**Abstract**

The study analyzed the heat transfer of water-based carbon nanotubes in non-coaxial rotation flow affected by magnetohydrodynamics and porosity. Two types of CNTs have been considered; single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). Partial differential equations are used to model the problem subjected to the initial and moving boundary conditions. Employing dimensionless variables transformed the system of equations into ordinary differential equations form. The resulting dimensionless equations are analytically solved for the closed form of temperature and velocity distributions. The obtained solutions are expressed in terms of a complementary function error. The impacts of the embedded parameters are graphically plotted in different graphs and are discussed in detail. The Nusselt number and skin friction are also evaluated. The temperature and velocity profiles have been determined to meet the initial and boundary conditions. An augment in the CNTs' volume fraction increases both temperature and velocity of the nanofluid as well as enhances the rate of heat transport. SWCNTs provides high values of Nusselt number compared to MWCNTs. For verification, a comparison between the present solutions and a past study is conducted and achieved excellent agreement.

**Keywords:** Nanofluids, Carbon nanotubes, Newtonian fluid, Magnetohydrodynamics, Heat transfer

## **1. Introduction**

The growing demand in manufacturing has led to a significant process of heat energy transfer in industry applications such as nuclear reactors, heat exchangers, radiators in automobiles, solar water heaters, refrigeration units and the electronic cooling devices. Enhancing the heating and cooling processes in industries will save energy, reduce the processing time, enhances thermal rate and increase the equipment's lifespan. Sivashanmugam [1] found that nanofluid emergence has improved heat transfer capabilities for processes in industries. Choi and Eastman [2] established the nanofluid by synthesizing nanoparticles in the conventional base fluid. To be specific, nanofluid is created by suspending nano-sized particles with commonly less than 100 nm into the ordinary fluids such as ethylene glycol, propylene glycol, water and oils [3]. Various materials from different groups can be used as the nanoparticles such as Al2O3 and CuO from metalic oxide, Cu, Ag, Au from metals, SiC and TiC from carbide ceramics, as well as TiO3 from semiconductors [4]. In addition, immersion of nanoparticles is a new way of enhancing thermal conductivity of ordinary fluids which directly improves their ability in heat transportation [5]. In line with nanofluid's contribution in many crucial applications, a number of research has been carried out to discover the impacts of various nanofluid suspension on the flow features and heat transfer with several effects including Sulochana et al. [6] considering CuO-water and TiO-water, Sandeep and Reddy [7] using Cu-water, and Abbas and Magdy [8] choosing Al2O3-water as their nanofluid.

Magnetohydrodynamics (MHD) is known as the resultant effect due to mutual interaction of magnetic field and moving electrical conducting fluid. Their great applications such as power generation system, MHD energy conversion, pumps, motors, solar collectors have drawn significant attention of several researcher for MHD nanofluid in convective boundary layer flow [9]. Benos and Sarris [10] studied the impacts of MHD flow of nanofluid in a horizontal cavity. Hussanan et al. [11] analyzed the transportation of mass and heat for MHD nanofluid flow restricted to an accelerated plate in a porous media. In this study, water-based oxide and non-oxide had been considered as the nanofluids. Prasad et al. [12] performed similar work as [11] concerning the radiative flow of nanofluid over a vertical moving plate. Anwar et al. [13] conducted the MHD nanofluid flow in a porous material with heat source/sink and radiation effects. Cao et al. [14] analyzed the heat transfer and flow regimes for a Maxwell nanofluid under MHD effect. While, Ramzan et al. [15] investigated for a radiative Jeffery nanofluid and Khan et al. [16] carried out for a Casson nanofluid with Newtonian heating.

