Viscosity robust weak Galerkin finite element methods for Stokes problems

Bin Wang; Lin Mu; Bin Wang; Lin Mu

doi:10.3934/era.2020096

Electronic Research Archive

2021, Volume 29, Issue 1: 1881-1895. doi: 10.3934/era.2020096

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Viscosity robust weak Galerkin finite element methods for Stokes problems

Bin Wang ^{1
,},
Lin Mu ^{2
,
,}

1.
Craft & Hawkins Department of Petroleum Engineering, Louisiana State University, 2245 Patrick F Taylor Hall, Baton Rouge, LA, 70803, USA
2.
Department of Mathematics, University of Georgia, Athens, GA, 30605, USA

Received: 01 April 2020 Revised: 01 July 2020 Published: 16 September 2020
Primary: 65N15, 65N30; Secondary: 35B45, 35J50, 35J35

In this paper, we develop a viscosity robust weak Galerkin finite element scheme for Stokes equations. The major idea for achieving pressure-independent energy-error estimate is to use a divergence preserving velocity reconstruction operator in the discretization of the right hand side body force. The optimal convergence results for velocity and pressure have been established in this paper. Finally, numerical examples are presented for validating the theoretical conclusions.

Keywords:

Citation: Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems[J]. Electronic Research Archive, 2021, 29(1): 1881-1895. doi: 10.3934/era.2020096

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Abstract

Nomenclature

$\overline{u}, \overline{v}$	velocity along measure coordinates [m/s]
${\mu }_{ternnf}$	enhanced dynamic viscosity [Pa s]
${\mu }_{hbnf}$	enhanced bihybrid dynamic viscosity [Pa s]
${\rho }_{ternnf}$	enhanced density [kg/m³]
${\rho }_{hbnf}$	enhanced bihybrid density [kg/m³]
${\alpha }_{ternnf}$	enhanced thermal diffusivity [m²/s]
${k}_{ternnf}$	enhanced thermal conductivity [W/(m K)]
${k}_{hbnf}$	enhanced bihybrid thermal conductivity [W/(m K)]
${\left(\rho {c}_{p}\right)}_{ternnf}$	improved heat capacity [J/K]
${\left(\rho {c}_{p}\right)}_{hbnf}$	improved bihybrid heat capacity [J/K]
$\overline{T}$	temperature [K]
${\overline{T}}_{\infty }$	ambient temperature [K]
${\overline{T}}_{w}$	wedge surface temperature [K]
${\overline{u}}_{w}$	velocity at the surface [m/s]
${\overline{\omega }}_{1}, {\overline{\omega }}_{2}, {\overline{\omega }}_{3}$	solid concentration of nanoparticles
${M}_{1}$	magnetic number
${P}_{r}$	Prandtl number
$\eta$	transformative variable
$F{'}$	dimensionless velocity
$\beta$	dimensionless temperature
${\lambda }_{1}$	wedge parameter

1. Introduction

In the present time, promising characteristics of nanofluids and their hybrid types ^[1,2] attained much interest of the researchers and engineers. Thus, Kudenatti et al. ^[3] analyzed the dynamics of Power Law Nanofluid (PLN) over a non-static wedge. The authors emphasized on the importance of MHD and their role in the controlling motion of the fluid. Later on, Akcay et al. ^[4] reported the behaviour of shear drag along a moving wedge by engaging the multiple effects of the significant physical controlling parameters. Also, analysis of the velocity and temperature distribution over the surface was major focus of the authors. The study of non-transient heat transport playing key role in many of the applied research areas and change the fluid movement under the multiple parameter ranges Therefore, Jafar et al. ^[5] provided an in-depth analysis of steady magnetohydrodynamic ^[6] boundary layer flow and discussed variations in boundary layer region due to increasing values of the parameters.

The study of variety of nanoparticles with enhanced characteristics attracted the researchers and engineers. Therefore, extensive efforts have been made to investigate the nanofluids characteristics from various physical aspects. In 2013, Ellahi ^[7] discussed the potential effects of MHD on non-newtonian nanoliquid. The model developed using thermal dependent viscosity and the model solutions computed via analytical scheme and analyzed the dynamics of the model for multiple appeared parameters. Irreversibility analysis of power-law nanoliquid for electro osmotic CPF (Couette-Poiseuille flow) reported in ^[8]. To improve the heat capability of the basic fluid, the authors preferred Al₂O₃ nanoparticles and examined interesting variations in entropy generation. In 2022, Bhatti et al. ^[9] emphasized on the study of magneto-nanoliquid using bihybrid nanoparticles. Diminishes in the fluid movement and concentration boundary layer with increasing magnetic, slip and Schmidt effects were core findings of the study. Besides these, nanoparticles extensively contribute in drug delivery systems. In this regard, a useful analysis provided by Bhatti et al. ^[10] and concluded that the study would be advantageous for biomedical engineering and those who are striving to investigate the dynamics of blood in arteries.

