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# Mathematical Study of MHD Micropolar Fluid Flow with Radiation and Dissipative Impacts over a Permeable Stretching Sheet: Slip Effects Phenomena

Pudhari Srilatha1, Ahmed M. Hassan2, B. Shankar Goud3,*, E. Ranjit Kumar4

1 Department of Mathematics, Institute of Aeronautical Engineering, Hyderabad, India
2 Faculty of Engineering, Future University in Egypt, New Cairo, Egypt
3 Department of Mathematics, JNTUH University College of Engineering, Science & Technology, Hyderabad, India
4 Department of Mathematics, Kakatiya Institute of Technology and Science, Telangana, India

* Corresponding Author: B. Shankar Goud. Email:

(This article belongs to the Special Issue: Computational and Numerical Advances in Heat Transfer: Models and Methods I)

Frontiers in Heat and Mass Transfer 2023, 21, 539-562. https://doi.org/10.32604/fhmt.2023.043023

## Abstract

The purpose of this research is to investigate the influence that slip boundary conditions have on the rate of heat and mass transfer by examining the behavior of micropolar MHD flow across a porous stretching sheet. In addition to this, the impacts of thermal radiation and viscous dissipation are taken into account. With the use of various computing strategies, numerical results have been produced. Similarity transformation was utilized in order to convert the partial differential equations (PDEs) that regulated energy, rotational momentum, concentration, and momentum into ordinary differential equations (ODEs). As compared to earlier published research, MATLAB inbuilt solver solution shows an extremely good correlation in exceptional instances. In exceptional instances, the present MATLAB inbuilt solver solution has a very excellent connection with the findings of the previously published investigations. A variety of flow field factors impact the Nusselt number, the wall couple shear stress, the friction factor, Sherwood numbers the dimensionless distributions discussed in detail. When the Eckert number rises, the temperature rises, and the Schmidt number falls, the concentration falls. Velocity increases with increases in the material factor but drops with increases in the magnetic parameter and the surface condition factor.

## Keywords

Nomenclature

 u,v x,y−Components of velocity kf Thermal conductivity uw Surface velocity (m/s) γ Spin gradient viscosity ζ Similarity variable D Mass diffusion coefficient of the fluid (m2/s) υ Kinematic viscosity (m2/s) g Gravitational acceleration Ec Eckert number μ Dynamic viscosity (kg m/s) ρ Density of the fluid (kg/m3) b Constant j Microinertia per unit mass (N/kg) S Surface condition parameter Sc Schmidt number M Magnetic parameter Cw Concentration of the fluid at the surface T Temperature across the thermal boundary layer (k) Pr Prandtl number Kr Chemical reaction parameter θ Dimensionless temperature cp Specific heat at constant pressure (J/Kg K) C Concentration of the fluid inside the boundary layer K Material parameter

1  Introduction

This study uses the micropolar method to examine how the presence of a magnetic field modifies the heat and mass transport properties of a moving flat surface immersed in a non-conducting electrical fluid to address this more complex challenge. The goal of this experiment is to investigate the effect of radiation on heat and mass transfer in a Micropolar MHD free convective fluid flow over a stretched porous sheet by injection and suction fluid through the sheet. According to reports, viscous dissipation also plays a contribution to the energy equation. To convert the controlling boundary layer equations into ordinary differential equations, similarity transformations are used. Afterward, the in-built solver in MATLAB is used to provide a numerical explanation of the equations. Several major influences on flow characteristics have been studied. When compared to other published papers, the findings demonstrate a significant level of coherence. Work in the fields of solar energy collecting, recovery of petroleum products, and the dynamics of fires in insulation may all benefit from this kind of study.

2  Mathematical Analysis

In this scenario, consider a stretching sheet that has a specified surface thermal gradient that is accountable for driving a continuous 2-D laminar movement of a non-compressible micropolar fluid at the ambient temperature T. It is thought that the movement is along the axis of x, which carried the plate ahead along the accelerative direction; the y-axis, on the other hand, is thought to be vertical to the x-axis. The direction of flow that has a changing magnetic field is connected in a traditional way to the y-axis via a connection that is formed in that axis.