One of the greatest discoveries in material science history is carbon nanotubes (CNTs), which was discovered by a Japanese researcher in the beginning of the 1990s. Since the discovery, due to the unique electronic structural and mechanical characteristics, CNTs are found as valuable nanoparticles, especially in nanotechnology field. CNTs are great conductance which is highly sought in medical applications. They have been used as drug carriers and have benefited cancer therapy treatments [17]. The high thermal conductivity of CNTs has attracted significant attention from many researchers, including Xue [18], Khan et al. [19] and Saba et al. [20]. CNTs are hollow cylinders of carbon atoms in the forms of metals or semiconductors. CNTs are folded tubes of graphene sheet made up of hexagonal carbon rings, and their bundles are formed. CNTs are classified into two types with respectively differ in the graphene cylinder arrangement which are single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). SWCNTs has one layer [21], while MWCNTs consist of more than one graphene cylinder layers [22]. Khalid et al. [23] studied the characteristics of flow and heat transfer for CNTs nanofluid affected by MHD and porosity effects. Acharya et al. [24] discussed a comparative study on the properties of MWCNTs and SWCNTs suspended in water with the imposition of magnetic field. The CNTs nanofluid flow induced by a moving plate was investigated by Anuar et al. [25] and a prominent effect on heat transfer and skin friction by SWCNTs was observed. Ebaid et al. [26] analyzed convective boundary layer for CNTs nanofluid under magnetic field

*Analysis of Heat Transfer in Non-Coaxial Rotation of Newtonian Carbon… DOI: http://dx.doi.org/10.5772/intechopen.100623*

effect. The closed form solution was derived using Laplace transform method and the findings showed increasing magnetic strength and volume fraction of CNTs had deteriorated the rate of heat transport. Aman et al. [27] improved heat transfer for a Maxwell CNTs nanofluid moving over a vertical state plate with constant wall temperature. The investigation of velocity slip of carbon nanotubes flow with diffusion species was conducted by Hayat et al. [28]. Recently, the heat transmission analysis for water-based CNTs was discussed by Berrehal and Makinde [29], considering the flow over non-parallel plates and Ellahi et al. [30] considering flow past a truncated wavy cone.

Due to its broad range of uses such as car brake system, manufacturing of glass and plastic films, gas turbines, and medical equipment's, numerous researchers have effectively studied heat transfer and fluid flow in a rotation system [31]. The impact of MHD and porosity on rotating nanofluid flow with double diffusion by using regular nanoparticles was discussed by Krishna and Chamkha [32]. More features of heat transfer affected by porosity and magnetic field for a rotating fluid flow were referred in Das et al. [33] and Krishna et al. [34, 35]. Kumam et al. [36] implemented CNTs in analyzing the flow behavior for a rotating nanofluid. The nanofluid was considered as an electrical conducting fluid moving in a channel under heat source/sink and radiation effects. More study on the heat propagation for a convective flow of nanofluid in a rotating system affected by CNTs with several effects and different geometries were presented by Imtiaz et al. [37], Mosayebidorcheh and Hatami [38] and Acharya et al. [39]. Interestingly, several researchers had recently concentrated their study on the non-coaxial rotation flow. Mixer machines in food processing industry, cooling pad of electronic devices and rotating propellers for aircraft have become great application to exemplify the noncoaxial rotating phenomenon in various industries. Mohamad et al. [40] presented the mathematical expression for heat transfer in non-coaxial rotation of viscous fluid flow. As the extension of the previous study, the heat and mass transfer effects (double diffusion) were considered by Mohamad et al. [41] and followed by Mohamad et al. [42] investigating porosity effect in double diffusion flow of MHD fluid. Ersoy [43] imposed a disk with non-torsional oscillation to study the convective non-coaxial rotating flow for a Newtonian fluid. Mohamad et al. [44, 45] worked on similar study considering the second grade fluid and Rafiq et al. [46] concerning the Casson fluid model. The time dependent flow of an incompressible fluid with MHD, chemical reaction and radiation effects under non-coaxial rotation was investigated by Rana et al. [47]. Subjecting to the same type of rotation, Mohamad et al. [48] studied the porosity and MHD consequences in mixed convection flow influenced by an accelerated disk. The study was improved by Noranuar et al. [49] including the effects of double diffusive flow. According to the review of non-coaxial rotation, it is clear that most of the study are subjected to the ordinary fluid. However, the study of non-coaxial rotation for nanofluid by using regular nanoparticles had been performed by Das et al. [50] and Ashlin and Mahanthesh [51] but then the study reporting the implementation of CNTs in non-coaxial rotation flow remains limited.

