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Computers, Materials & Continua
DOI:10.32604/cmc.2020.012677
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Article

Darcy-Forchheimer Hybrid Nano Fluid Flow with Mixed Convection Past an Inclined Cylinder

M. Bilal1, Imran Khan1, Taza Gul1,*, Asifa Tassaddiq2, Wajdi Alghamdi3, Safyan Mukhtar4 and Poom Kumam5

1Department of Mathematics, City University of Science and Information Technology, Peshawar, 25000, Pakistan
2Department of Basic Sciences and Humanities, College of Computer and Information Sciences, Majmaah University, Al-Majmaah, 11952, Saudi Arabia
3Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, 80261, Saudi Arabia
4Basic Sciences Department, Deanship of Preparatory Year, King Faisal University, Al Ahsa, 31982, Saudi Arabia
5Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand
*Corresponding Author: Taza Gul. Email: tazagul@cusit.edu
Received: 08 July 2020; Accepted: 01 October 2020

Abstract: This article aims to investigate the Darcy Forchhemier mixed convection flow of the hybrid nanofluid through an inclined extending cylinder. Two different nanoparticles such as carbon nanotubes (CNTs) and iron oxide images have been added to the base fluid in order to prepare a hybrid nanofluid. Nonlinear partial differential equations for momentum, energy and convective diffusion have been changed into dimensionless ordinary differential equations after using Von Karman approach. Homotopy analysis method (HAM), a powerful analytical approach has been used to find the solution to the given problem. The effects of the physical constraints on velocity, concentration and temperature profile have been drawn as well for discussion purpose. The numerical outcomes have been carried out for the drag force, heat transfer rate and diffusion rate etc. The Biot number of heat and mass transfer affects the fluid temperature whereas the Forchhemier parameter and the inclination angle decrease the velocity of the fluid flow. The results show that hybrid nanofluid is the best source of enhancing heat transfer and can be used for cooling purposes as well.

Keywords: Mixed convection; similarity transformation; HAM; hybrid nanofluid; CNTs; Darcy Forchhemier; inclined cylinder

1  Introduction

The study of the hybrid nanofluid in the existence of mass and heat transfer has received special attention from many scientists and researchers because of its essential role in the field of science and technology [1]. The convection of the hybrid nanofluid flow, together with heat and mass transfer, has several important applications in industry such as oil reservoir, suspension and colloidal solution, bioengineering, nuclear industries, polymer solution, paper production, geophysics, chemical industries and exotic lubricants etc. [26]. The fluid like kerosene oil, water, acetone, engine oil and ethylene glycol has low thermal conductivity. In the era of modern science and technology, the extensive need for thermal energy cannot be fulfilled through commonly used fluids. However, a significant enhancement in thermal characteristics was noted when these base liquids were synthesized with the addition of small sized particles [7]. Thus, this rise in the thermal properties of ordinary fluids developed the keen curiosity of scientists for further investigations. Numerous researches on nanoparticles and carbon nanotubes CNTs, both single walled carbon nanotubes and multi-walled carbon nanotubes (SWCNTs and MWCNTs) have been carried out by the researchers. CNTs are the allotropes of carbon with a nano cylindrical structure. The CNTs are frequently used in the energy sector and Nanoscience [8]. The mixture of copper oxide and water was examined by Animasaun [9]. The water based nanofluid fluid flow of CNTs was analyzed by Aman et al. [10]. The enhancement of the heat transfer rate using the nanofluid of carbon nanotubes was examined by Raza et al. [11]. The impact of the nanofluid using Arrhenius activation energy was examined by Muhammad et al. [12]. The water based iron oxide nanofluid flow was studied by Qasim et al. [13]. The nanofluid of magnetite-ferrium oxide images was examined by Hussanan et al. [14]. The water based aluminum oxide (images) nanofluid was studied by Sheikholeslami et al. [15]. The nanofluid flow through a stirring surface has been studied by Haq et al. [16]. The mixed convection flow of hybrid nanofluid consisting CNTs over a stretching inclined cylinder has tremendous use in the field of mechanical engineering. The mass and heat transfer of nanofluid through an inclined surface is affected due to the buoyancy forces. This type of phenomena plays an important role in the cooling of electronic devices, automobile demister, boilers, defroster system and in solar energy system [17]. The effects of slip flow over time dependent stretching sheet including mixed convection were explored by Makinde et al. [18]. Rashad et al. [19] minutely examined the micropolar fluid flow using double stratified medium. Turkyilmazoglu [20] scrutinized viscoelastic fluid flow with mixed convection over a stretching porous surface. Ashraf et al. [21] investigated the 3D Maxwell fluid flow with mixed convection over an extending inclined surface. The mixed convection flow with double effect of stratification of a Jeffrey fluid past on an extending inclined cylinder has been scrutinized by Hayat et al. [22]. The analytic solution of the nanofluid flow including natural convection over a linearly extending sheet has been presented by Hammad [23]. A survey article has been published by Buongiorno [24] on convection transport through nanofluid. The Darcy-Forchheimer model is the most well-known extension to Darcian flow usually in resemblance with the effects of inertia. The effect of inertia is considered by the insertion of squared terms of the velocity in the momentum equation known as Forchheimer modification. This new term has been named as Forchheimer factor by Muskat [25]. Mondal et al. [26,27] examined non-Darcy Forchheimer model in their research articles over a stretching surface. The Darcy Forchheimer (DF) flow over an upright surface has been studied by Anwar et al. [28]. In order to understand better the problems occurred in the field of physics, it is essential to involve non-Darcy effects in convective transport analysis. The Darcy Forchheimer mixed convective flow in porous media has been examined by Seddek [29]. The DF mixed convection flow of the hybrid nanofluid consisting CNTs through the impermeable inclined cylinder has been investigated minutely in this study. It will lead the researchers to new investigations. The mixed convection flow and heat transfer have several useful applications such as storage and food processing, underground disposing of nuclear wastes and geophysical system [30]. This versatility makes several applications to be studied through the fluid flow over an inclined cylinder for the enhancement of heat and mass transfer. Keep in view the importance and applications of this work, we have extended the idea of [17] and revealed this problem. The results have been achieved through HAM.