In 2023, Yasir et al. ^[11] performed thermal enhancement in bihybrid nanoliquids. The components of the working liquid taken as GO, Ag, AA7072 and MoS₂ and hybrid basic solvent H₂O/EG 50%/50%. The numerical simulation of the model done and analyzed the results of the interest and reported that the performed analysis would be expedient for electronic equipment cooling and heat exchangers devices. Abbas et al. ^[12] introduced a model for Sutterby nanofluid (SNF) by incorporating the influence of magnetic induction and Darcy resistance. The authors inspected that an increasing Eckert number corresponded to greater kinetic energy of the particles and thus an increase in temperature. Similarly, Murad et al. ^[13], Nisar et al. ^[14], Alsharari and Mousa ^[15] described deep knowledge about the characteristics of Casson-Carreau fluid directed to a stagnation point, changes in the nanoliquid performance due to slippery surface, activation energy, and buoyancy effects on copper/water nanofluid in the presence of an insulated obstacle. The studies showed that nanoliquids possessing outstanding thermal characteristics that ultimately increase their applications spectrum.

The transient heat transfer with hydrogen based nanoparticles and function fluid has tremendous characteristics and is extensively uses in multiple engineering disciplines. Thus, Mahmood and Khan ^[16] and Guedri et al. ^[17] analyzed the micropolar nanoliquid model and the effects of Al₂O₃ and Cu nanoparticles and on the heat transfer, shear drag and thermal transmission using basic fluid with hydrogen effects. The study supported findings that increasing the quantity of nanoparticles intensifies the shear drag coefficient and enhances the temperature of the fluid. Those researchers who are interesting in the field of nanofluids (see ^{[18,19,20,21,22]}) and their applications in various engineering zone paved their attention in the development of new innovative thermal transmission models and reported comparative or individual analysis. Some of the most latest and potential studies on various nanofluids described by the various scientists (for instance see ^{[23,24,25,26]}) using variety of tiny particles and basic functional fluid.

A comparative study of Sakiadis and Blasius flow reported by Klazly et al. ^[27]. The study revealed that shear drags rises for Sakiadis case and it drops for Blasius case. After that, Kumar et al. ^[28] reported MHD Blasius/Sakiadis flow of radiated Williamson fluid under effects of variable fluid characteristics. Devi et al. ^[29] focused on the investigation of 2D transient flow due to static sheet. Further, the authors integrated the significant effects of quadratic type radiation in chemically reactive fluid and interpreted the model results. The steady laminar boundary layer flow in the presence of Rosseland radiation was discussed by Pantokratoras et al. ^[30]. The study non-Newtonian Carreau, fluid along a static and a moving sheet reported in ^[31], examining wall shear drag and velocity profile for both Blasius and Sakiadis scenarios. Bataller ^[32] made efforts to analyze the surface convection effects on the boundary layer with thermal radiations effects. Hady et al. ^[33] discussed the heat transmission in single phase nanoliquids using shape factors effects. The effects of MHD and convective on BSF (Blasius and Sakiadis Flow) by using Cattaneo-Christov flux model over a sheet investigated in ^[34]. Similarly, the brief study of BSF using the effects of surface conditions and nanoparticles properties was reported by Krishna et al. ^[35]. They concluded that the Nusselt number increases for Sakiadis flow. Further, the heat transport characteristics of BSF MHD Maxwell fluid was analyzed by Sekhar ^[36].

The study of electrically conducting non-Newtonian fluid under the variations of physical quantities investigated by Pantokratoras et al. ^[37]. The two-dimensional BSF flow of MHD radiated Williamson fluid with chemically reacting species of variable conductivity explored in ^[28]. The study of free convection for BSF through porous media investigated in ^[38]. Oyem et al. ^[39] performed the analysis of BSF of 2D incompressible fluid with soret/dufour effects. Pantokratoras et al. ^[40] discussed the BSF for Riga-surface and graphically analyzed the velocity profile as well as skin friction for both cases.

Heat transfer investigation in boundary layer flows attained huge attention of the researchers in all times. Such flows have potential applications in designing of airplane wings, nuclear thermal plants, aerodynamics, applied thermal engineering, chemical engineering and many other applied research areas. The researchers made efforts to analyze the heat transfer through boundary layer using simple, mono nano and bihybrid nanofluids under the influence of additional physical aspects. However, no attempt has be made to report the comparative thermal transmission under the influence of induced magnetic field in three types of nanofluids (mono nano, bihybrid and ternary nanofluids) which is important to discuss in the field. For this, reason the present study for boundary layer flow past a wedge (Falkner Skan flow) with pores at the surface is conducted. Modelling of the problem completed via transformative equations and new effective thermo physical characteristics of ternary nanofluids. The results of the model will be advantageous for various engineering applications where enhanced heat transfer is essential and the parametric ranges were considered for achieving better results.

