Entropy generation applications in flow of viscoelastic nanofluid past a lubricated disk in presence
本站小编 Free考研考试/2022-01-02
Aamar Abbasi1, Waseh Farooq1, M Ijaz Khan2, Sami Ullah Khan3, Yu-Ming Chu,4, Zahid Hussain1, M Y Malik51Department of Mathematics, University of Azad Jammu and Kashmir Muzaffarabad, 13100 Pakistan 2Department of Mathematics and Statistics, Riphah International University I-14, Islamabad 44000, Pakistan 3Department of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan 4Department of Mathematics, Huzhou University, Huzhou 313000, China 5Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia
Abstract Entropy generation is the loss of energy in thermodynamical systems due to resistive forces, diffusion processes, radiation effects and chemical reactions. The main aim of this research is to address entropy generation due to magnetic field, nonlinear thermal radiation, viscous dissipation, thermal diffusion and nonlinear chemical reaction in the transport of viscoelastic fluid in the vicinity of a stagnation point over a lubricated disk. The conservation laws of mass and momentum along with the first law of thermodynamics and Fick’s law are used to discuss the flow, heat and mass transfer, while the second law of thermodynamics is used to analyze the entropy and irreversibility. The numbers of independent variables in the modeled set of nonlinear partial differential equations are reduced using similarity variables and the resulting system is numerically approximated using the Keller box method. The effects of thermophoresis, Brownian motion and the magnetic parameter on temperature are presented for lubricated and rough disks. The local Nusselt and Sherwood numbers are documented for both linear and nonlinear thermal radiation and lubricated and rough disks. Graphical representations of the entropy generation number and Bejan number for various parameters are also shown for lubricated and rough disks. The concentration of nanoparticles at the lubricated surface reduces with the magnetic parameter and Brownian motion. The entropy generation declines for thermophoresis diffusion and Brownian motion when lubrication effects are dominant. It is concluded that both entropy generation and the magnitude of the Bejan number increase in the presence of slip. The current results present many applications in the lubrication phenomenon, heating processes, cooling of devices, thermal engineering, energy production, extrusion processes etc. Keywords：entropy generation;viscoelastic fluid;lubricated surface;nonlinear thermal radiation;Joule heating;Keller box method
PDF (1403KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Aamar Abbasi, Waseh Farooq, M Ijaz Khan, Sami Ullah Khan, Yu-Ming Chu, Zahid Hussain, M Y Malik. Entropy generation applications in flow of viscoelastic nanofluid past a lubricated disk in presence of nonlinear thermal radiation and Joule heating. Communications in Theoretical Physics, 2021, 73(9): 095004- doi:10.1088/1572-9494/ac0c75
Nomenclature $\rho $density $ {T}_{\infty } $free stream temperature $ {\sigma }_{1} $Stefan–Boltzmann constant $ \mu $dynamical viscosity $ T\, $nanoparticle temperature $ n $power law index $ K $viscoelastic parameter $ {Rd}\, $radiation parameter $ {Nt} $thermophoresis parameter $ {Sc} $Schmidt number $ {\beta }_{c} $chemical reaction parameter $ {Ec} $Eckert number $ {Q}_{1}\, $power law lubricant $ \kappa $thermal conductivity $ {c}_{f} $heat capacity of fluid $ {D}_{B} $Brownian diffusion coefficient $ {q}_{r} $radioactive heat flux $ ({\alpha }_{1},{\alpha }_{2}) $normal stress moduli $ B\, $magnetic field $ {{Re}}_{{r}}^{-1/2}{{Nu}}_{{r}} $local Nusselt number $ (r,\theta ,z) $cylindrical coordinates $ P $pressure $ {\boldsymbol{S}} $stress tensor $ k* $mean absorption coefficient $ C $nanoparticle concentration $ m* $consistency index $ M $magnetic parameter $ {\theta }_{w} $temperature ratio $ {Re} $Reynolds number $ {\Pr } $Prandtl number $ {Nb} $Brownian motion parameter $ \gamma \, $slip parameter $ {\rho }_{p} $density of the particle $ {c}_{p} $heat capacity of the nanoparticle $ {D}_{T} $thermophoresis diffusion coefficient $ \mu $viscosity of nanofluid $ ({A}_{1},{A}_{2}) $kinematic tensors $ E\, $electric field $ \sigma $electric conductivity $ {{Sh}}_{{r}}{{Re}}_{{r}}^{-1/2} $local Sherwood number
1. Introduction
The second law of thermodynamics is assumed to be more applicable compared to the first law as it gives the sum of entropy generation and frictional rates successfully. This law can be efficiently used in various engineering and physical science fields as it reflects the quantitative analysis regarding the energy loss and irreversibilities within the system. Exergy helps to enhance the engineering process performances that account for the heat transportation phenomenon and examine the irreversibility based on entropy generation features. The implementation of entropy generation is based on the pattern of heat transfer and viscous dissipation. Bejan [1, 2] directed the basic concept of entropy generation for fluid flow within a system. Applications of the optimized phenomenon include various engineering, technological and engineering phenomena with appliances of nanofluids. Owing to its such promising and diverse applications, the scientific community has devoted special attention to this topic. Researchers have modeled different flow models for the entropy generation phenomenon amongst different fluids. The prospect of entropy generation associated with irreversibility with low Reynolds number is often presented, but it become undeniable at high Reynolds number because of hydrodynamics. Some dynamic research continuations are reported in the literature which present the entropy generation applications in various fluid flows. Rashidi et al [3] used a numerical approach to analyze the impact of magnetic field, slip parameter and temperature differences on entropy generation of a fluid with variable properties on a disk. Zahmatkesh et al [4] investigated the entropy generation in an axisymmetric stagnation point flow of a nanofluid over a stretching cylinder. Mehmood et al [5] carried out a theoretical study to investigate the irreversibility phenomena in a nanofluid along moving isothermal wavy surface. A statistical analysis for entropy generation in copper and aluminum nanoparticles confined by rotating disks was performed by Khan et al [6]. The phenomena of irreversibility in a Carreau nanomaterial over a sheet in the presence of Brownian motion and thermophoresis diffusion was discussed by Khan et al [7]. A