Original articles
Characteristics of thermophoresis and Brownian motion on radiative reactive micropolar fluid flow towards continuously moving flat plate: HAM solution

https://doi.org/10.1016/j.matcom.2021.08.004Get rights and content

Highlights

  • Mathematical model developed for a micropolar fluid flow through moving plate.

  • Characteristics of thermophoresis and Brownian motion are included in the model.

  • Utilizing Homotopy Analysis Method solutions have been obtained.

  • Validation of present results is conducted with previous published results.

  • Rapidly convergent and accurate solutions are achieved with HAM.

Abstract

The present objective of this study is to develop a mathematical model and rheological aspects by combining the micropolar fluid model to simulate the reactive flow from a continuously moving flat plate. The current flow model is formulated with employment Thermophoretic diffusion, Brownian motion and chemically reactive species. The focal micropolar based flow equations are transmitted into a non-dimensional form via similarity transformation for which Homotopy Analysis Method (HAM) is employed to obtain approximate analytical solutions. The role of numerous parameters on dimensionless flow phenomena are observed in very effective way utilizing graphical presentation. Various order of approximations for the convergence of HAM has been tabulated. Proper validation is conducted to show a strong relationship between our results with related published ones in literature with some limiting conditions. Furthermore, streamlines and isotherms are also plotted to know the pattern of the fluid flow. More so, the analysis shows that enhancement in Thermal Grashof number, solutal Grashof number and microrotation parameter rises the fluid spin rotation. Temperatures are suppressed with higher thermophoresis and Brownian parameters. Concentrations are elevated with higher thermophoresis parameter. Finally, the observations revealed that the presence of thermophoresis and Brownian motion is more effective to improve heat transportation phenomenon.

Introduction

The majority of fundamental fluid mechanics problems widely found in nature are convection flow and also complexity arising from nonlinearity in model. The physical properties and complex rheological characteristics of non-Newtonian materials is completely explained by using constitutive relation like microstructural features which classical Navier–Stokes fluid dynamic model cannot be accurately described. Various non-Newtonian fluid models are presented earlier and from these, micropolar fluid is the classification of non-Newtonian fluid because capable of sustaining both translational and rotatory motions. Micropolar fluids have the ability to support stress moments and body couples and are affected by the spin inertia. The most impressive theories in modern fluid mechanics titled micropolar fluid theory introduced by Eringen [11]. This type of flow received considerable attention due to the potential applications such as lubricants, liquid crystals, polymeric suspensions and many possible areas. Being, a very fashionable subject, simultaneous effect of heat and mass transfer on micropolar flows in the flat surface attract many investigators. Latest developments in micropolar are impressed by the work undertaken by Rees and Pop [29]. Later a technical note is presented by implementing radiation effects by Rees [27]. The magneto-hydrodynamic study of micropolar fluid flow through continuously moving plate has been reported by Rahman and Sattar [25]. Utilizing both Newtonian and non-Newtonian cases numerical solutions obtained for nonlinear stretching sheet by Bhargava et al. [6]. Novel analytical approach (DTM-Pade) implemented to study the effect of thermal radiation on magneto-micropolar fluid through continuously moving plate by Rashidi and Erfani [28], the results confirm that method is effective and convenient especially for boundary layer flows. The investigation dealing with temperature dependent convective flow of micropolar fluid due vertical flat plate analyzed numerically by Ramreddy and Pradeepa [26]. Micropolar fluid model induced by oblique moving plate carrying multi-physical consequences has been studied by Shamshuddin et al. [32]. Convergent series solution is implemented to study effect of thermal radiation on Marangoni convection flow of carbon–water​ nanofluid by Hayat et al. [14]. Recently, radiative micropolar fluid transport in non-Darcy porous medium with different effects was analyzed by Ferdows et al. [13]. Theoretical investigation has been conducted on non-Newtonian magnetized micropolar gas flow by Anwar Beg et al. [4]. They verified their solutions with different methods and obtained excellent correlation. Analytical study is conducted on the flow of conducting micropolar fluid by Anathaswamy et al. [2].

