Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2201 https://internationalpubls.com A Study of Bioconvective Williamson Fluid of an Exponential Stretching Sheet with Chemical Reaction M. R. Mishraa*, S. Singhb, D. Samalc a,b,cDepartment of Mathematics, O.P. Jindal University, Raigarh, 496001 India *Corresponding Author E-mail: mail_to_mrmishra@yahoo.co.in Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: The present study's primary objective is to look into how thermophoresis diffusion and Brownian motion impacts the results of heat radiation and microorganism bioconvection in non-Newtonian Williamson fluid flow via exponentially stretched sheets. Similarity transformations have been applied in this case to transform PDEs into their corresponding ODEs. R-K method with shooting technique is applied to find the solution in a numerical approach. Fluid velocity increases with mixed convection and decreases with increasing magnetic parameter. The temperature grows as thermophoresis and Brownian motion parameters rise. The velocity field is declined by the effect bioconvection Lewis number (Bio. LNo.). Keywords: Bioconvection, Williamson fluid, stretching sheet, chemical reaction. 1. Introduction The influence of heat radiation and the bio-microorganism phenomena in MHD Williamson fluid flow on an exponentially stretching sheet are investigated in this study. Because of it’s a variety of applications in the scientific and technological industrial sectors, as well as its growing capacity to transfer heat, researchers have taken an interest in the analysis over a stretching sheet of non- Newtonian fluid flow. In nuclear reactors, heat exchangers, solar systems, and other systems that include fluids and symmetrically extended sheets, the impact of HMT under the influence of chemical processes is crucial. [1–8]. A solution's microbe motility is the vital factor that drives the bioconvection process. Microbes react with different chemicals and density of other element by moving in certain directions. The variation among positive and negative microbe motility is caused by distortion in the stimulus's instruction for the opposite action. Because It's been demonstrated that bacteria have the potential to move up inside solution cells, gyrotactic microorganisms often migrate in response to a density difference. The only approach employed in bioremediation for removal of environmental pollutants from a location is the introduction of microorganisms. Researchers initially proposed a study on the importance of oxyntic and gyrotactic bacteria. [9-12] Due to its various applications in glass fiber, plastic film, paper processing, metal drawing, stretching sheets have drawn substantial interest in the last few years. Recently, making use of stretching sheets, several researchers delved at the MHD flow and its many consequences, including chemical reactions and viscous dissipation. [13-22] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2202 https://internationalpubls.com Chu et al. [23] shown the effects of thermal diffusion, activation energy (A.E.), Brownian motion, mobile microorganisms, and chemical reactions on the flow of a bioconvective MHD fluid related to stretching sheet. Mlamuli Dhlamini et. al. [24] developed the bio-convection flow’s mathematical model and calculated the A.E. for chemical reactions. Computational techniques and algorithms for the irreversibility investigation of a blood nanofluid passing over an interface with A.E. and squeezing were developed by Naresh Kumar et al. [25] in the field of biomedicine. Zafar et al. [26] studied chemical reactions and Prandtl nanofluid A.E. with bio-convection flow across a vertical surface. The viscosity that changes with temperature has been studied and the Arrhenius kinetic energy magnetized bioconvective nanofluid flow was numerically calculated by A. Shahid et. al. [27]. The effects of various thermal radiations as well as the Prandtl number on the MHD flow and thermal analysis of hybrid nanofluids were investigated by Sachin Shaw et. al. [28]. In a hybrid nanofluid flow (water based), researcher [29] used MHD and nonlinear radiation to simulate heat, entropy, and mass transfer. Using the bivariate spectral quasi-linearization approach, Oyelakin et al. [30] designed a Casson nanofluid that maximizes generation of entropy in unstable stagnating flows across a stretched sheet with Arrhenius A.E. and the chemical reaction. The industrial sector uses Williamson fluids' pseudo-plastic boundary layers for emulsion-coated polymer sheets used in extrusion and high molecular weight polymer materials for photographic films. Various non-Newtonian models [31-35] have been used to study the dynamics of pseudo-plastic fluids, including as the power law, Cross, Carreaus, Ellis, and other models. However, an immense amount of research has been done on the various ways that Williamson fluid moves through a stretching surface where MHD is present. Muhammad Imran Asjad et al. [36] investigated thermophoresis diffusion and Brownian motion in a Williamson fluid flow on a symmetrically stretched sheet. As a result, they identified that the Bio. LNo. Lb and Peclet number Pe caused the motile microbe profile to decline and the bioconvection Rayleigh number Rb to rise. Raising the suction/injection parameter s and the Williamson parameter We causes the coefficient of skin friction to decrease, whereas increasing the magnetic parameter M causes it to rise. R. Ahmed et al. [37] provided a depiction of this result. The rate of heat transmission increases as Pr grows, as shown by Ishak [16] and Goud et. al. [38]. The main goal of this context is to observe that, if adding heat radiation and bio-convection to Williamson magneto-hydrodynamics fluid flow using gyrotactic auto-motile microorganisms can lessen the chance of precipitation. These vital features give many contemporary technologies' heat exchange processes the necessary, favourable heat mobility, and their presence may be beneficial. [39-41]. 