An external electric field is not implicated. The flow geometry associated with this problem is shown in Fig. 1 under the further assumptions that the magnetic Reynolds amount is relatively small and that the magnetic field generated by the moving fluid has a negligible influence on the magnetic field around the liquid and the equations for the boundary layer that explain constant flow across a stretching sheet are as follows [28].

Figure 1: Flow geometry

Continuity:

ux+υy=0(1)

Momentum:

uux+υuy=(v+kρ)2uy2+kρNyσB02ρu+vkpu(2)

Angular momentum:

uNx+υNy=γjρ2Ny2kjρ(2N+Ny)(3)

Energy:

uTx+υTy=kfρCp2Ty2+(μ+k)ρCp(uy)2+σB02ρu2+16σT33kρCp2Ty2(4)

Concentration:

uCx+υCy=D2Cy2Kr(CC)(5)

The appropriate boundary measures to use for physical limits are the following:

u=uw=bx+αuy, υ=0, N=suy, T=Tw+αTy, C=cw+βCyat y=0u=0,N=0,T=T,C=Casy}(6)

In this case, the right-hand side of Eq. (4) shows terms for the impact of the Ohmic heating and viscous dissipation expressions.

Here γ=(μ+k2)j is presumed to be velocity spin gradient and is given by [28], here j=vb is referred to be the length of reference.

The predicted flow of heat flux, based on the Rosseland estimate [29] is

qr=4σ3αT4y(7)

In this hypothesize the thermal difference exclusive to the flow is insignificant and adequate for T4 to be stated as a linear arrangement of the temperatures. Expand T4 in the Taylor sequence extension approximately T obtains

T44T3T3T3(8)

Hence,

qry=16σT33α2Ty2(9)

In the process of converting Eqs. (2)(5) together with Eqs. (7) and (9) together into a set of ODEs, we make use of the similar translations and dimensional factors that are outlined in the following paragraphs, specifically:

ζ=(bv)y, u=bxf(ζ),υ=bvf(ζ)N=b3vxG(ζ), θ(ζ)=TTTwT,ϕ(ζ)=CCCwC}(10)

The changed equations are

(1+K)f+ff(f)2+Kg(M+Kp)f=0(11)

(1+K2)G+fGGfK(2G+f)=0(12)

(1+4R3)θ+Prfθ+(1+K)PrEc(f)2+PrEc(f)2=0(13)

ϕ+ScfϕScKrϕ=0(14)

The newly reshaped boundary circumstances are as follows:

f(ζ)=1+Af(ζ);f(ζ)=0;G(ζ)=sf(ζ);θ(ζ)=1+Bθ(ζ);ϕ(ζ)=1+Cϕ(ζ);}ζ=0;f(ζ)=0;G(ζ)=0;θ(ζ)=0,ϕ(ζ)=0}ζ(15)

Here “ ”signifies the dif. w.r. to “ ζ ” & the dimensionless quantities are

K=kμ,M=σB02ρb,Pr=ρvcpkf,Kp=bKpv,Ec=uw2Cp(TwT)Sc=vD,Kr=Krbα=Abv,β=Bbv,γ=Cbv}(16)

It is possible to determine the shear stress by applying the formula located at the surface.

τw=[(μ+k)(uζ)+kN]ζ=0=(μ+k)bxbvf(0)

By employing uw=bx is called characteristic velocity, the quantity of skin friction, designated by Cf, is to be specified as Cf=τwρuw2=(1+K)f(0)Rew.

Here local Reynolds number: Rew=uwxv.

The apparent, couple stress is precise as Mw=(γNζ)ζ=0=μuw(1+k2)g(0).