mathematical model was incorporated by Hayat et al [8] to discuss the entropy optimization rate and Bejan number in the electrically conducting flow of second grade fluid over a stretching cylinder. Total entropy under the influence of Brownian motion, thermophoresis diffusion, viscous dissipation and activation energy in the stagnation point flow of a Prandtl–Eyring nanofluid over a stretching sheet was also reported by Khan et al [9]. Hayat et al [10] reported the physical aspects of entropy generation in a flow of viscous fluid in the presence of thermal radiation and chemical reaction. In another attempt, Hayat et al [11] studied the entropy generation in an unsteady flow of viscous fluid with thermo-diffusion and diffusion-thermo effects over a rotating disk. A finite element method was used by Kumar et al [12] to explore the entropy generation in a nanofluid between two rotating disks. Qayyum et al [13] studied the simultaneous effects of thermal, porosity, viscous dissipation and Ohmic heating irreversibilities in the flow of a generalized Newtonian fluid characterized by Williamson over a stretching sheet. Sadiq and Hayat [14] explored the transport of a Casson nanofluid over a disk with entropy generation analysis. Exact solutions were calculated by Jawad et al [15] to discuss the entropy optimization in a magnetohydrodynamic bioconvective Casson nanofluid over a rotating disk. The heat and thermodynamic features were discussed in the biothermal frame under the influence of various parameters. Chu et al [16] used an optimal homotopy analysis method to characterize the impact of the generalized Fourier and Fick theories with double diffusion on entropy generation in the flow of a third grade nanofluid over a Riga plate. Shah et al [17] utilized the shooting method to compute the entropy optimization in the stagnation point flow of a viscous fluid impinging on a stretched Riga plate. Characteristic entropy generation rate in the flow of a non-Newtonian fluid with zero convective flux of nanoparticles was calculated and discussed by Rehamn et al [18]. Rashid et al [19] explored the impact of nonlinear thermal radiation and the slip condition in the irreversibility of a nanofluid over a nonlinear stretching sheet. Series solutions were calculated to report the impact of nonlinear thermal radiation and chemical reaction on entropy generation in the flow of a Powell–Eyring nanofluid in a porous medium by Tilli et al [20]. Qayyum et al [21] presented convective flow of a nanofluid with entropy generation and Bejan number on a curved stretching sheet with variable thickness. Entropy generation analysis in a Carreau liquid over a curved stretching sheet with nonlinear thermal radiation was reported by Raza et al [22]. To explore the effects of viscous dissipation, convective boundary conditions, partial slip, Lorentz force and Darcy–Forchheimer porous medium on the entropy generation and Bejan number in a viscous nanofluid over a curved stretching sheet, a theoretical study was carried out by Hayat et al [23]. Investigation of entropy generation for axisymmetric stagnation point flow of a nanofluid impinging on a cylinder was reported by Mohammadiun et al [24]. Bibi et al [25] focused on the variable thermal features of a Sisko nanofluid with entropy generation applications. Hussain et al [26] numerically analyzed the applications of entropy generation in the convective flow of a nanofluid by following the Darcy–Brinkman–Forchheimer model. Noreen and Ain [27] studied the entropy generation mechanism in the electroosmotic flow of a nanofluid on a porous saturated surface. Hayat et al [28] addressed the peristaltic pumping of a Sutterby nanofluid in the presence of the entropy generation phenomenon. Ullah et al [29] performed finite difference approximations for the power law model flow with entropy generation effects. Ahmad et al [30] examined the optimized features in a viscous fluid flow due to stretched configuration. The impact of thermal radiation and entropy generation for viscous nanoparticle flow has been addressed by Ahmad et al [31].
Lubricating oil is a fluid type used to reduce friction, wear and heat between various mechanical components which are in contact with each other. Lubricating materials are used in motorized vehicles. This process is usually termed as transmission fluid. Due to these applications and the significance of lubricants in industry and automobiles, many studies under different conditions over lubricated surfaces have been reported. Andersson and Rousselet [32] discussed the slip flow of a viscous fluid over a lubricated rotary disk. Santra et al [33] examined the lubrication aspects in a viscous fluid flow subject to stagnation point flow. Mahmood et al [34] carried out heat transfer analysis for the time-dependent flow of a viscous fluid over a lubricated disk. It was observed that the fall in temperature due to thermal diffusivity can be enlarged for a lubricated surface. In another attempt, Mahmood et al [35] investigated the heat transfer process for the axisymmetric stagnation point flow of a viscous fluid impinging on a disk.
This research communicates a mathematical model for the radiative flow of a viscoelastic nanofluid in the vicinity of the stagnation point over a lubricated disk. The thermal model is assessed in the presence of nonlinear thermal radiation, thermal diffusion and viscous dissipation. The optimized analysis for a viscoelastic fluid with lubrication applications is quite interesting and significant from an application point of view. To this end, this novel model captured such effects. The solution procedure is based on the Keller box method with excellent accuracy. A comprehensive analysis for flow parameters is presented.
2. Governing equations and flow configuration
We consider the steady two-dimensional axisymmetric flow of a viscoelastic nanofluid with power law as lubricant. Let the temperature and concentration at the surface of the disk be ${T}_{w}$ and ${C}_{w}$ respectively. For the mathematical formulation we use the cylindrical coordinate system $(r,\theta ,z).$ We consider the disk to be situated at $z=0$ and the fluid to lie in space $z\gt 0$ as shown in figure 1. The temperature of the fluid at free stream is ${T}_{\infty }$ and the concentration at free stream is ${C}_{\infty }.$
