ARTICLE

J. Cent. South Univ. (2019) 26: 2109-2118

DOI: https://doi.org/10.1007/s11771-019-4158-8

Entropy analysis of SWCNT & MWCNT flow induced by collecting beating of cilia with porous medium

Muhammad N ABRAR, Muhammad SAGHEER, Shafqat HUSSIAN

Department of Mathematics, Capital University of Science and Technology Islamabad, Pakistan

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: In this article, we considers the thermodynamics analysis of creeping viscous nanofluid flow in a horizontal ciliated tube under the effects of a uniform magnetic field and porous medium. Moreover, energy analysis is performed in the presence of an internal heat source and thermal radiation phenomena. The thermal conductivity of base fluid water is strengthened by considering the carbon nanotubes (CNTs). Mathematical formulation operated, results in a set of non-linear coupled partial differential equations. The governed differential system is transformed into an ordinary differential system by considering suitable similarity variables. Exact solutions in the closed form are computed for the temperature, momentum and pressure gradient profiles. In this study, special attention is devoted to the electrical conductivity of the CNTs. Streamlines patterns are also discussed to witness the flow lines for different parameters. Thermodynamics analysis shows that entropy of the current flow system is an increasing function of Brinkmann number, magnetic parameter, nanoparticle concentration parameter and Darcy number.

Key words: single-wall carbon nanotubes (SWCNT); multi-wall carbon nanotube (MWCNT); thermodynamics analysis; magnetic field; Darcy effect; thermal radiation

Cite this article as: Muhammad N ABRAR, Muhammad SAGHEER, Shafqat HUSSIAN. Entropy analysis of SWCNT & MWCNT flow induced by collecting beating of cilia with porous medium [J]. Journal of Central South University, 2019, 26(8): 2109-2118. DOI: https://doi.org/10.1007/s11771-019-4158-8.

1 Introduction

The mechanics of any machine involving a thermofluid depends on the structure and temperature of the functioning fluid. As energy transfer is irreversible, for the efficient performance of a system, it is essential to scrutinize the entropy of the system. The second law of thermodynamics is used to investigate the irreversibilities in a system in terms of entropy. Entropy defines a system’s thermal energy per unit temperature that is not available for performing work or the degree of disorder of a system and its surrounding. Entropy of any fluid system always increases with time. The second law investigation approach is effective and productive for reducing the system entropy. Entropy formation is connected with many processes such as geothermal power systems and solar power systems. BEJAN [1] described the knowledge of entropy in heat transfer and fluid flow systems. GORLA et al [2] presented second law analysis of heat transfer for a non-Newtonian fluid. AKSOY [3] investigated couple stress flow caused by the pressure gradient between parallel plates. SRINIVASACHARYA et al [4] presented entropy generation in a micropolar fluid. HUSSAIN et al [5] discussed entropy analysis for hybrid nanofluid in an open cavity. KAMRAN et al [6] addressed numerical study of Casson fluid with Ohmic heating. ABRAR et al [7, 8] presented entropy analysis in a cilia transport of nanofluid with Joule heating and fluid friction and they have recently contributed in this useful area of entropy analysis by considering different physical and engineering scenarios.

Energy crises are increasing rapidly, thus there is a massive need for new techniques that can increase the thermal efficiency of working fluid and reduces energy crises. To overcome the energy crises of wavy surfaces [9], micro channels, vibration phenomenon, etc have been developed by researchers and engineers to reinforce the heat transfer rate. In 1995, CHOI [10] initially introduced the term nanofluid to indicate that the heat transfer rate can be enhanced by improving the thermal conductivity of traditional base fluids. Actually, traditional base fluids, such as pure water, possess finite heat transfer strength due to minute thermal abilities. Conversely contrary, nanoparticles holds dramatically higher thermal strength. Moreover, nanofluids offer their potential applications in chemical, biological and earth engineering, for example, pharmaceutical transport [11], magnetic resonance technology [12], and safe drinking water [13]. On the basis of the above discussion, we have considered carbon nanotubes as nanoparticles. CNTs are an allotrope of carbon. They exhibit extraordinary strength and unique electrical properties, and they are efficient heat conductors. These marvelous properties of carbon nanotubes make CNTs interseting. Carbon nanotubes are subdivided into two unique classes: single-wall carbon nanotubes (SWCNT) and multi- wall carbon nanotubes (MWCNT). Following the report by CHOI, the subject of nanofluid with multiple effects has been discussed by many researchers. Some of these studies specially related to CNTs are reporeed by KHAN et al [14], ALY [15], AHMED et al [16], AKBAR et al [17] and ABRAR et al [18]. Some more explanatory articles on nanofluid are Refs. [19-27].

