# Simulation of Natural Convective Boundary Layer Flow of a Nanofluid Past a Convectively Heated Inclined Plate in the Presence of Magnetic Field

## Abstract

This paper deals with the numerical simulation of transient magnetohydrodynamics natural convective boundary layer flow of a nanofluid over an inclined plate. In the modeling of nanofluids, dynamic effects including the Brownian motion and thermophoresis are taken into account. Numerical solutions have been computed via the Galerkin-finite element method. The effects of angle of inclination, buoyancy-ratio parameter, Brownian motion, thermophoresis and magnetic field are taken into account and controlled by non-dimensional parameters. To compute the rate of convergence and error of the computed numerical solution, the double mesh principle is used. Similarity solutions are calculated and presented graphically for non-dimensional velocity, temperature, local rate of heat and mass transfer with pertinent parameters. The modified Nusselt number decreases with increasing inclination angle, buoyancy-ratio parameter, Brownian motion and thermophoresis parameter, whereas it increases with increasing Prandtl number. Validation of the results is achieved with earlier results for forced convective flow and non-magnetic studies. Such problems have several applications in engineering and petroleum industries such as electroplating, chemical processing of heavy metals and solar water heaters. External magnetic fields play an important role in electrical power generation, inclination/acceleration sensors, fine-tuning of the final materials to industrial specification because of their controlling behaviour on the flow characteristics of nanofluids.

## Keywords

Nanofluid MHD Inclined plate Similarity solution Convective boundary condition Finite element method Double mesh principle## List of symbols

## Roman

- \(B_o\)
Uniform magnetic field strength

- \(D_B\)
Brownian diffusion coefficient

- \(D_T\)
Thermophoresis diffusion coefficient

- \(E_N\)
Error

*f*Dimensionless stream function

- \(g_e\)
Acceleration due to gravity

*h*Dimensionless velocity function

- \(h_e\)
Step size

- \(h_f\)
Heat transfer coefficient

*k*Thermal conductivity

*Ln*Nanofluid Lewis number

*M*Dimensionless magnetic parameter

*Nb*Brownian motion parameter

*Nc*Convective heating parameter

*Nr*Buoyancy-ratio parameter

*Nt*Thermophoresis parameter

- \(Nu_x\)
Local Nusselt number

*Nur*Reduced Nusselt number

*Pr*Prandtl number

- \(Ra_x\)
Local Rayleigh number

- \(r^N\)
Rate of convergence

- \(Sh_{x,n}\)
Local nanoparticle Sherwood number

*Shrn*Reduced nanoparticle Sherwood number

*T*Fluid temperature

- \(T_f\)
Hot fluid temperature

- \(T_w\)
Fluid temperature at the wall

- \(T_\infty \)
Ambient temperature

*u*,*v*Velocity components along

*x*and*y*-directions

## Greek symbols

- \(\alpha _m\)
Thermal diffusivity

- \(\beta \)
Thermal expansion coefficient

- \(\delta \)
Acute angle of the plate to the vertical

- \(\mu \)
Viscosity of the nanofluid

- \(\nu \)
Kinematic viscosity of the fluid

- \(\phi \)
Dimensionless nanoparticle volume fraction

- \(\hat{\phi }\)
Nanoparticle volume fraction

- \(\hat{\phi }_w\)
Nanoparticle volume fraction at the wall

- \(\hat{\phi }_\infty \)
Ambient nanoparticle volume fraction

- \(\psi \)
Stream function

- \(\rho _f\)
Density of the nanofluid

- \(\sigma _{nf}\)
Electrical conductivity of the nanofluid

- \(\tau \)
Ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid, defined by \((\rho c_p)_p/(\rho c_p)_f\)

- \(\theta \)
Dimensionless temperature

## Subscripts

*f*Base fluid

*nf*Nanofluid

*p*Nanoparticle

- \(w,\infty \)
Condition at the surface and in the free stream, respectively

## Introduction

Nanoparticles provide a connection between molecular structure and bulk materials. When nanoparticles strategically deployed in the base fluids, the ensuing nanofluids have been verified to achieve remarkable enhancement in the properties of thermal conductivity, as introduced by Choi [8]. This has made nanofluids attractive in various areas of recent technology incorporating heat exchangers [15], aerospace cooling systems [20], and energy systems [17]. The two most common approaches to investigate the phenomena of heat and mass transfer characteristics are either the Tiwari and Das model [26] (which only requires momentum and energy equations and incorporates nanoparticle effects via a volume fraction parameter only) and the Buongiorno non-homogeneous model [6] (which introduces a separate equation for the nanoparticle concentration). Several researchers worked on these models including Hatami et al. [14], Goyal and Bhargava [12], Hamad et al. [13].