The velocity, temperature and concentration fields for axially symmetric flows are$\begin{eqnarray}\begin{array}{l}{\boldsymbol{V}}=[u\left(r,z\right),\,0,\,w\left(r,z\right)],\,\,T=T(r,z)\,{\rm{and}}\,\\ C=C\left(r,z\right).\end{array}\end{eqnarray}$The analysis is based on conservation laws. The equation of continuity and the momentum equation in presence of body force are respectively$\begin{eqnarray}{\rm{\nabla }}\,\cdot \,{\boldsymbol{V}}=0,\end{eqnarray}$$\begin{eqnarray}\rho \frac{{\rm{d}}{\boldsymbol{V}}}{{\rm{d}}t}=-{\rm{\nabla }}P+{\rm{\nabla }}{\boldsymbol{S}}+{\boldsymbol{J}}\times {\boldsymbol{B}},\end{eqnarray}$where $\rho $ stands for density, $P$ is pressure and $S\,$ is the stress tensor according to Revilin [36] for a second order fluid stress tensor, defined as [37]:$\begin{eqnarray}S=\mu {A}_{1}+{\alpha }_{1}{A}_{2}+{\alpha }_{2}{A}_{1}^{2},\end{eqnarray}$in which $\mu $ represents the viscosity of the nanofluid, ${A}_{1}\,$ and ${A}_{2}$ are kinematic tensors, ${\alpha }_{1}\,$ and ${\alpha }_{2}\,$ represent the normal stress moduli and the results for classical Newtonian fluids can be obtained if ${\alpha }_{1}={\alpha }_{2}=0.\,\,$ Following the work of Dunn and Fosdick [37], it is assumed that ${\alpha }_{1}\geqslant 0,\,\mu \geqslant 0,$ and ${\alpha }_{1}+{\alpha }_{2}=0$ are the thermodynamic constraints in the analysis. Moreover, according to Ohm’s law, ${\boldsymbol{J}}=\sigma ({\boldsymbol{E}}+{\boldsymbol{V}}\times {\boldsymbol{B}}),$ where ${\boldsymbol{E}}$ and ${\boldsymbol{B}}$ are electric and magnetic fields and $\sigma $ is electric conductivity. Heat transfer and nanoparticle concentration are analyzed using the first law of thermodynamics and Fick’s law. Brownian motion and thermophoresis diffusion are incorporated in a multi-phase Buongiorno model [11].$\begin{eqnarray}\begin{array}{l}{\left(\rho c\right)}_{f}\frac{{\rm{d}}T}{{\rm{d}}t}=k{\nabla }^{2}T+{\left(\rho c\right)}_{p}\left[{D}_{B}\left({\rm{\nabla }}T\,\cdot \,{\rm{\nabla }}C\right)\right.\\ \,+\left.\frac{{D}_{T}}{{T}_{\infty }}({\rm{\nabla }}T\,\cdot \,{\rm{\nabla }}T)\right]+{\boldsymbol{S}}:{\rm{\nabla }}{\boldsymbol{V}}+\frac{{\boldsymbol{J}}\,\cdot \,{\boldsymbol{J}}}{\sigma }-{\rm{\nabla }}.{q}_{r},\end{array}\end{eqnarray}$$\begin{eqnarray}\frac{{\rm{d}}C}{{\rm{d}}t}={D}_{B}{\nabla }^{2}C+\frac{{D}_{T}}{{T}_{\infty }}{\nabla }^{2}T-\beta {\left(C-{C}_{\infty }\right)}^{m},\end{eqnarray}$where ${\rho }_{p}$ is the density of the particle, $\kappa $ is thermal conductivity, ${c}_{p}$ is the heat capacity of the nanoparticle, ${c}_{f}$ is the heat capacity of the fluid, ${D}_{T}$ is the thermophoresis diffusion coefficient, ${T}_{\infty }$ is the ambient fluid temperature, ${D}_{B}$ is the Brownian diffusion coefficient, ${q}_{r}$ is radioactive heat flux, $T\,$ is the nanoparticle temperature and $C\,$ is the nanoparticle concentration.
2.1. Boundary conditions
It is assumed that the solid surface and lubricant of thickness $\xi (r)$ are maintained at the same temperature and concentration and the viscoelastic fluid under consideration occupies the space $z\gt \xi (r).$ The relevant boundary conditions for the present analysis are [39, 40]:$\begin{eqnarray}\left.\begin{array}{l}\begin{array}{l}\mathrm{Solid}\,\mathrm{surface}\,\left(z=0\right):\,U=W=0,T={T}_{w}\,\mathrm{and}\,\\ C={C}_{w}\end{array}\\ \begin{array}{l}\mathrm{Fluid}{\unicode{x02212}}\mathrm{fluid}\,\mathrm{interface}\,\left(z=\xi \left(r\right)\right):\,U=u,\\ W=w\,\mathrm{and}\,{\tau }_{RZ}={\tau }_{rz}\end{array}\\ \begin{array}{l}\mathrm{Free}\,\mathrm{stream}\,\left(z\to {\rm{\infty }}\right):\,u={u}_{e}=cr,\,w=-2cz,\\ T={T}_{{\rm{\infty }}}\,\mathrm{and}\,C={C}_{{\rm{\infty }}}\end{array}\end{array}\right\}\end{eqnarray}$
in which $U,W$ and ${\tau }_{RZ}$ are the velocity components and shear stress of the lubricant.
The governing boundary layer equations for a nanofluid over the thin lubricated layer are (Sajid et al [35], Tlili et al [38])$\begin{eqnarray}\displaystyle \frac{\partial }{\partial r}(ru)+\displaystyle \frac{\partial }{\partial z}(rw)=0,\end{eqnarray}$$\begin{eqnarray}\left.\begin{array}{l}u\displaystyle \frac{\partial u}{\partial r}+w\displaystyle \frac{\partial u}{\partial z}={u}_{e}\displaystyle \frac{\partial {u}_{e}}{\partial r}+\nu \displaystyle \frac{{\partial }^{2}u}{\partial {z}^{2}}-\displaystyle \frac{\sigma {B}_{0}^{2}\left(u-{u}_{e}\right)}{\rho }\\ \begin{array}{l}\,+{\alpha }_{2}\left(\displaystyle \frac{2u}{r}\displaystyle \frac{{\partial }^{2}u}{\partial {z}^{2}}+\displaystyle \frac{1}{r}{\left(\displaystyle \frac{\partial u}{\partial z}\right)}^{2}\right)+{\alpha }_{1}\left(u\displaystyle \frac{{\partial }^{3}u}{\partial r\partial {z}^{2}}\right.\\ \,+\left.\displaystyle \frac{\partial u}{\partial z}\displaystyle \frac{{\partial }^{2}w}{\partial {z}^{2}}+3\displaystyle \frac{\partial u}{\partial r}\displaystyle \frac{{\partial }^{2}u}{\partial {z}^{2}}+2\displaystyle \frac{\partial w}{\partial z}\displaystyle \frac{{\partial }^{2}u}{\partial {z}^{2}}+w\displaystyle \frac{{\partial }^{3}u}{\partial {z}^{3}}\right)\end{array}\end{array}\right\},\end{eqnarray}$$\begin{eqnarray}\left.\begin{array}{l}u\displaystyle \frac{\partial T}{\partial r}+w\displaystyle \frac{\partial T}{\partial z}=\displaystyle \frac{k}{{\left(\rho c\right)}_{f}}\displaystyle \frac{{\partial }^{2}T}{\partial {z}^{2}}+\displaystyle \frac{{\left(\rho c\right)}_{p}}{{\left(\rho c\right)}_{f}}\\ \begin{array}{l}\,\times \left[{D}_{B}\displaystyle \frac{\partial T}{\partial z}\displaystyle \frac{\partial C}{\partial z}+\displaystyle \frac{{D}_{T}}{{T}_{\infty }}{\left(\displaystyle \frac{\partial T}{\partial z}\right)}^{2}\right]+\displaystyle \frac{\mu }{{\left(\rho c\right)}_{f}}{\left(\displaystyle \frac{\partial u}{\partial z}\right)}^{2}\\ \,+\displaystyle \frac{{\alpha }_{1}}{{\left(\rho c\right)}_{f}}\left(u\displaystyle \frac{\partial u}{\partial z}\displaystyle \frac{{\partial }^{2}u}{\partial r\partial z}+w\displaystyle \frac{\partial u}{\partial z}\displaystyle \frac{{\partial }^{2}u}{\partial {z}^{2}}\right)\\ \,+\displaystyle \frac{\sigma {B}_{0}^{2}}{{\left(\rho c\right)}_{f}}{\left(u-{u}_{e}\right)}^{2}-\displaystyle \frac{1}{{\left(\rho c\right)}_{f}}\displaystyle \frac{\partial {q}_{r}}{\partial z}\end{array}\end{array}\right\},\end{eqnarray}$$\begin{eqnarray}u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}={D}_{B}\frac{{\partial }^{2}T}{\partial {z}^{2}}+\frac{{D}_{T}}{{T}_{\infty }}\frac{{\partial }^{2}C}{\partial {z}^{2}}-{K}_{1}{\left(C-{C}_{\infty }\right)}^{m}.\end{eqnarray}$The nonlinear radiative heat flux is$\begin{eqnarray}{q}_{r}=-\displaystyle \frac{4{\sigma }_{1}}{3k* }\displaystyle \frac{\partial {T}^{4}}{\partial z},\end{eqnarray}$in which the term ${\sigma }_{1}$ is the Stefan–Boltzmann constant, and $k* $ is the mean absorption coefficient.