2. Materials and methods

2.1. Model statement and physical configuration

Considered a steady incompressible and two-dimensional boundary layer flow through a wedge influenced by induced magnetic field. Let us supposed that, the velocities at the wedge surface and far from the wedge considered as ${\overline{\mathrm{u}}}_{w}\left(x\right) = {\overline{U}}_{w}{x}^{m}$ and ${\overline{\mathrm{u}}}_{e}\left(x\right) = {\overline{U}}_{\infty }{x}^{m}$ , respectively. Here, ${\overline{U}}_{w}$ , ${\overline{U}}_{\infty }$ are velocities and $m$ is a constant with $0\le m\le 1$ . As Shown in Figure 1.

Figure 1. Physical flow scenario of ternary nanofluid past a wedge.

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The governing equations according to the above considered model are as follows ^[41]:

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

(1)

$\frac{\partial {\overline{H}}_{1}}{\partial x}+\frac{\partial {\overline{H}}_{2}}{\partial y} = 0,$

(2)

$\overline{u}\frac{\partial \overline{u}}{\partial x}+\overline{v}\frac{\partial \overline{u}}{\partial y}-\frac{{\mu }_{ternnf}}{4\pi {\rho }_{f}}\left({\overline{H}}_{1}\frac{\partial {\overline{H}}_{1}}{\partial x}+{\overline{H}}_{2}\frac{\partial {\overline{H}}_{1}}{\partial y}\right) = \left({\overline{u}}_{e}\frac{{d\overline{u}}_{e}}{dx}-{\mu }_{ternnf}\frac{{\overline{H}}_{e}}{4\pi {\rho }_{f}}\frac{{d\overline{H}}_{e}}{dx}\right)+\frac{{\mu }_{ternnf}}{{\rho }_{ternnf}}\frac{{\partial }^{2}\overline{u}}{{\partial y}^{2}},$

(3)

$\overline{u}\frac{\partial {\overline{H}}_{1}}{\partial x}+\overline{v}\frac{\partial {\overline{H}}_{1}}{\partial y}-{\overline{H}}_{1}\frac{\partial \overline{u}}{\partial x}-{\overline{H}}_{2}\frac{\partial \overline{u}}{\partial y} = {\mu }_{e}\frac{{\partial }^{2}{H}_{1}}{{\partial y}^{2}},$

(4)

$\overline{u}\frac{\partial \overline{T}}{\partial x}+\overline{v}\frac{\partial \overline{T}}{\partial y} = {\alpha }_{ternf}\frac{{\partial }^{2}\overline{T}}{{\partial y}^{2}}.$

(5)

The applicable physical conditions for the current flow of ternary nanoliquid are considered as below:

$\left\{\begin{array}{l}\overline{u}{|}_{y = 0} = {\overline{u}}_{w}\left(x\right), \overline{v}{|}_{y = 0} = 0, \overline{T}{|}_{y = 0} = {\overline{T}}_{w}, \\ \frac{\partial {\overline{H}}_{1}}{\partial y}{|}_{y = 0} = {\overline{H}}_{2}{|}_{y = 0} = 0, \end{array}\right.$

(6)

$\left\{\begin{array}{l}\overline{u}\to {\overline{u}}_{e}\left(x\right) = {\overline{U}}_{\infty }{x}^{m}, \overline{T}\to {\overline{T}}_{\infty }, \\ {\overline{H}}_{1}{|}_{y\to \infty } = {\overline{H}}_{e}\left(x\right) = {\overline{H}}_{0}{x}^{m}.\end{array}\right. \text{when}~~ y\to \infty.$

(7)

2.2. Nanofluids characteristics

Thermophysical properties of nanofluids ^[42,43] are of key interest in the analysis nanoliquids thermal transport. Therefore, the following properties (Table 1) of ternary nanofluid taken for estimation of enhanced nanoliquid characteristics.

Table 1. Equations for nanofluid, hybrid nanofluid and ternary nanofluid.

