# Analytical Assessment of MHD Flow of Nanoliquid Subject to Thermal Radiation and Brownian Effect

Authors: Haroon Ur Rasheed; Ebenezer Bonyah; Soumaya Gouadria; Waris Khan; Ahmed Alshehri; Raees Khan
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

This manuscript studies the impact of the heat and mass flow and chemical reaction with electromagnetic field and heat flow of non-Newtonian Walter-B nanofluid via the uniform magnetic field. A mathematical model is used to simulate the arisen nonlinear partial differential equations (PDEs). By employing the suitable transformations, the system of PDEs is then transformed to a nonlinear system of ordinary differential equations (ODEs). The impact of the pertinent parameters on the velocity profile, energy, and concentration distribution has been discussed. These nonlinear coupled equations were addressed analytically by implementing an efficient and validated analytical method, where Mathematica 11.0 programming code is established for simulating the flow system. Stability and convergence analysis have been performed in order to improve the accuracy of the flow system. In order to gain physical insight, the effects of dimensionless parameters on flow fields are investigated. In addition, the impression of system parameters on skin-friction, heat transfer coefficient, and mass flow rate profiles is also debated.


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## Body

## 1. Introduction

Nanomaterials have been of great interest to engineers as well as to scientists during the previous decades. As a matter of fact, nanomaterials are known to enhance the thermal conductivity of base fluids. The first one who suggested the addition of carbon nanotubes and solid particles in base fluids was Maxwell [1]. The results achieved were motivational; however, some problems were raised such as pressure drop enhancement and pipe erosion. Then, Choi et al. [2] proposed the notion of carbon nanotubes and nanoscaled particles in base fluids and named them as nanofluids. Metals, such as Ag and Cu, and metal oxides, such as Al2O3 and CuO, are the most widely used nanoparticles. Improved nanofluids’ thermal conductivity has important applications not only in domestic heating but also in heat exchangers and cooling systems. Nanofluids, discovered by Choi [3], are colloids composed of nanoparticles and base fluid. Nanoparticles have thermal conductivity, typically greater in magnitude than base fluids and significantly smaller in size than 100 nm. The work of nanoparticles greatly improves the heat transfer efficiency of the base fluids. Basic fluids can be water, synthetic liquids, fats, lubricants and blood. Nanoparticles are synthetic materials with substantial use in biomedicine because of the special manner in which they interact with matter. Hybrid nanofluids are a particular type of nanofluid. Hybrid nanofluids are formed by the suspension in the base fluid of two or more kinds of nanoparticles with hybrid nanoparticles. Hybrid nanoparticle is a specific substance that integrates the physical and chemical characteristics of different materials at the same time and has been commonly used in the production of antitumor medicines. Some studies that discuss nanofluids and hybrid nanofluids are reported in the refs. [4–10]. Magnetic fluid flow due to the rotating body has potential applications in biomedical sciences, electronic devices, and aerodynamics [11]. Hafeez et al. investigated the flow of an Oldroyd-B fluid type’s magnetic fluid in a rotating system using modified Fourier’s law [12]. Reddy et al. reported the influence of hybrid nanoparticles in a swirling flow using an activation energy model [13]. Fluid flow investigation in biomicrofluidic systems is implemented by the electroosmosis process. The latter is the principal instrument for the stream activation in a wide scope of utilization. This electroosmotic transition takes place in such a way that, as long as the polar outer layer is connected to the electrolyte device, the counterparts of the electrolyte should be allowed to pass through the surface of the capacitor and ultimately to build a membrane with a high convergence of counteractors, which is commonly named after the Stern layer. In addition to the exterior diffuse coat, the Electric Double Layer (EDL) is produced in the area of the charged board. Applying the ambient electrical field to the electroosmotic flow (EOF), the functional particles in the dispersed area of the EOF are induced to travel and accomplish fluid displacement, which is typically known as the “EOF.” Propelled by the monumental use of the stream of the electroassimilation, various experiments are investigated [14–18]. Christopher et al. [19] discussed the chemical reaction consequence on the flow of hybrid nanoliquid on an SS with Cattaneo–Christov heat flux. Gowda et al. [20] examined the convective stream of second grade fluid on a coiled SS with Dufour and Soret effects. Alhadhrami et al. [21] pondered the LTNE impact on the flow of Casson liquid on an SS with a porous medium. Recently, Ali et al. [22–26] conferred the flow of different fluid past stretching surfaces with several influencing factors by considering different nanoparticles’ suspension. Another significant nonmechanical micropump is the electromagnetohydrodynamic (EMHD) which has different applications, some of which, fluid mixing and pumping along with flow control in microfluidic systems [27–29]. Lorentz force is generated because of an electric field force applied across the channel in the presence of a perpendicular magnetic field force. Reddy et al. reported the influence of hybrid nanoparticles in a swirling flow using an activation energy model [13]. Khan demonstrated the transportation of hybrid nanoparticles in convective flow due to a rotating plate [30]. Hayat et al. considered the viscous dissipation and Joule heating in the flow due to a rotating plate with variable thickness [31]. The following references [32–37] provide more about thermal radiation and activation energy.In this study, we explored the nature of stagnation point flow of nanoliquid driven by stretching surface. The role of Brownian motion and thermophoresis is also considered in modeling the flow system. The mechanisms of heat and mass flow are also are examined. The arisen nonlinear partial differential equations are altered to ODEs via transformations and then solved analytically [38–40]. Following are the key points of this investigation:
(i)
The main goal of this study is to examine the two-dimensional incompressible Walters-B nanofluid flow over a stretching sheet. Furthermore, the graphs are also used to discuss the variations in detailed profiles as a consequence of several dimensionless parameters(ii)
To study time subservient Walter-B fluid flow resulting from the impression of heat and mass transfer(iii)
Mathematical modeling of the fundamental flow equations comprises momentum, energy, and diffusion balances