The interfacial condition for shear stresses takes the form$\begin{eqnarray}\begin{array}{l}\mu \frac{\partial u}{\partial z}+{\alpha }_{1}\left(u\frac{{\partial }^{2}u}{\partial z\partial r}+\frac{\partial u}{\partial z}\frac{\partial u}{\partial r}-\frac{\partial w}{\partial z}\frac{\partial u}{\partial z}+w\frac{{\partial }^{2}w}{\partial {z}^{2}}\right)\\ \,=\,m* {\left(\frac{\partial U}{\partial z}\right)}^{n},\end{array}\end{eqnarray}$in which $\mu ,m* $ and $n$ are the dynamical viscosity, consistency and power law index, respectively.
2.2. Dimensionless formulation
To reduce the number of independent variables in the governing equations, we introduce the following dimensionless variables:$\begin{eqnarray}\begin{array}{l}\eta =z\sqrt{\displaystyle \frac{c}{\nu }},u=crf^{\prime} \left(\eta \right),w=-2\sqrt{c\nu }f\left(\eta \right),\\ \theta \left(\eta \right)=\displaystyle \frac{T-{T}_{\infty }}{{T}_{w}-{T}_{\infty }}\phi \left(\eta \right)=\displaystyle \frac{C-{C}_{\infty }}{{C}_{w}-{C}_{\infty }}.\end{array}\end{eqnarray}$The mass conservation equation is identically satisfied and (4)–(6) become$\begin{eqnarray}\begin{array}{l}f^{\prime\prime \prime} +2ff^{\prime\prime} -{f}^{{\prime} 2}+1+K\left(2f^{\prime} f^{\prime\prime \prime} -2f{f}^{iv}-{f}^{{\prime\prime} 2}\right)\\ \,+\,{M}^{2}\left(1-f^{\prime} \right)=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\left[\left\{1+\frac{4}{3}{Rd}{\left(1+\left({\theta }_{w}-1\right)\theta \right)}^{3}\right\}\theta ^{\prime} \right]}^{{\prime} }\\ \,+{\Pr }\left[\begin{array}{c}2f\theta ^{\prime} +{Nt}{\theta }^{{\prime} 2}+{Nb}\theta ^{\prime} \phi ^{\prime} +{Ec}{f}^{{\prime\prime} 2}\\ +{Ec}{M}^{2}\left(1-{f}^{{\prime} 2}\right)+K{Ec}\left(f^{\prime} {f}^{{\prime\prime} 2}-2ff^{\prime\prime} f^{\prime\prime \prime} \right)\end{array}\right]\,=\,0,\end{array}\end{eqnarray}$$\begin{eqnarray}\phi ^{\prime\prime} +2\mathrm{Sc}f\phi ^{\prime} +\frac{{Nt}}{{Nb}}\theta ^{\prime\prime} +\mathrm{Sc}{\beta }_{c}{\left(\phi \right)}^{n}=0,\end{eqnarray}$subject to boundary conditions [39, 40]:$\begin{eqnarray}\begin{array}{l}f^{\prime\prime} \left(0\right)\left\{1+4Kf^{\prime} \left(0\right)\right\}=\gamma {\left\{f^{\prime} \left(0\right)\right\}}^{2n},\\ \theta \left(0\right)=1=\phi \left(0\right),f\left(0\right)=0,\end{array}\end{eqnarray}$$\begin{eqnarray}f^{\prime} \left(\infty \right)=1,f^{\prime\prime} \left(\infty \right)=0,\theta \left(\infty \right)=0=\phi (\infty ).\end{eqnarray}$In the above set of equations and boundary conditions, the following dimensionless constants are involved: $K={\alpha }_{1}\mu /c\,\,$ is the viscoelastic parameter, $M={B}_{0}\sqrt{\sigma {u}_{e}/\rho c}$ is the magnetic parameter, ${Rd}=\tfrac{4{\sigma }_{1}}{k* }{T}_{\infty }$ represents the radiation parameter, ${\theta }_{w}=\tfrac{{T}_{w}}{{T}_{\infty }}$ stands for the temperature ratio, ${Nt}=\,\tfrac{\tau {D}_{T}({T}_{w}-{T}_{\infty })}{\nu {T}_{\infty }}$ is the thermophoresis parameter, Re is the Reynolds number, ${\Pr }=\nu /\alpha $ is the Prandtl number, ${Sc}=\nu /{D}_{B}$ is the Schmidt number, ${Nb}=\,\tfrac{\tau {D}_{B}({C}_{w}-{C}_{\infty })}{\nu }$ is the Brownian motion parameter, ${Ec}=\,\tfrac{{c}^{2}{r}^{2}}{{C}_{p}({T}_{w}-{T}_{\infty })}$ is the Eckert number, ${\beta }_{c}=\,\tfrac{\nu {K}_{1}}{{c}^{2}}$ is the chemical reaction parameter and $\gamma =\tfrac{\sqrt{\tfrac{\nu }{c}}}{\displaystyle \tfrac{\mu }{k}{\left(\tfrac{{Q}_{1}c}{\pi }\right)}^{\displaystyle \tfrac{1}{3}}}$ is the slip parameter. The flow rate ${Q}_{1}\,$ of the power law lubricant is specified as$\begin{eqnarray}{Q}_{1}={\int }_{0}^{\xi (r)}U(r,z)2\pi r{\rm{d}}z.\end{eqnarray}$The dimensionless expressions for the local Nusselt and Sherwood numbers are respectively$\begin{eqnarray}\begin{array}{l}{{Re}}_{r}^{-1/2}{{Nu}}_{r}=-\left(1+\frac{4}{3}{Rd}{\left(1+\left({\theta }_{w}-1\right)\theta \left(0\right)\right)}^{3}\right)\theta ^{\prime} \\ \,\times \left(0\right),{{Sh}}_{r}{{Re}}_{r}^{-1/2}\,=\,-\phi ^{\prime} \left(0\right).\end{array}\end{eqnarray}$
3. Solution methodology
In several engineering and industrial processes, the developed problem contains nonlinear differential equations. The researchers have developed some interesting numerical techniques to compute the desired solution [41–43]. The governing system of equations (15)–(17) subject to nonlinear boundary conditions is approximated for unknowns $f\left(\eta \right),\,\theta (\eta )$ and $\phi (\eta )$ using the second order accurate and efficient numerical scheme commonly known as the Keller box method, introduced by Cebeci and Keller and successfully used by several researchers [35, 43] for the approximations of two-point boundary value problems. The detailed procedure of the numerical method is stated below.
