1. Introduction
Nanofluids consist of solid particles called nanoparticles with higher thermal characteristics suspended in some base fluid. Moreover, convective heat transfer through nanoparticles has motivated many researchers for its industrial applications, pharmaceutical processes, domestic refrigerators, chillers, heat exchangers, electronic cooling system, and radiators, etc., [
1]. Nanofluids are considered as the finest coolants for its various industrial applications. Nanofluids exhibit promising thermos-physical properties e.g., they have small viscosity and density and large thermal conductivity and specific heat [
2]. As far as transportation of energy is concerned, the ideal features of nanofluids are the high thermal conduction and low viscosity [
3]. Choi and Eastman [
4] primarily examined the upsurge in thermal conductivity by submerging nanoparticles into the ordinary fluid. Because of these thermos-physical characteristics, nanofluids are considered as the finest coolants that can work at various temperature ranges [
5]. Sheikholeslami et al. [
6] found a numerical solution ferrofluid flow under the influence of applied magnetic field in a hot elliptic cylinder. It is examined by them that strong Lorentz force is a source in declining the temperature of the fluid. The water-based nanofluid flow with numerous magnetite nanoparticles amid two stretchable rotating disks is numerically studied by Haq et al. [
7]. Khan et al. [
8] numerically addressed the water and ethylene glycol based nanofluid flow containing copper nanoparticles with suction/injection effect between parallel rotating stretchable disks. Saidi and Tamim [
9] examined the pressure drop and heat transfer properties of nanofluid flow induce amid parallel stretchable disks in rotation by considering thermophoresis effects. Hayat et al. [
10] also found a series solution of Jeffrey nanofluid flow between two coaxial rotating stretchable disks having convective boundary condition. Pourmehran et al. [
11] numerically simulated the nanofluid flow between coaxial stretchable rotating disks.
Molecules of carbon atoms arranged in a cylindrical shape to form a structure called carbon nanotubes (CNTs). This arrangement of the molecule may be by rolling up of single sheet or by multiple sheets of graphene [
12]. The novel properties of CNTs are light weight and high thermal conductivity, which make them potentially useful. CNTs are not dangerous to the environment as they are composed of carbon atoms [
13]. The CNTs are the most desirous materials of the twenty-first century. Modern applications of CNTs are in microfabrication technique, pancreatic cancer test, and tissue engineering, etc., [
14]. The flow of nanofluid containing both types CNTs with thermal radiation and convective boundary condition effects is examined analytically by Imtiaz et al. [
15]. The water-based nanofluid flow containing CNTs of both categories under the impact of magneto-hydrodynamics (MHD) amid two parallel disks is studied by Haq et al. [
16]. Mosayebidorcheh et al. [
17] did heat transfer analysis with thermal radiation impacts of CNTs-based nanofluid squeezing flow between two parallel disks numerically via the least square method. Effects of thermal radiation in a magnetic field comprising both types of CNTs aqueous based nanofluid flow by two rotating stretchable disks are debated by Jyothi et al. [
18]. Transparent carbon nanotubes coating to obtain conductive transparent coating is analyzed by Kaempgen [
19]. Keefer et al. [
20] studied carbon nanotube-coated electrodes to improve the current electrophysiological techniques. Enzyme-coated carbon nanotube as a single molecule biosensor was reported by Besteman et al. [
21]. Some recent investigations featuring Carbon nanotubes amalgamated fluid flow may be found in [
22,
23,
24,
25,
26,
27,
28,
29,
30] and many therein.
Thermal energy transformation possesses significant importance in engineering applications such as fuel cell efficiency, biomedical applications including cooling of electronic devices, heat conduction in tissues, energy production, heat exchangers, and cooling towers etc., [
31]. Classical Fourier law of heat conduction was employed to describe the mechanism of heat transfer. But this model gives parabolic energy equation that is medium encountered initial disturbance instantly which is called “heat conduction paradox.” Cattaneo [
32] tackled this enigma by introducing the time needed for the conduction of heat via thermal waves at a limited speed which is known as thermal relaxation time. The modification in Fourier law gives hyperbolic energy equation for temperature profile. Christov [
33] further inserted Oldroyd’s upper convective derivative to maintain material invariant formulation. This upgraded model is known as Cattaneo- Christov heat flux model. The aqueous fluid flow by two rotating disks with the impact of CC heat flux is studied by Hayat et al. [
34]. Dogonchi et al. [
35] scrutinized the squeezed flow of nanofluid encompassing CC heat flux and thermal radiation effects. Lu et al. [
36] discussed the unsteady squeezing nanofluid flow between parallel disks comprising CNTs with CC heat flux model and HH reactions. The recent advance studies on CC heat flux is done by many researchers [
37,
38,
39,
40].