Dynamic viscosity
Nanofluid	${\mu }_{nf}={\mu }_{f}{\left[1-{\varpi }_{1}\right]}^{-2.5}.$
Hybrid nanofluid	${\mu }_{hbnf}={\mu }_{f}{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\right]}^{-1}.$
Ternary nanofluid	${\mu }_{ternnf}={\mu }_{f}{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}(1-{\varpi }_{3})\right]}^{-1}.$
Density
Nanofluid	${\rho }_{nf}=\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]{\rho }_{f}.$
Hybrid nanofluid	${\rho }_{hbnf}=\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}.$
Ternary nanofluid	${\rho }_{ternf}=\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}\right\}\right]+{\varpi }_{3}\frac{{\rho }_{p3}}{{\rho }_{f}}.$
Heat capacity
Nanofluid	$({\rho {c}_{p})}_{nf}=\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]{\left(\rho {c}_{p}\right)}_{f}.$
Hybrid nanofluid	$({\rho {c}_{p})}_{hbnf}=\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]+{\varpi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{p2}}{({\rho {c}_{p})}_{f}}.$
Ternary nanofluid	${\left(\rho {c}_{p}\right)}_{ternf}=\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\left(\rho {c}_{p}\right)}_{p1}}{({\rho {c}_{p})}_{f}}\right]+{\varpi }_{2}\frac{{\left(\rho {c}_{p}\right)}_{p2}}{({\rho {c}_{p})}_{f}}\right\}\right]+{\varpi }_{3}\frac{{\left(\rho {c}_{p}\right)}_{p3}}{({\rho {c}_{p})}_{f}}.$
Thermal conductivity
Nanofluid	${k}_{nf}=\frac{\left[\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)\right]}{[\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}({k}_{f}-{k}_{p1}\left)\right]}{k}_{f}.$
Hybrid nanofluid	${k}_{hbnf}=\frac{\left[\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)\right]}{[\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}({k}_{nf}-{k}_{p2}\left)\right]}{k}_{nf}.$
Ternary nanofluid	${k}_{ternnf}=\frac{\left[\left({k}_{p3}+2{k}_{hbnf}~~~ \right)-2{\varpi }_{3}\left({k}_{hbnf}-{k}_{p3}\right)\right]}{[\left({k}_{p3}+2{k}_{hbnf}~~~ \right)+{\varpi }_{3}({k}_{hbnf}-{k}_{p3}\left)\right]}{k}_{hbnf}.$

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To acquire to desired heat transfer model, the following similarity equations were used along with mathematical operations:

$\left.\begin{array}{l}\overline{u} = \frac{\partial \overline{\psi }}{\partial y} = {\overline{\mathrm{U}}}_{\infty }{x}^{m}{F}^{{'}}, \overline{v} = -\frac{\partial \overline{\psi }}{\partial x} = -{\left(\frac{\left(\overline{m}+1\right){\nu }_{f}\overline{U}\left(x\right)}{2x}\right)}^{\frac{1}{2}}\left(F+\frac{\overline{m}-1}{\overline{m}+1}\right)\eta {F}^{{'}}, \\ \overline{\psi } = {\left(\frac{2{\nu }_{f}xU\left(x\right)}{\left(m+1\right)}\right)}^{0.5}F, \beta = \frac{\overline{T}-{\overline{T}}_{\infty }}{{\overline{T}}_{w}-{\overline{T}}_{\infty }}, \eta = {\left(\frac{\left(m+1\right)U\left(x\right)}{2{\nu }_{f}x}\right)}^{0.5}, \\ {\overline{H}}_{1} = {\overline{H}}_{0}{x}^{m}{G}^{{'}}, {\overline{H}}_{2} = -{\overline{H}}_{o}{\left(\frac{2{\nu }_{f}{x}^{\overline{m}-1}}{\left(\overline{m}+1\right){\overline{U}}_{\infty }}\right)}^{0.5}\left\{\overline{m}G+0.5\left(\overline{m}-1\right)\eta {G}^{{'}}\right\}.\end{array}\right\}$

(8)

Finally, the following heat transport model obtained which includes the characteristics of ternary nanofluid.

${F}^{{'}{'}{'}}+\frac{\frac{{\rho }_{ternnf}}{{\rho }_{f}}}{\frac{{\mu }_{ternnf}}{{\mu }_{f}}}\left[{F}^{{'}{'}}F+2m{\left(m+1\right)}^{-1}\left(1-{F}^{{'}2}\right)+\frac{{2\delta }_{1}}{\left(m+1\right)}\left(m{G}^{{'}2}-m{G}^{{'}{'}}G-m\right)\right] = 0,$

(9)

${M}_{1}{G}^{{'}{'}{'}}+{G}^{{'}{'}}F-2m{\left(m+1\right)}^{-1}{F}^{{'}{'}}G = 0,$

(10)

$\frac{{k}_{ternnf}}{{k}_{f}}{\beta }^{{'}{'}}+\frac{Pr{\left(\rho {c}_{p}\right)}_{ternnf}}{{\left(\rho {c}_{p}\right)}_{f}}{\beta }^{{'}}F = 0,$