Reduction to first order
First, we reduce the given system into a corresponding system of first order equations by introducing the following new variables.
In this section the graphical results are presented for temperature profile $\theta (\eta )$ and concentration profile $\phi (\eta )$ for both the slip case $\gamma =1.0$ and no slip case $\gamma =\infty $ and several values of ${\theta }_{w}.$ Several tables are also presented to analyze the impact of penetrating parameters on engineering interest quantities like the Nusselt and Sherwood numbers for several cases involved in nonlinear transport. Since this model is based on theoretical flow constraints, all flow parameters kept constant ranges like $K=0.1,\,\mathrm{Nt}=0.5,\,\beta =5,\,K=0.1,\,{\beta }_{c}=1, \mathrm{Rd}=0.1,\,K=0.1,\,\Pr =1,\, {\theta }_{w}=2.0,\,\mathrm{Nb}=0.5,$ $\mathrm{Sc}=1$ and $\mathrm{Ec}=0.1.$
4.1. Heat and mass transfer analysis
Figure 2 is plotted to highlight the response of temperature profile $\theta (\eta )$ against the increasing values of the Eckert number, thermophoresis parameter, Brownian motion parameter and magnetic parameter for different values of slip parameter $\gamma .$ Figure 2(a) shows that the rate of heat transfer increases with increasing viscous dissipation in the fluid. Furthermore, the temperature rise is large for the no slip case which corresponds to the rough disk. It is observed that temperature jumps can be controlled by encountering lubrication, as due to the lubrication aspects the friction between the rough surface and the moving fluid decreases and as a result a fall in temperature is reported. From figures 2(b) and (c) a temperature rise can be observed with increasing Brownian motion and thermophoresis diffusion of nanoparticles in the ionic liquid. It is clear that due to lubrication the increasing trend at the lubricated layer is smooth but at the rough surface the rise in temperature is enormous. In figure 2(d) the temperature of the nanofluid corresponds to an inverse trend with the strength of magnetic field. It is observed that the strength of the magnetic field reduces the thermal boundary layer thickness over both lubricated and rough disks. The temperature profile for rising values of Prandtl number $\Pr $ and radiation parameter $\mathrm{Rd}$ is reported for both linear and nonlinear radiative phenomena over a lubricated surface in figure 3. The temperature distribution is a decreasing function of the Prandtl number and an increasing function of the radiation parameter. The decrease in temperature due to the Prandtl number can be controlled by incorporating nonlinear radiation phenomena which correspond to ${\theta }_{w}\gt 1.$ The heat transfer rate is large in the case of nonlinear radiative flux so the temperature profile rises largely due to the radiation parameter in the case of nonlinear radiative flow. Figure 4 is a graphical description of the concentration profile with increasing chemical reaction constant ${\beta }_{c},$ thermophoresis parameter $\mathrm{Nt},$ Brownian motion parameter $\mathrm{Nb}$ and magnetic parameter $M$ for both slip flow and inverse slip flow. It is observed that chemical reaction increases the concentration of nanoparticles in the suspension and the concentration at the lubricated layer is large, but with distance from the surface the effects of chemical reaction disappear. The increase in thermophoresis diffusion boosts the concentration of nanoparticles in the nanofluid and on the lubricated disk, but these effects are small in magnitude. Both the Brownian motion and magnetic force parameters reduce the concentration at the lubricated surface as well as at the rough surface.
Figure 2.
New window|Download| PPT slide Figure 2.Temperature profile for (a) Eckert number $\mathrm{Ec}$, (b) thermophoresis parameter $\mathrm{Nt}$, (c) Brownian motion parameter $\mathrm{Nb}$ and (d) magnetic parameter $M$ with other parameters $\mathrm{Rd}=K=0.1,\,\Pr =1.0,\,{\beta }_{c}=1=\mathrm{Sc},$ and ${\theta }_{w}=2.0.$
Figure 3.
New window|Download| PPT slide Figure 3.Temperature profile for (a) Prandtl number $\Pr $ and (b) radiation parameter $\mathrm{Rd}$ with other parameters $M=K=0.1=\mathrm{Ec},\,\mathrm{Nt}\,=0.5,\,\beta =5.0,\,{\beta }_{c}=1=\mathrm{Sc}$ and $\mathrm{Nb}=0.5.$
Figure 4.
