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BY 4.0 license Open Access Published by De Gruyter Open Access July 18, 2022

Numerical simulation of nanofluid flow between two parallel disks using 3-stage Lobatto III-A formula

  • Hammad Alotaibi EMAIL logo and Khuram Rafique
From the journal Open Physics


The development of nanofluid technology has become a key research area in physics, mathematics, engineering, and materials science. Nowadays, in many industrial applications, nanofluids are widely used to enhance thermophysical properties such as thermal diffusivity, thermal conductivity, and convective heat transfer. Scientists and engineers have established interests in the direction of flow problems developed via disk-shaped bodies. There are various logics to discuss flow phenomenon due to rotating bodies, but its applications include in thermal power engineering system, gas turbine rotors, air cleaning machines, aerodynamics, etc. Nowadays manufacturing industries have inaugurated to select liquid based on heat transfer properties. Therefore, this article focuses on studying the laminar incompressible nanofluid between two parallel disks. Mathematical formulations of the law of conservation of mass, momentum, and heat transfer are investigated numerically. By using suitable similarities, the flow equations are converted into nonlinear ordinary differential equations. The resulting equations were solved numerically via MATLAB software. The effects of physical parameters of interest, such as Reynolds number, magnetic factor, Brownian parameter, and thermophoresis parameter on normal velocity, streamwise velocity, temperature, and concentration profiles are computed and presented using the graphs. The results revealed that the energy profile significantly rises, and the profile moves closer to the upper disk by enhancing the Brownian motion and thermophoresis parameter. The dynamics behind this is that by increasing the Brownian motion, the boundary layer wideness increases which increases the temperature. Moreover, streamwise velocity increases for large values of Reynolds number. Besides, the thermophoresis profile increases for large values of the thermophoresis factor. It could be observed that shear stress at nonporous/porous disk is adjusted by selecting a suitable value of injection velocity at the porous disk. Also, normal velocity decreases by increasing the parameter M.


B 0

constant magnetic field


nanoparticle concentration


Brownian diffusion coefficient


thermophoresis diffusion coefficient


Lewis number


Hartmann number


Brownian motion parameter


thermophoresis parameter


Prandtl number




Reynolds number

T m

mean temperature of the fluid


temperature of the fluid


axial velocity component


kinematic viscosity


constant suction/injection velocity


radial velocity component


fluid density


dynamic viscosity


Boltzmann constant

1 Introduction

In recent years, there has been a growing interest and substantial improvement in nanofluids due to their enormous potential to enhance heat transfer which has vast applications in the field of biochemical, engineering, biomedical sciences, and industry. The word “nanofluid” alludes to a fluid interruption of tiny rigid atoms having dimensions regularly of 1–100 nm in a regular liquid. In the late 1995s, nanofluid was introduced by Choi and Eastman [1], and they checked that the thermal conductivity of the fluids upgrade by the incorporation of these nanoparticles in base fluids. To feature the immense impacts of thermophoretic as well as Brownian movement dissemination of nanoparticles, a scientific model was introduced by Buongiorno [2].

Numerous analysts utilized this model to dissect the movement of the Nanofluids in different geometries. Several methods have been previously proposed that aim to enhance the thermal conductivity. For example, Sheikholeslami et al. [3] discussed the study of natural convection heat exchange in a nanofluid-packed field with an elliptic internal cylinder. Dogonchi et al. [4] analyzed nanofluid flow between parallel disks by considering the heat flux model. Khan et al. [5] worked on the Nanofluid flow between two parallel plates and examined the heat and mass transfer effect. Sheikholeslami and Abelman [6] discussed the heat transfer for nanofluid flow numerically. Hosseinzadeh et al. [7] explained nanofluid fluid flow analytically between analogous sheets. Rafique et al. [8] examined the effects of thermophoresis and Brownian motion on nanofluid flow for a slanted plate. Noor et al. [9] presented mixed convection micropolar nanofluid flow toward a vertically enlarging surface. Mabood et al. [10] analyzed nanofluid boundary layer flow and heat transfer along with a nonlinearly stretching sheet.

