# Unsteady MHD free convection flow past a vertical permeable flat plate in a rotating frame of reference with constant heat source in a nanofluid

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## Abstract

The unsteady magnetohydrodynamic flow of a nanofluid past an oscillatory moving vertical permeable semi-infinite flat plate with constant heat source in a rotating frame of reference is theoretically investigated. The velocity along the plate (slip velocity) is assumed to oscillate on time with a constant frequency. The analytical solutions of the boundary layer equations are assumed of oscillatory type and they are obtained by using the small perturbation approximations. The influence of various relevant physical characteristics are presented and discussed.

## Keywords

Nusselt Number Local Nusselt Number Skin Friction Coefficient Nanoparticle Volume Fraction Thermal Boundary Layer Thickness## List of symbols

*A*_{i},*B*_{i}Constants

*B*_{0}Constant applied magnetic field (Wb m

^{−2})*C*_{p}Specific heat at constant pressure (J kg

^{−1}K^{−1})*E*Electric field (kJ)

*g*Gravity acceleration (m s

^{−2})*J*Current density

*M*Dimensionless magnetic field parameter

*n*Dimensionless frequency

*Nu*Local Nusselt number

*Nur*Reduced Nusselt number

*Pr*Prandtl number

- \( \bar{q}_{w} \)
Dimensional heat flux from the plate

*Q*Dimensional heat source (kJ s

^{−1})*Q*_{H}Dimensionless heat source parameter (kJ s

^{−1})*R*Dimensionless rotation parameter

- \( \text{Re}_{x} \)
Local Reynolds number

*S*Dimensionless suction parameter

*t*Dimensionless time (s)

*T*Local temperature of the nanofluid (K)

*T*_{w}Wall temperature (K)

*T*_{∞}Temperature of the ambient nanofluid (K)

*u*,*v*,*w*Dimensionless velocity components (m s

^{−1})*U*_{0}Characteristic velocity (m s

^{−1})*w*_{0}Mass flux velocity

## Greek symbols

*α*Thermal diffusivity (m

^{2}s^{−1})*β*Thermal expansion coefficient (K

^{−1})- ε
Dimensionless small quantity (≪1)

- \( \phi \)
Solid volume fraction of the nanoparticles

*κ*Thermal conductivity (m

^{2}s^{−1})*μ*Dynamic viscosity (Pa s)

*ν*Kinematic viscosity (m

^{2}s^{−1})*θ*Dimensionless temperature

*σ*Electrical conductivity (m

^{2}s^{−1})- \( \bar{\tau }_{w} \)
Skin friction or shear stress

- Ω
Constant rotation velocity

## Superscript

- –
Dimensional quantities

## Subscripts

*f*Fluid

*s*Solid

*nf*Nanofluid

## 1 Introduction

The reported breakthrough in substantially increasing the thermal conductivity of fluids by adding very small amounts of suspended metallic or metallic oxide nanoparticles (Cu, CuO, Al_{2}O_{3}) to the fluid [14, 27], or alternatively using nanotube suspensions [10, 38] conflicts with the classical theories [5, 6, 7, 8, 13, 17, 21, 29, 30], of estimating the effective thermal conductivity of suspensions. A very small amount (less than 1% in terms of volume fraction) of copper nanoparticles was reported to improve the measured thermal conductivity of the suspension by 40% [14, 27], while over a 150% improvement of the effective thermal conductivity at a volume fraction of 1% was reported by Choi et al. [10] for multi-walled carbon nanotubes suspended in oil. The comprehensive references on nanofluid can be found in the recent book by Das et al. [12] and in the review papers by Trisaksri and Wongwises [35], Wang and Mujumdar [37], and Kakaç and Pramuanjaroenkij [23]. There have been published quite many numerical studies on the modeling of natural convection heat transfer in nanofluids, namely Khanafer et al. [25], Roy et al. [34], Jou and Tzeng [22], Ho et al. [18, 19], Congedo et al. [11], and Ghasemi and Aminossadati [15]. These studies have used traditional finite-difference and finite-volume techniques with the tremendous call on computational resources that these techniques necessitate. Abu-Nada [1] studied numerically the heat transfer characteristics of flow over a backward facing step using nanofluids. Abu-Nada et al. [3] investigated the heat transfer enhancement in a differentially heated enclosure using variable thermal conductivity and variable viscosity of Al_{2}O_{3}–water and CuO–water nanofluids. Khan and Pop [24] analyzed the development of the steady boundary layer flow, heat transfer and nanoparticle volume fraction over a linear stretching surface in a nanofluid. Ahmad and Pop [4] studied the steady mixed convection boundary layer flow past a vertical flat plate embedded in a porous medium filled with nanofluids using different types of nanoparticles as Cu (copper), Al_{2}O_{3} (alumina) and TiO_{2} (titania). Kuznetsov and Nield [26] studied the classical problem of free convection boundary layer flow of a viscous and incompressible fluid (Newtonian fluid) past a vertical flat plate to the case of nanofluids. In both of these papers the authors have used the nanofluid model proposed by Buongiorno [9]. Although this author discovered that seven slip mechanisms take place in the convective transport in nanofluids, it is only the Brownian diffusion and the thermophoresis that are the most important when the turbulent flow effects are absent. However, we will use here the nanofluid model proposed by Tiwari and Das [36], which was used by many researchers, such as, Abu-Nada [1], Oztop and Abu-Nada [32], Abu-Nada and Oztop [2], Muthtamilselvan et al. [31], etc.

