Insight into the dynamics of fluid conveying tiny particles over a rotating surface subject to Cattaneo–Christov heat transfer, Coriolis force, and Arrhenius activation energy

https://doi.org/10.1016/j.camwa.2021.04.006Get rights and content

Highlights

Abstract

This article addressees the dynamics of fluid conveying tinny particles and Coriolis force effects on transient rotational flow toward a continuously stretching sheet. Tiny particles are considered due to their unusual characteristics like extraordinary thermal conductivity, which are significant in advanced nanotechnology, heat exchangers, material sciences, and electronics. The main objective of this comprehensive study is the enhancement of heat transportation. The governing equations in three dimensional form are transmuted in to dimensionless two-dimensional form with implementation of suitable scaling transformations. The variational finite element procedure is harnessed and coded in Matlab script to obtain numerical solution of the coupled non-linear partial differential problem. It is observed that higher inputs of the parameters for magnetic force and rotational fluid cause to slow the primary as well as secondary velocities, but the thermophoresis and Brownian motion raise the temperature. However, thermal relaxation parameter reduces the nanofluid temperature. The velocities for viscosity constant case are faster than that for the variable viscosity, but temperature and species concentration depict opposite behavior.

Introduction

Historically, the literature reveals that Roman Poet Titus Lucretius Carus who lived between 99BC and 55BC discovered the motion of tiny particles in a sunbeam. Lucreci et al. [1] identified the mingling motion of dust particles in sunbeams, which cause to a significant difference in temperature or/and pressure (i.e. air currents). The Dutch biologist, chemist, and physiologist Jan Ingenhousz observed irregular motion of tiny particles of coal dust on the surface of alcohol in the year 1785. They started the broad work on the photosynthesis. Further, in the year 1827, the Scottish botanist Robert Brown observed the random motion of tiny particles suspended in gas or liquid. Einstein [2] explained the Brownian motion with the help of kinetic theory. In a note on the presence of rotational movement over a sphere surface, Aydiner [3] commented that all the parts of the rotational relaxation during the process are exponential in nature when the influence of the inertia of the dipoles is insignificant. Tsekov and Lensen [4] once remarked that the migration of living cells obeys the laws of Brownian motion, which is a universal movement of matter. Furthermore, Maxwell [5] proposes the innovative idea of adding solid particles to heat transfer fluids to rise their thermal conductivity thermodynamic parameter further discoveries that are put forth by Choi and Eastman [6] in 1995. Nanofluids are essentially a combination of nano-sized objects contained in a fluid named a base fluid that increases the thermal characteristics due to the collaboration of these nanomaterials. Nanofluids can be used to cool the engines of automobiles, air conditioners, high-flux devices, washing machines, high-power microwave ovens, heavy-power laser diode arrays, and numerous welding systems. In addition, major advances in nano engineering have opened up the possibility of using magnetized nanomaterials to treat brain tumors, pharmacological treatments, artificial heart surgery, artificial lungs, cancer therapy, etc. Advanced nanotechnology has proposed some useful tools focused on the interaction of nanomaterials in order to boost fossil fuel consumption and mitigate environmental crises. Buongiorno [7] examines the two unusual slip phenomena specifically, Brownian motion and thermophoresis influence for enhancement of the convective rate of thermal energy transmission. Khan and Pop [8] provided nanoliquids flow through the Buongiorno method, which had passed over the stretching surface. Bhatti and Rashidi [9] investigated the Brownian process and heat diffusion in the distribution of tangent hyperbolic nanofluids submerged in the porous medium.

Due to Earth's rotation, every body on the earth experiences Coriolis force. Therefore, the examination of the heat transport fluid flow problems in the rotating frame is considerable importance due to its universal presence in natural phenomena and wide range of industrial applications such as oceanography, astrophysics, rotating machinery, gas turbine rotors, thermal power stations, food handling, rotating MHD generators, rotating drum separators for liquid metal, and petroleum industry. While the influence of Coriolis forces upon turbidity currents is acknowledged in several reviews [10], [11], [12]. The principal endeavor toward this path was made by Wang [13]. He utilized perturbation approach to present series solutions of the developed nonlinear problem. Mustafa et al. [14] analyzed the rotation in Maxwell fluid which is caused by the movement of the surface. The impact of magnetohydrodynamics (MHD) in rotating liquid is studied by Takhar et al. [15]. Turkyilmazoglu [16] studied the axisymmetric motion of a fluid flow around the sphere rotation. The influence of dimensionless rotating parameter on 3D rotating flow considered by Abbas et al. [17] and Butt et al. [18] examined the unsteady Casson flow in the rotatory frame incited by the deforming sheet. Recently, published research articles on rotating flow are mentioned in References [19], [20], [21], [22].

