Since the discovery of the carbon nanotubes (CNTs), there is an increasing interest in their applications in industry and medical fields. Attempts of using such CNTs as drug carriers and in cancer therapy in the presence of a magnetic field are now undertaken because of their direct impacts on increasing the thermal conductivity of base fluids. Two types of CNTs are well known for the researchers, the single-walled CNT (SWCNTs) and the multi-walled CNTs (MWCNTs); however, the subject of which one is more effective in treatment of cancer deserves more investigations. The present article discusses the effect of such types of CNTs on the flow and heat transfer of nanofluids in the presence of a magnetic field. Exact analytical solution for the heat equation has been obtained by using the Laplace transform, where the solution is expressed in terms of a new special function, the generalised incomplete gamma function. The effects of various parameters on the fluid velocity, temperature distribution, and heat transfer rates have been introduced. Details of possible applications of the current results in the treatment of cancer have been also discussed.
In the beginning of the 1990s, the Japanese researcher Ligima discovered what is currently known as carbon nanotubes (CNTs). The name CNTs arises from their long, resonating edifice with the ramparts formed by one-atom-thick sheets of carbon, called graphene. These sheets are rolled at specific and discrete angles, and the combination of the rolling angle and radius decides the nanotube properties; for example, whether the individual nanotube shell is a metal or semiconductor. Nanotubes are categorised as single-walled nanotubes (SWNTs) and multi-walled nanotubes (MWNTs). Due to the importance of the nanoparticles/CNTs in enhancing the thermal conductivity of the base fluids since the first investigation of Choi [1], many authors [2–14] have considered the flow and heat transfer of the regular nanoparticles and the CNTs-suspended nanofluids. Hamad [9] examined the natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field, where four types of nanofluids were considered. Ebaid et al. [6] have extended the work of Hamad [9] by considering the velocity slip effect in both cases of stretching and shrinking sheets. In Ref. [11], the authors investigated the fluid flow and heat transfer of CNTs along a flat plate with the Navier slip boundary. They have used water, kerosene, and engine oil as base fluids. Their study may be the first in that direction, and they have concluded that the slip parameter reduces the skin friction and increases the heat transfer rate. There is an increase in skin friction and heat transfer rates as the CNT volume fraction increases. The engine oil-based CNTs have higher heat transfer rates than water- and kerosene-based CNTs. Finally, they have declared that kerosene-based CNTs have higher skin friction than engine oil- and water-based CNTs. Cylindrical CNTs are found to have special thermal properties with very high thermal conductivities. The diameter of CNTs ranges from 1 to 100 nm and has lengths in micrometer. The authors [15, 16] reported that the thermal conductivity of single-walled CNT (SWCNT) is about 6600 W/mK, while for multi-walled CNT (MWCNT) is about 3000 W/mK. The convective heat transfer of MWCNT-based nanofluids in a straight tube under a constant wall heat flux condition has been investigated by Kamali and Binesh [17]. In medical applications, the magnetic field in addition to these nanoparticles/CNTs plays the role of a corner stone in the recent treatment of cancer [18]. In such medical application, the nanoparticles/CNTs are injected into the blood vessel nearest to the tumour along with placing a magnet close to the tumour. These particles act like heat sources in the presence of the applied magnetic field of alternating nature, where the cancer tissues are destroyed if the temperature reaches 42–45 °C. The objective of the present study is to extend the work of Hamad [9] by considering the CNTs instead of the regular nanoparticles. Hence, we consider in this article the effect of magnetic field on the flow and heat transfer of CNTs-suspended nanofluids. The same governing equations as in Ref. [9] shall be used in the present study; however, a new analytical approach based on the Laplace transform is used to establish a new exact analytical solution for the heat transfer of CNTs-suspended nanofluids.