The present paper deals with a theoretical study for the problem of unsteady MHD free convection flow of a nanofluid past an oscillatory moving vertical permeable semi-infinite flat plate with constant heat source in a rotating frame of reference with a constant suction velocity at the plate. We will use here the nanofluid model proposed by Tiwari and Das [36]. The aim is to investigate the influence of solid volume fraction parameter *ϕ* on the flow and heat-transfer characteristics for various nanoparticles considered. The mathematical analysis and the corresponding solutions have been presented in the form of Ganapathy [16].

## 2 Governing equations and the boundary conditions

*ɛ*is a small constant parameter (\( \varepsilon \ll 1 \)) and

*U*

_{0}is the characteristic velocity. We consider that initially (\( \bar{t} < 0 \)) the fluid as well as the plate are at rest but for \( \bar{t} \ge 0 \) the whole system is allowed to rotate with a constant velocity Ω about the \( \bar{z} \)-axis. A uniform external magnetic field

*B*

_{0}is taken to be acting along the \( \bar{z} \)-axis. We consider the case of a short circuit problem in which the applied electric field

*E*= 0, and also assume that the induced magnetic field is small compared to the external magnetic field

*B*

_{0}. This implies a small magnetic Reynolds number for the oscillating plate (see Liron and Wilhelm [28]). The surface temperature is assumed to have the constant value

*T*

_{ w }while the ambient temperature has the constant value

*T*

_{∞}, where

*T*

_{ w }>

*T*

_{∞}. The conservation equation of current density \( \nabla \cdot {\mathbf{J}} = 0 \) gives

*J*

_{ z }= constant, where \( {\mathbf{J}}(J_{x} ,J_{y} ,J_{z} ) \). Since the plate is electrically nonconducting, this constant is zero. It is assumed that the plate is infinite in extent and hence all physical quantities do not depend on \( \bar{x} \) and \( \bar{y} \) but depend only on \( \bar{z} \) and \( \bar{t} \), that is \( \partial \bar{u}/\partial \bar{x} = \partial \bar{u}/\partial \bar{y} = \partial \bar{v}/\partial \bar{x} = \partial \bar{v}/\partial \bar{y} = 0 \), etc. It is further assumed that the regular fluid and the suspended nanoparticles are in thermal equilibrium and no slip occurs between them. Following the nanofluid model proposed by Tiwari and Das [36] along with the Boussinesq and boundary layer approximations, the boundary layer equations governing the flow and temperature are,

*T*is the local temperature of the nanofluid and

*Q*is the additional heat source. On the other hand,

*β*

_{ f }and

*β*

_{ s }are the coefficients of thermal expansion of the fluid and of the solid, respectively,

*ρ*

_{ f }and

*ρ*

_{ s }are the densities of the fluid and of the solid fractions, respectively, while \( \rho_{nf} \) is the density of the nanofluid, \( \mu_{nf} \) is the viscosity of the nanofluid, \( \alpha_{nf} \) is the thermal diffusivity of the nanofluid, and \( (\rho C_{p} )_{nf} \) is the heat capacitance of the nanofluid, which are defined as (see Oztop and Abu-Nada [32])

*κ*

_{ f }and

*κ*

_{ s }are the thermal conductivities of the base fluid and of the solid, respectively. The thermo-physical properties of the base fluid (water), copper and titania which were used for code validation are given in Table 1. We consider the solution of Eq. (1) as

Thermo-physical properties [32]

Physical properties | Water | Copper (Cu) | Titanium oxide (TiO |
---|---|---|---|

C | 4,179 | 385 | 686.2 |

ρ (kg/m | 997.1 | 8,933 | 4,250 |

κ (W/m K) | 0.613 | 400 | 8.9538 |

β × 10 | 21 | 1.67 | 0.9 |

*w*

_{0}represents the normal velocity at the plate which is positive for suction (\( w_{0} > 0 \)) and negative for blowing or injection (\( w_{0} < 0 \)). Thus, we introduce the following dimensionless variables:

*ν*

_{ f }is the kinematic viscosity of the fluid part of the nanofluid. Using 8, Eqs. 2–4 can be written in the following dimensionless form:

*ν*

_{ f }/

*α*

_{ f }is the Prandtl number,

*S*is the suction (

*S*> 0) or injection (

*S*< 0) parameter,

*M*is the magnetic parameter,

*R*is the rotation parameter and \( Q_{H} \) is the heat source parameter, which are defined as:

*U*

_{0}is defined as

*z*. However, this expansion of the solution is meaningful only if the reduced equations are ordinary differential equations of independent variable

*z*. In fact, the solutions of \( \chi_{1} ,\chi_{2} ,\theta_{1} \) and \( \theta_{2} \) are time dependent and are not consistent with the assumption. In addition, the corresponding boundary conditions can be written as:

*A*

_{1}and

*m*

_{ i }(

*i*= 1–4) are given in “Appendix”.