2  Mathematical Formulation

The mixed convection Darcy-Forchhimier fluid flow over an inclined stretching cylinder has been examined in this research. The flow has been considered steady and axisymmetric. The physical sketch of the flow has been illustrated in Fig. 1. The analysis of heat and mass transfer has been considered for the hybrid nanofluid consisting CNTs and iron oxide. The analysis of the hybrid nanofluid flow past an extended cylinder has been considered. In the coordinate system, the x-axis and r-axis are considered along the axial and normal direction to the cylinder respectively. After using the boundary layer approximations, the laws have been reduced to the form as follows [17]. The basic flow equations are:

images

Figure 1: Geometry of the problem

images

images

images

images

The physical conditions for the governing equations are:

images

The velocity components u and v have been taken along the axial and normal directions of the cylinder. images, images and images demonstrate the dynamic, kinematic and density of hybrid nanofluid. images and imagesshow the thermal expansion coefficient and the concentration expansion coefficient respectively. Whereas images and images represents the convective fluid temperature and concentration respectively.

The appropriate transformations are:

images

images

By using Eq. (6) in Eqs. (1)(5), we get

images

images

images

The transformed conditions for nonlinear differential equations are as follows:

images

images is the thermal conductivity and images is the volumetric heat capacity of hybrid nanofluid as stated in [31]:

images

images

images

images

images

images

where

images

The volumetric concentrations of images and CNTs have been denoted by images and images. Each and every abbreviation has been defined individually. Furthermore, images and images imply the thermal conductivities of images and CNTs. images is the density, images is the viscosity and images is the specific heat of the images. images, images, and images at constant pressure indicate specific heat capacities and densities of images and CNTs. The Deborah number, Prandtl number and Schmidt number have been denoted by images, Pr and Sc. images and imagesdenote temperature Grashof number and mass Grashof number respectively. Biot numbers images and images are for the heat and mass transfer.

The physical constraints have been defined in [17]:

images, images, images, images,

images

images, images, images, images.

The Sherwood number images, the local Nusselt number images and skin friction coefficient have been expressed in dimensional form as follows:

images

In which surface mass flux, surface heat flux and surface shear stress have been represented byimages, images and images:

images

Local Sherwood number images, the local Nusselt number images and Skin friction coefficient images are:

images

The local Reynolds number is images

3  Problem Solution

The current problem has been solved by using HAM technique that was initiated by Liao [3234]. BVPh 2.0 package [3542] has been used for the convergence of the modeled problem. The initial approximations for velocity images, temperature images and concentration images are given as:

images

The linear operators images, images and images presented as:

images

The solved form of images, images and images are:

images

The series introduced by Taylor’s has been used as:

images

images

images

Now

images

The equations have been further concluded in the form of a system as:

images

4  Results and Discussion

This study aims to use the hybrid nanofluids flow over a stretching cylinder for the rapid heating and cooling applications in the field of thermal engineering. The hybrid nanofluid contains solid particles of the images , images and base liquid of images. The solid particles disperse in the base liquid and as a result the hybrid nanofluid is prepared. The analytical solution of the system has been obtained through the Homotopy analysis method (HAM). The influence of the constraints has been demonstrated in Figs. 2–16. Fig. 2 reveals the influence of the Farchemmier’s parameter images on velocity profile. When images increases, it enhances the transfer of mass in fluid flow which results in a decrease in the velocity of fluid flow. Fig. 3 intends to perceive images effects on the velocity field. The rising value ofimages results in an increase in the fluid kinematic viscosity and consequently declines the velocity of the hybrid nanofluid. The kinematic viscosity of images is greater than the CNTs. Thus, the influence of images is comparatively large using the iron oxide.