The natural convection exerts a significant influence on the heat and mass transfer analysis in the problems of nanofluids. In most of the fluid flow processes, transport phenomena occur due to the combined effect of heat and mass transfer. This is because of buoyancy effects arising from density variation, which is due to variation in temperature and/or concentration of particles. The classical problem, which involves natural convective flow of a regular fluid over a vertical plate, was first investigated theoretically by Pohlhausen [22]. Thereafter, Bejan [5] incorporated the effect of Prandtl number on boundary layers in natural convective fluid flow problems. An extension of the classical problem [22] to incorporate the effect of heat and mass transfer was investigated by Khair and Bejan [16]. Later, Aziz and Khan [4] numerically investigated the free convective boundary layer flow of a nanofluid over a vertical plate. Their analysis showed that the flow pattern, heat and mass transfer analysis strongly influenced by the pertinent parameters.

Lately, the problems of free convection fluid flow over a plate for different values of inclination angle were frequently encountered in engineering devices such as solar water heaters and inclination/acceleration sensors. Most of the researchers [1, 3, 7] observed that fluid flow through the medium was favoured in case of an inclined plate as inclination to the vertical reduces the drag force. A generalized formulation was explained by Ali et al. [2] for the combined effect of chemical reaction and radiation on MHD free convective flow of viscous fluid over an inclined plate. They found that the flow features not only depend on the magnitude of inclination but also on the distance from the leading edge. Later, Narahari et al. [19] has studied the effect of free convective flow of a nanofluid over an isothermal inclined plate and observed that the thickness of the momentum boundary layer decreases with an increase in angle of inclination whereas the temperature and nanoparticle volume fraction increase with increasing inclination angle.

The study of flow analysis and heat transfer under the influence of an applied magnetic field is considered a significant research topic due to its numerous scientific, industrial and biological applications such as crystal growth, cooling of metallic plates, production of magnetorheostatic materials known as smart fluids, metal casting and liquid metal cooling blankets for fusion reactors. The rate of heat transfer can be controlled by MHD flow in electrically conducting fluid and hence desired cooling effect can be achieved. The different types of thermal boundary conditions were used by Sathiyamoorthy and Chamkha [23] to study steady state, laminar, 2D natural convective flow in the presence of an inclined magnetic field in a square enclosure filled with liquid Gallium. Recently, Goyal and Bhargava [9] numerically investigated the MHD viscoelastic nanofluid flow past a stretching sheet with heat source/sink and partial slip. It was observed from the study that modified Nusselt number is directly proportional to Brownian motion and thermophoretic parameters and indirectly proportional to all other parameters.

As, the study of a convectively heated inclined plate plays an important role in many processes such as manufacturing of tetrapacks, glass fibres, plastic and rubber sheets, solidification of casting. An efficient manufacturing of such materials incorporates various physical phenomena including the implementation of magnetohydrodynamics (MHD), thermal and mass diffusion effects at nanoscale level. To improve the interpretation of the inter-disciplinary transport phenomena in such type of systems, a robust approach is provided with the help of mathematical model. Hence, motivated by this, the present study focused to develop a mathematical model for natural convective boundary layer flow of a nanofluid past a convectively heated inclined plate in the presence of Magnetic field.

The Buongiorno nanofluid model approach [6] is used which emphasizes the Brownian motion and thermophoresis effects. This approach also introduces a separate equation for nano-particle species diffusion. By using the suitable similarity transformation for velocity, temperature and nanoparticle concentration, the equations governing for flow, heat and mass transfer were transformed to a set of ordinary differential equations. The resulting equations subjected to the boundary conditions were solved numerically using conventional finite element method (FEM). The numerical investigation is carried out for different thermophysical parameters, namely: the magnetic parameter, buoyancy-ratio parameter, convective heat parameter, Prandtl number, nanofluid Lewis number, Brownian motion parameter, and thermophoresis parameter. The obtained results are validated by comparing with work of other authors that has reported in literature. The rates of heat and nano-mass transfer were computed and were shown in both tabular and graphical formats.