The aforementioned literature survey (
Table 1) reveals that unsteady nanofluid flow containing CNTs with CC heat flux under the influence of hall current between two rotating stretchable disks is not yet discussed. Additional impacts like HH reactions and thermal stratification of the presented mathematical model may be considered as added features toward the novelty of the problem. The problem is solved numerically by using the bvp4c function of MATLAB software.
2. Problem Formulation
Consider an axisymmetric unsteady MHD water base nanofluid flow between continuously stretchable disks with hall current effect amid non-conducting rotating disks at
and
. The disks rotate at constant angular velocities
and
about its axis. Magnetic field
that is uniformly distributed is applied in the normal direction of the disks (
Figure 1). Furthermore, the stretching rates of the disks are
and
. Temperature
refers to the temperature of upper disk while the disk’s temperature at
is
in a thermally stratified medium.
For isothermal cubic autocatalysis, a model for homogeneous and heterogeneous reactions with reactants as chemical species are
and was proposed by Merkin and Chaudary [
41] and is given by:
The continuity equation is
The momentum equations are
The relevant energy equation is
where
represents the temperature,
the specific heat and
the heat flux. Heat flux in perspective of Cattaneo–Christov expression is satisfied.
Here,
is the thermal relaxation time and
is the thermal conductivity. Utilizing the incompressibility condition, we arrive at
Eliminating
from Equations (9) and (7), we get
As
is the velocity vector, we obtain the following governing equations after applying the boundary layer theory:
The associated boundary conditions are
Here, is the reference temperature. are the dimensional constant with dimension .
Thermo-physical properties of CNTS are represented in mathematical form as follows:
Table 2 represents the thermos-physical characteristics of CNTs and H
2O.
Following transformation are used to convert the above nonlinear partial differential equations to dimensionless ordinary differential equations.
Equation (11) is satisfied automatically, Equations (12) to (17) are transformed into the following form:
with transformed boundary conditions
where
By assuming the chemical species alike, we take diffusion coefficient of both species equal, so that
. And thus we have
we get from Equations (30) and (31)
Differentiating Equation (26), we get
5. Outcomes with Discussion
In this section the impact of different parameters on velocity and temperature profile, drag force coefficient, and Nusselt number is described in the form of graphs and tables. In order to acquire the required outcome we fix the different flow parameters such as ,
5.1. Radial and Axial Velocity Profile
In
Figure 2,
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7,
Figure 8 and
Figure 9, the radial velocity
and axial velocity profiles
is depicted for
, parameters, scaled Stretching
and
and nanoparticle volume fraction
. The solid line ( ) and the dashed line (----) represent the single wall carbon nanotubes and multiwall carbon nanotubes respectively.
Figure 2 and
Figure 3 show that the magnitude of radial
and axial velocity
reduces for incremental value of
. The fact is that for increasing values of Reynolds number causes the increase in resistive forces which reduces the motion of fluid. Magnitude of
for multiwall carbon nanotubes is higher as compared with single wall carbon nanotubes.
takes on negative values near the lower disks because upper disks are moving faster than the lower disks.
Figure 4 depicts that
escalates in the vicinity of the lower disk and declines in the vicinity of the upper disks by enhancing the value of
, while the behavior of
remain same throughout the system as shown in
Figure 5. But by the increase in the value of
,
increases in the vicinity of the lower disks and decreases in the vicinity of the upper disks, (see
Figure 6), and
shows decrease in magnitude throughout the system, (see
Figure 7).
Figure 8 shows that
reduces by the increase of nanoparticle volume fraction and magnitude of
is smaller for MWCNTs.
is decreasing near the lower disk and enhancing near the upper disks by increasing
, while the amplitude of
is higher for MWCNTs than SWCNTs. This effect is shown in
Figure 9.