The physics of energy transfer in rapidly moving machines and engines with lubricant insides is playing a very useful role in daily life. Therefore, it has been an active area of research these days which helps today’s practical life. The study of the heat transfer phenomenon is necessary for these systems for consistent and safe working of such machines. Magnetohydrodynmaic (MHD) has attracted a lot of interest due to its variety of applications. Some noteworthy innovative uses of MHD are in tests of measured atomic combination where a magnetic field is utilized to restrict poles of sizzling plasma, to create energy where fluid metals are compelled over a magnetic field, etc. Besides, MHD standards are utilized for shuttle impetus and light-ion-beam-driven inertial constraint. Some amazing endeavors have been made to examine the impact of MHD on different flow conditions. The extensive literature on the MHD flows in the existence of applied magnetic fields occurs nowadays. Likewise, Azimi and Riazi [11] attempted analytical methods for solving MHD convective and slip flow due to a spinning disk. Makinde et al. [12] probed the flow over a disk by considering different nanoparticles. Turkyilmazoglu [13] discussed the particular result of MHD viscid fluid by a revolving disk. Furthermore, Rafique et al. [14] described the properties of energy and species exchange in MHD flow for slanted sheets. Rafique et al. [15] probed the MHD stream of nano liquid along with the slanted sheet. Recently, Iqbal et al. [16] considered a wavy stretchy surface for nanoliquid flow by incorporating magnetic impacts. Furthermore, Iqbal et al. [17] utilized hybrid nanoliquid for the stretchy wavy geometry by taking magnetic effect.

In the same vein, the fluid movement over parallel disks has picked up great significance because of its extensive variety of technical and industrial uses, for example, laptop packing devices, semiconductor engineering procedures, thrust bearings, air turbine motors, biomechanics, etc. Elcrat [18] initially demonstrated the hypothesis of the non-rotational liquid wave over settled permeable disks with constant subjective drag or injunction. Ibrahim [19] has examined viscid unstable flow among adjacent disks. Vajravelu et al. [20] have investigated the squeezing flow of nanoliquid and energy exchange rate through parallel disks. Moreover, Bhatta et al. [21] discussed nanofluid unsteady flow with slip impacts. Hayat et al. [22] discussed the energy exchange rate for nanofluid flow for disks. Recently, Mustafa extended the von Karman problem of the infinite rotating disk by using Buongiorno’s model. Sobamowo et al. [23] discussed nanofluid flow between two corresponding disks. Recently, Ghaffari et al. [24] examined two stretchy disks for power law nanoliquid via shooting technique. They discussed energy exchange phenomenon. Aziz et al. [25] probed three-dimensional (3D) flow for nanofluid by rotating disk. Elakkiyapriya and Anjali [26] examined the nanofluid flow because of the spinning disk. Hatami et al. [27] investigated 3D nanofluid with the help of an analytical method. Yin et al. [28] discussed nanofluid with heat transfer over a disk. Rafique et al. [29] investigated energy and mass exchange via numerical simulation of nanofluid. Alotaibi et al. [30] treated Casson nanofluid numerically over a heated extended surface. For higher order chemical reactions with heat source (sink) impacts, Alotaibi and Eid [31] proposed a model to study the flow of Darcy–Forchheimer’s 3D permeable nanofluid through convective heating of porous extended surfaces considering the effect of magnetic fields and nonlinear radiation. Furthermore, by using the Keller box technique following the finite difference method in the MATLAB program, Alotaibi and Rafique [32] investigated mixed convection heat and mass transfer of non-homogenous nanofluid around a vertical Riga plate in the presence of MHD effect. Different researchers [33,34,35,36] contributed valuable research work in the latest research area by considering different fluid models.

Engineers and scientists aim to explore further investigations of flow characteristics between disks due to wide range of applications of such flow in different varieties of field such as lubrication, heat and mass exchange, biomechanics, rotating machinery, etc. To our knowledge, there is no investigation so far that has been devoted to study the proposed model for nanofluid between two parallel disks by incorporating Brownian movement and thermophoresis impacts. Therefore, the principle aim of this article is to present a sufficient understanding of the mechanism of heat transfer as well as mass exchange in nanofluids flow behavior through porous and non-porous parallel disks. Here we restrict attention to nanofluid flow over parallel disks by taking Brownian motion and thermophoresis effects to enhance the heat transfer performance of viscous fluid [37]. It is known that various methods are available for the numerical solution of the considered problem, but we employed the Lobatto formula for the solution. Because this method is built by finite difference technique in bvp4c via Matlab software. This method is very easy for the numerical solutions of complicated ordinary differential equations (ODEs) of flow problems. In addition, this research pays intention to the heat as well as mass transportation which is very helpful in industry, engineering viewpoint, and practical life via the hydraulic engine, generators, electric motors, etc. This work has potential application in the implications of many devices including electronics, air conditioning, power, and chemical. It is expected that the results recovered in this study will not only helpful to give the information about industrial and engineering uses but also compliment on the published work.

