.3$		$2.89300$
					$0.1$	$2.17630$
					$0.2$	$2.24850$
$0.6$	$0.5$	$0.4$	$0.7$	$0.8$	$0.3$	$2.35410$

Table 4.

Effects of numerous parameters on Nusselt number $(N u_{x})$ .

$Rd$	$α$	$Ec$	$Q_{t}$	$Q_{e}$	$ϕ$	$λ$	$N u_{x}$
$0.1$	$0.5$	$0.4$	$0.3$	$0.4$	$0.006$	$0.8$	$18.5553$
$0.2$							$19.6156$
$0.3$							$20.6759$
	$0.3$						$22.6219$
$0.2$	$0.4$	$0.3$	$0.1$	$0.4$	$0.006$	$0.8$	$21.4521$
	$0.5$						$20.5633$
		$0.1$					$16.1522$
$0.2$	$2.0$	$0.2$	$0.1$	$0.4$	$0.006$	$0.8$	$16.1312$
		$0.3$					$16.1008$
			$0.2$				$22.4210$
$0.2$	$2.0$	$0.3$	$0.4$	$0.4$	$0.006$	$0.8$	$23.3229$
			$0.6$				$24.2213$
				$0.1$			$25.5329$
$0.2$	$2.0$	$0.3$	$0.1$	$0.3$	$0.006$	$0.8$	$25.3289$
				$0.5$			$25.1767$
					$0.006$		$19.3256$
$0.2$	$2.0$	$0.3$	$0.1$	$0.4$	$0.016$	$0.8$	$20.1263$
					$0.026$		$21.2523$
						$0.1$	$24.1281$
$0.2$	$2.0$	$0.3$	$0.1$	$0.4$	$0.7$	$0.2$	$25.1561$
						$0.3$	$26.1231$

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

Sabir Ali Shehzad

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Rheological behavior of magnetized ZnO – SAE 50 nanolubricant over Riga plate: A theoretical study

Abstract

Keywords

Introduction

Problem’s formulation

Navier Stokes’ equation

Energy equation

Solution methodology

Zeroth order deformation

Mth order deformation

Convergence

Results and discussion

Flow enactment of nanolubricant (ZnO – SAE50)

Thermal enactment of nanolubricant (ZnO – SAE50)

Validation of results

Concluding remarks

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

References