## Problem Formulation

### Governing Equations and Boundary Conditions

*u*and

*v*are the velocity components parallel and perpendicular to the plate, respectively, \(B_o\) is uniform magnetic field strength, \(\hat{\phi }\) is the local solid volume fraction of the nanoparticles, \(\beta \) is volumetric thermal expansion coefficient of the base fluid, \(D_B\) is the Brownian diffusion coefficient, \(D_T\) is the thermophoretic diffusion coefficient, and

*T*is the local temperature.

Continuity, momentum, thermal energy, and nanoparticle species equations for nanofluid are represented by Eqs. (1)–(4), respectively. The terms (from left to right) in the right side of Eq. (2) represent the stress component due to viscosity, the convective acceleration and the force due to the magnetic field. The first and second terms in the square bracket in (2) represent the positive (upward) buoyancy term due to the thermal expansion of the base fluid and the negative (downward) buoyancy term due to the variation in densities of the nanoparticles and the base fluid, respectively. The terms in the left hand side of Eq. (3) are the convection terms due to temperature. On the other hand, the terms on right side (left to right) represent the heat enthalpy, diffusion of thermal energy due to Brownian diffusion and thermophoretic effect. A similar interpretation could be given to the terms on the right hand side of Eq. (4).

### Similarity Transformations

**Nusselt and Sherwood number evaluation**

## Numerical Solution with Finite Element Solution

### Variational Formulation

### Finite Element Formulation

Comparison of *Nur* of regular fluid for various values of *Pr* with \(Ln=10\), \(Nb=Nt=Nr=10^{-5}\), \(M=0\)

Comparison of results for *Nur* and *Shrn* when \(Nt=0.1\), \(Nc=10\), \(Ln=10\), \(M=0,\) \(\delta =0\)

| | \(Pr=1\) | \(Pr=5\) | ||||||
---|---|---|---|---|---|---|---|---|---|

| | | | ||||||

Aziz and Khan [4] | Present results | Aziz and Khan [4] | Present results | Aziz and Khan [4] | Present results | Aziz and Khan [4] | Present results | ||

0.1 | 0.1 | 0.3396 | 0.3395 | 0.9954 | 0.9955 | 0.3807 | 0.3807 | 1.0608 | 1.0609 |

0.2 | 0.3366 | 0.3364 | 0.9828 | 0.9830 | 0.3773 | 0.3770 | 1.0482 | 1.0484 | |

0.3 | 0.3334 | 0.3331 | 0.9697 | 0.9699 | 0.3739 | 0.3737 | 1.0351 | 1.0352 | |

0.4 | 0.3301 | 0.3297 | 0.9559 | 0.9563 | 0.3702 | 0.3699 | 1.0214 | 1.0217 | |

0.5 | 0.3267 | 0.3266 | 0.9414 | 0.9415 | 0.3665 | 0.3663 | 1.0071 | 1.0075 | |

0.3 | 0.1 | 0.2939 | 0.2938 | 1.0435 | 1.0437 | 0.3306 | 0.3301 | 1.1101 | 1.1102 |

0.2 | 0.2918 | 0.2917 | 1.0317 | 1.0317 | 0.3282 | 0.3280 | 1.0985 | 1.0988 | |

0.3 | 0.2896 | 0.2896 | 1.0195 | 1.0199 | 0.3258 | 0.3255 | 1.0866 | 1.0870 | |

0.4 | 0.2872 | 0.2869 | 1.0067 | 1.0069 | 0.3232 | 0.3231 | 1.0741 | 1.0743 | |

0.5 | 0.2848 | 0.2844 | 0.9934 | 0.9935 | 0.3206 | 0.3202 | 1.0611 | 1.0612 | |

0.5 | 0.1 | 0.2530 | 0.2525 | 1.0584 | 1.0585 | 0.2855 | 0.2852 | 1.1263 | 1.1266 |