It is possible to describe the local surface temperature flow according to Fourier’s law, which may be summarized as follows:

qw(x)=kf(Nζ)ζ=0=kf(TwT)bvθ(0)

The following expression may be used to calculate the component of the surface heat flux transfer: h(x)=qw(x)(TwT)=kfbvθ(0).

In addition, the Nusselt number can be expressed using the formula as follows: Nux=xh(x)kf=bvxθ(0) also we can be written as NuxRew=θ(0).

Likewise, local mass flux indicated as Jw=D(Cζ)ζ=0.

Also, a quantity of Sherwood number stated Shx=Jw(x)D(CwC)=bvxϕ(0) and it changes to

ϕ(0)=ShxRew.

3  Method of Solution

One may find an answer by recasting the system of coupled nonlinear ODEs (11)(14) and with constraints (15) as an initial value problem. We set

y1=f,y2=f,y3=f,y4=g,y5=g,y6=θ,y7=θ,y8=ϕ,y9=ϕ, now the set of ODEs is transformed into the following arrangement:

y3=(y1  y2(y2)2((M+Kp)  y2)+(K  y5))/(1+K)

y5=((y1  y5)(y2  y4)(K (2  y4+y3)))/(1+K2)

y7=((Pr(y1  y7)+(1+K)PrEc(y3)2+PrEc(y2)2))/(1+4R3)

y9=(sc(y1  y9)ScKry8)

The boundary constraints are

y1(0)=fw,y2(0)=1+A  fa(3),y4(0)sy3(0),y6(0)=1+B  fa(7),y8(0)=1+C  fa(9)} and y2()=0,y4()=0y6()=0,y8()=0}

The technique used in MATLAB’s built-in algorithm may be utilized to assimilate the preceding differential equations. This technique is carried performed repeatedly until the results achieve the 106 the desired degree of accuracy.

4  Results and Discussion

The exploration into the study of micropolar fluid movement with slip conditions, and thermal radiation is now being performed as a component of the examination. Through the use of numerical techniques, flow fields were examined, together with friction factor & quantity of Nusselt number, spanning a range of possible estimates for the characterizing parameters.

This section decides to investigate the diverse range of physical qualities that different embedding configurations have on flow region characteristics. These flow regional profiles are depicted in the pictures. Typical trends of the nondimensional flow fields are shown in Figs. 2a2d, respectively, over a wide range of values of the magnetic field component. The values of f(ζ) and G(ζ) diminish when the magnetic field component M is enriched, whereas the values of θ(ζ) and ϕ(ζ) continue to rise. When M increases, the opposite force, which is known as the Lorenz force, also increases. This causes the flow to become even more slow. Subsequently, particle velocity and microrotation are decelerated by a huge M, this makes perfect sense from a physics perspective. As a direct result of this, the parameter M was responsible for controlling both the velocity & the temperature.

Figure 2a: f(ζ) vs. M

Figure 2b: G(ζ) vs. M

Figure 2c: θ(ζ) vs. M

Figure 2d: θ(ζ) vs. M

Figs. 3a3d demonstrate the f(ζ), G(ζ),θ(ζ) & ϕ(ζ) outlines fluctuating values of the material factor K. The graph illustrates unmistakably that rising K boosts velocity. As a micro concentration substance enhances, with diverse values of K. Because of this, micro concentration influences the flow pattern. Since K has a positive impact on the width of the boundary level. The graph demonstrates that when the K upsurges and angular rotation at the boundary reduces. Hence, an increase in particle concentration causes a reduction in microrotation near the boundary. An increase in the worth of material property is seen to be accompanied by a drop in temperature. Immediately before the boundary, the temperature spikes and remains high for some time before commencing a slow drop. As the value of the material factor grows, concentration declines. The concentration profile is rather insensitive to this variable’s adjustment.

