Abstract

In this study, we aim to deal with the flow behavior betwixt a pair of rotating circular plates filled with Carreau fluid under the suspension of nanoparticles and motile gyrotactic microorganisms in the presence of generalized magnetic Reynolds number. The activation energy is also contemplated with the nanoparticle concentration equation. The appropriate similarity transformations are used to formulate the proposed mathematical modeling in the three dimensions. The outcomes of the torque on both plates, i.e., the fix and the moving plate, are also contemplated. A well-known differential transform method (DTM) with a combination of Padé approximation will be implemented to get solutions to the coupled nonlinear ordinary differential equations (ODEs). The impact of different nondimensional physical aspects on velocity profile, temperature, concentration, and motile gyrotactic microorganism functions is discussed. The shear-thinning fluid viscosity decreases with shear strain due to its high velocity compared to the Newtonian and shear-thickening case. The impact of Carreau fluid velocity for shear-thinning , Newtonian case , and shear-thickening cases on axial velocity distribution has been discussed in tabular form and graphical figures. For the validation of the current methodology, a comparison is made between DTM-Padé and the numerical shooting scheme.

1. Introduction

A nanofluid is a substance that contains nanosized particles suspended in the base fluid. These nanoparticles are usually made of oxides, metals, carbon nanotubes, or carbides, while base fluids are often taken as water, oil, and ethylene glycol. The potential use of nanofluids can be found in many applications in industry, i.e., heat transfer, especially in pharmaceutical processes, microelectronics, engine cooling (vehicle heat management), hybrid-power engines, chiller, fuel cells, heat exchanger, grinding, and boiler flue gas temperature reduction. The non-Newtonian fluids have significant applications in science and engineering, such as coating of wire and blade, textile dying, papers, manufacturing of plastics, the flow of biological liquids, and food processing. In industrial applications, the flow of non-Newtonian viscoelastic Carreau nanofluids is essential to increase energy performance. For example, it is used to release polymer sheets from the die or in plastic film drawing. Non-Newtonian fluids have drawn considerable interest among researchers and scientists due to their broad applications. Examples include apple sauce, chyme, photographic emulsion, dirt, soaps, blood, and shampoos at low shear stress that may be noted as non-Newtonian fluids. In the chemical engineering industry, the viscosity of fluid plays a significant role. Viscosity is dependent on the shear rate in the case of generalized Newtonian fluids. The idea of the generalized non-Newtonian fluid was introduced by Bird et al. [1]. Fluid flow past a rigid surface has also been investigated, and literature suggests that the microlevel surface forces are essential and that fluid layering increases viscosity. They help to increase the viscosity of fluid due to fluid coating. In the power-law model, the limitation is that when the shear rate is extremely low or high, the viscosity is not adequately addressed. To tackle this problem, the model of the Carreau fluid model is presented [2]. The Carreau fluid model gains the attention of many researchers for several years because of their essential characteristics. The flow on the magnetized permeable shrinking sheet and radiation due to heat was discussed by Yahya et al. [3]. Eid et al. [4] analyzed flow over a nonlinear permeable stretching sheet due to Carreau fluid under the influence of chemical reaction. Santoshi et al. [5] numerically studied the Carreau nanofluids in three dimensions on a stretching sheet in addition to considering mass slip and nonlinear thermal radiation. The effects of the internal energy in the porous von Karman model in steady electrical Carreau fluid under ohmic heating and transverse magnetohydrodynamics were studied by Khan et al. [6]. They considered a cylindrical coordinate system, and similarity transformations are applied to obtain the ordinary differential equation for the proposed problem and implement a well-known shooting scheme to achieve the results. Bilal et al. [7] investigated flow features of Carreau fluid by conferring the stimulating traits of thermal stratification. Appropriate use of MHD and infinite shear rate viscosity flow equation are modeled. They concluded that the thermal stratification characterizes fluid flow’s thermal distribution, and Carreau fluid acts in a reverse direction for shear-thickening and shear-thinning liquids. Khan et al. [8] examined the mass and heat transfer for convection in non-Newtonian Carreau nanofluid on a cylinder in the presence of temperature-dependent thermal conductivity. They considered a well-known model, i.e., Buongiorno’s model, which contains the Brownian and thermophoresis parameters. The key finding of this study was that the temperature profile and concentration of nanoparticle were increasing functions for the thermophoresis parameter in shear-thickening and shear-thinning fluids. Some recent critical studies related to the current topic are given in [9–13].