0.2 | 0.2513 | 0.2512 | 1.0471 | 1.0471 | 0.2836 | 0.2835 | 1.1152 | 1.1153 | |

0.3 | 0.2495 | 0.2492 | 1.0353 | 1.0355 | 0.2816 | 0.2818 | 1.1037 | 1.1040 | |

0.4 | 0.2477 | 0.2475 | 1.0230 | 1.0233 | 0.2796 | 0.2794 | 1.0918 | 1.0919 | |

0.5 | 0.2458 | 0.2456 | 1.0102 | 1.0105 | 0.2775 | 0.2771 | 1.0794 | 1.0795 |

Errors and rate of convergence for the finite element method (FEM)

\(h_e\) | \(f'\) | \(\theta \) | \(\phi \) | |||
---|---|---|---|---|---|---|

Error | Rate | Error | Rate | Error | Rate | |

0.128 | 2.0186e−2 | 1.1664 | 7.0795e−2 | 1.5609 | 4.0550e−2 | 1.4903 |

0.064 | 8.9935e−3 | 1.2898 | 2.3994e−2 | 1.6565 | 1.4433e−2 | 1.5759 |

0.032 | 3.6779e−3 | 1.4059 | 7.6112e−3 | 1.7482 | 4.8414e−3 | 1.6549 |

0.016 | 1.3880e−3 | 1.5135 | 2.2656e−3 | 1.8108 | 1.5374e−3 | 1.7238 |

0.008 | 4.8616e−4 | 1.5849 | 6.4578e−4 | 1.9468 | 4.6547e−4 | 1.8136 |

0.004 | 1.6206e−4 | 1.6328 | 1.6751e−4 | 2.0004 | 1.3242e−4 | 1.9091 |

0.002 | 5.2259e−5 | – | 4.1868e−5 | – | 3.5258e−5 | – |

Owing to the nonlinearity of the system an iterative scheme has been used to solve it iteratively. The system of equations is linearized by incorporating known functions \(\displaystyle {\bar{f},~\bar{f'},~\bar{\theta },~\bar{\phi }}\) which are calculated using the approximate values of variables \(\displaystyle {f,~f',~\theta ,~\phi }\) at node \(\displaystyle {i}\) on previous iteration, as given in Eq. (50). The whole system is solved by using a Gaussian elimination method and the whole procedure is executed in MATLAB. This gives a new set of values of unknowns \(\displaystyle {f,~f',~\theta ,~\phi }\) and the process continues until the required accuracy of \(\displaystyle {1\times 10 ^{-5}}\) is achieved.

## Validation of the Numerical Procedure

For validation purpose, results were compared with previously reported results in the literature. The results for the regular fluid at different values of \(\displaystyle {Pr}\) were compared with those reported by Bejan [5], Kuznetsov and Nield [17] and Narahari et al. [19], has been captured in Table 1. Closer correlation has been achieved as compared to results computed by [5, 17, 19]. On the other hand, Table 2 shows the excellent correlation between the current FEM computation and the earlier results of Aziz and Khan [4] on the modified Nusselt and nanoparticle Sherwood number under the influence of the various parameters such as \(\displaystyle {Nb,~Nr,~Pr}\).

## Double Mesh Principle

## Computations and Discussion

In Figs. 5, 6 and 7, the influence of the plate inclination from the vertical, \(\displaystyle {\delta }\), ranging from 0 to \(\displaystyle {\pi /4}\), on the velocity \(\displaystyle {f'(\eta )}\), temperature \(\displaystyle {\theta (\eta )}\) and nanoparticle volume fraction \(\displaystyle {\phi (\eta )}\) profiles are depicted, respectively. It is observed from Fig. 5 that within the hydrodynamic boundary layer, the velocity of the fluid is diminished with an augmentation of inclination angle. This is because to the plate’s alignment via the thermal buoyancy term, \(\displaystyle {g[-(\rho _p-\rho _{f_\infty })(\hat{\phi }-\hat{\phi }_\infty )+(1-\hat{\phi }_\infty )\rho _{f_\infty } \beta (T-T_\infty )]\cos \delta }\), which is arising in the momentum equation (2). As the value of \(\displaystyle {\delta }\) increases, the corresponding value of \(\displaystyle {\cos \delta }\) decreases. This causes the buoyancy effect to be vanished with increasing the plate inclination. Consequently, the driving force to the fluid attenuates, resulting in decrease of velocity of the fluid. A similar type of trend has been found by Alam et al. [1] in case of velocity profile. On the other hand, a depletion in buoyancy effect will enhance thermal and species (nanoparticle) diffusion, which is shown in Figs. 6 and 7.