New window|Download| PPT slide Figure 4.Concentration profile for (a) chemical reaction parameter $\mathrm{Ec}$, (b) thermophoresis parameter $\mathrm{Nt}$, (c) Brownian motion parameter $\mathrm{Nb}$ and (d) magnetic parameter $M$ with other parameters $\mathrm{Rd}=K=0.1,\,\Pr =1.0=\mathrm{Sc},\,{\theta }_{w}=2.0$ and $\mathrm{Ec}=0.1.$
4.2. Analysis for engineering quantities
The engineering interest quantities like Nusselt number ${{Re}}_{r}^{-1/2}{{Nu}}_{r}$ and Sherwood number ${{Re}}_{r}^{-1/2}{{Sh}}_{r}$ are computed and discussed for both the slip and no slip flow cases in table 1. Furthermore, these quantities are also computed for linear radiation phenomena as well as for nonlinear radiation phenomena in table 2. The influence of the second grade parameter on ${{Re}}_{r}^{-1/2}{{Nu}}_{r}$ is opposite in behavior for the slip and no slip cases. It is observed that due to lubrication, the second grade parameter increases the Nusselt number which is an important application of lubricant in industry. On the rough surface both the Nusselt number and Sherwood number decrease. The magnetic force increases both ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Nu}}_{r}$ and ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Sh}}_{r}$ on the lubricated surface as well as on the usual surface. The thermophoresis diffusion decreases both the Nusselt number and Sherwood number with large magnitude over the lubricated surface. The Brownian motion parameter also decreases the Nusselt number while increasing the Sherwood number for $\gamma =1.0$ and $\gamma =\infty .$ An increase in the radiation parameter increases both the Nusselt number and Sherwood number. An increase in the Prandtl number increases the Nusselt number while decreasing the Sherwood number for both the lubricated disk and usual disk. ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Sh}}_{r}$ increases with increasing mass diffusivity yet decreases with increasing chemical reaction parameter, and the decreasing trend is small due to lubrication. From table 2 it can be observed that for nonlinear radiation the magnitude of the Nusselt number is large compared to the linear radiation phenomena. Furthermore, the magnitude of the Sherwood number is large for linear radiative flow.
Table 1. Table 1.Variation of ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Nu}}_{r}$ and ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Sh}}_{r}$ when ${\theta }_{w}=2.0.$
Table 2. Table 2.Variation of ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Nu}}_{r}$ and Sherwood number ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Sh}}_{r}$ when $\gamma =1.0.$
The mathematical relation under boundary layer approximation for the volumetric rate of entropy generation under nonlinear thermal radiation, viscous dissipation and Lorentz force is given as$\begin{eqnarray}\begin{array}{l}{S}_{G}=\displaystyle \frac{k}{{T}_{\infty }^{2}}\left(1+\displaystyle \frac{16\sigma * {T}^{3}}{3kk* }\right){\left(\displaystyle \frac{\partial T}{\partial z}\right)}^{2}\\ \,+\displaystyle \frac{1}{{T}_{\infty }}\left[\mu {\left(\displaystyle \frac{\partial u}{\partial z}\right)}^{2}+{\alpha }_{1}\left(u\displaystyle \frac{\partial u}{\partial z}\displaystyle \frac{{\partial }^{2}u}{\partial r\partial z}+w\displaystyle \frac{\partial u}{\partial z}\displaystyle \frac{{\partial }^{2}u}{\partial {z}^{2}}\right)\right]\\ \,+\displaystyle \frac{RD}{{C}_{\infty }}{\left(\displaystyle \frac{\partial C}{\partial z}\right)}^{2}+\displaystyle \frac{RD}{{C}_{\infty }}\left(\displaystyle \frac{\partial C}{\partial z}\displaystyle \frac{\partial T}{\partial z}\right)+\displaystyle \frac{\sigma {B}_{0}^{2}}{{T}_{\infty }}{\left({u}_{e}-u\right)}^{2},\end{array}\end{eqnarray}$The characteristic entropy generation rate can be defined as$\begin{eqnarray}{N}_{G}=\displaystyle \frac{{S}_{G}}{{S}_{0}},\end{eqnarray}$where ${S}_{0}=\tfrac{k{\left({T}_{w}-{T}_{\infty }\right)}^{2}}{{T}_{\infty }^{2}{r}^{2}}$
After using the dimensionless variables, we have$\begin{eqnarray}\begin{array}{l}{N}_{G}={Re}\left[1+{Rd}{\left(\left({\theta }_{w}-1\right)+1\right)}^{3}\right]{\theta }^{{\prime} 2}\\ \,+\frac{\mathrm{PrEc}}{{\rm{\Omega }}}{Re}\left\{{f}^{{\prime\prime} 2}+K\left(f^{\prime} {f}^{{\prime\prime} 2}-2ff^{\prime\prime} f^{\prime\prime \prime} \right.\right\}\\ \,+{Re}{\left(\frac{L}{{\rm{\Omega }}}\right)}^{2}{\alpha }_{1}{\phi }^{{\prime} 2}+{Re}\frac{L}{{\rm{\Omega }}}{\alpha }_{1}\theta ^{\prime} \phi ^{\prime} \\ \,+\frac{{Re}}{{\rm{\Omega }}}{PrEc}{M}^{2}{\left(1-f^{\prime} \right)}^{2}\end{array}\end{eqnarray}$in which ${\rm{\Omega }}=\tfrac{{T}_{w}-{T}_{\infty }}{{T}_{\infty }},\,{\alpha }_{1}=\tfrac{RD{C}_{\infty }}{k}$ and $L=\tfrac{{C}_{w}-{C}_{\infty }}{{C}_{\infty }}.$
The Bejan number can be defined as$\begin{eqnarray}\begin{array}{l}\mathrm{Be}\\ =\frac{{Re}\left[1+{Rd}{\left(\left({\theta }_{w}-1\right)+1\right)}^{3}\right]{\theta }^{{\prime} 2}++{Re}{\left(\displaystyle \frac{L}{{\rm{\Omega }}}\right)}^{2}{\alpha }_{1}{\phi }^{{\prime} 2}+{Re}\displaystyle \frac{L}{{\rm{\Omega }}}{\alpha }_{1}\theta ^{\prime} \phi ^{\prime} }{{N}_{G}}.\end{array}\end{eqnarray}$Graphical analysis is carried out through figures 5–10 for increasing values of viscoelastic parameter $K,$ magnetic parameter $M,$ Eckert number $\mathrm{Ec},$ radiation parameter $\mathrm{Rd}$ and chemical reaction parameter ${\beta }_{c}$ for various values of $\gamma .$ The analysis is presented to report the comparative results for both entropy generation number ${N}_{s}$ and Bejan number $\mathrm{Be}$ on the lubricated disk as well as on the rough disk. From figure 5 it is clear that the second grade parameter decreases both ${N}_{s}$ and $\mathrm{Be}$ on the surface of the disk and as we move away from the surface the entropy generation number increases. On the other hand, the Bejan number shows an opposite trend to ${N}_{s}$ against viscoelasticity. The Bejan number at the surface of disk increases and also decreasing amplitude is small at the thin lubricated layer. Due to the strong efficiency of the lubricant an increase in Bejan number is observed. The magnetic force strongly increases both the entropy generation number ${N}_{s}$ and Bejan number $\mathrm{Be}.$ The large magnitude of Bejan number over the lubricated surface corresponds to the strong irreversibility effects in the fluid. The Eckert number, which corresponds to more heat generated due to friction between the layers of the fluid, causes a remarkable increase in entropy generation number and Bejan number. The Eckert number increases the entropy generation number and also increases the Bejan number at the surface of the disk. Due to lubrication the magnitude of the Bejan number increases and this is the advantage of the lubricant in many industrial phenomena. The increase in thermophoresis diffusion decreases the entropy generation number at the surface of the disk and increases it away from the lubricated disk. A similar trend can be reported for the Bejan number at the surface of the lubricated disk. However, the effects of lubrication are dominant due to lubrication when the Bejan number has large magnitude. A large decrease in entropy generation number is observed for increasing values of Brownian motion parameter over the lubricated disk while the Bejan number rises with large magnitude. An increase in thermal radiation also increases both the entropy generation number and Bejan number over the lubricated disk and rough disk.