Inspiring from the above literature, new study is essential to explore more findings on non-coaxial rotation of CNTs nanofluid. Therefore, the investigation of MHD non-coaxial rotating flow of CNTs nanofluid due to free convection in a porous medium become the primary focus of the current study. Water base fluid is chosen to suspend nanoparticle of SWCNTs and MWCNTs. The exact solutions for velocity and temperature distributions are attained by solving the problem analytically using the Laplace transform method. The results are illustrated in several graphs and tables for further analysis of various embedded parameters.

## **2. Problem formulation**

The incompressible time-dependent carbon nanofluid instigated by non-coaxial rotation past a vertical disk with an impulsive motion is considered as illustrated in **Figure 1**, where *x* and *z* are the Cartesian coordinates with *x*-axis is chosen as the upward direction and *z*-axis is the normal of it. The semi-finite space *z*>0 is occupied by nanofluid that composed by constant kinematic viscosity *υnf* of SWCNTs and MWCNTs suspended in water and acts as an electrically conducting fluid flowing through a porous medium. The disk is placed vertically along the *x*-axis with forward motion and a uniform transverse magnetic field of strength *B*<sup>0</sup> is applied orthogonal to it. The plane *x* ¼ 0 is considered as rotation axes for both disk and fluid. Initially, at *t* ¼ 0, the fluid and disk are retained at temperature *T*<sup>∞</sup> and rotate about *z*<sup>0</sup> -axis with the same angular velocity Ω. After time *t*>0, the fluid remains rotating at *z*<sup>0</sup> -axis while the disk begins to move with velocity *U*<sup>0</sup> and rotates at *z*-axis. Both rotations have a uniform angular velocity Ω. The temperature of the disk raises to *Tw* and the distance between the two axes of rotation is equal to *ℓ*. With above assumptions, the usual Boussinesq approximation is applied, and the nanofluid model proposed by Tiwari and Das [52] is used to represent the problem in the governing equations, express as

$$\begin{split} \rho\_{\rm nf} \frac{\partial F}{\partial t} + \left(\rho\_{\rm nf} \boldsymbol{\Omega} \dot{\boldsymbol{i}} + \sigma\_{\eta f} \boldsymbol{B}\_0^2 + \frac{\mu\_{\eta f}}{k\_1}\right) &= \mu\_{\eta f} \frac{\partial^2 F}{\partial \mathbf{z}^2} + (\rho \boldsymbol{\beta}\_T)\_{\eta f} \mathbf{g}\_{\rm x} (T - T\_{\infty}) \\ &+ \left(\rho\_{\eta f} \boldsymbol{\Omega} \dot{\boldsymbol{i}} + \sigma\_{\eta f} \boldsymbol{B}\_0^2 + \frac{\mu\_{\eta f}}{k\_1}\right) \boldsymbol{\Omega} \boldsymbol{\ell}, \end{split} \tag{1}$$

$$\left(\rho \mathbf{C}\_{\mathrm{p}}\right)\_{\mathrm{vf}} \frac{\partial T}{\partial t} = k\_{\mathrm{vf}} \frac{\partial^2 T}{\partial \mathbf{z}^2}. \tag{2}$$

The corresponding initial and boundary conditions are

$$F(z,0) = \Omega \ell; T(z,0) = T\_{\circ \circ}; z > 0,$$

$$F(0,t) = U\_0; T(0,t) = T\_w; t > 0,\tag{3}$$

$$F(\infty, t) = \Omega \ell; T(\infty, t) = T\_{\circ \circ}; t > 0,$$

in which *F* ¼ *f* þ *ig* is the complex velocity; *f* and *g* are (real) primary and (imaginary) secondary velocities respectively, *T* is the temperature of nanofluid and *U*<sup>0</sup> is the characteristic velocity. The following nanofluid constant for dynamic