In any engineering procedure, there are many ways to increase the rate of heat transfer, one of which is to use porous media in heat transfer appliances. Porous materials occur everywhere in daily life, in technology and in nature. A material or structure must posses two properties to be called a porous medium: 1) it must contain spaces, so-called voids or pores, free of solids, imbedded in the solid or semi-solid matrix. 2) It must be permeable to a variety of fluids, i.e., fluids should be able to penetrate through one face of a sample of material and emerge on the other side. The use of porous material has been the core subject of multiple studies and widely appreciated. This appreciation is because of the fact that porous structures are highly acceptable in many engineering applications, such as thermal insulation, oil flow, ground water and power stations. Porous media are specially applied in engineering devices, in which cooling or heating is required [28].

The purpose of this study is to examine the entropy generation analysis of a viscous nanofluid influenced by metachronical waves in a horizontal channel. The effects of a porous medium and thermal radiation are also considered in the momentum and energy equations respectively. The next sections comprises a philosophical analysis of the problem. The closed form exact solution is presented in Section 3. The physical understanding of the analysis from figures and numerical tables is presented in Section 4. The last section summarizes the consequence of this study.

2 Mathematical computations

We investigate the axisymmetric flow of a viscous nanofluid in a ciliated flexible horizontal tube. When the group of cilia operate together, metachronal waves are produced, which move in the direction of effective stroke. These wave have wavy and beating motion. Flow is developed as a consequences of these wavy or beating motion of metachronal waves and because of the no slip condition fluid is moving with the wave speed (c). Further, we also considere the effects of a uniform magnetic field, porous medium, internal heat source coefficient and thermal radiation. We introduce the cylindrical coordinate systemto present the geometry of the problem. Figure 1 influences the systematic view of the governed peristaltic model.

The shape of the cilia tips should obey the following pattern [8]:

        (1)

where e is the mean radius of the tube; ε is the wave amplitude; Z₀ is the reference position of cilia; λ is the the wavelength; c is the wave speed. No slip condition suggests that the cilia tips and fluid adjacent to cilia tips have the same velocity, so the radial and axial velocities are given as follows:

Figure 1 Geometry of problem

 (2)

Combining Eqs. (1) and (2), the radial and axial velocities of the cilia are given as:

              (3)

Flow is unsteady in the fixed coordinates and it becomes steady in a wave frame so the following variables are meaningful to convert the flow from fixed frame to wave frame.

                       (4)

The derived fundamental system of equations after invoking the above variables are of the form [7]:

                          (5)

        (6)

      (7)

               (8)

here and are the velocity elements in the radial and axial directions; B₀ is the total magnetic field; σ_nf is the electric conductivity of nanofluid; K is the permeability parameter; ρ_nf is the effective density; μ_nf is the effective dynamic viscosity; (ρc_p)_nf is the heat capacitance; σ^* the Stefan-Boltzmann constant; Q₀ is the internal heat source; α_nf is the effective thermal diffusivity; k_f is the thermal conductivity of the fluid fraction; k_s is the thermal conductivity of the solid nanoparticle; and k_nf is the effective thermal conductivity, given in Refs. [15] and [18]:

      (9)

Thermophysical properties of the base fluid water and CNTs nanoparticles are presented in Table 1.