Figure 5.
New window|Download| PPT slide Figure 5.Variation of (a) entropy generation number $Ns$ and (b) Bejan number $\mathrm{Be}$ against $K$ when $n=0.5,\,M=1={\Pr }={Sc}={Ec}={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Nt}=0.5\,$ and ${Nb}=0.3.$
Figure 6.
New window|Download| PPT slide Figure 6.Variation of (a) entropy generation number $Ns$ and (b) Bejan number $\mathrm{Be}$ against $M$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,={Ec}={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Nt}=0.5\,$ and ${Nb}=0.3.$
Figure 7.
New window|Download| PPT slide Figure 7.Variation of (a) entropy generation number $Ns$ and (b) Bejan number $\mathrm{Be}$ against $\mathrm{Ec}$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,=M={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Nt}=0.5\,$ and ${Nb}=0.3.$
Figure 8.
New window|Download| PPT slide Figure 8.Variation of (a) entropy generation number $Ns$ and (b) Bejan number $\mathrm{Be}$ against $\mathrm{Nt}$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,=M={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Ec}=0.1\,$ and ${Nb}=0.3.$
Figure 9.
New window|Download| PPT slide Figure 9.Variation of (a) entropy generation number $\mathrm{Ns}$ and (b) Bejan number $\mathrm{Be}$ against $\mathrm{Nb}$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,=M={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Ec}=1.0\,$ and ${Nt}=1.0.$
Figure 10.
New window|Download| PPT slide Figure 10.Variation of (a) entropy generation number $\mathrm{Ns}$ and (b) Bejan number $\mathrm{Be}$ against $\mathrm{Rd}$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,=M={\beta }_{c}$, ${Nt}=0.1,\,{\theta }_{w}=2.0,\,{Ec}=0.1\,$ and ${Nb}=0.3.$
5. Main findings
The radiative stagnation point flow of a viscoelastic nanofluid over a lubricated disk has been addressed. The impact of entropy generation, viscous dissipation and thermal diffusion are also incorporated. The numerical simulations are presented by following the Keller box scheme. The novel aspects of the current results are summarized as:The Brownian motion and magnetic force parameters reduce the concentration at the lubricated surface as well as at the rough surface. The magnitudes of the local Nusselt number and local Sherwood number are enhanced over a lubricated disk compared to a rough disk. The presence of the second grade parameter over the lubricated surface increases the local Nusselt number and Sherwood number, while on a traditional surface this behavior is contrary. A rise in entropy generation number is observed with increasing fraction between the layers of fluid and the Bejan number has large magnitude due to lubrication. A decrease in entropy generation number is observed for thermophoresis diffusion and Brownian motion when lubrication effects are dominant. With growth in thermal radiation phenomena, the entropy generation number and Bejan number increase.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia for funding this work through the Research Groups Program under grant number R. G. P-1/75/42.
BejanA1996 Method of entropy generation minimization, or modeling and optimization based on combined heat transfer and thermodynamics Rev. Gén. Therm. 35 637646 DOI:10.1016/S0035-3159(96)80059-6 [Cited within: 1]
RashidiM MKavyaniNAbelmanS2014 Investigation of entropy generation in MHD and slip flow over a rotating porous disk with variable properties Int. J. Heat Mass Transfer 70 892917 DOI:10.1016/j.ijheatmasstransfer.2013.11.058 [Cited within: 1]
ZahmatkeshRMohammadiunHMohammadiunMBonabM D2019 Investigation of entropy generation in nanofluid’s axisymmetric stagnation flow over a cylinder with constant wall temperature and uniform surface suction-blowing Alex. Eng. J. 58 14831498 DOI:10.1016/j.aej.2019.12.003 [Cited within: 1]
MehmoodAKhanSIqbalM S2019 Entropy generation due to a moving isothermal wavy surface in nanofluid Int. J. Exergy 28 317332 DOI:10.1504/IJEX.2019.099291 [Cited within: 1]
KhanM IQayyumSHayatTAlsaediA2018 Entropy generation minimization and statistical declaration with probable error for skin friction coefficient and Nusselt number Chin. J. Phys. 56 15251546 DOI:10.1016/j.cjph.2018.06.023 [Cited within: 1]
HayatTKhanS AKhanM IAlsaediA2019 Optimizing the theoretical analysis of entropy generation in the flow of second grade nanofluid Phys. Scr. 94 085001 DOI:10.1088/1402-4896/ab0f65 [Cited within: 1]
KhanM IAlsaediAQayyumSHayatTKhanM I2019 Entropy generation optimization in flow of Prandtl–Eyring nanofluid with binary chemical reaction and Arrhenius activation energy Colloid Surface A 570 117126 DOI:10.1016/j.colsurfa.2019.02.060 [Cited within: 1]
HayatTJavedSKhanM IKhanM IAlsaediA2019 Physical aspects of irreversibility in radiative flow of viscous material with cubic autocatalysis chemical reaction Eur. Phys. J. Plus 134 124 DOI:10.1140/epjp/i2019-12564-7 [Cited within: 1]
HayatTSajjadLKhanM IKhanM IAlsaediA2019 Salient aspects of thermo-diffusion and diffusion thermo on unsteady dissipative flow with entropy generation J. Mol. Liq. 282 557565 DOI:10.1016/j.molliq.2019.02.072 [Cited within: 2]
KumarRSethG SBhattacharyyaA2019 Entropy generation of von Karman’s radiative flow with Al2O3 and Cu nanoparticles between two coaxial rotating disks: a finite-element analysis Eur. Phys. J. Plus. 134 597 DOI:10.1140/epjp/i2019-13086-0 [Cited within: 1]
QayyumSKhanM IMasoodFChuY MKadrySNazeerM2020 Interpretation of entropy generation in Williamson fluid flow with nonlinear thermal radiation and first-order velocity slip Math. Meth. Appl. Sci. DOI:10.1002/mma.6735 [Cited within: 1]