Buongiorno [7] claims that out of several slip transport features, the thermophoresis and Brownian movement attain prime importance. Thermophoresis phenomena spotted owing to mounting of reaction of non-identical particles, this phenomenon is because relocation change from massive structure molecules to a comprehensive temperature slope. The Brownian moment situation spotted owing to persistence bombardment of the molecules in neighborhood medium. This kind of gesture is the result of the strike with accessible aeriform/liquid molecules. Hayat et al. [15] analytically reported accelerated radiative flow of micropolar nanofluid additionally contains thermophoresis and Brownian motion effectiveness. Patel and Singh [23] imposed Brownian motion and Thermophoretic consequences on micropolar fluid encountered in stretched porous surface. Sabir et al. [31] established numerical solution for a stretchable plate on steady micropolar fluid utilizing the effects of thermophoresis and Brownian moment, observing that within the fluid medium thermophoresis is behind for carrying the molecules. Species reaction is also of considerable significance in flowing fluid along with the combined impact of heat and mass transfer. Such studies have vast applications in many industrial processes and also in physiological flows. Reaction is said to be homogeneous when it occurs uniformly through a given phase. Otherwise, reaction is called heterogeneous. Again, chemical reaction is of first order if rate of reaction is directly proportional to the concentration. Many investigators predicted the interesting properties of chemical reactions on micropolar fluid under different circumstances are Chamber and Young [8] described the effects of homogeneous first order chemical reactions in the neighborhood of a flat plate for destructive and generative reactions. Khan et al. [18] reported the comparative study with the effects of homogeneous–heterogeneous reaction considering Cason fluid. Khan and Alzahrani [16] studied the impacts of activation energy and binary chemical reaction in stagnation point flow of Walter-B nanofluid numerically. Abbas et al.  [1] deliberated entropy optimized second order slip of a nanofluid assuming activation energy. Chaudhary and Jha [9] employed chemical reaction effects to micropolar fluid model to predict the heat transfer prospective in a vertical plate assuming slip flow regime. Sheri and Shamshuddin [35], [36] solved magnetohydrodynamic micropolar fluid flow utilizing finite element method with chemical reaction with an extension of Hall current. The interesting numerical simulations were based on Galerkin finite element method with excellent accuracy. A chain of research, Das [10], Shamshuddin et al. [34], Anjanna and Nagaraju [3], Fatunmbi and Agbolade [12], Tripathy et al. [37], Mishra et al. [20], Shamshuddin et al. [33] and Anwar Beg et al. [5] have designed models which are compelled by thermal distribution micropolar fluid of a chemical reaction. In these reported problems, the fluid reaction is formulated by considering destructive type of reactive species which has an application in combustion and identification of critical parameter value. Further studies of micropolar fluid dynamics includes [21], [22].

In the present analysis, the novelty of the flow model is formulated with employment Thermophoretic diffusion, Brownian motion in heat equation and chemically reactive species in mass equation in thermal radiative flow of micropolar fluid through flat plate which is continuously moving. Previous results obtained on the considered fluid have motivated the present study as result of their significant to the technology and industrial advancement. The reduced problem is solved adopting homotopy analysis method to achieve approximate analytical solutions. Convergence of the series solution up to 40th order of approximation is presented in table as convergence of HAM solution strongly depends upon the convergence control parameters, ħ curves also been represented graphically. The obtained results are in good agreement with previous results Wang [38], Mabood et al. [19], Khan and Pop [17] and Poornima and Reddy [24]. The present result clears a realistic fact that chemical reaction reduces the species concentration of the fluid flow. So, the importance dependence in various physiological functions is clearly shown.

Section snippets

Basic model

Two-dimensional mixed convection flow of micropolar fluid over a continuously moving semi-infinite flat plate is considered in the current analysis. In addition, the behavior of thermophoresis and Brownian motion are considered based on the flow model. The novel Thermal radiation–conduction, chemical reaction with thermophoresis and Brownian motion features are also taken into account. The chemical reaction features are homogeneous destructive first order. In case of mathematical modeling,the

Solution via HAM

The description of the HAM for the transformed equations (8)–(11) with Eq. (12) for the initial guess solutions for fη, hη, θη and ϕη are expressed as; f0(η)=1eη,h0(η)=0,θ0(η)=eη1+Bi,ϕ0(η)=eη,and the appropriate operators used are: Lf=d3fdη3dfdη,Lh=d2hdη2h,Lθ=d2θdη2θ,Lϕ=Lϕ=d2ϕdη2ϕ,with properties LfC1+C2eη+C3eη=0,LhC4eη+C5eη=0,LθC6eη+C7eη=0,LϕC8eη+C9eη=0,where Cii=19 are constants. Since the method is very familiar therefore the detail of involvement mathematical computations,

Convergence criterion of the methodology and results discussion

The controlling parameters for the criterion to converge the HAM on which the solution strongly depends are ħf,ħθ and ħϕ. The proper values of these parameters can be obtained by plotting their corresponding ħ-curves. The sketching of ħf,ħθ and ħϕ curves at the 20th order approximations as shown in Fig. 2, Fig. 3, Fig. 4. It is clearly noted from these Fig. 2, Fig. 3, Fig. 4 that the range of ħ is 1.5<ħf<0.5. However, the convergence for 40th order of approximations is displayed in Table 1.

Concluding remarks

The present investigation conveys the study of non-unidirectional flow of micropolar liquid with the existence of Thermophoresis and Brownian motion in flow with heat radiation and chemical reaction past continuously moving flat. Analytical solutions are provided for the simplified reduced core equations. The major results of the study are outlined as follows:

  • Rising G and Gm boosts the flow velocity and heat transport in the system due to heat reaction is stirred to propel fluid particles,

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