**Figure 1.** *Physical model of the problem.*

*Analysis of Heat Transfer in Non-Coaxial Rotation of Newtonian Carbon… DOI: http://dx.doi.org/10.5772/intechopen.100623*

viscosity *μnf* , density *ρnf* , heat capacitance *ρCp* � � *nf* , electrical conductivity *<sup>σ</sup>nf* , thermal expansion coefficient *β<sup>T</sup>* ð Þ*nf* and thermal conductivity *knf* can be used as

$$\begin{aligned} \mu\_{\text{nf}} &= \frac{\mu\_{f}}{(1-\phi)^{2}}, \rho\_{\text{nf}} = (1-\phi)\rho\_{f} + \phi\rho\_{\text{CNT}}, \\ \left(\rho\text{C}\_{\text{P}}\right)\_{\text{nf}} &= (1-\phi)\left(\rho\text{C}\_{\text{P}}\right)\_{\text{f}} + \phi\left(\rho\text{C}\_{\text{P}}\right)\_{\text{CNTs}}, \\ \frac{\sigma\_{\text{nf}}}{\sigma\_{\text{f}}} &= 1 + \frac{3\left(\frac{\sigma\_{\text{CNTs}}}{\sigma\_{\text{f}}} - 1\right)\phi}{\left(\frac{\sigma\_{\text{C}NTs}}{\sigma\_{\text{f}}} + 2\right) - \phi\left(\frac{\sigma\_{\text{C}NTs}}{\sigma\_{\text{f}}} - 1\right)}, \\ (\mathbb{\mu}\_{\text{T}})\_{\text{nf}} &= \frac{(1-\phi)(\rho\text{\theta}\_{\text{T}})\_{\text{f}} + \phi(\rho\text{\theta}\_{\text{T}})\_{\text{CNTs}}}{\rho\_{\text{nf}}}, \\ \frac{\mathbf{k}\_{\text{nf}}}{\mathbf{k}\_{\text{f}}} &= \frac{\mathbf{1} - \phi + 2\phi \frac{\mathbf{k}\_{\text{C}NNs}}{\mathbf{k}\_{\text{C}NNs} - \mathbf{k}\_{\text{f}}} \ln\left(\frac{\mathbf{k}\_{\text{C}NNs} + \mathbf{k}\_{\text{f}}}{2\mathbf{k}\_{\text{f}}}\right)}{\mathbf{1} - \phi + 2\phi \frac{\mathbf{k}\_{\text{C}}}{\mathbf{k}\_{\text{C}NNs} - \mathbf{k}\_{\text{f}}} \ln\left(\frac{\mathbf{k}\_{\text{C}NNs} + \mathbf{k}\_{\text{f}}}{2\mathbf{k}\_{\text{f}$$

where the subscripts *f* is for fluid and *CNTs* is for carbon nanotubes. Meanwhile, *ϕ* is the solid volume fraction of nanofluid. The constants in Eq. (4) are used based on the thermophysical features in **Table 1**.

Introducing following dimensionless variables

$$F^\* = \frac{F}{\Omega \ell} - \mathbf{1}, z^\* = \sqrt{\frac{\Omega}{\nu}} \mathbf{z}, t^\* = \Omega \mathbf{t}, T^\* = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}}. \tag{5}$$

Using Eqs. (4) and (5), the governing equations in Eqs. (1)–(3) reduce to (excluding the \* notation to simplify the equations)

$$\frac{\partial F}{\partial t} + d\_1 F = \frac{1}{\phi\_1} \frac{\partial^2 F}{\partial x^2} + \phi\_3 GrT,\tag{6}$$

$$\frac{\partial T}{\partial t} = \frac{1}{a\_1} \frac{\partial^2 T}{\partial \mathbf{z}^2} \tag{7}$$

and the conditions take the form

$$\begin{aligned} \mathbf{F}(\mathbf{z}, \mathbf{0}) &= \mathbf{0}, \mathbf{T}(\mathbf{z}, \mathbf{0}) = \mathbf{0}; \mathbf{z} > \mathbf{0}, \\ \mathbf{F}(\mathbf{0}, \mathbf{t}) &= \mathbf{U} - \mathbf{1}, \mathbf{T}(\mathbf{0}, \mathbf{t}) = \mathbf{1}; \mathbf{t} > \mathbf{0}, \\ F(\boldsymbol{\omega}, t) &= \mathbf{0}, T(\boldsymbol{\omega}, t) = \mathbf{0}; t > \mathbf{0}, \end{aligned} \tag{8}$$