SadiqM AHayatT2020 Characterization of marangoni forced convection in Casson nanoliquid flow with Joule heating and irreversibility Entropy 22 433 DOI:10.3390/e22040433 [Cited within: 1]
JawadMSaeedAKhanAIslamS2020 MHD bioconvection Darcy–Forchheimer flow of Casson nanofluid over a rotating disk with entropy optimization Heat Transfer DOI:10.1002/htj.21973 [Cited within: 1]
ChuY MShahFKhanM IFarooqSKadrySAbdelmalekZ2020 Investigation of viscous dissipation and entropy generation in third grade nanofluid flow over a stretched riga plate with Cattaneo–Christov Double Diffusion (CCDD) model Phys. Scr. 95 115004 DOI:10.1088/1402-4896/abbaf5 [Cited within: 1]
ShahFKhanM IChuY MKadryS2020 Heat transfer analysis on MHD flow over a stretchable Riga wall considering Entropy generation rate: a numerical study Numer. Methods Partial Differ. Equ. DOI:10.1002/num.22694 [Cited within: 1]
RehmanS UHaqR UKhanZ HLeeC2016 Entropy generation analysis for non-Newtonian nanofluid with zero normal flux of nanoparticles at the stretching surface J. Taiwan Inst. Chem. Eng. 63 226235 DOI:10.1016/j.jtice.2016.03.006 [Cited within: 1]
RashidMKhanM IHayatTKhanM IAlsaediA2019 Entropy generation in flow of ferromagnetic liquid with nonlinear radiation and slip condition J. Mol. Liq. 276 441452 DOI:10.1016/j.molliq.2018.11.148 [Cited within: 1]
TliliIShahmirNRamzanMKadrySKimJ YNamYLuD2020 A novel model to analyze Darcy Forchheimer nanofluid flow in a permeable medium with entropy generation analysis J. Taibah Univ. Sci. 14 916930 DOI:10.1080/16583655.2020.1790171 [Cited within: 1]
QayyumSHayatTAlsaediA2019 Optimization of entropy generation in motion of magnetite–Fe3O4 nanoparticles due to curved stretching sheet of variable thickness Int. J. Numer. Method H 29 33473365 DOI:10.1108/HFF-12-2018-0782 [Cited within: 1]
HayatTQayyumSAlsaediAAhmadB2020 Entropy generation minimization: Darcy–Forchheimer nanofluid flow due to curved stretching sheet with partial slip Int. Commun. Heat Mass Transfer 111 104445 DOI:10.1016/j.icheatmasstransfer.2019.104445 [Cited within: 1]
MohammadiunMDibaiee BonabM HDarabiMHejaziS2020 Investigation of entropy generation in axisymmetric stagnation point flow of nano fluid impinging on the cylinder with constant wall temperature Alex. Eng. J. 21 5834 [Cited within: 1]
BibiFHayatTFarooqSKhanA AAlsaediA2021 Entropy generation analysis in peristaltic motion of Sisko material with variable viscosity and thermal conductivity J. Therm. Anal. Calorim. 143 363375 DOI:10.1007/s10973-019-09125-4 [Cited within: 1]
HussainSMehmoodKSagheerMAshrafA2019 Mixed convective magnetonanofluid flow over a backward facing step and entropy generation using extended Darcy–Brinkman–Forchheimer model J. Therm. Anal. Calorim. 138 31833203 DOI:10.1007/s10973-019-08347-w [Cited within: 1]
NoreenSAinQ2019 Entropy generation analysis on electroosmotic flow in non-Darcy porous medium via peristaltic pumping J. Therm. Anal. Calorim. 137 19912006 DOI:10.1007/s10973-019-08111-0 [Cited within: 1]
HayatTNisarZAlsaediAAhmadB2021 Analysis of activation energy and entropy generation in mixed convective peristaltic transport of Sutterby nanofluid J. Therm. Anal. Calorim. 143 18671880 DOI:10.1007/s10973-020-09969-1 [Cited within: 1]
UllahHHayatTAhmadSAlhodalyM S2021 Entropy generation and heat transfer analysis in power-law fluid flow: finite difference method Int. Commun. Heat Mass Transfer 122 105111 DOI:10.1016/j.icheatmasstransfer.2021.105111 [Cited within: 1]
MahmoodKSajidMAliNJavedT2016 Heat transfer analysis in the time-dependent slip flow over a lubricated rotating disc Int. J. Eng. Sci. Technol. 19 19491957 DOI:10.1016/j.jestch.2016.07.009 [Cited within: 1]
MahmoodKSajidMAliNJavedT2018 Heat transfer analysis in the time-dependent axisymmetric stagnation point flow over a lubricated surface Therm. Sci. 22 24832492 DOI:10.2298/TSCI160203257M [Cited within: 3]
DunnJErnestFosdickR L1974 Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade Arch. Ration. Mech. Anal. 56(1974)191252 DOI:10.1007/BF00280970 [Cited within: 2]
TliliIWaqasHAlmaneeaAKhanS UImranM2019 Activation energy and second order slip in bioconvection of Oldroyd-B nanofluid over a stretching cylinder: a proposed mathematical model Processes 7 914 DOI:10.3390/pr7120914 [Cited within: 1]
SajidMAhmadMAhmadITajMAbbasiA2015 Axisymmetric stagnation-point flow of a third-grade fluid over a lubricated surface Adv. Mech. Eng. 7 18 DOI:10.1177/1687814015591735 [Cited within: 2]
MahmoodKSajidMAliNJavedT2017 MHD mixed convection stagnation point flow of a viscous fluid over a lubricated vertical surface Ind. Lubr. Tribol. 69 527535 DOI:10.1108/ILT-02-2016-0025 [Cited within: 2]
BejawadaS GKhanZ HHamidM2021 Heat generation/absorption on MHD flow of a micropolar fluid over a heated stretching surface in the presence of the boundary parameter Heat Transfer Asian Res. DOI:10.1002/htj.22165 [Cited within: 1]
SheikholeslamiMHamidMHaqR UShafeeA2020 Numerical simulation of wavy porous enclosure filled with hybrid nanofluid involving Lorentz effect Phys. Scr. 95 115701 DOI:10.1088/1402-4896/abbcf3
KhanZ HUsmanMZubairTHamidMHaqR U2020 Brownian motion and thermophoresis effects on unsteady stagnation point flow of Eyring–Powell nanofluid: a Galerkin approach Commun. Theor. Phys. 72 125005 DOI:10.1088/1572-9494/abb7d5 [Cited within: 2]
,4, Zahid Hussain1, M Y Malik51Department of Mathematics, 
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