In previous decades, the study of magnet fields in fluid flow grabbed substantial attention because magnetohydrodynamics (MHD) is frequently used in many areas such as crystal growth process, pumping, agriculture, and polymer industry. MHD was recently identified as very useful in biotechnology as it is used in multiple testing processes for diseases. Recently, Lu et al. [14] studied mathematical models for the axisymmetric steady magnetohydrodynamic flow of Carreau nanofluids across radially stretched surfaces under nonlinear radiation of heat and chemical reaction. The additional feature of the problem, which makes it unique, is the generation/absorption of heat connected with new applied zero mass flux conditions. The flow due to boundary conditions due to convection with Carreau nanofluid with a magnetic field is studied by Wakif et al. [15] in addition to jump and slip conditions on a stretching cylinder. Khan et al. [16, 17] examined the flow induced by non-Newtonian Carreau fluid on a stretching cylinder with a magnetic field.

Further, the flow on a stretching cylinder affected by homogenous and heterogeneous conditions was examined by them and applied to convective boundary conditions numerically. This study mainly aims at the direct influence of homogenous and different reactions of Carreau fluid on a stretching cylinder with a magnetic field. A practical method for two-dimensional Carreau nanofluid for a nonsimilar solution with a magnetic field (applied) and mixed convection was analyzed by Sardar et al. [18]. They showed that increasing the buoyancy parameter boosts both skin friction and Nusselt number. In the presence of infinite shear rate viscosity, the stagnation point and the MHD flow of a Carreau fluid are also detected. Salahuddin et al. [19] examined the generalized slip effects of the magnetic field on Carreau nanofluid for a linear stretching cylinder with reactive species. Bhatti et al. [20] investigated the peristaltic motion of small particles suspended in a Carreau nanofluid with constant density. Laminar flow in two-dimensional past a stretching cylinder covered with the porous surface was studied with the effects of the magnetic field by Bovand et al. [21]. This article presented the findings of a numerical study of the circulatory cylindrical fluid flow under the influence of magnetohydrodynamics. Another numerical analysis is made by Amanulla et al. [15]. They studied the two-dimensional steady convective boundary layer flow over the surface of an isothermal sphere with the radial magnetic field and slip conditions. They perceived that growth in momentum slip parameters allows the skin friction coefficient to be reduced. In contrast, the local Nusselt number declined as the Carreau fluid parameter increases. Rudraswamy et al. [22] examined the Carreau fluid flow in a two-dimensional stretchable surface with the magnetic field. They concluded with a statement that when Brownian motion parameter and thermophoretic parameters are enhanced, the temperature of the fluid increases. Kumar et al. [23] investigated the Cattaneo–Christov heat flux model with Carreau fluid under the magnetic field on a varying thickness of the melting surface. Akbar et al. [24] investigated the flow of stagnation point for Carreau fluid on a wrinkled sheet in two dimensions. They presented a dual solution for the problem under consideration, and the graphs for various parameters in the equations are drawn. Shit et al. [25] investigated the nanofluid flow with MHD and thermal radiation. They studied the system of energy efficiency through the Bejan number.

Due to the complicated nature of the chemical reactions system, restricting to binary type alone becomes more comfortable and straightforward. A chemical reaction requires a quantity of activation energy to start. In the Arrhenius equation, a reaction’s activation energy is obtained by describing constant rate changes with the temperature. A chemical reaction is a chemical change in which reactants with varying properties may yield one or more products. Several industries need chemical reactions as an essential phase in the manufacturing process. Traditionally these types of responses occur in chemical reactors and are generally constrained to mass transfer. The response is sufficient when reagent volume and energy inputs enhance, and waste is reduced; then the optimum product can be obtained. Bestman [26] applied the model of binary reaction in the Arrhenius equation to produce a chemical reaction. The effects of energy of activation and binary chemical change on a two-dimensional radiative magnetohydrodynamics boundary layer flow for nanofluid on a vertical plate were discussed by Anuradha and Yegammai [27]. They scrutinized that the temperature distribution was accelerated, and the nanoparticle concentration profile decelerates under the effects of the heat generation, viscous dissipation, and MHD. Irfan et al. [28] discussed the Carreau nanofluid time-dependent flow for Arrhenius activation energy by using properties of binary chemical reaction with mixed convection. Khan et al. [29] explored the incompressible flow past a stretchable sheet of the Carreau–Yasuda model. Kumar et al. [30] analyzed the effects of nonlinear radiation with heat transfer attributes and activation energy. Khan et al. [29] reported a new nanofluid relation which examines the characteristics of the energy of activation with mixed convection Carreau nanofluid. They studied radiation and magnetic field parameters on both the entropy generation and the Bejan number.

