Heat and mass transport phenomena of nanoparticles on timedependent flow of Williamson fluid towards heated surface
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Abstract
An enhancement in the thermal conductivity of conventional base fluids has been a topic of great concern in recent years. An effective way to improve the heat transfer rate of conventional base fluids is the suspension of solid nanoparticles. In this framework, a theoretical study is performed to analyse the heat and mass transfer performance in the timedependent flow of nonNewtonian Williamson nanofluid towards a stretching surface. There exist several studies focusing on the flow of Williamson fluid by assuming zero infinite shear rate viscosity. Nonetheless, there is a lack of knowledge regarding mathematical formulation for twodimensional flow of the Williamson fluid by taking into account the impacts of infinite shear rate viscosity. In the current review, the Buongiorno model for nanofluids associated with Brownian motion and thermophoretic diffusion is employed to describe the heat transfer performance of nanofluids. The thermal system is composed of flow velocity, temperature, and nanoparticles concentration fields, respectively. The governing dimensionless equations are solved numerically by Runge–Kutta Fehlberg integration method. The numerical results are compared with published results and are found to have an excellent agreement. Effects of numerous dimensionless parameters on velocity, temperature, and nanoparticle concentration field together with the skin friction coefficient and rates of heat and mass transfer are presented with the assistance of graphical and tabular illustrations. With this analysis, we reached that the thermal boundary layer thickness as well as the nanofluids temperature has higher values with increase in thermophoresis and Brownian motion. It is further observed that the rate of heat transfer is significantly raised with an increment in Prandtl number and unsteadiness parameter.
Keywords
Williamson nanofluids Unsteady flow Buongiorno’s model Heated stretching surfaceList of symbols
 \((u,\,w)\)
Components of velocity
 \((x,\,y)\)
Space coordinates
 \(T\)
Dimensional temperature of the fluid
 \(T_{w}\)
Shear stress at the surface
 \(T_{\infty }\)
Temperature of the fluid in free stream
 \(C\)
Nanoparticles volume fraction
 \(C_{w}\)
Nanoparticles volume fraction at the surface
 \(C_{\infty }\)
Nanoparticles volume fraction in free stream
 \(D_{\text{B}}\)
Brownian diffusion coefficient
 \(D_{\text{T}}\)
Thermophoretic diffusion coefficient
 \(U_{w}\)
Velocity of the stretching surface
 \(a,c\)
Positive constants
 t
Time
 k
Thermal conductivity
 f
Dimensionless velocity
 We
Weissenberg number
 A
Unsteadiness parameter
 Pr
Prandtl number
 Sc
Schmidt number
 \(C_{fx}\)
Skin friction coefficient
 \(Nu_{x}\)
Nusselt number
 \(Sh_{x}\)
Sherwood number
 \(\;q_{w}\)
Surface heat flux
 \(\;q_{m}\)
Surface mass flux
 \(Nt\)
Thermophoretic parameter
 \(Nb\)
Brownian motion parameter
 \(Re\)
Local Reynolds number
Greek letters
 \(\nu\)
Kinematic viscosity
 \(\varGamma\)
Relaxation time
 \(\beta^{*}\)
Ratio of viscosities
 \(\mu_{\infty }\)
Infinite shear rate viscosity
 \(\mu_{0}\)
Zero shear rate viscosity
 \(\;\tau\)
Effective heat capacities ratio
 \((\rho c)_{f}\)
Heat capacity of the fluid
 \((\rho c)_{p}\)
Heat capacity of nanoparticles
 \(\tau_{w}\)
Surface shear stress
 \(\rho\)
Fluid density
 \(\psi\)
Stream function
 \(\theta\)
Dimensionless temperature
 \(\varphi\)
Dimensionless concentration
 \(\eta\)
Dimensionless variable
 \(\alpha_{m}\)
Thermal diffusivity
 \(C_{p}\)
Specific heat capacity
Subscripts/superscripts
 \(w\)
Surface conditions
 \(\infty\)
Ambient conditions
1 Introduction
At present, the world is confronting a noteworthy problem of low heat transfer rate of base fluids, which limits the effectiveness of heat transfer performance in heat exchangers. The most regular working liquids are water, ethanol and ethylene–glycol blend. To cope up this problem, recently, engineers and scientists have shown their great concern in improving the thermal properties of energy transmission fluids and their heat transfer performance for industrial applications. This innovation aims at enhancing the thermal conductivities and the convective heat transfer of fluids through suspensions of ultrafine nanoparticles in the base fluids. This mixture is known as “nanofluid,” which was first employed by Choi [1]. Nanofluids are the fluids that possess 100 nm or less size of nanoparticles such as metals, oxides and nitrides together with usual base fluids like water, engine oil and toluene. Considering higher thermal conductivity of nanoparticles contrasted with base liquids, nanofluids have tremendous applications in almost every field of science, technology and biomedicine, viz. better coolants in PCs and nuclear reactors, cancer therapy, wire drawing and quenching in metal foundries, lubricants, heat exchangers. It is experimentally verified that the nanoparticles may be of the shape like spherical, rodlike, tubular. It was an amusing start by Choi [1] to ponder experimentally and uncover to the society about the improvement in thermal conductivity of liquids with nanoparticle. In this work, he utilized the nanoparticles for the first time to improve the thermal conductivity of working fluids. He explained numerous experimental and numerical studies in the literature to know how the thermal conductivity is improved.