## 2. Problem Formulation

Consider the magnetohydrodynamic MHD flow of a Walter-B non-Newtonian nanofluid with heat transfer transient through a two-dimensional conduit. The impact of Brownian and thermophoretic effects is considered into the account. Additionally, the thermal radiation, Joule heating, viscous dissipation, and heat generation/absorption characteristics are taken into consideration. The fluid is electrically conductive in nature along with uniform magnetic fieldB0 in normal direction. Based on these assumptions with the approximations of the boundary layer, the basic equations of Walter-B fluid reduced as follow [31]:
(1)∂u¯∂x¯+∂v¯∂y¯=0,(2)u¯∂u¯∂x¯+v¯∂u¯∂y¯=ue∂ue∂x¯+ν∂2u¯∂y¯2−k0ρfu¯∂3u¯∂x¯∂y¯2+v¯∂3u¯∂y¯3+∂u¯∂x¯∂2u¯∂y¯2−∂u¯∂y¯∂2u¯∂x¯∂y¯+σB02ρfue−u¯+gβTT¯−T¯∞−gβCC¯−C¯∞,(3)u¯∂T¯∂x¯+v¯∂T¯∂y¯=α∂2T¯∂y¯2−1ρcf16σ∗3k∗∂∂y¯T¯3∂T¯∂y¯+σB02u¯2ρcf+τDB∂C¯∂y¯∂T¯∂y¯+DTT¯∞∂T¯∂y¯2+Q0ρcfT¯−T¯∞,(4)u¯∂C¯∂x¯+v∗∂C¯∂y¯=DB∂2C¯∂y¯2+DTT¯∞∂2T¯∂y¯2−kr2e−Ea/kT∗T¯T¯∞mC¯−C¯∞,with boundary postulates [31]
(5)u¯=uwx¯=c1x¯,v=0,−k∂T¯∂y¯=T¯s−T¯h1,−DB∂C¯∂y¯=C¯s−C¯h2:y¯=0u¯=ue=c2x¯,T¯⟶T¯∞,C¯⟶C¯∞:y¯⟶∞,whereas u¯,v¯ are velocity component in x¯− and y¯−direction, ue is the free stream velocity, DB Brownian motion, DT thermophoretic coefficient, g gravitational acceleration, C¯ fluid concentration, T¯ temperature; α=k/ρcf thermal diffusion coefficient, ν=μ0/ρf kinematic viscosity, σ∗ is the Stefan-Boltzmann constant, k∗ absorption coefficient, βC¯ denote the solutal expansion coefficient, βT¯ thermal expansion coefficient, τ heat capacity ratio, σ electrical conductivity, Q0 heat generation/absorption, Ea activation energy, T¯s surface temperature, C¯ssurface concentration, h1 heat transfer coefficient, h2 mass transfer coefficient, m fitted rate constant, kr reaction rate, T¯∞ ambient temperature, C¯∞ ambient concentration, and k0 short memory coefficient.Adopting the local transformation similarity [31]:
(6)u¯=c1x¯F′ς,v¯=−c1ν1/2Fς,ς=c1νy¯,θ=T¯−T¯∞T¯s−T¯∞,φ=C¯−C¯∞C¯s−C¯∞,After incorporating Equation (6), one gets the dimensional Equations (2)–(5) into dimensionless form:
(7)F′′′−F′2+FF′′+WeF′′2−2F′F′′′+FF′′′′+MA−F′+A2+λθ+Nφ=0,(8)1+34R1+θf−1θ3θ′′+PrFθ′+Nbθ′φ′+Ntθ′2+MEcF′2+Sθ=0,(9)φ′′+ScFφ′+NtNbθ′′−Scσ1+δ1θmφexp−E11+δ1θ=0,(10)F=0,F′=1,θ′=−α1−θ,φ′=−β1−φ:ς=0andF′=A,θ=0,φ=0:ς=∞where We=k0c1/μ0 is the Weissenberg number, λ=gβT¯/c12x¯T¯s−T¯∞ mixed convection parameter, M=σB02/ρfc1 magnetic parameter, N=gβC¯/c12x¯C¯s−C¯∞ ratio of thermal to concentration buoyancy forces, Nt=τDT/T∞∗νT¯s−T¯∞ thermophoresis force, Nb=τDB/νC¯s−C¯∞ Brownian motion, δ1=T¯s−T¯∞/T¯∞ temperature difference, R=4σ∗T∞¯3/kk∗ radiation factor, S=Q0/ρcfc1 heat generation factor, Ec=u¯w2/cfT¯s−T¯∞ Eckert number, B1=h1/kv/y thermal Biot number, B2=h2/DBv/y Biot number, Pr=μcp/k Prandtl number, Sc=ν/DB Schmidt number, σ=kr2/c1 dimensionless reaction rate, A=c2/c1 ratio of constants, θF=T¯w/T¯∞ temperature ratio parameter, and E1=Ea/κT¯∞ activation energy.The physical quantities for the engineering practical and usefulness are the local skin friction, temperature gradient, and concentration gradient. The dimensionless form of these quantities is, respectively, expressed as follows(11)RexCfx=1−3BdFdςd2Fdς2ς=0,(12)NuxRex=−1+43R1+Tf−1T3dθdςς=0,One can obtain theCfx in nondimensional form by substituting Equation (12) in Equation (12):
(13)ShxRex=−dφdςς=0,Rex=c1x2/ν is the local Reynold number.