Consider the steady laminar two-dimensional flow of an incompressible viscous nanofluids with CNTs past a linearly semi-infinite stretching sheet under the influence of a constant magnetic field. As in Ref. [9], the flow is finally governed by the following system of nonlinear differential equations:
(1) f ′ ′ ′ ( η ) + α [ f ( η ) f ′ ′ ( η ) − ( f ′ ( η ) ) 2 ] − γ f ′ ( η ) = 0, (1)
(2) τ θ ′ ′ ( η ) + f ( η ) θ ′ ( η ) = 0, (2)
(3) α = ( 1 − ϕ ) 2.5 [ 1 − ϕ + ϕ ( ρ CNT / ρ f ) ] , γ = M ( 1 − ϕ ) 2.5 , (3)
(4) τ = 1 Pr ( k n f k f ) 1 [ 1 − ϕ + ϕ ( ρ C p ) CNT / ( ρ C p ) f ] . (4)
The primes denote the differentiation with respect to a similarity variable η·f and θ are the dimensionless stream function and temperature, respectively; ϕ is the solid volume fraction of the CNTs; ρs and ρf are densities; (ρCp)f and (ρCp)CNT are the heat capacitances; M is the magnetic parameter; Pr is the Prandtl number; and knf is the thermal conductivity defined as follows [13]:
(5) k n f k f = 1 − ϕ + 2 ϕ ( k CNT k CNT − k f ) ln ( k CNT + k f 2 k f ) 1 − ϕ + 2 ϕ ( k f k CNT − k f ) ln ( k CNT + k f 2 k f ) , (5)
where kf and kCNT are the thermal conductivities, and ()f and ()CNTs denote the basic fluid and solid fractions, respectively. The flow is subject to the boundary conditions:
(6) f ( 0 ) = 0, f ′ ( 0 ) = 1, f ′ ( ∞ ) = 0, (6)
(7) θ ( 0 ) = 1, θ ( ∞ ) = 0. (7)
The exact solution for the stream function f(η) is well known as
(8) f ( η ) = 1 β ( 1 − e − β η ) , β = α + γ . (8)
Inserting (8) into (2) yields
(9) τ θ ′ ′ ( η ) + 1 β ( 1 − e − β η ) θ ′ ( η ) = 0. (9)
In the next section, we discuss the effectiveness of the Laplace transformation in obtaining θ(η) in a closed integral form and then in an exact form. In a subsequent section, numerical results shall be derived to analyse various phenomena related to the dynamics of CNTs-suspended nanofluids. Let us begin with the construction of the analytical solution of the heat transfer (9).
Suppose that t= e – βη ; hence, we have
(10) d d η ( ♢ ) = − β t d d t ( ♢ ) , (10)
(11) d 2 d η 2 ( ♢ ) = β 2 [ t 2 d 2 d t 2 ( ♢ ) + t d d t ( ♢ ) ] . (11)
Therefore, (9) becomes
(12) m t θ ′ ′ ( t ) + ( m − 1 + t ) θ ′ ( t ) = 0, m = β 2 τ , (12)
subject to the following set of boundary conditions
(13) θ ( 0 ) = 0, θ ( 1 ) = 1. (13)
Applying the Laplace transform to (12), we have
(14) s ( m s + 1 ) Θ ′ ( s ) + [ ( m + 1 ) s + 1] Θ ( s ) = 0, (14)
where Θ(s) is the Laplace transform of θ(t). Integrating (14), we obtain
(15) Θ ( s ) = c s ( m s + 1 ) 1 / m , (15)
where c is a constant of integration. On applying the inverse Laplace transform to (15), we have
(16) θ ( t ) = c m 1 m Γ ( 1 m ) ∫ 0 t μ 1 m − 1 e − ( 1 m ) μ d μ . (16)
It is clear from this equation that the boundary condition θ(0)=0 is automatically satisfied. Besides, the other boundary condition θ(1)=1 gives c by
(17) c = m 1 m Γ ( 1 m ) ∫ 0 1 μ 1 m − 1 e − ( 1 m ) μ d μ . (17)
Therefore, θ(t) is given in a closed form as
(18) θ ( t ) = ∫ 0 t μ 1 m − 1 e − ( 1 m ) μ d μ ∫ 0 1 μ 1 m − 1 e − ( 1 m ) μ d μ . (18)
Performing the integrations arise in (18), we get the following exact solution for θ(t) in terms of the generalised incomplete gamma function:
(19) θ ( t ) = Γ ( 1 m , 0, t m ) Γ ( 1 m , 0, 1 m ) , (19)
which can be finally expressed in terms of η as
(20) θ ( η ) = Γ ( 1 m , 0, e − β η m ) Γ ( 1 m , 0, 1 m ) . (20)
It may be important here to point out not only the simplicity of the current analytical procedure if compared with that followed by Hamad [9] but also the very simple expression of the exact solution given by (20). Of course, such simplicity in the exact solution form facilitates the derivation of the numerical results, where little inputs have to be inserted into a software program, MATHEMATICA, for example. Accordingly, a very short time is expected to obtain the outputs of calculations. The formulas (8) and (20) for the fluid velocity and the temperature distribution, respectively, shall be used in the next section to analyse the effects of various parameters on the physical phenomena of practical interest.