2 Problem formulation

This numerical investigation is conducted for nanofluids between two analogous disks. The higher disk is porous, while the other is nonporous. Furthermore, the flow of nanofluid is driven with a continuous pressure gradient having velocity U . The lower disk is located at z = L and the upper disk is at z = L with center at z = 0 as shown in Figure 1.

Figure 1 
               Physical sketch of the proposed problem.
Figure 1

Physical sketch of the proposed problem.

The governing equations for the flow of nanofluids between parallel disks by following Ashraf and Wehgal are [37]:

(1) u r + u r + w z = o ,

(2) u u r + w u z = 1 δ f p r + ϑ 2 u r 2 + 2 u z 2 + 1 r u r u r 2 σ B 0 2 ρ u ,

(3) u w r + w w z = 1 δ f p r + ϑ 2 w r 2 + 2 w z 2 + 1 r w r ,

(4) u T r + w T z = α 2 T r 2 + 1 r T r + 2 T z 2 + τ D B c r T r + c z T z + D t T m T r 2 + T z 2 ,

(5) u c r + w c z = D B 2 c r 2 + 1 r c r + 2 c z 2 + D t T m 2 T r 2 + 1 r T r + 2 T z 2 .

The boundary settings for lower and upper disks by following Ashraf et al. [38] are given as

(6) w ( r , L ) = 0 , w ( r , L ) = 2 v

u ( r , L ) = 0 , u ( r , L ) = 0 .

where 2 v denotes uniform and constant suction/injection velocity magnitude at the upper disk. Moreover, the lower disk is taken as nonporous. In view of Ashraf and Wehgal [37], the boundary settings for temperature and concentration are given as

(7) T = T 1 , C = C 1 at η = 1 ,

T = T 2 , C = C 2 at η = 1 .

Here we use the Von Karman [39] and Elko [40] similarity transformations to reduce the above partial equations into ordinary differential equations.

(8) ψ ( r , η ) = v r 2 2 f ( η ) , θ = T T 1 T 2 T 1 , φ = c c 1 c 2 c 1 .

by eliminating the pressure term, we get:

(9) f iv = 2 L ϑ f f + σ B 0 2 μ L 2 f .

By using the following dimensionless variables into Eq. (9):

f ( η ) = F ( z ) v , w here η = z L .

Therefore, Eq. (9) becomes:

(10) f iv 2 Rf f M 2 f = 0 .

Eqs. (4) and (5) become

(11) θ + Pr Nb θ φ + PrNt θ 2 2 fR Pr θ = 0 ,

(12) 2 fR Le + Nt Nb θ = 0 ,

where M = σ L B 0 2 μ , R = ρ vL μ Pr = ν D B , Le = ν α , Nb = τ D B ( c w c h ) ν , and Nt = τ D t ( T w T h ) T m ν

With associate boundary conditions:

(13) f ( 1 ) = 0 , f ( 1 ) = 1 , θ ( 1 ) = 1 , θ ( 1 ) = 0 ,

(14) f ( 1 ) = 0 , f ( 1 ) = 0 , ( 1 ) = 1 , ( 1 ) = 0 .

3 Numerical solution

In order to find the numerical solution of Eqs. (10)–(12) subject to the boundary conditions (13)–(14), we employ the Lobatto formula. This method can be depicted in MATLAB 2010 by a built-in function named “bvp4c.” This is a finite difference method executed in three stage Lobatto III-A formula. This is also known via the collocation method with fourth-order accuracy. This numerical technique is very helpful to solve all kinds of complicated nonlinear ODE’s. For this, we transform our governing nonlinear ODEs into a first-order ODEs system by introducing:

y 1 = f , y 2 = f , y 3 = f , y 4 = θ , y 5 = ϕ

For “bvp4c” function, we use a routine named “NANO_ODE,” which has the system of 1st order ODEs as mentioned above. “NANO_BC” is used for boundary conditions and “NANO_GUESS” is used for the initial guess which fulfilled our boundary conditions.