## 3. Convergence and Stability Analysis

The series solutions developed by the homotopic analysis method (HAM) comprise the convergence control parametersℏF,ℏθ and ℏφ [36–38]. These convergence control parameters are effective at regulating and controlling the series solution convergence region. The admissible values are attained by the flat parts of the ℏ−curves. Figure 1 displays the acceptable ranges of ℏF,ℏθ and ℏφ are −1.4≤ℏF≤−0.4, −1.2≤ℏθ≤−0.5 and −0.7≤ℏφ≤−0.3. Table 1 display the convergence region for the approximate solutions.Figure 1
(a)ℏ−curves for Fςandθς functions. (b) ℏ−curves for φς function.
(a)(b)Table 1
The allowable ranges for convergence solutions.
Approximate solutionsAuxiliary parametersConvergent intervalsFςℏF−1.4≤ℏF≤−0.4θςℏθ−1.2≤ℏθ≤−0.5φςℏφ−0.7≤ℏφ≤−0.3


## 4. Discussions

This section discusses the graphical implications of the physical dimensionless quantities on the relevant profiles. By selecting appropriate similarity variables, the equations that reflect the specified flow are converted first into ODEs. To clearly understand the behavior of flow profiles, an analytical scheme is adopted, which is strategized and debated using graphs.Figure2 portrays the sway of M on velocity F′ς portraits of stagnation point flow of nanofluids. It is noticed that velocity distributions diminish with higher values of magnetic parameter M. In reality, the fluid viscosity increases as M apply to any fluid due to which magnetic field power increase. In consequences, Lorentz force is produced which slows down the fluid flow significantly. It is observed from Figure 3 that the velocity sketches are enhance in the entire flow domain of nanofluid with the increasing values of ratio parameter A. Figure 4 reveals the sway of Weissenberg number We on velocity sketches of stagnation point flow of nanofluid. It is pointed out that in the intensification in F′ς field with step up values ofWenumber. The sketches of velocity fields with augmented values of mixed convection parameter λ in the flow domain are illustrated through Figure 5. This is owing to the fact that the increasing values of λ increase the buoyancy force. Consequently, F′ς profiles increase. It is detected from Figure 6 that the F′ς profiles of the nanofluid intensify in the fluid region with higher estimation of ratio of thermal to concentration buoyancy forces N.Figure 2
Plot ofF′ς via diverse M.Figure 3
Plot ofF′ς via diverse A.Figure 4
Plot ofF′ς via diverse We.Figure 5
Plot ofF′ς via diverse λ.Figure 6
Plot ofF′ς via diverse N.The distribution ofθς temperature of nanofluid with step up values of M is presented in Figure 7. It is observed that boundary layer thickness improves with incrementing values of M. This is due to the fact that Lorentz force which acts against the fluid flow direction, hence the enhancement in θς profiles with increasing values of M in the entire fluid flow. Attributes’ features of radiation parameter R on θς are exposed in Figure 8. It is perceived from this plot that the θς sketches enhance in the entire flow of nanofluid with increasing data of R. The sketches of θς field with augmented values of Nb in the flow region are describes through Figure 9. The energy sketches are enlarged with increasing data of Nb. This owes due to the motion of nanoparticles in the base fluid that is influenced by the motion of particles, and it moves from hot surface region and is related to the size and agglomeration of molecules. It is observed in Figure 10 that θς intensify subject to increasing