Continuous development in present time has inspired researchers in the examination of the exchange of heat system. Thusly, the assessment of the movement of heat characteristics in various fundamental conditions has expanded physical significance as a result of their noteworthy beneficial in energy creation, cooling of nuclear reactors, biomedical advancements, etc. In different related situation of heat transfer process, perhaps the best rule is classic Fourier law which utilized in classical physics science [23]. One of its rule flaws of parabolic equation of heat is that particular unsettling influence and at the same time the idea of determinism is challenged over the whole entire medium. The Cattaneo's thermal relaxation is the adjusted type of Fourier's thermal Conduction Law [24]. Hyperbolic structure energy function occurs within the sight of Cattaneo articulation. In the presence of Oldroyd's supper-convected model, Christov [25] extended Cattaneo law to achieve the product invariance of the framework by adding relaxation time. In this way another heat flux model called Cattaneo–Christov was presented. Hayat et al. [26] worked on the transfer of heat in the out floe of the stagnation point which depends upon the heat flux of the Cattaneo-Christov. More work on Cattaneo-Christov theory is carried out [27], [28], [29].

The procedure of chemically reacting systems involving the chemical reactions and Arrhenius activation energy has been given a great deal of consideration because of its different applications in cooling of an atomic reactor, chemical engineering, geothermal repositories, and recovery of warm oil. Also, activation energy can be characterized as the least required energy that reactants must secure before a compound reaction can happen. The theoretical meaning and application of Arrhenius activation energy described by Menzinger [30]. Bestman [31] first considered the natural convection of the binary mixture in a permeable porous space along chemical reaction. Further, Bestman [32] investigated heat transfer rate in fluid flow utilization the Arrhenius activation energy. Recently, Zeeshan et al. [33] explored the energy activation in Couette-Poiseuille flow in a horizontal channel. More work on activation energy is carried out [34], [35], [36].

In the previously mentioned literature, less consideration is paid towards the dynamics of fluid conveying tiny particles over a rotating surface subject to Cattaneo–Christov heat transfer, variable viscosity, Coriolis force, and Arrhenius activation energy. The novelties of the current study have been a focus on six perspectives. Firstly, to address the heat and mass transportation of nanofluid flow. Secondly, to analyze the impact of the variable viscosity. Thirdly, to investigate the Coriolis force impacts on dynamics of the fluid. Fourthly, to study the effect of chemical reaction along with activation energy. Fifthly, the temperature distribution is associated with the Cattaneo-Christov heat flux model. Six, the finite element approach for this elaborated problem. It solves the boundary value problems adequately, rapidly, and precisely [37]. As far as the authors come to know, these aspects of the problem not considered in the existing studies. This comprehensive study's main objective is the enhancement of heat transportation with consideration of variable viscosity and Cattaneo-Christov thermal diffusion. A three-dimensional geometrical problem, which converted into two-dimensional transformation. The obtained non-linear partial differential formulation is solved using finite element discretization. The variational Glerikan procedure is coded and simulated in the Matlab environment. The validation of the numerical results verified in the face of previously available data for limiting cases. Furthermore, pictorial representations of some principal findings with a detailed discussion are also present.

This investigation provides scientific answers to following related research questions:

  • 1.

    What is the effect of Lorentz and Coriolis forces on the dynamics of fluid velocity, temperature, and nanoparticles volume fraction when the fluid viscosity is constant and variable?

  • 2.

    To observe the effect of Lorentz and Coriolis forces on the skin friction factor, mass transfer rate, and heat transfer rate when the fluid viscosity is constant and variable?

  • 3.

    How does thermal relaxation, Brownian motion, and thermophoresis of tiny particles affect the heat transfer?

  • 4.

    Explore the combine influence of thermophoresis and Brownian motion on the heat and mass transfer rate when the fluid viscosity is constant and variable?

  • 5.

    Assess the effect of chemical reaction, Lewis number, and activation energy on characteristics of nanoparticles concentration in fluid?

Section snippets

Physical model and mathematical formulation

Unsteady transient three-dimensional MHD viscous and an incompressible nanofluid flow over an extending sheet along with a rotating frame are considered. Physically, we assume that the whole framework is at rest in the time t<0; however, for t0, the sheet is stretched along x-direction at z1=0 with angular velocity (Ω1). The concentration distribution is created through a species type that incorporates the chemical reactions and Arrhenius activation energy. The mass and heat transfer component

Finite-element method solutions

The set of PDEs (partial differential equations) (14)–(17) cannot be solved analytically due to highly non-linearity. The transformed set of non-linear PDEs (14)–(17) is solved numerically utilizing the variational finite element method along with boundary conditions (Eq.18). This technique is an excellent numerical computational strategy valuable to solve the different real word and engineering analysis problems such as, fluids with heat transfer, Structural Engineering, Bio-materials,

Results and discussion

This segment provides some significant outcomes through solution of the boundary value problem as finally constituted in Eqs. (14) to (18). A variational Galerkin technique is employed with finite element discretization. An extensive computational proceeding is performed to perceive the responses of velocity components (f(ξ,η), h(ξ,η)), temperature θ(ξ,η) and concentration ϕ(ξ,η) with the varying inputs of influential parameters. Moreover, the results for Nusselt number as well as the

Conclusions

This theoretical and computational work addresses the time-dependent three-dimensional MHD rotational flow of nanofluids across a stretching sheet with Cattaneo- Christov heat flux, chemical reaction, variable viscosity, and activation energy. The transformed two-dimensional partial differential formulation is solved by variational Galerkin procedure for two viscosity cases (constant and variable). The influential parameters are varied to conceive their impacts on velocities, coefficients of

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