Figure 3a: f(ζ) vs. M

Figure 3b: G(ζ) vs. K

Figure 3c: θ(ζ) vs. K

Figure 3d: ϕ(ζ) vs. K

Figs. 4a and 4b exhibit the velocity and microrotation patterns, respectively, as a consequence of the factor (s). It is clear that changing the parameters will result in a reduction in the velocity. The comparable velocity is indeed greater when there is no turn relative to other cases. As s becomes greater, the growth rate accelerates exponentially.

Figure 4a: f(ζ) vs. s

Figure 4b: G(ζ) vs. s

Fig. 5 portrays the contribution of the Schmidt number (Sc) on the concentration profile. Based on this graph, we can see that as Sc enhances, ϕ(ζ) declines.

Figure 5: ϕ(ζ) vs. Sc

Fig. 6 illustrates how the Eckert number plays a role in the thermal gradients of the system. It has been found that when the Eckert number increases, there is an accompanying rise in the temperature of the fluid located above the sheet. Even when Ec increases, the liquid region becomes much more effective at loading energy that has been lost due to dissipation as a result of viscosity and elastic properties. This is the case even while Ec is growing.

Figure 6: θ(ζ) vs. Ec

Fig. 7 depicts the variations that take place in the concentration profile as a consequence of the various reaction factors. The saturation curve is demonstrated to have a declining slope the greater “Kr” is made to be.

Figure 7: ϕ(ζ) vs. Kr

Fig. 8 is an illustration of temperature gradients for a variety of different radiation input variables (R). It has been shown that R, which stands for R, makes the thermal field more intense. This is due to an elevation in R leading to an intensification in the size of the temperature boundary layer, which brings about the aforementioned effect.

Figure 8: θ(ζ) vs. R

Figs. 9a9d exhibit how the suction/injection component has a consequence on the geometries of the flow fields (d). Increasing the value of the suction parameter causes a general smoothing out of the profiles. These profiles become narrower as fw gets higher because the heating forces the fluid closer to the sheet’s surface, which leads to a smaller quantity of flow underneath the surface. This is why the profiles get narrower as fw gets higher.

Figure 9a: f(ζ) vs. fw

Figure 9b: G(ζ) vs. fw

Figure 9c: θ(ζ) vs. fw

Figure 9d: ϕ(ζ) vs. fw

Due to the reduction in thermal conductivity of the fluid caused by an enhancement in Pr, the heat transfer rate increases as Pr is increased. The energy profile has been deemed unreliable for this same reason and has thus been deprecated. This phenomenon is shown in Fig. 10.

Figure 10: θ(ζ) vs. Pr

Figs. 11a11d depict the flow profiles vs. varying the velocity slip factor A. Here, the velocity and microrotation of the fluid are lower in the presence of velocity slip than in the absence of slip (A=0), and these decrease as the slip parameter grows whereas A. Meanwhile, As thermal and concentration profiles increase with the enhancement of A. A consequence of imposing the velocity slip condition, the hydrodynamic boundary layer is seen to thin. Nonlinear stretch sheets provide a source of momentum, which is then transmitted to a micropolar fluid. However, as one moves farther from the sheet, the profiles overlap and the slip’s influence diminishes.

Figure 11a: f(ζ) vs. A

Figure 11b: G(ζ) vs. A

Figure 11c: θ(ζ)vs. A

Figure 11d: ϕ(ζ)vs. A

Fig. 12 displays the effect of changing the thermal slip factor on the temperature field in the boundary layer. When B rises, so does the size of the thermal boundary surface enhances. Because of this, the temperature profile becomes more uniform and heat transmission from the sheet to the fluid slows down. When the concentration slip factor C is large, the concentration curves exhibit the same behavior as shown in Fig. 13. As a matter of fact, slip tends to lessen the fluid flow, which in turn serves to decrease the net molecular movement, which in turn lessens the temperature and concentration gradients.

Figure 12: θ(ζ) vs. B

Figure 13: ϕ(ζ) vs. C

Figs. 14a14d depict the impact of the permeability factor on the flow characteristics across the boundary layer. When Kp increases, the velocity of the fluid declines, and the microrotation temperature and concentration curves increase.

