Abstract
For innovations in manufacturing and engineering scientific fields, the devices (electrical and computer systems) with large thermal effectiveness are needed. As a result, their thermal efficiency has become a very hot problem for many canvassers. With the novelty of this analysis, a mathematical study is performed to estimate the Darcy-Forchheimer flow of viscous magnetized fluid with Arrhenius activation energy and bioconvection effects through a variable thick surface of a rotating disk. The impact of thermal conductivity, heat source, and nonlinear thermal radiation is considered. The higher-order velocity slip impacts are also scrutinized. The system of partial differential equations (PDEs) and specific boundary restrictions is altered into a system of ODEs by adopting the suitable similarity transformations. The reduced ODE’s system is tackled with the aid of shooting scheme under (bvp4c) built-in tool commercial software MATLAB. Moreover, the effects of different parameters over velocity components, thermal conductivity, concentration, and microorganism’s fields are also examined. The confirmation of our findings is also explained through tables and graphical results. The results revealed that the radial velocity increases with the growing estimations of mixed convection parameter. The second-order velocity slip in radial direction causes a decrement in the estimation of axial velocity. Temperature distribution increases with a larger temperature ratio parameter. The concentration field of species and microorganism profile is reduced via a Brownian motion parameter and Peclet number, respectively.
1. Introduction
1.1. Literature Survey
Nanofluid is a fluid constructed with nanosized (1 to 100 nm) materials or molecules, named “nanomaterials” and otherwise “nanoparticles.” Such substances are designed colloidal suspension of nanopowders in a continuous phase fluid. Nanoparticles utilized in nanofluids are constructed with oxidations, metals, and carbon nanotubes including carbides. Continuous phase fluids include ethanol glycol oil as well as water. Nanofluids have novel characteristics that build them potentially helpful in a broad range of heat transformation applications such as fuel cells, microprocessors, hybrid powered engines, pharmacological mechanisms, cooling equipment, chillers, refrigerators, and heat radiators in grinding, among others. Nanofluids are combinations of nanoparticles and the base fluids that can generate numerous heterogeneous nanofluids that are described for their thermophysical characteristics (thermal diffusivity and thermal conductivity, as well as viscosity) as a cooling system in heat transmission in comparison to base fluid, which increased with growing volumetric fraction of nanomaterials. Choi and Eastman [1] were firstly introduced to the basic idea of nanoparticles in constant phase fluids. Buongiorno [2] addressed nanoparticle study by developing a model to evaluate the thermal properties of continuous phase fluids. He predicted that the increased conductance of constant phase fluid would be caused by the tiny-sized and lower volume fraction of supporting nanoelements. Nanotechnology is extremely important in numerous fields including chemical as well as metallurgical operating systems, transportation, macroscopic objects, cancer treatment, and electricity generation. Eid and Mabood [3] reported the suspending magnetohydrodynamic (MHD) flowing of micropolar dusty nanoparticles impinging on a permeability expanding sheet. Umar et al. [4] studied numerically the 3-dimensional incompressible Eyring-Powell nanofluid flow across a stretched surface including velocity slip as well as activation energy. Muhammad et al. [5] explored numerical simulations for 3-D Eyring–Powell (EP) nanoliquid under the nonlinear thermal radiation with changed heat and solutal fluxes. Rasool et al. [6] researched the Marangoni convective Casson-type flow of nanofluid impacted by the existence of Lorentz forces introduced into the modeling by an organized arrangement of magnets in the shape of the Riga configuration. Mahanthesh et al. [7] described the effect of quadratic thermal radiation as well as convection on the boundary layer 2-phase flowing of a dusty flow of nanoliquid through a vertical surface. Aaiza et al. [8] evaluate the magneto-nanoliquid flow across a channel in the presence of different nanoparticle shape factors. Hussanan et al. [9] investigate the thermal transportation phenomenon in micropolar water-based fluid. Khalid et al. [10] scrutinized the exact solution of nanofluid flow. Ali et al. [11] discussed the Brinkman-type nanofluid flow through vertical surface. Zin et al. [12] introduced the free convection behavior in Jeffery fluid through vertical porous surface.