images

Figure 2: The effects of images on velocity profileimages when images

images

Figure 3: The effects of images on velocity profileimages when images

Fig. 4 has been sketched to reveal the influences of images (angle of inclination) on both CNTs and images nanofluid with velocity images. The increase in the value of the parameter images decreases the velocity field. In fact, decreasing the effect of gravity decreases velocity profile. The decreasing effect of gravity decreases velocity profile. The essential performance of the velocity field images with the rising values of images has been shown in Fig. 5. As images is the ratio of relaxation to retardation time, thus, the increase in the values of images (i.e., extending the relaxation time) provides some additional resistance to the flow field. That’s why fluid velocity decreases.

imagesimages

Figure 4: The effects of images on velocity profileimages when images

imagesimages

Figure 5: The effects of images on velocity profileimages when images

The additional resistance due to relaxation time also enhances the fluid temperature presented in Fig. 6. Fig. 7 shows the effects ofimages parameter on temperature distribution. The fluid temperature decreases while increasing images. In fact, the kinematic viscosity of the fluid increases which drops the temperature normally.

imagesimages

Figure 6: The effects of images on temperature distributionimages when images

imagesimages

Figure 7: The effects of images on temperature distribution images when images

Figs. 8 and 9 demonstrate the variation of temperature distribution with Biot numbers images and images respectively. Both Biot numbers consist of heat and mass transfer coefficient images andimages respectively. Thus, the increase in the values of images and images enriches the thermal boundary layer and concentration field. Which results in an increase in the fluid temperature.

imagesimages

Figure 8: The effects of images on temperature distribution images when images

imagesimages

Figure 9: The effects of images on temperature distribution images when images

Figs. 10 and 11 describe images and images (volume fraction parameters) of the mentioned nanoparticles that affect the temperature distribution. The volume fraction constraints images and images boost up the temperature field. It has been noticed that the adequate amount of volume fraction can increase thermal property of the base fluid and consequently the temperature of the fluid increases.

imagesimages

Figure 10: The effects of images on temperature distribution images when images

imagesimages

Figure 11: The effects of images on temperature distribution images when images

Fig. 12 establishes the changes in the temperature field versus images. The increasing value of imagesreduces the thermal diffusivity, consequently it drops the fluid temperature. Fig. 13 illustrates the Schmidt number images and its effects on the concentration distribution respectively. It has been observed that Schmidt number is used to increase the thickness of associated boundary layer and concentration.

imagesimages

Figure 12: The effects of images on temperature distribution images when images

imagesimages

Figure 13: The effects of images on concentration distribution images when images

Table 1: Thermo-physical properties of water, images and images nanoparticles

images

Tab. 1 shows the thermophysical properties of the base fluid and nanoparticles. Tab. 2 shows the effect of the parameters images and images. It has been observed that skin friction increases due to the resistivity created by the mentioned parameters.

Table 2: Presents the numerical outcomes of skin friction images

images

Tab. 3 shows the effects of images and images. It has been noticed that both parameters jointly increase Nusselt number due to the rise in the numerical values of these parameters during the thermal process. Moreover, SWNCTs offer excellence behavior to MWCNTs.

Table 3: Exhibits the numerical outcomes of Nusselt number images

images

Tab. 4 shows the way images behaves to Sherwood number. It is mainly found that as the Sherwood number decreases, it results in increasing in Schmidt number. The dominant impact of the SWCNTs is still observed on MWCTs.

Table 4: Shows the numerical outcomes of Sherwood number images

images

5  Conclusion

In this work, we have addressed the Darcy Forchhemie’r hybrid nanofluid flow past a stretched and inclined cylinder. The solid nanoparticles of the CNTs and iron oxide have been used for the preparation of hybrid nanofluid. The main findings are as below:

•   It has been noticed that increase in the values of the curvature parameter results in an increase in the profiles of the temperature, concentration and velocity of the hybrid nanofluid.

•   The Biot numbers are used to improve the concentration and temperature transfer rates.

•   In temperature distribution, the role of images is more dominant than images.

•   The variation in images (mixed convection parameter) increases the velocity profile while the rise in the Skin friction coefficient decreases the velocity field.

•   The use of images is more significant to increase the thermal efficiency of the base fluid as compared to the common fluid.

Funding Statement: This research has received no specific funding.

Conflict of Interest: The authors of this research have no conflict of interest.

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