(11)

$\frac{{k}_{ternnf}}{{k}_{f}} = \left[\begin{array}{l}\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}*\frac{\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}{\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}*\\ \frac{\left({k}_{p3}+2{k}_{hbnf}~~~ \right)-2{\varpi }_{3}\left({k}_{hbnf}-{k}_{p3}\right)}{\left({k}_{p3}+2{k}_{hbnf}~~~ \right)+{\varpi }_{3}({k}_{hbnf}-{k}_{p3})}\end{array}\right].$

The above-described model is supported by the following boundary conditions over the wedge:

$F = 0, {F}^{{'}} = {\lambda }_{1}, G = 0, {G}^{{'}{'}} = 0, \beta = 1~~ at~~ \eta = 0,$

(12)

${F}^{{'}} = 1, G = 1, \beta \to 0~~when~~ \eta \to \infty .$

(13)

The embedded quantities are ${\delta }_{1} = \frac{{\overline{\mu }}_{e}{{\overline{H}}_{0}}^{2}}{4\pi {\rho }_{f}{{\overline{U}}_{\infty }}^{2}}, Pr = {\nu }_{f}{\alpha }_{f}^{-1}$ and ${\lambda }_{1} = \frac{{\overline{U}}_{w}}{{\overline{U}}_{\infty }}$ . Moreover, the significant (skin friction and Nusselt number) formulas described by the following expressions (Table 2):

Table 2. Skin friction and Nusselt number expressions for nano, hybrid and ternary nanofluids.

Skin friction formulas
Nanofluid	$\frac{{\left[1-{\varpi }_{1}\right]}^{-2.5}}{\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]}{F}^{{'}{'}}\left(0\right).$
Hybrid nanofluid	$\frac{{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\right]}^{-1}}{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}}{F}^{{'}{'}}\left(0\right).$
Ternary nanofluid	$\frac{{\left[{\left(1-{\varpi }_{1}\right)}^{2.5}{\left(1-{\varpi }_{2}\right)}^{2.5}\left(1-{\varpi }_{3}\right)\right]}^{-1}}{\left(1-{\varpi }_{3}\right)\left[\left\{\left(1-{\varpi }_{2}\right)\left[\left(1-{\varpi }_{1}\right)+{\varpi }_{1}\frac{{\rho }_{p1}}{{\rho }_{f}}\right]+{\varpi }_{2}\frac{{\rho }_{p2}}{{\rho }_{f}}\right\}\right]+{\varpi }_{3}\frac{{\rho }_{p3}}{{\rho }_{f}}}{F}^{{'}{'}}\left(0\right).$
Nusselt number formulas
Nanofluid	$\left\|\left[\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}\right]{\beta }^{{'}}\left(0\right)\right\|.$
Hybrid nanofluid	$\left\|\left[\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}*\frac{\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}{\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}\right]{\beta }^{{'}}\left(0\right)\right\|.$
Ternary nanofluid	$\left\|\left[\begin{array}{l}\frac{\left({k}_{p1}+2{k}_{f}\right)-2{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}{\left({k}_{p1}+2{k}_{f}\right)+{\varpi }_{1}\left({k}_{f}-{k}_{p1}\right)}\frac{\left({k}_{p2}+2{k}_{nf}~~ \right)-2{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}{\left({k}_{p2}+2{k}_{nf}~~ \right)+{\varpi }_{2}\left({k}_{nf}-{k}_{p2}\right)}\\ \frac{\left({k}_{p3}+2{k}_{hbnf}~~~ \right)-2{\varpi }_{3}\left({k}_{hbnf}-{k}_{p3}\right)}{\left({k}_{p3}+2{k}_{hbnf}~~~ \right)+{\varpi }_{3}\left({k}_{hbnf}-{k}_{p3}\right)}\end{array}\right]{\beta }^{{'}}\left(0\right)\right\|.$

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2.3. Mathematical analysis

The current model is coupled and contains high nonlinearities, and an exact solution is not feasible. Therefore, the RKF-45 (see ^[44,45,46]) implemented for the analysis of the model using the software MATHEMATICA 13.0 and the impacts of physical constraints on the heat transport performances of mono, hybrid and ternary nanofluids. The adopted technique is applicable for initial value problems and the desired initial value problem was obtained after using the following transforms in the system:

$\left[{\zeta }_{1}, {\zeta }_{2}, {\zeta }_{3}, {\zeta }_{3}^{{'}}, {\zeta }_{4}, {\zeta }_{5}, {\zeta }_{6}, {\zeta }_{6}^{{'}}, {\zeta }_{7}, {\zeta }_{8}, {\zeta }_{8}{'}\right] = \left[F, {F}^{{'}}, {F}^{{'}{'}}, {F}^{{'}{'}{'}}, G, {G}^{{'}}, {G}^{{'}{'}}, {G}^{{'}{'}{'}}, \beta , {\beta }^{{'}}, {\beta }^{{'}{'}}\right].$

(14)

The system was then arranged in the following way to reduce it into respective initial value problems.