Bioconvection flow is the flow of macroscopic convection of the fluid due to the density gradient generated by the collective swimming of motile microorganisms. Due to the swimming of these motile microorganisms in a particular direction, the density of the fluid increases, which results in the phenomenon of bioconvection. Slip effects and the movement of motile gyrotactic microorganisms on an exponentially stretching sheet were examined by Nayak at el. [31]. They discussed the temperature profile, velocity profile, nanoparticle concentration profile, and gyrotactic microorganism profile associated with different physical parameters. The thermal radiation effects with activation energy over a stretching cylinder with Oldroyd-B fluid due to the flow of microorganisms under the impact of the magnetic field were examined by Tlili et al. [32]. The thermal developmental nanofluid flow propagating on an extended surface was analyzed by Abdelmalek et al. [33] with additional activation energy characteristics, second-order, and viscous dissipation slip. Atif et al. [34] inspected the flow of the stratified micropolar MHD fluid with nanoparticles that contains gyrotactic microorganisms. The process of ohmic heating and radiated heat has also been taken into consideration. Khan et al. [35] determined the suspension of nanoparticles in Sisko nanofluid, utilizing the idea of microorganisms with activation energy and chemical reaction. Bhatti et al. [36] demonstrated the movement of motile gyrotactic microorganisms past a stretched surface filled with nanofluid. They applied the Successive Local Linearization Method (SLLM) for solving the modeled formulation and showed that the current method is stable and gives excellent results compared with similar methods. Nagendramma et al. [37] developed a mathematical model for Casson fluid through a slandering sheet in the existence of microorganisms for Cattaneo–Christov thermal flux and nonuniform heat source/sink. References [38–42] depict the flow in different geometries with different fluid models.

After investigating the above literature, it is found that no efforts are devoted to determining the analysis of activation energy for the Carreau non-Newtonian nanofluid with gyrotactic microorganism through a pair of rotating circular disks with generalized magnetic Reynolds number. In the present formulation, we considered the induced magnetic field in the axial and tangential direction. The Carreau nanofluid with gyrotactic microorganisms is suspended betwixt the set of rotating circular disks. The effects of activation energy are also contemplated with nanoparticle concentration. A well-known differential transform method (DTM) with Padé approximant is employed to obtain the solutions of the given coupled highly nonlinear ordinary differential equations. The effects of all the parameters associated with velocity profile, temperature profile, induced magnetic field, nanoparticle concentration, and motile microorganism functions are discussed in detail via graphs and tables.

2. Mathematical and Physical Modeling

Let us contemplate the flow in three-dimensional axisymmetry in a squeezed lubricant film of a Carreau nanofluid betwixt a set of two circular rotating parallel plates with finite length. The coordinate system is chosen as cylindrical polar axis with the associated velocity field . The heights of two parallel rotating circular plates are considered and taken as at time . The lower disk is stationary and upper disk is moving towards fix disk. The velocity of upper disk is denoted by . The axis indicates the symmetry axis in which the plates are rotating. The external magnetic field on the moving plate is applied with axial and azimuthal components, i.e.,

in which , are the dimensionless quantities which make dimensionless and , are the magnetic permeability’s of the squeeze film and the medium external to the disks, respectively. For the liquid metals, where denotes the permeability of free space. on the lower plate is assumed to be zero here. In present investigation, induced magnetic field having the component is generated by the magnetic field (applied) in a squeezed film between the plates. Figure 1 shows the geometrical coordinates. The upper and lower plate are held at fixed temperature and concentration . The fluid is electrically conducting under the suspension of nanoparticles and gyrotactic microorganisms.

Figure 1 

The schematic diagram of nanoparticles between parallel finite plates in the presence of microorganisms.