value of Nt parameter. In reality, the thermophoretic forces and the nanoscale solid particles in the base liquid produce warm surfaces in the boundary layer region. Thus, θς of nanofluid in the entire fluid flow boosts. The effect of thermal Biot number B1 on θς is portrayed in Figure 11. It is cognized that temperature profiles of nanofluid increase with rising values of B1. Figure 12 explains the effect of heat generation parameterS on θς thermal curves. The energy field of the nanofluid increases with larger estimation of S. Physically, upsurging values of S add extra thermal energy to the entire flow domain. Variation in θς energy profiles is shown in Figure 13. The thermal field θς sketches enhance in the entire flow of nanofluid with increasing data of Ec. Figure 14 discloses the sway of Pr on θς sketches of nanofluid. It is noticed that thermal diffusivity decreases with the incrementing values of Pr. Hence, θς curves decline.Figure 7
Impact ofM on temperature field.Figure 8
Impact ofR on temperature field.Figure 9
Impact ofNb on temperature field.Figure 10
Impact ofNt on temperature field.Figure 11
Impact ofB1 on temperature field.Figure 12
Impact ofS on temperature field.Figure 13
Impact ofEc on temperature field.Figure 14
Impact ofPr on temperature field.Figure15 describes the power of E1 on φς concentration distribution nanofluids. It is pointed out that φς augmented with upsurging values of E1. Figure 16 shows the role of Sc on φς. The improving values of Sc diminish the concentration profiles. The lower values of Sccorrespond to the uppermost φς of nanoscale materials. In an increment in theSc, there is a decline in theφςdue to the mass diffusion. The impression of B2 on φς is revealed in Figure 17. The increasing data of B2 enhances φς profiles. One can observe that φς is the increasing function of B2. Figure 18 displays the variation of Cfx for diverse values of M and A. As expected, both parameter the surface drag force with larger estimation of these factors. Figure 19 shows the variation of Nux for unlike values of Nb and Nt. It has been detected from this plot that Nux gets dwindled for higher values of these parameters. Figure 20 shows the deviation in mass flow rate coefficient Shx. As witnessed, mass transfer rate increases for increasing data of Nb and Sc.Figure 15
Nanoparticle volume fraction viaE1.Figure 16
Nanoparticle volume fraction viaSc.Figure 17
Nanoparticle volume fraction viaB2.Figure 18
Results ofM and A via Cfx.Figure 19
Results ofNb and Nt via Nux.Figure 20
Results ofSc and Nb via Shx.


## 5. Conclusions

From this study, the following conclusions can be drawn:(i)
It has been noticed that boundary layers inF′ς profiles diminish for incrementing data of M and We, whereas increasing the A, N, and λ caused the F′ς profiles to upsurge(ii)
The thermal field curves and heat transfer rates got boost due to augmentation inM, R, S, and Ec along with Nb, Nt, and B1(iii)
With the upsurge in thePr, the fluid thermal energy and related thickness dwindle(iv)
The fluid concentration curves increase owing to increaseB2, E1, and Nt. Moreover, increasing Nb and Sc caused the temperature profiles to diminish(v)
The surface drag force coefficientCfx enhances with higher data of M and A parameters(vi)
It has been found that the heat transfer coefficientNux decays via increasing values of Nb and Nt parameters(vii)
Uprising of mass flow rateShx is detected for the increasing values of Sc and Nb parameters



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