${F}^{{'}{'}{'}} = -\frac{\frac{{\rho }_{ternnf}}{{\rho }_{f}}}{\frac{{\mu }_{ternnf}}{{\mu }_{f}}}\left[{F}^{{'}{'}}F+2m{\left(m+1\right)}^{-1}\left(1-{F}^{{'}2}\right)+\frac{{2\delta }_{1}}{\left(m+1\right)}\left(m{G}^{{'}2}-m{G}^{{'}{'}}G-m\right)\right],$

(15)

${{G}^{{'}{'}}}^{{'}} = -\frac{1}{{M}_{1}}\left[{G}^{{'}{'}}F-2m{\left(m+1\right)}^{-1}{F}^{{'}{'}}G\right],$

(16)

${\beta }^{{'}{'}} = -\frac{1}{\frac{{k}_{ternnf}}{{k}_{f}}}\left[\frac{Pr{\left(\rho {c}_{p}\right)}_{ternnf}}{{\left(\rho {c}_{p}\right)}_{f}}{\beta }^{{'}}F\right].$

(17)

Now, the system takes the below appropriate form for further computation:

${\zeta }_{3}^{{'}} = -\frac{\frac{{\rho }_{ternnf}}{{\rho }_{f}}}{\frac{{\mu }_{ternnf}}{{\mu }_{f}}}\left[{\zeta }_{3}{\zeta }_{1}+2m{\left(m+1\right)}^{-1}\left(1-{\zeta }_{2}^{2}\right)+\frac{{2\delta }_{1}}{\left(m+1\right)}\left(m{\zeta }_{5}^{2}-m{\zeta }_{5}^{{'}}{\zeta }_{4}-m\right)\right],$

(18)

${\zeta }_{6}^{{'}} = -\frac{1}{{M}_{1}}\left[{\zeta }_{6}{\zeta }_{1}-2m{\left(m+1\right)}^{-1}{\zeta }_{3}{\zeta }_{4}\right],$

(19)

${\zeta }_{8}^{{'}} = -\frac{1}{\frac{{k}_{ternnf}}{{k}_{f}}}\left[\frac{Pr{\left(\rho {c}_{p}\right)}_{ternnf}}{{\left(\rho {c}_{p}\right)}_{f}}{\zeta }_{8}{\zeta }_{1}\right].$

(20)

Accuracy of the technique is achieved by setting its tolerance up to ${10}^{-6}$ and the step size $0.01$ . The authenticity of the results is subject to the asymptotic behavior of the temperature profile and it must satisfy the boundary conditions for the velocity and temperature distributions. These are obvious from the plotted results which gives the correctness of the model results.

3. Results and discussion

The physical results of the model representing the flow of ternary, nano and bihybrid nanoliquids over a nonstationary wedge are demonstrated in this section. It is commendable to mention here that the three types of nanoparticles namely Al₂O₃, Cu and Ag taken for the formation of resultant nanoliquid and H₂O is taken as working base solvent due its good solvent characteristics, density and high heat capacity. Further, ${\overline{\omega }}_{1}, {\overline{\omega }}_{2}$ and ${\overline{\omega }}_{3}$ are the associated concentration of the nanoparticles in the basic solvent. Further, the concentrations of nanoparticles were taken up to 20% and feasible value of Prandtl number for water is fixed at $6.2$ . This section further classified into three subsequent subsections which represent the model results for the velocity, ternary nanoliquid temperature, shear drag and Nusselt number. The study validation is also presented in subsection 4.4.

3.1. Impacts of the flow parameters on $F{'}\left(\eta \right)$ and $G{'}\left(\eta \right)$

demonstrating the velocity $F{'}\left(\eta \right)$ changes under potential effects of $m,$ moving wedge number ${\lambda }_{1}$ and magnetic number ${\delta }_{1}$ . The presentation of the velocity distribution shows that the fluid movement increases by enlarging the values of $m$ and ${\lambda }_{1}$ . Physically, when the wedge moves then the neighboring fluid layer on the wedge surface gain the velocity of the wedge. As a consequence the particles moves rapidly. Then the frictional forces with the successive fluid layer reduces and the fluid velocity increases. After, $\eta = 3.0$ the velocity of the fluid reaches to its maximum speed and then moves with the speed of free stream. These results elucidated in Figures 2(a) and 2(b), respectively. On the other side, the magnetic number reduces the fluid motion and almost negligible variations observed in Figure 2(c).