*C*

_{ f }and the local Nusselt number

*Nu*, which are defined as

It should be mentioned that in the absence of the nanoparticles (*ϕ* = 0), the relevant results obtained are in agreement with the results reported by [33] without mass transfer.

## 3 Results and discussion

*t*in the presence of a rotating frame of reference has been performed in this paper. The effects of nanoparticles on the velocity and the temperature profiles as well as on the skin friction coefficient and the local Nusselt number are discussed numerically. We have chosen here

*n*= 10,

*nt*=

*π*/2,

*Pr*= 6.2 and

*ɛ*= 0.02, while ϕ,

*M*,

*R*,

*Q*

_{ H }and

*R*are varied over a range, which are listed in the figures legends. In order to highlight the important features of the flow and the heat transfer characteristics, the numerical values are plotted in Figs. 2 3, 4, 5, 6, 7, 8. These figures show the velocity profiles (Figs. 2, 4, 5), the temperature profiles (Figs. 3, 6), the variation of the skin friction coefficient (Fig. 7) and the variation of the reduced Nusselt number (Fig. 8) for different values of the physical parameters.

Figure 2a, b are the graphical representation of the velocity profiles *χ* for various values of the Cu nanoparticles volume fraction for *M* = 0 (no magnetic field) and for *M* = 10, respectively, when *S* = 1, *R* = 0.1 and *Q* _{ H } = 10. It can be seen that the momentum boundary layer thickness decreases with the increase in *ϕ* and also the presence of the magnetic field leads to more thinning of the boundary layer. The effects of the nanoparticle volume fraction for *S* = 0 and for *S* = 1.5 are presented in Figs. 2a, b, respectively. From these figures it results in that increase of the nanoparticle volume fraction leads to the increase of the thermal boundary layer thickness. Also the thermal boundary layer for Cu-water is greater than for pure water (*ϕ* = 0). This is because copper has high thermal conductivity and its addition to the water based fluid increases the thermal conductivity for the fluid, so the thickness of the thermal boundary layer increases. Furthermore, it can be observed that the thermal boundary layer thickness becomes thinner in the case of suction (*S* > 0).

Figure 3 shows the variation of the temperature profiles for different values of the rotation parameter *R* for Cu-water (see Fig. 3a) and for TiO_{2}-water (see Fig. 3b) in the absence/presence of nanoparticles. It is observed that the temperature profiles across the boundary layer increase with the decrease of *R*. Figure 4a, b show the effects of the rotation parameter *R* on the velocity, respectively for *ϕ* = 0 (regular fluid) and ϕ ≠ 0 (nanofluids). It is seen that the velocity profiles increase when *R* decreases. Figure 5 depicts the temperature profiles for various values of the heat generation parameter *Q* _{ H }. It is noted from this figure that the temperature profiles increase with a decreasing of heat generation parameter *Q* _{ H }. Figure 7a, b show the variation of the skin friction coefficient *C* _{ f } or the shear stress with the magnetic parameter *M* for some values of the rotation parameter *R* and the heat generation parameter *Q* _{ H } for two different values of the nanoparticle volume fraction. It is observed that *C* _{ f } increases with the increase in *R* and *Q* _{ H }. It is also seen that for high value of ϕ, the values of the skin friction coefficient become higher. Furthermore, it is seen that when the magnetic field parameter *M* increases, it leads to the increase of the skin friction coefficient. Finally, Fig. 8 illustrates the variation of the heat transfer rate or the reduced Nusselt number *Nur* with *S* for various values of the heat generation parameter *Q* _{ H } and for two different values of ϕ. It is noticed that the heat transfer rates increase with the increase in *Q* _{ H } and *ϕ*, and the changes in the heat transfer rates increase with the increase in *S*.

## 4 Conclusions

In the present study, we have theoretically studied the effects of the metallic nanoparticles on the unsteady MHD convective flow of an incompressible fluid along an oscillating vertical permeable semi-infinite plate in the presence of a rotating frame of reference. We have investigated the way in which the velocity and temperature profiles as well as the surface skin friction and the surface heat flux depend on the nanoparticle volume fraction parameter. It is shown that the inclusion of the nanoparticles into the base fluid is capable to change the flow pattern for the problem under consideration.

## Notes

### Acknowledgments

The authors wish to express their very sincerely thanks to the reviewers for the valuable comments and suggestions.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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## Copyright information

**Open Access**This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.