Brownian motion is the haphazard motion of nanoparticles inside the base fluid due to of the continuous collision of nanoparticle with the molecules of base fluid. This motion of the particles is described by parameter \(\displaystyle {Nb}\), known as Brownian motion coefficient. Figures 8, 9 and 10 illustrate the effect of \(\displaystyle {Nb}\) on velocity \(\displaystyle {f'(\eta )}\), temperature \(\displaystyle {\theta (\eta )}\) and concentration \(\displaystyle {\phi (\eta )}\) profiles. With an increase in \(\displaystyle {Nb}\), the randomness of the nanoparticles increased and as a result, nanoparticles move more chaotically, causing more collisions in the system and vice versa. This increase in number of collisions and velocity result an increase in heat transfer properties, and thus, the value of temperature increases. Simultaneously, the increase in \(\displaystyle {Nb}\) has an adverse effect on the concentration of nanoparticles along the wall. The nanoparticles start moving away from the boundary into the fluid by increasing the random motion of nanoparticles which causing a decrease in the value of concentration of nanoparticles along the wall.

Variation of *Nur* and *Shrn* for *Pr*, *Nb* and *Nr* for \(\delta =\pi /4\), \(Nc=10\), \(Nt=0.5\), \(M=0.1\), \(Ln=10\)

| | \(Pr=1\) | \(Pr=5\) | \(Pr=10\) | |||
---|---|---|---|---|---|---|---|

| | | | | | ||

0.1 | 0.1 | 0.2120 | 0.7253 | 0.2183 | 0.7418 | 0.2191 | 0.7441 |

0.2 | 0.2076 | 0.7094 | 0.2136 | 0.7252 | 0.2144 | 0.7273 | |

0.3 | 0.2030 | 0.6926 | 0.2086 | 0.7076 | 0.2094 | 0.7097 | |

0.4 | 0.1981 | 0.6748 | 0.2034 | 0.6890 | 0.2042 | 0.6909 | |

0.5 | 0.1929 | 0.6559 | 0.1979 | 0.6692 | 0.1986 | 0.6710 | |

0.6 | 0.1874 | 0.6355 | 0.1921 | 0.6479 | 0.1927 | 0.6496 | |

0.3 | 0.1 | 0.1840 | 0.8280 | 0.1897 | 0.8420 | 0.1905 | 0.8439 |

0.2 | 0.1816 | 0.8143 | 0.1871 | 0.8280 | 0.1879 | 0.8299 | |

0.3 | 0.1791 | 0.8000 | 0.1845 | 0.8133 | 0.1852 | 0.8151 | |

0.4 | 0.1766 | 0.7849 | 0.1817 | 0.7978 | 0.1825 | 0.7995 | |

0.5 | 0.1739 | 0.7689 | 0.1789 | 0.7814 | 0.1796 | 0.7831 | |

0.6 | 0.1710 | 0.7519 | 0.1758 | 0.7640 | 0.1765 | 0.7656 | |

0.5 | 0.1 | 0.1581 | 0.8502 | 0.1632 | 0.8639 | 0.1639 | 0.8658 |

0.2 | 0.1564 | 0.8371 | 0.1614 | 0.8506 | 0.1620 | 0.8524 | |

0.3 | 0.1546 | 0.8235 | 0.1594 | 0.8366 | 0.1601 | 0.8384 | |

0.4 | 0.1527 | 0.8091 | 0.1574 | 0.8219 | 0.1580 | 0.8236 | |

0.5 | 0.1507 | 0.7939 | 0.1552 | 0.8064 | 0.1559 | 0.8081 | |

0.6 | 0.1486 | 0.7778 | 0.1530 | 0.7900 | 0.1536 | 0.7916 |

Variation of *Nur* and *Shrn* for *M*, *Nt* and \(\delta \) for \(Nc=10\), \(Pr=5.0\), \(Nb=Nr=0.5\), \(Ln=10\)

| \(\delta \) | \(M=1.0\) | \(M=3.0\) | \(M=5.0\) | |||
---|---|---|---|---|---|---|---|