4 Results and discussion

This section contains the discussions and interpretations of our findings in graphical and tabular form. Numerous figures as well as tables are prepared for numerical interpretation. Table 1 exhibits the statistical results of θ ( 1 ) and ϕ ( 1 ) . Table 1 elucidated the mathematical values of energy and species exchange at the bottom disk for the variation in Brownian motion and thermophoresis. In same vein from Table 1, we can perceive that the values of heat exchange growths for the values of Nb, and this due to the irregular movement of particles which causes the collisions between fluid particles in return kinetic energy boost up while, opposite in the case of the mass exchange rate. Furthermore, the values of θ ( 1 ) and ( 1 ) increase with the increase in the values of thermophoresis ( Nt ). Physically, it can be argued that boundary layer thickness enhances by increasing the strength of N b . To check the validity of our solution, we compare Figures 2 and 3 with the already published literature by Ashraf [37]. Table 2 presents the statistical standards of skin friction at the lower disk for the case of hydrodynamic flows ( M = 0 ) and MHD flows ( M = 1 ) . Skin friction increases with the decrease in the values of Reynolds number in both the cases ( i . e . , M = 1 , M = 0 ) . From this observation we can conclude that in particular engineering applications of the problem under study, by selecting a suitable value of injection velocity at the porous disk, shear stress at nonporous/porous disk may be adjusted.

Table 1

Values of the heat transfer rate θ ( 1 ) and mass transfer rate ( 1 ) for different values of Nb and Nt , when M = 1 , R = 10 , Le = 2

R M Le Nb Nt θ (−1) (−1)
−10 1.0 2 0.1 0.3 −1.829 −2.014
0.5 −1.520 −3.051
1 −1.193 −3.164
1.5 −0.0928 −3.191
2 −0.714 −3.197
0.5 0 −1.664 −3.137
0.1 −1.614 −3.096
0.2 −1.566 −3.068
0.3 −1.520 −3.051
0.4 −1.475 −3.044
Figure 2 
                  f′(η) versus R.
Figure 2

f′(η) versus R.

Figure 3 
                  f(η) versus R.
Figure 3

f(η) versus R.

Table 2

Values of skin friction f ( 1 ) against altered values of R, when M = 0 and when M = 1

R M f″(−1)
0 0 1.5
−3 2.644
−6 3.529
−9 4.233
0 1 1.597
−3 2.653
−6 3.516
−9 4.213

In the same manner, we discussed the effects of Reynolds number (R) on f ( η ) and f ( η ) in Figures 2 and 3, respectively. On the other hand, the influence of Magnetic constraint ( M ) on streamwise velocity and normal velocity are discussed (Figures 4 and 5). Now, we probe the impact of Reynolds number R and graphical representation of streamwise and normal velocity profiles. Figure 2 displays the parabolic shape against R = 0 . As the lower disk is porous, the profile irregularly moves in its direction subsequently growing in boundary layer width for the higher disk and declining for the bottom disk. Furthermore, the extreme velocity is lifted in the direction of the lower disk. We may additionally argue that an enhancement in the value of Reynolds number R intensifies these properties driving the main part of the flow toward the lower disk and for R , boundary layer thickness δ tends to zero. From Figure 2, it can be noted that streamwise velocity rises by increasing R . The behavior of normal velocity f ( η ) against different values of Reynolds number 10 R 0 is shown in Figure 3. It can be perceived that for growth in R , normal velocity f ( η ) increases throughout from the lower disk to upper disk. Numerical results of streamwise velocity and normal velocity have been presented graphically for the variation in M (Figures 4 and 5).

Figure 4 
                  f′(η) versus 
Figure 4

f′(η) versus M .

Figure 5 
                  f(η) versus 
Figure 5

f(η) versus M .