The word activation energy plays a significant role in chemical reactions. The scientist Svante Arrhenius was the principal who initiated the description on activation energy in 1889 and that it is the minimum amount of energy required to initiate chemical reactions to a state in which they could experience material change. The application includes compound construction, food processing, transportation structures, a geothermal stored, and businesses. Heat and mass distribution aspects manage various mechanisms that were reacting synthetically which include the species composite reactions as well as codification strength having application in oil production and geothermal configuration. In actuality, it is the least amount of energy required to transform the reactants into substances. Activation energy can take the shape of kinetic and potential energy. It is essential to generate theoretical observations in addition to experimental contributions to measure the flow influence of activation energy. There have been very several theoretical efforts on this discussion. Indeed, the relationship between mass transfer and chemical reaction becomes tiresome. Such interactions can be observed both within the fluid and in mass transmission during the production process and employ of reactants at various levels. Bestman [13] clarified that viscoelastic fluid flow of thermal and mass moved in which restricts divider movements in its own plane utilizing a simplistic scientific theory of twofold material reaction with Arrhenius actuation energy. In this inspection, he obtained an analytical curriculum of action for the problem by employing an irritation method. Guo et al. [14] examined the kinetic model and thermodynamic of oxidant pyrolysis of microalgae wastes using a double distribution activation energy concept and simulating annealing. Hayat et al. [15] evaluated the effect of activation energy on entropy generation (EG) in a 3-dimensional magnetohydrodynamic (MHD) rotational flowing of nanofluids containing a binary chemical process. Araújo et al. [16] scrutinized the kinetic modeling and Arrhenius activation energy distributions in complicated systems with Hopfield Neural Network-based system. Elangovan and Natarajan [17] reported the primary treatment influences on qualitative characteristics, hydration diffusivity, and Arrhenius activation energy of solar drying gourd. The numerical analysis of unsteady Maxell nanofluid is simulated by Bilal et al. [18]. The effect of magneto nanofluid flow over disk with viscous dissipation is analyzed by Saeed et al. [19].
Microorganism molecules have been widely used in the production of manufacturing and industrial products such as ethanol, waste-derived biofuel, and fertilizers. They are also utilized in water therapeutic facilities. These microorganisms generate hydrogen gas and biofuel, a favorable dispatchable energy source. As a result, we must investigate the swimming structures and mass transmission properties of microorganisms in order to make their applications more successful, profitable, and widespread for the benefit of humanity. Bioconvection is the production of various types of irregular fluid structures at the microscopic level therefore of the unexpected swimming of self-propelled microorganisms found in water or those certain denser fluids. Natural hypotheses such as searching for nutrients, oxygen for breathing, and improving light absorption for photosynthesis influence the swimming of such microorganisms. Platt in 1961 [20] introduced the term “bioconvection” to illustrate the methodology of improvement of manners in depth suspensions of motile microorganisms at constant temperatures, in comparison to those reported under convectional conditions. Kuznetsov [21] proposed the concept of biothermal convection caused by temperature gradients as well as microorganism swimming. Tlili et al. [22] scrutinized the effect of bioconvection micropolar nanoliquid flow including gyrotactic motile microorganisms on thermal and solutal stratifications at the boundary layer. Al-Mubaddel et al. [23] examined Sisko-based nanofluid under the bioconvection radiation flow with specific thermal and solutal fluxes. Abbasi et al. [24] illustrated the flow of viscoelastic nanoparticles containing gyrotactic motile microorganisms across a rotating stretched disk under convective and zero mass flux conditions. Shehzad et al. [25] scrutinized the bioconvection of a Maxwell-based nanoliquid above an isolated rotational disk under the effect of double diffusional Cattaneo–Christov (C-C) concepts. Aziz et al. [26] evaluated the effects of motile microorganisms on unstable Williamson nanoliquid caused by a bidirectional accelerating surface. Alizadeh and Ganji [27] analyzed the two-phase thermosyphon utilizing RSM. The thermal transportation features and thermal resistance are discussed by Alizadeh and Ganji [28]. Some important research work about fluid flow and heat transfer can be studied in [29–31].