Figure 2. The velocity profile

${F}^{{'}}$ against different values of parameters (a)

$m$ , (b)

${\lambda }_{1}$ , and (c)

${\delta }_{1}$ .

DownLoad: Full-Size Img PowerPoint

and organized to analyze the tangential velocity distribution $G{'}\left(\eta \right)$ for increasing ${\lambda }_{1}, {M}_{1}$ and ${\delta }_{1}$ . The tangential velocity of the fluid varies in very interesting way while the wedge moves and the reciprocal Hartmann parameter ${M}_{1}$ enlarges. It is inspected that $G{'}\left(\eta \right)$ declines prominently near the wedge surface (); however, these variations become slow and finally reaches to the its maximum limit i.e., ${G}^{{'}}\left(\eta \right) = 1$ after which the fluid moves with constant velocity. indicates that the reciprocal Hartmann parameter () is effective in increasing the velocity component $G{'}\left(\eta \right)$ and is maximum fluctuation is observed around $\eta = 0$ . The 3D representation of the results in Figures 3(a) and 3(b) given in Figures 3(c) and , respectively. demonstrating the impacts of ${\delta }_{1}$ which shows that the velocity drops when the parameter ${\delta }_{1}$ enhances and maximum drop is noticed for nanoliquids. supports the 3D pictorial view of the ${\delta }_{1}$ variations on the profile of $G{'}\left(\eta \right)$ .

Figure 3. The velocity profile G

${'}$ against (a), (c)

${\lambda }_{1}$ and (b), (d)

${M}_{1}$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. The velocity profile G' against

${\delta }_{1}$ .

DownLoad: Full-Size Img PowerPoint

3.2. Impacts of the flow parameters on temperature

This subsection represents the comparative enhanced heat transfer for Al₂O₃/H₂O (mono nanoliquid), (Al₂O₃-CuO)/H₂O (bihybrid nanoliquid) and (Al₂O₃-CuO-Ag)/H₂O (ternary nanoliquid) due to variations in ${\delta }_{1}, {\lambda }_{1}$ and ${M}_{1}$ . For this, Figure 5 furnished.

Figure 5(a) indicates that when the magnetic forces enlarges, the temperature of the fluids also increases. However, higher temperatures are observed for ternary nanoliquid than bihybrid and mono nanoliquids. Physically, ternary nanoliquid (Al₂O₃-CuO-Ag)/H₂O comprises the thermal conductivity and heat capacity of three particles which increase thermal conductivity of ternary nanoliquid which enhance it heat transport property. Thus, in ternary nanoliquid the heat transfer is greater than in the other nanofluids. Similarly, the temperature decreases with increasing ${\lambda }_{1}$ () but the behavior is greater in the ternary nanofluid. Further, thermal boundary layer thickness increase for larger ${\delta }_{1}$ and no significant contribution of the reciprocal Hartmann number (Figure 5c) was noticed in thermal enhancement of the nanofluids.

Figure 5. The temperature trends via (a)

${\delta }_{1}$ (b)

${\lambda }_{1}$ and (c)

${M}_{1}$ .

DownLoad: Full-Size Img PowerPoint

3.3. Impacts of the flow parameters on shear drag and Nusselt number

The study of shear drags and thermal gradient (Nusselt number) in nanofluids is essential from various engineering field and industrial purposes. Therefore, this subsection fill this requirement under the variations of the model parameters. For this, Figures 5 and 6 plotted for the shear drags and Nu trends.

and present the shear drag in the mono nanofluid, bihybrid nanofluid and ternary nanofluid against ${\delta }_{1}$ and ${\lambda }_{1}$ , respectively. In both the cases, the greatest shear drag effects were in the ternary nanoliquids. Physically, composite density of (Al₂O₃-CuO-Ag) nanoparticles make the resultant fluid denser than that of mono (Al₂O₃) and bihybrid (Al₂O₃-CuO) nanoparticles. Due to this reason maximum shear drag observed on the wedge surface.

Figure 6. The changes of skin friction against (a)

${\delta }_{1}$ and (b)

${\lambda }_{1}$ .

DownLoad: Full-Size Img PowerPoint

Figure 7. The changes of Nu with increasing (a)

${\delta }_{1}$ and (b)

${\lambda }_{1}$ .