Many engineering and technological applications of nanofluids have motivated and encouraged many researchers in early decades of twentieth century to investigate the several aspects of flow and heat transfer of nanofluids over various surfaces. In 1993, Masuda et al. [2] presented the work to enhance the thermal conductivity of fluid particles. Later on, Eastman et al. [3] have exposed that the thermal conductivityimproved ethylene glycolbased nanofluid has raised up to 60% when CuO nanoparticles of volume fraction 5% are added to base fluid. The most important and widely used mechanisms in industrial applications are the thermophoresis and Brownian movement phenomena. Therefore, Buongiorno [4] exhibited that the homogeneous models tend to predict the nanofluid heat transfer coefficient, while the distribution impact is totally negligible because of the nanoparticle size. Thus, Buongiorno developed an alternative model to clarify the unusual convective heat transfer improvement in nanofluids and thus wipe out the weaknesses of the homogeneous and dispersion models. On the basics of his findings, he proposed a twocomponent fourequation nonhomogeneous equilibrium model for convective transport in nanofluids. The effects of heat transfer on the flow of nanofluids in a twosided liddriven heated square cavity have been scrutinized by Tiwari and Das [5]. Experimental study [6] has described that the nanofluid requires 5% volumetric portion for a compelling warmth exchange upgrades. The Buongiorno’s model has been utilized by Kuznetsov and Nield [7] to investigate the effects of thermophoresis and Brownian movement on the natural convection flow in the presence of nanoparticles over a vertical plate. Khan and Pop [8] inspected the heat and mass transfer features in free convection flows of nanofluid over a porous stretched surface. Transient hydromagnetic rotating flow of a nanofluid with free convection was analysed by Hamad and Pop [9]. The threedimensional flow of an electrically conducting nanofluid along with heat transfer in a rotating system has been examined by Sheikholeslami [10]. He utilized the wellknown fourthorder Runge–Kutta numerical scheme to solve the governing problem. After that, loads of articles have been reported on improvement in heat transfer rate in flow of nanofluids over different geometries, like Dhanai et al. [11], Hashim and Khan [12], Aman et al. [13], Khan et al. [14] and Dogonchi and Ganji [15].
 1.
Mathematical modelling for twodimensional timedependent flow of nonNewtonian Williamson fluid.
 2.
The impacts of nonzero infinite shear rate viscosity in constitutive relation of Williamson fluid are taken.
 3.
The influence of suspended nanoparticles on the thermal conductivity enhancement during the flow of Williamson fluid with heat transfer are studied in this article.
 4.
The numerical simulations of the governing equations for Williamson nanofluid flow have been conducted by employing the Runge–Kutta Fehlberg technique which has been proved for its accuracy over the years.
In view of abovestated points, a comprehensive analysis is presented to examine the timedependent flow of Williamson nanofluids caused by a stretching surface. With the assistance of boundary layer approximations, the conservation equations for the twodimensional flow of Williamson liquid have been modelled. Numerical simulations of the governing momentum and energy along with concentration equations are made via Runge–Kutta Fehlberg scheme along with shooting technique. Finally, the influence of diverse physical parameters such as unsteadiness parameter, ratio of viscosities, mass transfer parameter, Brownian motion parameter, thermophoresis parameter, Weissenberg number and Prandtl number on the flow, heat and mass transfer has been explored.
2 Modelling of the physical problem
2.1 Problem statement and governing equations
The governing equations for the flow regime by using Boussinesq approximations for the current problem are expressed as:
2.2 Physical boundary conditions
2.3 Nondimensional problem
2.4 Engineering coefficients
3 Implementation of numerical method
3.1 Validation of numerical computations
Comparison for the numerical values of \( f^{\prime\prime}(0)\) when \(We = \beta^{ * } = 0.\)
Comparison for the numerical values of \( \;\theta^{\prime}(0)\) when \(We = \beta^{ * } = Nt = Nb = A = 0\)
4 Discussion of graphical results
In the current problem, the numerical solution for nanofluid velocity, temperature and concentration is derived to describe the flow behaviour of Williamson fluid towards a stretching surface. For this aim, we discuss the numerical results in terms of nondimensional velocity, temperature and nanoparticles concentration for different model parameters, like, unsteadiness parameter, viscosity ratio parameter, Weissenberg number, Brownian motion parameter, thermophoresis parameter and Lewis number.
4.1 Impacts of unsteadiness parameter
4.2 Impacts of Weissenberg number
The Weissenberg number plays a vital role on the profiles of nondimensional temperature, as shown in Fig. 3b. It is known from the graphs that, as the Weissenberg number increases, the fluid temperature is found to rise significantly. This is due to that the increase in the Weissenberg number means the rise in relaxation time, which, in turn, results in the increase in nondimensional fluid temperature. The variation of nanoparticle concentration profile for growing values of Weissenberg number is portrayed in Fig. 3c. One can easily observe that, for larger values of \(We\), the nanoparticle concentration and associated boundary layer thickness increases.
4.3 Impacts of Brownian motion parameter
4.4 Impacts of thermophoresis parameter
4.5 Impacts of Prandtl number
4.6 Impacts of Lewis number
Numerical values of \(Re_{x}^{1/2} C_{fx}\) for distinct values of \(A,\)\(\beta^{ * }\) and \(We\)
A  \(\beta^{ * }\)  \(We\)  \(Re_{x}^{1/2} C_{fx}\) 