Taking the higher-order velocity slip consider, this communication extends the bioconvection nanofluid flow through a stretching disk. The Darcy-Forchheimer porous medium is considered. The heat transfer is incorporated in the presence of nonlinear thermal radiation, joule heating, thermophoresis, and Brownian diffusion. The significance of Arrhenius activation energy is considered. The dimensionless system is tackled by utilizing a shooting scheme with the bvp4c function of MATLAB. The effects of flow controlling parameters are analyzed.
2. Mathematical Description
2.1. Flow Analysis
Here, our main purpose is to scrutinize the steady three-dimensional, incompressible, and axially symmetric laminar flow of nanofluid including bioconvection with motile microorganism over a stretching disk surface. The schematic view of flow problem and the system of coordinates is depicted in Figure 1; here, the surface of the disk is stretchable through the rate of and moving along own axis under an angular frequency . Due to the stretchable and angular frequency as well as electrical conduction in the presence of electromagnetic field, the flow is produced. The suitable boundary conditions are applied at the disk surface. Here, no suction/injection is present at the disk surface, that is, . The velocity components denoted as are along the axis, respectively. The temperature of fluid , nanoparticle concentration of fluid , bioconvection fluid , ambient temperature , and ambient volumetric concentration as well as ambient swimming organisms are also presented at the surface of a stretching disk.
The governing partial deferential equations are expressed as [32, 33]
2.2. With Boundary Restrictions
when .
In the above governing equations, the velocity components are along directions, represent the kinematic viscosity; is the electrical conductivity of momentum; is the fluid density of microorganism; demonstrate the porous space permeability; is the magnetic field strength; illustrate the coefficient of nonuniform inertia; is the gravity; volume expansion coefficient is indicated by ; represents the microorganism diffusion parameter; exemplify the power law index; chemotaxis constant is expressed as ; the cell swimming speed is identified as ; display the drag coefficient; are the basic temperature, volumetric concentration, and swimming bioconvection of fluid; signify the thermal conductivity; represents specific heat capacity; the heat capacitance ratio is symbolized by ; represents thermophoresis diffusion coefficient; stand for ambient temperature; is the fitted rate constant; represent the Brownian motion coefficient; the dynamic viscosity is , are ambient concentration and ambient density microorganism, respectively; the rate of chemical reaction is ; are linear/nonlinear thermal-based and nanoparticle concentration expansions; the exponential function is ; the angular frequency is ; the activation energy is symbolized by ; the stretching constant is ; expressed the Stefan Boltzmann constant; and are the first- and second-order slip coefficients, respectively; and and are the heat mass transfer coefficient and the mass transfer coefficient, respectively.
2.3. Similarities
The similarity variables are
In above transformations, radius is ; the power law index is ; the thickness power law index is ; the components of velocity (radial, tangential, and axial) are symbolized by , , and ; and is the constant of nondimensional radius. The temperature variable conductivity is addressed as
2.4. Reduced Equations
The governing dimensionless equations after applying the suitable similarity transformation are expressed as with
The following are currently considered for introducing the innovative similarity transformation to modify the origin from to , that is,
We get through boundary conditions
2.5. Dimensionless Prominent Parameters
Now, the dimensionless prominent parameters are given in Table 1.
2.6. Physical Quantities
Here, the local skin friction coefficient (LSF) , local Nusselt number (LNN) , local Sherwood number (LSW) , and density motile microorganism (LMN) are represented as follows:
Local skin friction coefficient
Here, are expressed as
Here, is identified as
Finally, local skin friction coefficient
Here, denoted the shear stress along the radial direction, represent the shear stress along the direction of tangential, and denoted the shear stress.