| | | | | | ||

0.1 | 0 | 0.1979 | 0.8541 | 0.1500 | 0.6858 | 0.1266 | 0.5903 |

\(\pi /12\) | 0.1956 | 0.8446 | 0.1480 | 0.6768 | 0.1249 | 0.5819 | |

\(\pi /6\) | 0.1885 | 0.8156 | 0.1418 | 0.6490 | 0.1198 | 0.5561 | |

\(\pi /4\) | 0.1759 | 0.7636 | 0.1312 | 0.5997 | 0.1111 | 0.5105 | |

0.3 | 0 | 0.1829 | 0.8692 | 0.1387 | 0.6958 | 0.1169 | 0.5986 |

\(\pi /12\) | 0.1808 | 0.8596 | 0.1368 | 0.6867 | 0.1153 | 0.5901 | |

\(\pi /6\) | 0.1743 | 0.8300 | 0.1310 | 0.6585 | 0.1105 | 0.5642 | |

\(\pi /4\) | 0.1626 | 0.7769 | 0.1211 | 0.6086 | 0.1024 | 0.5186 | |

0.5 | 0 | 0.1696 | 0.8886 | 0.1284 | 0.7089 | 0.1081 | 0.6097 |

\(\pi /12\) | 0.1676 | 0.8787 | 0.1267 | 0.6996 | 0.1067 | 0.6012 | |

\(\pi /6\) | 0.1615 | 0.8483 | 0.1213 | 0.6709 | 0.1022 | 0.5751 | |

\(\pi /4\) | 0.1507 | 0.7939 | 0.1121 | 0.6204 | 0.0946 | 0.5294 |

The buoyancy-ratio parameter \(\displaystyle {Nr}\) is defined as the ratio of the variation of the fluid density (due to the variation of the concentration) to the variation of the density of the nanofluid (due to the variation of temperature). Figures 14, 15 and 16 present the behavior of buoyancy ratio parameter \(\displaystyle {Nr}\) on the velocity \(\displaystyle {f'(\eta )}\), temperature \(\displaystyle {\theta (\eta )}\) and nanoparticle volume fraction \(\phi (\eta )\) profiles. It is observed from these figures that an increase in the Buoyancy-ratio parameter increases the magnitude of the dimensionless temperature and nanoparticle concentration while decreases the magnitude of the dimensionless velocity of the nanofluid.

Variation of *Nur* and *Shrn* for *Ln*, *Nt* and *Nr* for \(\delta =\pi /4\), \(Nc=10\), \(Nb=0.5\), \(M=0.1\), \(Pr=5.0\)