Figure 14a: f(ζ) vs. Kp

Figure 14b: G(ζ) vs. Kp

Figure 14c: θ(ζ) vs. Kp

Figure 14d: ϕ(ζ) vs. Kp

Under such precise circumstances for varying values of the Pr, we have assessed by matching our numerical answer to the earlier study of [3032]. The comparison demonstrates that there is an astounding level of consistency, as can be shown in Table 1.

Table 2 elucidates the friction upsurges with enriched values of M,K,s Kp fw,and A and the contradictory consequence is established in the couple stress. Nusselt number boosts with an increase of fw & A, and the opposite effect is found with a rise of M, K, s Kp & B. Sharewood number enhances with K, fw and reverses a trend observed M, s, Kp, A, C. The results shown in Table 3 illustrate that a rise in Pr results in a higher Nusselt number, but an opposite pattern is seen with Ec and R. The findings presented in Table 4 indicate that an increase in Sc and Kr causes a boost in the Sharewood quantity.

5  Conclusions

This work examines the impact that a slip impact has on the rate of mass transport along a radially expanding surface. The relevance of a changing magnetic field is also included in the analysis. The governing flow problem might be simplified with the use of a similar conversion model. Combining the shooting method with the bvp4c solver allows for the numerical solution of the resulting nonlinear ODEs. When compared to other published papers, the findings demonstrate a significant level of coherence. Work in the fields of solar energy collecting, recovery of petroleum products, and the dynamics of fires in insulation may all benefit from this kind of study. Key inferences may be made from the numerical data and the visual representation, including the following:

•   Velocity enhances when K values are raised, but declines as M and s quantities rise.

•   Microrotation rises with both M and s factor but declines with a larger K.

•   Raising the M, K, and Ec values raises the temperatures. Thus, by modifying these variables, a higher temperature may be attained.

•   The concentration upsurges as M enhances, but the concentration diminutions as K and Sc enhance.

•   Increases in M & K, the rise in the outcome of the friction and couple stress nevertheless lower θ(0) & ϕ(0).

•   Couple stress and θ(0) enhance with enhancement of sand the opposite effect is found in friction factor and ϕ(0).

•   As fw increases the result in friction & ϕ(0), θ(0).

•   With a rise in R, the result in θ(0) rises whereas a fall in Pr diminishes the result. The value of ϕ(0) rises when Sc and Kr increase.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm their contribution to the paper as follows: study conception and design: B.S. Goud and P. Srilatha; data collection: B.S. Goud and P. Srilatha; analysis and interpretation of results: E.R. Kumar and P. Srilatha; draft manuscript preparation: B.S. Goud and A.M. Hassan. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: No data was used for the research described in the article.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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APA Style
Srilatha, P., Hassan, A.M., Goud, B.S., Kumar, E.R. (2023). Mathematical study of MHD micropolar fluid flow with radiation and dissipative impacts over a permeable stretching sheet: slip effects phenomena. Frontiers in Heat and Mass Transfer, 21(1), 539-562. https://doi.org/10.32604/fhmt.2023.043023
Vancouver Style
Srilatha P, Hassan AM, Goud BS, Kumar ER. Mathematical study of MHD micropolar fluid flow with radiation and dissipative impacts over a permeable stretching sheet: slip effects phenomena. Front Heat Mass Transf. 2023;21(1):539-562 https://doi.org/10.32604/fhmt.2023.043023
IEEE Style
P. Srilatha, A.M. Hassan, B.S. Goud, and E.R. Kumar "Mathematical Study of MHD Micropolar Fluid Flow with Radiation and Dissipative Impacts over a Permeable Stretching Sheet: Slip Effects Phenomena," Front. Heat Mass Transf., vol. 21, no. 1, pp. 539-562. 2023. https://doi.org/10.32604/fhmt.2023.043023

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