Local Nusselt number
Here, is symbolized as
Finally, the local Nusselt number
Local Sherwood number
Here, is denoted as
Finally, the local Sherwood number
Density motile microorganism number
Here, is identified as
Finally, the density motile microorganism number
3. Numerical Scheme
Significantly, the method of finding the exact solution for momentum, temperature, nanoparticle concentration, and bioconvection equations through corresponding initial conditions is very complicated and doubtful about the significance of results. Researchers have attempted to mathematically research the nanoliquid flowing past the stretching disk. Chemical processes, including activation energy, are also used to research the characteristics of mass transformation. Choose the initial guesses, and the dimensionless highly linear governing equations ((20))–((25)) with the related boundary conditions ((26))-((27)) are numerically integrated by utilizing the computational software MATLAB through built-in rule bvp4c (shooting method). The bvp4c method is a powerful way of resolving an initial value problem and a well-known methodology to find more than one solution. In order to solve these equations, first, we converted the higher-order differential equation into a first-order system by using the following technique.
We identify new variables as
4. Result and Discussion
The aim of this portion is to envisage variations in velocity components, thermal field, concentration of nanoparticles, and microorganism profile due to interesting involved parameters introduced during the flow of bioconvective nanofluid that are demonstrated in Figures 2–10. Figure 2 explains the effects of second-order slip parameter and stretching rate to angular frequency. The escalating estimations of stretching rate to frequency exaggerate the axial velocity component. From this scenario, it can be detected that axial velocity of nanofluid decays via larger variations in second-order velocity slip parameter. Figure 3 communicates the impacts of second-order velocity slip as well as magnetic parameter versus tangential velocity. It can be observed that tangential velocity boosts up via larger second-order velocity slip parameter. Here, we also observe that the larger estimations in magnetic parameter diminishes the tangential velocity of nanofluid flow. In terms of physics, the magnetic parameter is associated to the Lorentz force, which is a resistive force to the fluid flow. As the magnetic parameter increases, the resistance forces increase, and the velocity decreases. Figure 4 demonstrates the features of mixed convection parameter as well as bioconvection Rayleigh number over a radial velocity component. It is noticed that velocity is improved by growing the estimations of mixed convection parameter while it depresses via a greater bioconvection Rayleigh number. The features of buoyancy ratio parameter and second-order velocity slip parameter versus radial component of velocity are elaborated through Figure 5. The radial velocity is a decreasing function of second-order velocity slip parameter and buoyancy ratio parameter. Physically, for a given buoyancy impact, bioconvection inhibits the up movement of solid particles that arise in nanofluid; however, for a higher buoyancy impact, the fluid resists the fluid, resulting in fluid movement decline.
Figure 6 examines the behavior of thermal conductivity and temperature ratio parameter via thermal distribution of nanomaterials. It is witnessed that temperature field upsurges due to an increment in thermal conductivity and temperature ratio parameter. Figure 7 portrays the impression of Prandtl number and thermal Biot number versus temperature field. It is mentioned that improving Prandtl number reduces the temperature distribution. From this communication, we analyzed that enhancing the thermal Biot number escalates thermal field. Physically, when the Prandtl number rises, the thermal diffusivity diminishes. Fluid temperature drops as a result of the lower thermal diffusivity. Figure 8 presents the trend of the Brownian motion coefficient and Prandtl number against solutal field. Here, concentration reduces via a greater amount of Brownian motion coefficient as well as Prandtl number. Micromixing and heat conduction in the nanofluid are aided by enhancing the Brownian motion parameter, causing the temperature to rise and the nanoparticles to scatter more widely.