5.2. Tangential Velocity Profile
Tangential velocity
decreases by escalating the value of
because increasing magnetic field exerts a retarding force which slows the motion of the particles within the fluid.
Figure 10 depicts that the tangential velocity has smaller magnitude for MWCNTs as compared to SWCNTs.
Figure 11 depicts that tangential velocity decreases for increasing value of
and its value is smaller for MWCNTs.
Figure 12 shows that as stretching rate increases at the upper disk it causes a decrease of tangential velocity.
increases for incremental values of hall current parameter
and magnitude of tangential velocity profile is more increasing for MWCNTs as compared with SWCNTs as shown in
Figure 13.
Figure 14 depicts the relationship between Ω and
. It represents that the tangential velocity is an escalating function of rotation parameter.
Figure 15 and
Figure 16 depict that for increasing
the amplitude of
increases and it decreases for increasing Reynolds number.
5.3. Dimensionless Temperature Distribution
The dimensionless temperature distribution for different values of relaxation parameter is depicted for both MWCNTs and SWCNTs in
Figure 17. The figure shows that higher rate of thermal relaxation parameter causes the increase in temperature profile. Results shows that temperature profile is more increasing for MWCNTs than SWCNTs.
Figure 18 shows that temperature decreases by increasing nanoparticle volume fraction and temperature profile shows more decreasing behavior for MWCNTS as compared to SWCNTs. Effect of Reynolds number, Prandtl number, stratification parameter, unsteadiness parameter
, stretching parameter
at lower disk on temperature profile is shown in
Figure 19,
Figure 20,
Figure 21,
Figure 22 and
Figure 23. Results are plotted both for MWCNTs and MWCNTs.
Figure 19 shows that for positive values of
there is an increase in temperature profile, and it shows that multi-walled carbon nanotubes have higher temperature distribution for increasing Reynolds number as compared to single-walled carbon nanotubes. Similarly, graph is plotted for negative values of Reynolds number. It is revealed that on decreasing the value of Reynolds number, temperature profile also decreases and shows more decreasing behavior for MWCNTs than SWCNTs.
Figure 20,
Figure 21 and
Figure 22 portray the variation of temperature profile which decreases for incremental values of
,
, and
this decreasing behavior is observed more for SWCNTs as compared with MWCNTs.
Figure 23 depicts for increasing value of Prandtl number temperature profile decreases. The decrease in temperature by augmentation of Prandtl number is consistent with the physical expectation, as by increasing Prandtl number fluid possesses lower thermal diffusivity which causes the thickness of thermal boundary layer to decrease.
5.4. Concentration Profile
Figure 24 demonstrate the analysis of concentration profile. For various estimates of homogeneous reaction parameter
there is decay in concentration profile. Similar results are obtained for heterogeneous reaction parameter
in
Figure 25. Concentration field is observed for Schmidt number in
Figure 26. As it is momentum to mass diffusivity ratio, so smaller the value of mass diffusivity, stronger the value of Schmidt number, which causes the reduction of the concentration of the fluid.
Comparison of
and
with Stewartson [
42] for several estimates of
by considering all extra terms as zero is depicted in
Table 3. An excellent synchronization is achieved in this case. This substantiates our mathematical model and presented results.
5.5. Drag Force Coefficient and Heat Transfer Rate
Influence of Hartmann number
, Hall current parameter m, stretching parameter
and
and Reynolds number on Skin friction coefficients for MWCNTs and SWCNTs at both disks is portrayed in
Table 4. Skin coefficient friction decrease by increasing the value of Hall current parameter m and Hartmann number
at lower and upper disk for both MWCNTs and SWCNTs, while increasing behavior for
and scaled stretching parameter
for disk at
and stretching parameter
for the disk at
for both MWCNTs and SWCNTs.
Table 5 is erected to depict the impact of numerous parameters on heat transfer rate. It is gathered that rate of heat transfer is a decreasing function of unsteadiness parameter and Prandtl number at lower disk for both MWCNTS and SWCNTs, while it is a decreasing function of Reynolds number at lower disk and increasing function of Reynolds number at upper disk for both MWCNTs and SWCNTs.