#### **Table 1.**

*Thermophysical features of water, SWCNTs, and MWCNTs.*

where

$$\begin{split} d\_1 &= \left(\mathbf{i} + M^2 \phi\_2 + \frac{\mathbf{1}}{\phi\_1 \mathbf{K}}\right), a\_1 = \frac{Pr\phi\_4}{\lambda}, M = \frac{\sigma\_f B\_0}{\Omega \rho\_f}, \frac{\mathbf{1}}{K} = \frac{\nu\_f}{k\_1 \Omega}, \\ \mathbf{Pr} &= \frac{\nu\_f \left(\rho C\_p\right)\_f}{k\_f}, Gr = \frac{\mathbf{g}\_x \beta\_{Tf} (T\_w - T\_\infty)}{\Omega^2 \ell}, U = \frac{U\_0}{\Omega \ell}. \end{split} \tag{9}$$

At this point, *d*<sup>1</sup> and *a*<sup>1</sup> are constant parameters, *M* is the magnetic parameter (magnetic field), *K* is the porosity parameter, Pr is Prandtl number, *Gr* is Grashof number and *U* is the amplitude of disk. Besides that, the other constant parameters are

$$\lambda = \frac{\mathbf{k\_{nf}}}{\mathbf{k\_{f}}}, \phi\_{1} = (\mathbf{1} - \phi)^{2.5} \left( (\mathbf{1} - \phi) + \frac{\Phi \mathbf{p\_{CNTs}}}{\rho\_{\rm f}} \right),$$

$$\Phi\_{2} = \left( \mathbf{1} + \frac{\Im\left(\frac{\sigma\_{\rm CNTs}}{\sigma\_{\rm f}} - \mathbf{1}\right) \phi}{\left(\frac{\sigma\_{\rm CNTs}}{\sigma\_{\rm f}} + 2\right) - \phi \left(\frac{\sigma\_{\rm CNTs}}{\sigma\_{\rm f}} - \mathbf{1}\right)} \right) \frac{\mathbf{1}}{\left( (\mathbf{1} - \phi) + \frac{\Phi \mathbf{p\_{CNTs}}}{\rho\_{\rm f}} \right)},\tag{10}$$

$$\Phi\_3 = \frac{(\mathbf{1} - \boldsymbol{\phi}) + \frac{\boldsymbol{\Phi}(\boldsymbol{\rho}\boldsymbol{\beta})\_{\mathrm{CNTs}}}{\left(\boldsymbol{\rho}\boldsymbol{\beta}\right)\_{\mathrm{f}}}}{(\mathbf{1} - \boldsymbol{\phi}) + \frac{\boldsymbol{\Phi}\boldsymbol{\rho}\_{\mathrm{CNTs}}}{\boldsymbol{\rho}\_{\mathrm{f}}}},\\ \Phi\_4 = (\mathbf{1} - \boldsymbol{\phi}) + \frac{\boldsymbol{\Phi}(\boldsymbol{\rho}\mathbf{C}\_{\mathrm{p}})\_{\mathrm{CNTs}}}{\left(\boldsymbol{\rho}\mathbf{C}\_{\mathrm{p}}\right)\_{\mathrm{f}}}.$$