Figure 9 shows the nature of bioconvection Lewis number and Peclet number over a microorganism’s field. It is concluded that the microorganism field diminishes by increasing the variations of bioconvection Lewis number and Peclet number. The bioconvection Lewis number has an converse relationship with microorganism diffusivity. As the bioconvection Lewis number rises, the diffusivity decreases, and the microorganism profile drops. The features of microorganism Biot number and bioconvection Rayleigh number are mentioned in Figure 10. It is noted that larger magnitudes of microorganism Biot number and bioconvection Rayleigh number increases the microorganism’s profile. Here, it was analyzed that good validation in results between published literature and current outcomes is presented in Table 2.
5. Conclusion
Computational analysis is conducted on bioconvective viscous nanofluid flow past a stretching disk with higher-order slips and nonlinear thermal radiation. The main outcomes are listed as follows: (i)Axial velocity component escalates versus a larger amount of stretching rate to angular frequency(ii)Tangential velocity is increases against a second-order slip parameter(iii)The increment in radial velocity has been analyzed along with increments in mixed convection parameter(iv)The larger slip parameter and buoyancy ratio parameter reduce the radial velocity of nanofluid(v)Thermal field of species rises against temperature-dependent thermal conductivity(vi)Temperature distribution is increased via the temperature ratio parameter(vii)Greater values of the thermal Biot number boost the temperature field(viii)Concentration is reduced against the Brownian motion parameter(ix)The microorganism’s profile boosts via the microorganism’s Biot number while it diminishes against the Peclet number(x)The microorganism’s field is depressed against the bioconvection Lewis number
Nomenclature
| : | Velocity components (m·s−1) |
| : | Coordinates of system (m) |
| : | Kinematic viscosity |
| : | Inertia coefficient |
| : | Porous permeability |
| : | Chemotaxis constant |
| : | Electrical conductivity |
| : | Gravitational acceleration |
| : | Volume expansion coefficient |
| : | Thermophoretic diffusion coefficient |
| : | Microorganism coefficient |
| : | Capacity ratio |
| : | Stefan Boltzmann number |
| : | Chemical reaction coefficient |
| : | Coefficient of activation energy |
| : | Ambient microorganisms |
| : | Cell swimming speed |
| : | Surface microorganisms |
| : | Chemical reaction parameter |
| : | Chemical reaction parameter |
| : | Darcy Forchheimer parameter |
| : | Mixed convection parameter |
| : | Magnetic parameter |
| : | Buoyancy ratio parameter |
| : | Bioconvection Rayleigh number |
| : | Bioconvection Lewis number |
| : | Temperature ratio parameter |
| : | Radial directional shear stress |
| : | Tangential directional shear stress |
| : | Peclet number |
| : | Microorganism difference parameter |
| : | Stretching ratio to angular frequency |
| : | Radial direction first-order velocity slip |
| : | LSN |
| : | LMN |
| : | Fluid density |
| : | Nanoparticle density |
| : | Microorganism density |
| : | Thermal conductivity |
| : | Specific heat |
| : | Heat capacity |
| : | Ambient temperature |
| : | Magnetic field strength (N·m−1·A−1) |
| : | Brownian diffusion coefficient |
| : | Mean absorption coefficient |
| : | Coefficient of velocity slips |
| : | Heat, mass, microorganism transfer coefficient |
| : | Brownian motion parameter |
| : | Ambient concentration |
| : | Surface temperature |
| : | Surface concentration |
| : | Reynolds number |
| : | Prandtl number |
| : | Thermal radiation parameter |
| : | Temperature difference parameter |
| : | Activation energy parameter |
| : | Thermophoresis parameter |
| : | Eckert number |
| : | Schmidt number |
| : | Radial direction second-order velocity slip |
| : | Tangential direction first-order velocity slip |
| : | Tangential direction second-order velocity slip |
| : | Thermal Biot number |
| : | Mass Biot number |
| : | Microorganism Biot number |
| : | Skin friction coefficient |
| : | LNN |
| : | Nusselt number |
| : | Shear stress |
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors proclaim that they have no competing interests.