Table 1 Thermophysical properties of CNTs and water given in Refs. [18, 29]

Propose the following dimensionless variables:

     (10)

By considering the above non-dimensional variables in Eqs. (5)-(8), and the limitations of creeping flow are also enforced here, i.e. (Re<<1), and the wavelength is assumed to be very large (i.e. λ→∞). The obtained differential system is:

                                  (11)

     (12)

    (13)

where

        (14)

where M is the magnetic parameter; f is the solid volume fraction; Gr is the Grashof number; Da is the Darcy number; Ω is the internal heat source parameter. The boundary conditions are given as:

             (15)

A non-equilibrium situation emerges as a result of exchange of momentum, temperature and magnetic effects within the fluid and at the walls which causes a continuous entropy generation. The volumetric entropy generation term (S_G) can be calculated as follows:

Equation (15) reflects the contribution of three distinct factors causing the entropy generation. These factors are heat transfer (E_H), thermal radiation (E_Rn) and the magnetic field (E_M). Entropy basically gives the degree of disorder of the system and its surroundings and the rate of dimensionless

entropy formation is:

     (16)

where

              (17)

where Λ is the dimensionless temperature difference; Br is the Brinkmann number. To determine the irreversibility distribution, the Bejan number (Be) introduced by BEJAN [1] is the ratio of the heat transfer irreversibility to the total irreversibility and is given as:

                      (18)

3 Exact solutions

In this section, we present the closed form solutions to the ordinary differential Eqs. (12) and (13) with Eq. (15) as boundary condition. The constitutive boundary layer equations for the considered flow analysis incorporates continuity, momentum and energy equations. These equations are coupled nonlinear partial differential equations. To convert the governing system of partial differential equations to a system of ordinary differential equations, we introduce similarity variables. The resulting system is a linear, 2nd order non-homogeneous ordinary differential system. The exact solution of ordinary differential system presented in Eqs. (19), (20) and (22) is acquired by the program DSolve in MATHEMATICA software. The general solution of non-homogeneous differential equations is comprised of complementary and particular parts. For the complementary solution, DSolve chooses the Bessel functions, because the homogeneous part of Eqs. (12) and (13) corresponds to the standard form of Bessel equation of order zero. Once we are successful in obtaining the complementary solution, DSolve straightforwardly proceeds for the particular solution. The interpretation for temperature, velocity and pressure gradient is computed as:

(19)

(20)

the flow rate is described as:

                        (21)

now substituting Eq. (20) into Eq. (21) and then we have solution of dp/dz

       (22)

the mean flow rate can be calculated as:

                       (23)

where the expression for (Ψ_i, i=4, …, 9) are given as:

  (24)

4 Results and discussion

Here we highlight the dynamics of various pertinent flow parameters on streamlines, velocity field, temperature field, pressure gradient, and entropy number.

4.1 Streamlines patterns

An interesting part of peristaltic motion, called trapping, is presented with respect to change in the various parameters. An internal circulating bolus is formed during the peristaltic transport which is forced to move in the direction of the waves. The trapping phenomenon for variation in α and Gr is shown in Figures 2(a), (b) and 3(a), (b). With an increase in α and Gr, the number of bolus increases and the size of bolus reduces when considering SWCNT and MWCNT, respectively.

4.2 Velocity variation

The consequence of the Darcy number (Da) on the momentum profile is depicted in Figure 4(a), which shows that the fluid velocity increases with an increase in Da. Physically, an increases in Da enhances the permeability of the medium, which corresponds to greater permeability and thus increases the over all velocity. The variation of magnetic parameter (M) on w(r, z) is graphed in Figure 4(b). An enhancement in M reduces the fluid velocity for both SWCNT and MWCNT. Physically, an increase in M accelerates the strength of the Lorentz force, which is a resistive force, therefore more resistance is offered to the fluid motion which consequently reduces the fluid velocity.

4.3 Temperature variation

The variation of Ω and Rn on θ(r, z) is depicted in Figures 5(a) and (b). When Ω is kept at zero, the over all heat transfer rate is zero, which is obvious from Eq. (19), whereas increasing behavior of fluid energy is found for Ω>0 for both SWCNT and MWCNT (see Figure 5(a)). Figure 5(b) displays the effect of Rn on the energy profile. The fluid temperature decreases rapidly as there is an increase in the radiation parameter. Physically, an increase in the radiative parameter Rn increases the mean absorption coefficient (k^*) which reduces the fluid temperature significantly.