| | \(Ln=1\) | \(Ln=5\) | \(Ln=10\) | |||
---|---|---|---|---|---|---|---|

| | | | | | ||

0.1 | 0.1 | 0.2105 | 0.2710 | 0.1897 | 0.6148 | 0.1839 | 0.8201 |

0.2 | 0.2025 | 0.2607 | 0.1868 | 0.6025 | 0.1821 | 0.8071 | |

0.3 | 0.1938 | 0.2495 | 0.1837 | 0.5895 | 0.1801 | 0.7934 | |

0.4 | 0.1842 | 0.2374 | 0.1805 | 0.5757 | 0.1781 | 0.7789 | |

0.5 | 0.1736 | 0.2241 | 0.1770 | 0.5608 | 0.1759 | 0.7636 | |

0.6 | 0.1617 | 0.2095 | 0.1733 | 0.5447 | 0.1736 | 0.7472 | |

0.3 | 0.1 | 0.1986 | 0.2437 | 0.1763 | 0.6191 | 0.1704 | 0.8333 |

0.2 | 0.1902 | 0.2336 | 0.1734 | 0.6067 | 0.1686 | 0.8203 | |

0.3 | 0.1810 | 0.2228 | 0.1704 | 0.5937 | 0.1667 | 0.8066 | |

0.4 | 0.1709 | 0.2116 | 0.1672 | 0.5798 | 0.1647 | 0.7922 | |

0.5 | 0.1598 | 0.1999 | 0.1638 | 0.5649 | 0.1626 | 0.7769 | |

0.6 | 0.1477 | 0.1879 | 0.1601 | 0.5488 | 0.1604 | 0.7607 | |

0.5 | 0.1 | 0.1875 | 0.2235 | 0.1641 | 0.6283 | 0.1581 | 0.8502 |

0.2 | 0.1788 | 0.2133 | 0.1613 | 0.6159 | 0.1564 | 0.8371 | |

0.3 | 0.1693 | 0.2031 | 0.1583 | 0.6027 | 0.1546 | 0.8235 | |

0.4 | 0.1589 | 0.1930 | 0.1552 | 0.5888 | 0.1527 | 0.8091 | |

0.5 | 0.1477 | 0.1836 | 0.1519 | 0.5738 | 0.1507 | 0.7939 | |

0.6 | 0.1358 | 0.1754 | 0.1484 | 0.5577 | 0.1486 | 0.7778 |

## Conclusion

- 1.
Amplifying the strength of magnetic field \(\displaystyle {M}\) attenuates the thickness of the momentum boundary layer and expands the thermal and nano-volume fraction boundary layer. The application of an external magnetic field produces Lorentz drag force which retards the fluid motion. By customizing the external magnetic field, the transfer of heat can be controlled. In the field of ’smart’ cooling devices, widespread growth is based on this idea.

- 2.
With an augmentation in the magnetic parameter \(\displaystyle {M}\), the magnitude of heat and nano-mass transfer rates decrease as a consequence of intensified Lorentz drag force.

- 3.
Strengthening the thermophoresis \(\displaystyle {Nt}\) and Brownian motion \(\displaystyle {Nb}\) parameters, the rate of heat and nano-mass transfer decrease for an increase in the value of magnetic field parameter \(\displaystyle {M}\). The heat and mass transfer rates can be altered by taking different combinations of base fluid and nanoparticles. This idea can be implemented for numerous industrial applications involving inclined/vertical plates (production of glass fibres, plastic products, tetrapacks etc.) in adjusting the heat transfer rates.

- 4.
With the mounting values of an angle of inclination \(\displaystyle {\delta }\), the width of the momentum boundary layer decays whereas the reverse effect occurs for temperature and concentration boundary layers.

- 5.
The use of a convective boundary condition instead of a constant temperature or a constant heat flux makes this approach novel. As, the convective heating parameter \(\displaystyle {Nc}\) enhances the rate of heat transfer at the surface of the plate. This effect finds application in case of heat exchangers where the convection in the fluid past the solid surface influenced the conduction in the solid surface.

- 6.
The excellent accuracy of the computed FEM results was shown with the help of double mesh principle.