DownLoad: Full-Size Img PowerPoint

The analysis of thermal gradient at the surface is important for heating/cooling purposes. Thus, the set of and show the trends of Nusselt number on the surface for increasing values of ${\delta }_{1}$ and ${\lambda }_{1}$ , respectively. For both parametric variations, the Nusselt number is greatest in the ternary nanoliquid. Physically, the sum of thermal conductivities of the nanoparticles boosts the thermal conductivity of ternary nanofluid. Therefore, maximum heat transmitted on the wedge surface. However, for bihybrid and mono nanoliquids, the heat transport mechanism is slow due to weak thermal conductivity.

3.4. Study validation

The study validation successfully performed and is presented in this subsection. In order make the current model compatible with the existing models in open science literature, it is essential to set the values ${\phi }_{j}$ for $j = \mathrm{1, 2}, 3$ , ${\lambda }_{1}$ and ${\delta }_{1}$ such that these tends to zero. Then, the model results for shear drag $F{'}{'}\left(0\right)$ computed at multiple stages of the parameter $m$ . These results given in Table 3 and it is cleared that the current results aligned with the results of Ahmed et al. ^[47], Watanabe ^[48] and Adnan et al. ^[49]. This shows that the results obtained from the study are correct and can be replicated in future studies.

Table 3. Validation of the present model results with the previous efforts made by different researchers.

$\boldsymbol{m}$	Results from various studies for $\boldsymbol{F}\boldsymbol{{'}}\boldsymbol{{'}}\left(0\right)$
$\boldsymbol{m}$	present results	Ahmed et al. ^[47]	Watanabe ^[48]	Adnan et al. ^[49]
0.0	0.4695999960	0.46959	0.46960	0.469590
0.0141	0.5046143299	0.504614	-----	0.504614
0.0435	0.5689777817	0.568977	0.56898	0.568977
0.0909	0.6549788596	0.654978	0.65498	0.654978
0.1429	0.7319985706	0.731998	0.73200	0.731998
0.2000	0.8021256343	0.802125	0.80213	0.802125
0.3333	0.9276536249	0.927653	0.92765	0.927653
0.5000	1.0389035229	1.038903	1.03890	1.038903

| Show Table

DownLoad: CSV

4. Conclusions

The study of thermal enhancement in ternary nanofluids over a moving wedge amid an induced magnetic field is presented. The basic model was transformed into a simplified form via defined transformative rules and improved properties of the ternary nanofluid. To investigate the influence of the model parameters on the dynamics of fluid, numerical analysis was conducted and the results were portrayed multiple parametric values. It is concluded that:

• The fluids movement (nanofluid, hybrid nanofluid and ternary nanofluid) can be intensified by increasing the values of $m$ and ${\lambda }_{1}$ as $0.1, 0.2, 0.3$ and $1.0, 1.2, 1.3$ , respectively.

• The magnetic field strongly opposed the fluid velocity over the wedge surface.

• The tangential velocity $G{'}\left(\eta \right)$ diminishes rapidly when ${\lambda }_{1}$ and ${\delta }_{1}$ increase as $1.0, 1.2, 1.3$ .

• The ternary nanofluid has high capacity to transmit the heat with increasing ${\delta }_{1}$ , while the transmission in the nanofluid and hybrid nanoliquid is slow.

• The thermal gradient in ternary nanoliquid was 65%, for the hybrid nanofluid, it was 45% and for the common nanofluid it was 35% which shows that ternary nanofluids are excellent for thermal transport applications.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through large group Research Project under grant number RGP2/16/44.

Conflict of interest

The authors declare no conflict of interest.

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Viscosity robust weak Galerkin finite element methods for Stokes problems

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Abstract

1. Introduction

2. Materials and methods

2.1. Model statement and physical configuration

2.2. Nanofluids characteristics

2.3. Mathematical analysis

3. Results and discussion

3.1. Impacts of the flow parameters on $F{'}\left(\eta \right)$ and $G{'}\left(\eta \right)$

3.2. Impacts of the flow parameters on temperature

3.3. Impacts of the flow parameters on shear drag and Nusselt number

3.4. Study validation

4. Conclusions

Acknowledgments

Conflict of interest

References

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Electronic Research Archive

Viscosity robust weak Galerkin finite element methods for Stokes problems

Related Papers:

Abstract

1. Introduction

2. Materials and methods

2.1. Model statement and physical configuration

2.2. Nanofluids characteristics

2.3. Mathematical analysis

3. Results and discussion

3.1. Impacts of the flow parameters on F′(η) F{'}\left(\eta \right) and G′(η) G{'}\left(\eta \right)

3.2. Impacts of the flow parameters on temperature

3.3. Impacts of the flow parameters on shear drag and Nusselt number

3.4. Study validation

4. Conclusions

Acknowledgments

Conflict of interest

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3.1. Impacts of the flow parameters on $F{'}\left(\eta \right)$ and $G{'}\left(\eta \right)$