Figure 2 Streamlines for SWCNT

Figure 3 Streamlines for MWCNT

Figure 4 Variation of Da and M on w(r, z)

4.4 Pressure gradient variation

Figures 6 (a) and (b) show the influence of Darcy parameter (Da) and flow rate (Q) on the pressure gradient profiles. The pressure gradient is a physical quantity that characterize in which direction and at what rate the pressure raises the most rapidly. A uniform oscillating behavior is found for increasing values of Da and Q (for both SWCNT and MWCNT). Figure 6(a) shows that with higher values of Darcy number, the permeability of the medium is also enhanced, which correspondingly enhances the pressure gradient profile. Figure 6(b) shows that the pressure profile is a decreasing function of flow rate.

Figure 5 Variation of internal heat source and thermal radiation on temperature field

Figure 6 Variation of Da on dp/dz (a) and variation of Q on dp/dz (b)

4.5 Entropy variation

We present multiple plots to reflect the variation of pertinent parameters on the entropy profile. Figure 7(a) shows that the entropy of the fluid system increases with an increase in the magnitude of the Brinkmann number (Br). As Br increases, energy transfer influences the viscosity of fluid within the tube, therefore increases the total entropy. Figure 7(b) shows that the fluid entropy increases for the greater values of the magnetic number (M). Moreover, maximum entropy is seen at the central portion of the tube, because, in this location, the velocity is at its extreme, thus, the contribution to MHD flow is also at its maximum. The effect of nanoparticles concentration (f) on E_g is plotted in Figure 7(c). The maximum entropy is noted at the boundaries and at the center of tube, because the concentration of CNTs is minimum at these regions. The effect of Darcy number (Da) on E_g is plotted in Figures 7(d), wherein the total entropy of the system increases for variation in Darcy number.

Tables 2 and 3 represents the numerical values of solid volume fraction of nanoparticle and Brinkmann number. With an increase in solid volume fraction and Brinkmann number the irreversibility distribution parameter Bejan number decreases.

5 Conclusions

1) Streamlines patterns for variation in Grash of number shows that the Bolus number increases, whereas the Bolus size decreases for MWCNT.

2) The velocity field is found to be an increasing function of Darcy number and decreasing function of magnetic parameter for both SWCNT and MWCNT.

Figure 7 Variation of Br on E_g (a), variation of M on E_g (b), variation of φ on E_g (c) and variation of Da on E_g (d)

Table 2 Bejan profile for both SWCNT and MWCNT for variation in increasing solid volume fraction (f) while setting Ω=0.5, Rn=Da=Br=0.3, Λ=0.1

Table 3 Bejan profile for both SWCNT and MWCNT for variation in increasing Brinkmann number (Br) while setting Ω=0.5, Rn=Da=0.3, Λ=0.1, f=0.2

3) An increase in the internal heat source parameter increases the fluid energy, whereas the fluid energy reduces with an increment in thermal radiation parameter for both SWCNT and MWCNT.

4) The pressure gradient profile increases with an increment in Darcy parameter whereas it decreases with an increment in flow rate for both SWCNT and MWCNT.

5) Entropy number is an increasing function of Brinkmann number, magnetic parameter, nanoparticle concentration parameter and Darcy number for both SWCNT and MWCNT.

6) Bejan number is found to be a decreasing function of Brinkmann number and nanoparticle concentration parameter for both SWCNT and MWCNT.

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(Edited by YANG Hua)

中文导读

单壁碳和多壁碳纳米管内多孔纤毛脉动诱导流体流动的熵分析

摘要：对均匀磁场和多孔介质作用下水平纤毛管内黏性纳米流体流动的热力学进行分析。另外，对存在内部热源和热辐射的情况也进行了能量分析。加入碳纳米管(CNTs)增强了基流水的热导率。通过公式转换得到一组非线性耦合偏微分方程。通过相似变量的转换，将控制微分系统转化为一个常微分系统。计算了温度、动量和压力梯度分布的封闭形式的精确解。本研究主要研究了碳纳米管的导电性，讨论了不同流线的模式。热力学分析表明，电流系统的熵随着Brinkmann数、磁性参数、纳米粒子浓度参数和Darcy数的增大而增大。

关键词：单壁碳纳米管；多壁碳纳米管；热力学分析；磁场；达西效应；热辐射

Received date: 2019-05-08; Accepted date: 2019-07-16

Corresponding author: Muhammad N ABRAR, PhD; Tel: +92-51-111-555; E-mail: nasirabrar10@gmail.com, muhammad.nasir@ cust.edu.pk; ORCID: 0000-0003-4088-2522