#### **3. Exact solution**

Next, the system of equations in Eqs. (6)–(8) after applying Laplace transform yield to the following form

$$\frac{d^2}{dz^2}\overline{F}(\mathbf{z},q) - (\phi\_1 q + d\_2)\overline{F}(\mathbf{z},q) = -d\_3 Gr \overline{T}(\mathbf{z},q),\tag{11}$$

$$
\overline{F}(\mathbf{0}, q) = (U - \mathbf{1}) \frac{\mathbf{1}}{q}, \overline{F}(\infty, q) = \mathbf{0}, \tag{12}
$$

$$\frac{d^2}{dz^2}\overline{T}(z,q) - (a\_1q)\overline{T}(z,q) = 0,\tag{13}$$

$$\overline{T}(\mathbf{0}, q) = \frac{1}{q}, \overline{T}(\mathbf{\stackrel{\bullet}{\cdot}}, q) = \mathbf{0}. \tag{14}$$

Then, Eqs. (11) and (13) are solved by using the boundary conditions, Eqs. (12) and (14). After taking some manipulations on the resultant solutions, the following Laplace solutions form

$$\overline{F}(z,q) = \overline{F}\_1(z,q) - \overline{F}\_2(z,q) - \overline{F}\_3(z,q) + \overline{F}\_4(z,q) + \overline{F}\_5(z,q) - \overline{F}\_6(z,q), \tag{15}$$

$$\overline{T}(z,q) = \frac{1}{q} \exp\left(-z\sqrt{a\_1q}\right),\tag{16}$$

*Analysis of Heat Transfer in Non-Coaxial Rotation of Newtonian Carbon… DOI: http://dx.doi.org/10.5772/intechopen.100623*

where

$$\begin{split} \overline{\mathbf{F}}\_{1}(\mathbf{z},\mathbf{q}) &= \frac{\mathbf{U}}{\mathbf{q}} \exp\left(-\mathbf{z}\sqrt{\phi\_{1}\mathbf{q}+\mathbf{d}\_{2}}\right), \overline{\mathbf{F}}\_{2}(\mathbf{z},\mathbf{q}) = \frac{1}{\mathbf{q}} \exp\left(-\mathbf{z}\sqrt{\phi\_{1}\mathbf{q}+\mathbf{d}\_{2}}\right), \\ \overline{\mathbf{F}}\_{3}(\mathbf{z},\mathbf{q}) &= \frac{\mathbf{a}\_{4}}{\mathbf{q}} \exp\left(-\mathbf{z}\sqrt{\phi\_{1}\mathbf{q}+\mathbf{d}\_{2}}\right), \overline{\mathbf{F}}\_{4}(\mathbf{z},\mathbf{q}) = \frac{\mathbf{a}\_{4}}{\mathbf{q}-\mathbf{a}\_{3}} \exp\left(-\mathbf{z}\sqrt{\phi\_{1}\mathbf{q}+\mathbf{d}\_{2}}\right), \\ \overline{\mathbf{F}}\_{5}(\mathbf{z},\mathbf{q}) &= \frac{\mathbf{a}\_{4}}{\mathbf{q}} \exp\left(-\mathbf{z}\sqrt{\mathbf{a}\_{1}\mathbf{q}}\right), \overline{\mathbf{F}}\_{6}(\mathbf{z},\mathbf{q}) = \frac{\mathbf{a}\_{4}}{\mathbf{q}-\mathbf{a}\_{3}} \exp\left(-\mathbf{z}\sqrt{\mathbf{a}\_{1}\mathbf{q}}\right) \end{split} \tag{17}$$

are defined, respectively. The exact solutions for the temperature and velocity are finally generated by utilizing the inverse Laplace transform on Eqs. (15) and (16). Hence, it results

$$F(\mathbf{z},t) = F\_1(\mathbf{z},t) - F\_2(\mathbf{z},t) - F\_3(\mathbf{z},t) + F\_4(\mathbf{z},t) + F\_5(\mathbf{z},t) - F\_6(\mathbf{z},t) \tag{18}$$