2.1. Rheological Model for Carreau Fluid

The Carreau fluid model is defined as [43]

in which represents the stress tensor, represents the zero shear rate viscosity, represents the limiting constant viscosity at infinite shear rate viscosity, represents the constant of time, represents the power law index, represents the first Rivlin–Ericksen tensor, and represents the second invariant rate of strain tensor which is defined as

Assume that is zero, and by using binomial series approximation of first order of (2), we have

The Carreau fluid model in component form is given in Appendix.

2.2. Problem Formulation

The governing flow equations by taking the above assumptions, the MHD squeezed film regime for Carreau fluid model, the equation of continuity, and momentum in direction can be read aswhere , , are denoted by fluid density, viscosity, stress tensor, and pressure, respectively. The equation of magnetic field can be read aswhere represents the electrical conductivity.

The energy equation for the proposed problem is read aswhere , , , , are the temperature, concentration, mean fluid temperature, Brownian diffusivity, the proportion of the effected heat capacitance of the nanoparticle to the base fluid, and thermophoretic diffusion coefficient, respectively, where is the thermal conductivity and is the heat capacity of the nanofluid.

The concentration of nanoparticle equation with activation energy readswhere , , , and represent the Boltzmann constant, reaction rate, rate constant, and activation energy, respectively.

The conservation of microorganism readswhere chemotaxis constant is combined with maximal speed of cell swimming and is denoted by ( is considered as a constant) and diffusivity of microorganisms is denoted by .

The initial and boundary conditions for (5)–(15) with our assumptions are the same as considered in [44]

2.3. Similarity Transformations

The set of similarity transformations are introduced as

Now substituting the abovementioned similarity transformation in (5)–(15), the following highly coupled nonlinear ordinary differential equations (ODEs) with unit-spaced variable are obtained aswhere squeezed Reynolds number is represented by , rotational Reynolds number is denoted by , magnetic field strength in axial and azimuthal direction is represented by , respectively, magnetic Reynolds number is represented by , Weissenberg number is represented by , Brownian motion parameter is represented by , thermophoresis parameter is represented by , Prandtl number is represented by , Schmidt number is represented by , chemical reaction parameter is represented by , temperature difference is represented by , bioconvection Schmidt number is represented by , Peclet number is represented by , and represents constant number. They are defined aswhere is the Batchelor number.

The corresponding boundary mentioned in (16) and (17) is reduced aswhere represents the axial and represents the tangential velocity, represents the axial and represents the tangential induced magnetic field components, represents the temperature profile, represents nanoparticles concentration, represents the motile density microorganism profile, and denotes the angular velocity and the range of the velocity betwixt the rotating plates is , which is useful to analyze various flows characteristics of rotating plates which revolve in the same or opposite directions.

The dimensionless torque can be measured on the moving disk by

The radius of the disk is denoted by .

Using (18) in (28), it becomeswhere the nondimensional torque on moving disk by fluid is denoted by , and gradient of the tangential velocity on the moving disk is .

Similarly, the torque in dimensionless form on the lower (fixed) plate is obtained by the same calculation at ; it becomes

3. Solution of the Problem: Differential Transform Method (DTM)

The nonlinear dimensionless ordinary differential equations (19)–(25) with boundary conditions (28) are elucidated with DTM. The DTM produces an analytical result based on Taylor series expansion in polynomial form. The differential transform method can easily be applied to linear or nonlinear problems and reduces the size of computational work. With this method, exact solutions may be obtained without cumbersome work, and it is a useful tool for analytical and numerical solutions. In the past decades, DTM has been successfully applied in many models of fluid dynamics, nanofluid dynamics in biotechnology, heat transfer, burgers equations, applications to nonlinear oscillators, plane Couette fluid flow problem, free vibration analysis, micropolar fluid flow, and non-Newtonian nanofluids flow analysis [45–48]. The proposed Padé approximation helps to enhance the convergence rate of the solutions of the truncated series. The DTM solutions can not satisfy the given boundary conditions at infinity without the use of Padé approximations. It is, therefore, essential to use DTM-Padé to afford an effective way to deal with infinite boundary value problems. Mathematica (12v) software has been used to obtain approximations. The complete procedure of this method is described by Zhang et al. [49]. Taking differential transform of each term of (19)–(25), the following transformations are obtained:where the transformed functions of are , respectively, and are expressed as