## References

- 1.Alam, M., Rahman, M., Sattar, M.: On the effectiveness of viscous dissipation and joule heating on steady magnetohydrodynamic heat and mass transfer flow over an inclined radiate isothermal permeable surface in the presence of thermophoresis. Commun. Nonlinear Sci. Numer. Simul.
**14**, 2132–2143 (2009)CrossRefGoogle Scholar - 2.Ali, F., Khan, I., Samiulhaq, S.S.: Conjugate effects of heat and mass transfer on MHD free convection flow over an inclined plate embedded in a porous medium. Abbott D, ed. PLoS ONE
**8**(6), e65,223 (2013). https://doi.org/10.1371/journal.pone.0065,223 CrossRefGoogle Scholar - 3.Anghel, M., Hossain, M.A., Zeb, S., Pop, I.: Combined heat and mass transfer by free convection past an inclined flat plate. Int. J. Appl. Mech. Eng.
**2**, 473–497 (2001)zbMATHGoogle Scholar - 4.Aziz, A., Khan, W.: Natural convective boundary layer flow of a nanofluid past a convectively heated vertical plate. Int. J. Therm. Sci.
**52**, 83–90 (2012)CrossRefGoogle Scholar - 5.Bejan, A.: Convection Heat Transfer. Wiley, New York (1984)zbMATHGoogle Scholar
- 6.Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf.
**128**, 240–250 (2006)CrossRefGoogle Scholar - 7.Chen, C.H.: Heat and mass transfer in MHD flow by natural convection from a permeable, inclined surface with variable wall temperature and concentration. Acta Mech.
**172**, 219–235 (2004)CrossRefzbMATHGoogle Scholar - 8.Choi, S.U.S.: Enhancing thermal conductivity of fluids with nanoparticles in developments and applications of non-Newtonian flows. In: Siginer, D.A., Wang, H.P. (eds.) ASME FED 231/MD, vol. 66, pp. 99–105 (1995)Google Scholar
- 9.Goyal, M., Bhargava, R.: Numerical solution of MHD viscoelastic nanofluid flow over a stretching sheet with partial slip and heat source/sink. ISRN Nanotechnol.
**2013**, 1–11 (2013)CrossRefGoogle Scholar - 10.Goyal, M., Bhargava, R.: Boundary layer flow and heat transfer of viscoelastic nanofluids past a stretching sheet with partial slip conditions. Appl. Nanosci.
**4**, 761–767 (2014a)CrossRefGoogle Scholar - 11.Goyal, M., Bhargava, R.: Numerical study of thermodiffusion effects on boundary layer flow of nanofluids over a power law stretching sheet. Microfluid. Nanofluidics
**17**, 591–604 (2014b)CrossRefGoogle Scholar - 12.Goyal, M., Bhargava, R.: Thermodiffusion effects on boundary layer flow of viscoelastic nanofluids over a stretching sheet with viscous dissipation and non-uniform heat source using hp-finite element method. Proc. IMechE Part N J. Nanoeng. Nanosyst.
**230**, 124–140 (2014c)Google Scholar - 13.Hamad, M.A.A., Pop, I., Ismail, A.I.M.: Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate. Nonlinear Anal. Real World Appl.
**12**, 1338–1346 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Hatami, M., Jing, D., Song, D., Sheikholeslami, M., Ganji, D.: Heat transfer and flow analysis of nanofluid flow between parallel plates in presence of variable magnetic field using HPM. J. Magn. Magn. Mater.
**396**, 275–282 (2015)CrossRefGoogle Scholar - 15.Huminic, G., Huminic, A.: Application of nanofluids in heat exchangers: a review. Renew. Sust. Energy Rev.
**16**(8), 5625–5638 (2012). https://doi.org/10.1016/j.rser.2012.05.023 CrossRefzbMATHGoogle Scholar - 16.Khair, K.R., Bejan, A.: Mass transfer to natural convection boundary layer flow driven by heat transfer. Int. J. Heat Mass Transf.
**30**, 369–376 (1985)Google Scholar - 17.Kuznetsov, A., Nield, D.: Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci.
**49**, 243–247 (2010)CrossRefGoogle Scholar - 18.Margenov, S., Vulkov, L., Vulkov, L.G., Wasniewski, J.: Numerical Analysis and Its Applications: 4th International Conference, NAA 2008 Lozenetz, Bulgaria, June 2008. Revised Selected Papers. Springer (2008)Google Scholar
- 19.Narahari, M., Akilu, S., Jaafar, A.: Free convection flow of a nanofluid past an isothermal inclined plate. Appl. Mech. Mater.
**390**, 129–133 (2013)CrossRefGoogle Scholar - 20.Narvaez, J.A., Veydt, A.R., Wilkens, R.J.: Evaluation of nanofluids as potential novel coolant for aircraft applications: the case of de-ionized water-based alumina nanofluids. ASME J Heat Transf.
**136**(051), 702 (2014)Google Scholar - 21.Natesan, S., Jayakumar, J., Vigo-Aguiar, J.: Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers. J. Comput. Appl. Math.
**158**, 121–134 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Pohlhausen, E.: Der warmeaustausch zwischen festen korpern und flussigkeiten mit kleiner reibung und kleiner warmeleitung. J. Appl. Math. Mech.
**1**, 115–121 (1921)zbMATHGoogle Scholar - 23.Sathiyamoorthy, M., Chamkha, A.: Effect of magnetic field on natural convection flow in a liquid gallium filled square cavity for linearly heated side wall(s). Int. J. Therm. Sci.
**49**, 1856–1865 (2010)CrossRefGoogle Scholar - 24.Shanthi, V., Ramanujam, N., Natesan, S.: Fitted mesh method for singularly perturbed reaction–convection–diffusion problems with boundary and interior layers. J. Appl. Math. Comput.
**22**, 49–65 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 25.Siddiqa, S., Hossain, M., Saha, S.C.: The effect of thermal radiation on the natural convection boundary layer flow over a wavy horizontal surface. Int. J. Therm. Sci.
**84**, 143–150 (2014)CrossRefGoogle Scholar - 26.Tiwari, R.K., Das, M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf.
**50**, 2002–2018 (2007)CrossRefzbMATHGoogle Scholar

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