$$T(z,t) = \text{erfc}\left(\frac{z}{2}\sqrt{\frac{a\_1}{t}}\right) \tag{19}$$

with

F1ð Þ¼ z, t U <sup>2</sup> exp z ffiffiffiffiffiffiffiffiffiffi ϕ1d4 � � <sup>p</sup> erfc <sup>z</sup> 2 ffiffiffiffiffi ϕ1 t r <sup>þ</sup> ffiffiffiffiffiffiffi d4t p ! þ U <sup>2</sup> exp �<sup>z</sup> ffiffiffiffiffiffiffiffiffiffi ϕ1d4 � � <sup>p</sup> erfc <sup>z</sup> 2 ffiffiffiffiffi ϕ1 t r � ffiffiffiffiffiffiffi d4t p !, F2ð Þ¼ z, t 1 <sup>2</sup> exp z ffiffiffiffiffiffiffiffiffiffi ϕ1d4 � � <sup>p</sup> erfc <sup>z</sup> 2 ffiffiffiffiffi ϕ1 t r <sup>þ</sup> ffiffiffiffiffiffiffi d4t p ! þ 1 <sup>2</sup> exp �<sup>z</sup> ffiffiffiffiffiffiffiffiffiffi ϕ1d4 � � <sup>p</sup> erfc <sup>z</sup> 2 ffiffiffiffiffi ϕ1 t r � ffiffiffiffiffiffiffi d4t p !, F3ð Þ¼ z, t a4 <sup>2</sup> exp z ffiffiffiffiffiffiffiffiffiffi ϕ1d4 � � <sup>p</sup> erfc <sup>z</sup> 2 ffiffiffiffiffi ϕ1 t r <sup>þ</sup> ffiffiffiffiffiffiffi d4t p ! þ a4 <sup>2</sup> exp �<sup>z</sup> ffiffiffiffiffiffiffiffiffiffi ϕ1d4 � � <sup>p</sup> erfc <sup>z</sup> 2 ffiffiffiffiffi ϕ1 t r � ffiffiffiffiffiffiffi d4t p !, F4ð Þ¼ z, t a4 <sup>2</sup> exp a3t <sup>þ</sup> <sup>z</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ1ð Þ a3 þ d4 � � <sup>p</sup> erfc <sup>z</sup> 2 ffiffiffiffiffi ϕ1 t r <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ a3 <sup>þ</sup> d4 <sup>t</sup> p ! þ a4 <sup>2</sup> exp a3t � <sup>z</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ1ð Þ a3 þ d4 � � <sup>p</sup> erfc <sup>z</sup> 2 ffiffiffiffiffi ϕ1 t r � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ a3 <sup>þ</sup> d4 <sup>t</sup> p !, F5ð Þ¼ z, t a4erfc <sup>z</sup> 2 ffiffiffiffi a1 t � � r , F6ð Þ¼ z, t a4 <sup>2</sup> exp a3t <sup>þ</sup> <sup>z</sup> ffiffiffiffiffiffiffiffi a1a3 <sup>p</sup> ð Þerfc <sup>z</sup> 2 ffiffiffiffi a1 t r <sup>þ</sup> ffiffiffiffiffiffi a3t <sup>p</sup> � � þ a4 <sup>2</sup> exp a3t � <sup>z</sup> ffiffiffiffiffiffiffiffi a1a3 <sup>p</sup> ð Þerfc <sup>z</sup> 2 ffiffiffiffi a1 t r � ffiffiffiffiffiffi a3t <sup>p</sup> � �, (20)

where

$$\mathbf{d}\_2 = \boldsymbol{\phi}\_1 \mathbf{d}\_1, \mathbf{d}\_3 = \boldsymbol{\phi}\_1 \boldsymbol{\phi}\_3, \mathbf{d}\_4 = \frac{\mathbf{d}\_2}{\boldsymbol{\phi}\_1}, \mathbf{a}\_2 = \mathbf{a}\_1 - \boldsymbol{\phi}\_1, \mathbf{a}\_3 = \frac{\mathbf{d}\_2}{\mathbf{a}\_2}, \mathbf{a}\_4 = \frac{\mathbf{d}\_3 \mathbf{G} \mathbf{r}}{\mathbf{a}\_2 \mathbf{a}\_3}.\tag{21}$$
