Entropy optimized MHD nanomaterial flow subject to variable thicked surface

doi:10.1016/j.cmpb.2019.105311

Computer Methods and Programs in Biomedicine

Volume 189, June 2020, 105311

https://doi.org/10.1016/j.cmpb.2019.105311 Get rights and content

Highlights

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Here entropy generation in viscous fluid flow over a variable thicked surface is addressed.
•
Electrical conducting fluid is considered.
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Heat generation/absorption, dissipation and Joule heating effects are considered.
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Brownian and thermophoresis diffusion effects are further accounted.

Abstract

Here we investigate the irreversibility aspects in magnetohydrodynamics flow of viscous nanofluid by a variable thicked surface. Viscous dissipation, Joule heating and heat generation/absorption in energy expression is considered. Behavior of Brownian diffusion and thermophoresis are also discussed. The nanoliquid is considered electrical conducting under the behavior of magnetic field exerted transverse to the sheet. Using similarity variables the nonlinear PDEs are altered to ordinary one. The obtained system are computed through Newton built in shooting method. Significant behavior of various involving parameters on entropy generation rate, velocity, concentration, Bejan number and temperature are examined. Gradient of velocity and heat transfer rate are numerically computed through tabulated form. Velocity field is augmented versus power index (n). Temperature and velocity profiles have opposite characteristics for larger approximation of Hartmann number. Concentration profile has similar impact against Brownian diffusion variable and Lewis number. Entropy optimization is boost up via rising values of Brinkman and Hartmann numbers. Bejan number is declined for increasing value of Hartmann number.

Introduction

The suspension of nano-size solid particles (1-100nm) in traditional (base) liquids is known as nanoliquids. Nanomaterials have more thermal efficiency than compared to traditional liquids. Thermal efficiency of traditionally working materials is improved through enclosure of nanoparticles into it. Nanomaterials have widespread applications in numerous fields including nuclear reactors chilling, energy stowing structures, micro-manufacturing, thermal power plants and heat exchangers etc. Initially Choi [1] experimentally discussed nanoparticles and designated their influence in traditional liquids concerning thermal conduction efficiency of the traditionally working materials extra-ordinary. Buongiorno [2] described a model which revealed that thermal efficiency of traditional liquids is enhanced through Brownian diffusion and thermophoresis factors. Hayat et al. [3] scrutinized the Brownian diffusion and thermophoresis behaviors in third grade nanoliquid with nonlinear heat flux subject to rotating disk. Liu and Liu [4] scrutinized the Maxwell fluid flow by a stretchable surface with variable thickness. Magnetohydrodynamic flow of Powell-Eyring nanoliquid with thermophoresis and Brownian movement phenomena subject to variable thickness surface is illustrated by Hayat et al. [5]. Temperature dependent thermal conductivity impact in hydro-magnetic convective flow of non-Newtonian liquid with heat generation/absorption and variable thickness is exemplified by Awais et al. [6]. Variable thermal conductivity and heat flux effect in Jeffrey liquid flow with Cattaneo–Christov heat flux subject to a variable thicked surface is deliberated by Hayat et al. [7]. Some other advancement about nanomaterials and variables thickness are highlighted in Refs [8], [9], [10], [11], [12], [13], [14].

Entropy optimization is used to enhance the system performance. Entropy generation is caused due to heat fluxes, Joule heating, dissipation and mass fluxes etc. For higher entropy optimization in a system then there are more probabilities of irreversibilities therefore proficiency of our system reduces. To make the systems for good productivity we decrease the entropy optimization of the system. Irreversibility of an isolated system will never reduce however it is diminished in non-isolated system. Second law of thermodynamics provides the systematic tools and entropy generation for reduction of resistance. It helps to progress the significance of several electronic devices and engineering improvement. Initially Bejan [15] modified the entropy optimization analysis technique. Gibanov et al. [16] deliberated the magnetohydrodynamic natural convection flow of nanoliquid with entropy generation rate. Characteristics of induced magnetic field and irreversibility in viscous nanoliquid over a stretchable surface are reported by Iqbal et al. [17]. Irreversibility exploration in Ree-Eyring nanoliquid flow with activation energy and heat generation/absorption subject to rotating disks is exemplified by Hayat et al. [18]. Hosseinzadeh et al [19]. scrutinized the irreversibility impact in ethylene glycol based CNTs with heat flux. Impact of irreversibility in Prandtl–Eyring nanoliquid with heat flux and activation energy through stretching of sheet is illustrated by Khan et al. [20]. Numerical simulation of MHD convective flow of viscous liquid with entropy rate is deliberated by Alkanhal et al. [21]. Characteristics of heat flux and irreversibility in Jeffrey nanoliquid over a stretchable sheet is exemplified by Hayat et al. [22]. Some relevant investigation about entropy optimization problems is highlighted in Refs. [23], [24], [25].

In current discussion we scrutinize behavior of magnetohydrodynamic flow of viscous nanoliquid by a variables thicked surface. The Brownian movement and thermophoresis behaviors are the key factor in nanoliquid flow. Joule heating, dissipation and heat generation/absorption in heat equation are deliberated. Irreversibility investigation is also scrutinized. Nonlinear expression are converted ordinary one by similarity variables. The obtained nonlinear expression are solved through ND-solve technique. Influence of different engineering variables on entropy generation, temperature, Bejan number velocity and concentration are graphically examined. Computational results of velocity gradient and Nusselt number are also examined through tables.