0.0  0.3  2.0  0.46805 
0.7  –  –  0.497807 
1.4  –  –  0.513561 
2.0  –  –  0.523043 
0.2  0.0  –  0.474937 
–  0.2  –  0.714153 
–  0.4  –  0.83401 
–  0.6  –  0.92544 
–  0.8  –  1.00168 
–  0.3  1.0  0.715694 
–  –  2.0  0.479109 
–  –  3.0  0.345023 
–  –  4.0  0.270663 
Numerical values of \(Re_{x}^{  1/2} Nu_{x}\) and \(Re_{x}^{  1/2} Sh_{x}\) for distinct values of \(A\), \(\beta^{ * }\)\(Pr\), \(Nt\), \(Nb\;\) and \(Le\) when \(We = 2.0\)
\(A\)  \(\beta^{ * }\)  \(Pr\)  \(Nt\)  \(Nb\;\)  \(Le\)  \(Re_{x}^{  1/2} Nu_{x}\)  \(Re_{x}^{  1/2} Sh_{x}\) 

0.0  0.3  0.72  0.1  0.2  1.0  0.915093  0.671930 
0.7  –  –  –  –  –  1.18310  0.926204 
1.4  –  –  –  –  –  1.39156  1.10673 
2.0  –  –  –  –  –  1.54809  1.23959 
0.2  0.2  –  –  –  –  1.04339  0.794434 
–  0.4  –  –  –  –  1.06648  0.815047 
–  0.6  –  –  –  –  1.08310  0.830594 
–  0.8  –  –  –  –  1.96080  0.843116 
–  0.3  1.0  –  –  –  1.19642  0.949338 
–  –  3.0  –  –  –  1.90716  1.99536 
–  –  5.0  –  –  –  2.13629  2.83588 
–  –  7.0  –  –  –  2.17731  3.57517 
–  –  0.5  0.2  –  –  0.808876  0.351777 
–  –  –  0.4  –  –  0.795233  − 0.123751 
–  –  –  0.6  –  –  0.782066  − 0.579199 
–  –  –  0.8  –  –  0.769339  − 1.01574 
–  –  –  0.5  0.1  –  0.800539  − 1.606290 
–  –  –  –  0.3  –  0.776840  0.0632418 
–  –  –  –  0.5  –  0.753927  0.396419 
–  –  –  –  0.7  –  0.731779  0.538701 
–  –  –  –  0.2  1.0  0.788590  − 0.35391 
–  –  –  –  –  5.0  0.758201  1.50967 
–  –  –  –  –  10.0  0.747420  2.75713 
–  –  –  –  –  15.0  0.742076  3.67417 
5 Conclusions

The most practical outcome of this study was that the fluid velocity was significantly enhanced by higher viscosity ratio parameter.

Temperature of the nanofluids was considerably promoted by the thermophoresis phenomenon.

Heat transfer rate was elevated by higher values of Lewis number.

Velocity, temperature, and concentration profiles were depressed by increasing the unsteadiness parameter.

Larger values of Brownian motion parameter created an enhancement in temperature profile due to higher thermal conductivity of the liquid.
Notes
Acknowledgements
The authors gratefully acknowledge the anonymous reviewers for their valuable suggestions and comments to progress the superiority of this article.
Compliance with ethical standards
Conflict of Interest
We declare that we have no financial and personal interest of any nature in any product.
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