Figure 4 shows that the streamwise velocity profile decreases to close the lower disk by increasing the magnetic field parameter ( M ) and increases near the upper disk. Physically, the magnetic parameter is a measure of the magnetic field strength within the fluid. The Lorentz force appears due to the incorporation of the magnetic field in the fluid which opposes the fluid velocity. The magnetic effect near the porous disk decreases the stream velocity because of large Lorentz forces and the stream velocity increases toward the upper nonporous disk because of the small strength of Lorentz force. Figure 5 shows that velocity distribution declines with the rise in M because the magnetic field factor starts to slow down the velocity at all points of the flow field. It is due to an opposing force (Lorentz force) which tends to resist fluid flow. Also, with the increase in magnetic impact, boundary layer width decreases. The impacts of Brownian motion and thermophoresis parameters Nb and Nt on the temperature field θ ( η ) are discussed in the Figures 6 and 7, respectively.

Figure 6 
                  θ(η) versus 
Figure 6

θ(η) versus Nb .

Figure 7 
                  θ(η) versus 
Figure 7

θ(η) versus Nt .

The strength of Brownian motion and thermophoresis will be higher on the temperature field against Nb and Nt . The energy profile θ ( η ) significantly rises and the profile moves closer to the upper disk as Nb and Nt increase. The physics behind this is that by increasing the Brownian motion, the boundary layer wideness increases which increases the heat (because Brownian motion strength increases the temperature due to random movement).

Brownian motion and thermophoresis parameters ( Nb , Nt ) impact on the nanoparticles thermophoresis is illustrated in Figures 8 and 9, respectively. The concentration profile φ ( η ) decreases with the increase in the value of the Brownian movement constraint. From Eq. (12), it can be confirmed that the expression 1 Nb , shows direct correspondence with the nanoparticle concentration. Therefore, the concentration profile diminishes against the growth of Nb , while influence of Nt shows direct relation with φ ( η ) , thus the profile is increased. Physically, we can see that by increasing the thermophoretic effect due to the temperature gradient, a larger flux will be generated which in turn raises the concentration.

Figure 8 
                  ϕ(η) versus 
Figure 8

ϕ(η) versus Nb .

Figure 9 
                  ϕ(η) versus 
Figure 9

ϕ(η) versus Nt .

5 Conclusion

This article presented numerical simulations of nanofluid flow between a porous disk and a nonporous disk. We applied suitable similarity transformations to reduce the complex governing equations into a set of nonlinear ODEs. The numerical solutions are obtained by using the Lobatto method. Numerical simulations demonstrated the influence of physical parameters of interest, such as Reynolds number, magnetic factor, Brownian parameter, and thermophoresis parameter on normal velocity, streamwise velocity, temperature, and concentration profiles. Our results are in excellent agreement with the existing numerical literature results. For highly nonlinear differential equations, the Lobatto method is one of the most powerful, effective, and efficient tools to derive the solutions. The investigation of current work reveals that the heat and mass transfer along with flow phenomenon were influenced by physical factors. Additionally, this heat and mass transfer phenomenon provide useful information which can be utilized in paper production processes and in electronic devices. The main outcomes drawn from the above analysis are given as:

  • The energy profile significantly rises for large values of N b .

  • Streamwise velocity enhanced against higher values of Reynolds number.

  • Normal velocity decreases by increasing parameter M .

  • θ ( 1 ) and ( 1 ) increases by increasing the values of thermophorsis ( Nt ).

  • The concentration profile shows direct correspondence against thermophoretic effects.

Future work will focus on extending this study by selecting a suitable value of injection velocity at the porous disk shear stress at nonporous/porous disk. Moreover, we could consider different types of fluids such as hybrid nanoliquid, Williamson nanoliquid, Casson nanoliquid etc., with different numerical techniques.


The authors are thankful for the Taif University research supporting project number (TURSP-2020/304), Taif University, Saudi Arabia.

  1. Funding information: This work was supported by Taif University research supporting project (Grant numbers. TURSP-2020/304).

  2. Author contributions: conceptualization, validation, and formal analysis: Hammad Alotaibi; methodology, software, and investigation: Khuram Rafique; writing – original draft preparation and writing – review and editing: Hammad Alotaibi and Khuram Rafique. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.


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Received: 2022-03-15
Revised: 2022-06-03
Accepted: 2022-06-09
Published Online: 2022-07-18

© 2022 Hammad Alotaibi and Khuram Rafique, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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