Figure 1 is depicted to get information about the effect of the magnetic parameter M on the velocity of nanofluids suspended with SWCNTs/MWCNTs. It is clear from this figure that the fluid velocity decreases with the increase in the magnetic parameter. This is correct also for the nanofluids suspended with regular nanoparticles such as copper (Cu), sliver (Ag), alumina (Al2O3), and titanium oxide (TiO2) as shown in Ref. [9]. However, the velocities of CNTs-suspended nanofluids are faster than those of regular nanoparticles-suspended nanofluids, and this conclusion can be easily reached through Figure 1 of the present study and Figure 1 obtained in Ref. [9], where the same values of the physical parameters have been chosen. Also, Figure 1 shows that the MWCNTs-suspended nanofluids have higher velocities than the SWCNTs-suspended nanofluids and this may be of benefits in industrial applications as well as medical applications. The preceding discussion tells us that the CNTs may be faster than the nanoparticles in reaching the cite of tumour.
Effect of M on velocity distribution f′(η) at ϕ=0.1.
The effect of the magnetic parameter on the variation of temperature is displayed in Figure 2. This figure shows an increase in the fluid temperature with the increase in the magnetic parameter. Accordingly, the magnetic field may act as a heat source in the presence of CNTs/nanoparticles. The main notice here is that the SWCNTs nanofluids are of higher temperature than the MWCNTs nanofluids. However, nanofluids with any of the two types of CNTs are of higher temperature than nanofluids with all types of regular nanoparticles. We learn from these last results that the use of CNTs in cancer therapy may be more effective than all types of nanoparticles, this is due to the high temperature for nanofluids suspended with CNTs.
Effect of M on temperature distribution θ (η) at ϕ=0.1.
In the field on nanofluids, a well-known result is the increase in temperature with adding more nanoparticles to the base fluids. This is because with increasing ϕ the thermal conductivity of the base fluid increases. Such result is also confirmed for the CNTs through Figure 3. In addition, this figure declares the higher temperature of SWCNTs nanofluids over the MWCNTs nanofluids, which confirms the same conclusion of Figure 2. The results presented in Figure 4 for the effect of the solid volume fraction ϕ on the variation of the velocity reveal that f′(η) is an increasing function in ϕ. Moreover, the MWCNTs nanofluids are of higher velocities than the SWCNTs nanofluids, and this agrees with the outputs of Figure 1. The heat transfer rate (–θ′(0)) (the reduced Nusselt number, Nu x / Re x 1 / 2 ) is one of the main characteristics that gains practical interest in the field of nanofluid mechanics. It is clear from Figure 5 that the heat transfer rates decrease with the increase in the volume fraction of the CNTs and M. Such change in the reduced Nusselt number is found to be lower for higher values of ϕ, and this change decreases with the increase in M.
Effect of ϕ on temperature distribution θ (η) at M=1.