Section snippets

Mathematical formulation

Consider two dimensional and incompressible magnetohydrodynamic nanomaterials by nonlinear stretchable surface. Heat expression is deliberated through Joule heating, dissipation and heat generation/absorption. Brownian diffusion and thermophoresis characteristics are also accounted. Physical featured of irreversibility are discussed. Considered the sheet is to be in x-axis which is along the stretchable surface and y-axis is normal to it. Let $u = U_{w} = a {(x + b)}^{n}$ stretching sheet velocity in which a is

Surface drag force

Velocity gradient $(C_{f_{x}})$ is $C_{f_{x}} = \frac{τ_{w}}{ρ U_{w}^{2}}$ shear stress τ_w is $τ_{w} = {μ (\frac{\partial u}{\partial y}) |}_{y = A_{1} {(x + b)}^{\frac{1 - n}{2}}}$ one can find $C_{f_{x}} {R e}_{x}^{1 / 2} = \sqrt{\frac{n + 1}{2}} f^{″} (0) .$

Nusselt number

Temperature gradient (Nu_x) is $N u_{x} = \frac{(x + b) q_{w}}{k (T_{w} - T_{\infty})}$ heat flux q_w is $q_{w} = - {k (\frac{\partial T}{\partial y}) |}_{y = A_{1} {(x + b)}^{\frac{1 - n}{2}}}$ we have $N u_{x} {R e}_{x}^{- 1 / 2} = - \sqrt{\frac{n + 1}{2}} θ^{'} (0) .$ in which ${R e}_{x} = \frac{U_{w} (x + b)}{ν}$ shows local Reynold number.

Entropy generation equation

Entropy generation is defined as $\begin{matrix} S_{g} & = & \frac{k}{T_{\infty}^{2}} {(\frac{\partial T}{\partial y})}^{2} + \frac{μ}{T_{\infty}} {(\frac{\partial u}{\partial y})}^{2} + \frac{σ B_{0}^{2}}{T_{\infty}} u^{2} + \frac{R D}{T_{\infty}} (\frac{\partial T}{\partial y} \frac{\partial C}{\partial y}) \\ + \frac{R D}{C_{\infty}} {(\frac{\partial C}{\partial y})}^{2}, \end{matrix}$ we have $N_{G} = α_{1} Θ^{' 2} + B r F^{″ 2} + (\frac{2}{n + 1}) H_{a}^{2} B r F^{' 2} + L Θ^{'} Φ^{'} + \frac{L}{α_{1}} Φ^{'^{2}},$ Taking $F (η) = f (η - α) = f (ξ),$ $Θ (η) = θ (η - α) = θ (ξ)$ and $Φ (η) = ϕ (η - α) = ϕ (ξ)$ we have $N_{G} = α_{1} θ^{' 2} + B r f^{″ 2} + (\frac{2}{n + 1}) H_{a}^{2} B r f^{' 2} + L θ^{'} ϕ^{'} + \frac{L}{α_{1}} ϕ^{' 2} .$ Bejan number (Be) is $B e = \frac{Entropy generation due to heat and mass transfer}{Total entropy generation}$ or $B e = \frac{α_{1} θ^{' 2} + L θ^{'} ϕ^{'} + \frac{L}{α_{1}} ϕ^{' 2}}{α_{1} θ^{' 2} + B r f^{″ 2} + (\frac{2}{n + 1}) H_{a}^{2} B r f^{' 2} + L θ^{'} ϕ^{'} + \frac{L}{α_{1}} ϕ^{' 2}}}$ where $N_{G} (= \frac{2 T_{\infty} {(x + b)}^{1 - n} ν}{a k (T_{w} - T_{\infty}) (n + 1)} S_{g})$ denotes the entropy generation rate, $α_{1} (= \frac{(T_{w} - T_{\infty})}{T_{\infty}})$

Discussion

Here behaviors of physical variables on velocity, entropy rate, temperature, Bejan number and concentration are examined. Gradient of velocity and temperature are discussed in this section.

Physical quantities of interest

Significant behavior of various involving parameters like (H_a), (α), (n) and (Nt) on velocity gradient $(C_{f_{x}})$ and Nusselt number (Nu_x) is illustrated in Tables 1 and 2. For rising values of (H_a) and (α) both $(C_{f_{x}})$ and (Nu_x) are increased. Heat transfer rate and velocity gradient have opposite impact versus (n). Nu_x is diminished against larger approximation of (Nt).

Conclusions

The main observations of the present study are:

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Velocity is decreased against rising values of (H_a) and (α).
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For higher estimation of (n) the f′(ξ) boosts up.
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θ(ξ) have similar characteristics of (Nt) and (Nb).
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Temperature has reverse effect versus (Br) and (α).
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Concentration is augmented via (Nt).
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ϕ(ξ) is declined for rising values of (Nb) and (Le).
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N_G augmented when increment occurs in (Br) and (H_a).
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(Br) and (L) have opposite effect on Bejan number.
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Cf_x is boost up via (α) and (H_a).
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Temperature

Acknowledgment

This research was funded by the General Project of Science and Technology Plan of Beijing Municipal Education Commission (grant no. KM202010037003).

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Entropy optimized MHD nanomaterial flow subject to variable thicked surface

Highlights

Abstract

Introduction

Section snippets

Mathematical formulation

Surface drag force

Nusselt number

Entropy generation equation

Discussion

Physical quantities of interest

Conclusions

Acknowledgment

App. Math. Lett.

Result. Phy.

Result. Phy.

Int. J. Heat Mass Tran.

Int. J.Mechan. Sci.

Ener. Conver. Manag.

J. Colloid Interface Sci.

Physica B

J. Comp. Des. Eng.

J. Mol. Liq.

J. Magnet. Magnet. Mater.

J. Mol. Liq.