2. Mathematical Formulation Here x- and y-axes were assumed to be normal with velocity / 0 x l wU a e= , and we studied steady incompressible MHD fluid flow through an exponentially stretching sheet in the present context. A magnetic field can be found in the flow zone that acts in the y-direction. Microorganisms and nanoparticles are gently dispersed throughout the liquid. When thermal radiation is taken into account, microorganism movement causes bioconvection to occur. ,U V are the fluid velocity for 2D fluid flow. The Governing equations are. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2203 https://internationalpubls.com 0x yU V+ = (1) ( ) ( )( ) ( )( ) ( )( ) 2 02 1 1 x y yy y yy p f m f UU VU U U U M U C T T C C N N                 + = +  − +  − − − − − − − −   (2) ( ) 21 r T x y yy B y y y p q D UT VT T D T C T C y T        + = − + +        (3) ( )( ) ( )/2 / n Ea kT T x y yy r yy D UC VC DC K C C T T e T T −    + = − − + (4) Bioconvection equation yy n c yx y N N D dW C y UN V C N   −     =  + (5) with the associated boundary conditions ( ,0) , ( ,0) ( ), ( ,0) , ( ,0) , ( ,0)w w w wU x U V x x T x T C x C N x N= = − = = = (6) 0, , ,U T T C C N N as y  → → → → → At this point, let’s consider ( ) ( ) ( ) ( ) ( )/ / 2 / 2 / 2 / 2 0 0 0 0 0, ( ) , , , x l x l x l x l x l w w w wU a e x V e T T T e C C C e N N N e   = = − = + = + = + (7) By Rosseland approximation * 4 1 4 3 , r T q k y   =  and using the Taylor series, we have 3 4 4 4 3T T T T = − where T , is the ambient temperature [42], Eq. (3) can be written as ( ) * 3 2 1 16 3 T x y yy B y y y p T D UT VT T D T C T k C T           + = + + +         (8) Consider the following similarity transformation [40]   2 / / /0 0 0, ( ), ( ) ( ) 2 2 x l x l x ly a e a e U a e f V f f l l         = = = − + (9) / 2 / 2 / 2 0 0 0( ), ( ), ( )x l x l x lT T T e C C C e N N N e       = + = + = + The similarity transformation has been applied in equations (1) to (5): The same solution is found for Equation (1); the reduced ODE equations for equations (2) to (5) are given below in Equations (10) to (13) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2204 https://internationalpubls.com Dimensionless momentum equation: ( )22 0 = f Mf f ff Nr Rb    −  −  +  + − − (10) Dimensionless energy equation: ( ) 4 1 Pr ( ) 0 3 K f f Nb Nt        +  +  −  +  +  =    (11) Dimensionless concentration equation: ( ) 11 0 E n m Nt f Cr f e Sc Nb         − +    +  − −  − + +  =    (12) Dimensionless bioconvection equation: ( ) ( ) ( ) ( ) ( ) ( ) ( )1Pr ( )( ) ( ) 0 Lb f LbPrf Pe                  +  −  − + +  =  (13) The associated boundary conditions become into ( ) ( ) ( ) ( ) ( )0 1, 0 , 0 1, 0 1, 0 1, 0, = = = = = f f s at    = − (14) ( ) ( ) ( ) ( ) ( )0, 0, 0, 0, 0 f f as     →   →  →  →  → →  Along stretching surface, the shear stress w , the thermal flux wq , the mass flux mq and the motile microorganisms flux nq are as follows: ( ) 2 02 w y y y U U    =   = +    , * 3 * 0 4 w y T T q R yk   =    = − +      , (15) 00 ,m B n N yy C N q D q D y y ==      = − = −        The expression for xf C , xNu , xSh and xNn are as follows 2x w f w C U   = , ( ) w x w xq Nu k T T = − , ( ) m x B w xq Sh D C C = − and ( ) n x N w xq Nn D N N = − (16) The non-dimensional form of xf C , xNu , xSh and xNn are as follows ( ) ( )( ) 21 0 2 0 xf x C Re f f   = +      , ( ) 4 1 3 0x xNu Re R    = +    − (17) ( )0x xSh Re − = , ( )0x xNn Re − = Here, w x xU Re  = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2205 https://internationalpubls.com 3. Numerical Solution Approach Methods The FDM, FVM, and FEM are the fundamental discretization techniques. However, during calculation, these methods required much more for the finding of the unknowns, but the R-K method with shooting techniques is a suitable method to solve flow problems related to ODEs. To put it briefly, boundary value issues are adequately, quickly, and exactly solved using the R-K technique. Because of its relative simplicity this numerical approach is therefore frequently used in applied science's nonlinear analysis. 4. Results and Discussions Numerical solutions are found for the physical interpretations of the non-dimensional formulation of steady nanofluid MHD flow caused by the exponential stretched sheet in the presence of a bioconvection equation and chemical reaction with the related boundary conditions. Figure 1 demonstrates the velocity distribution for many values of the magnetic field parameter M. It is noticed that when M increases, the velocity decreases. This indicates that high resistance to fluid motion and a high viscosity are produced by Lorentz's force, which results in a drop in velocity. Figure 1: Repercussions of M values on ( )f  Figure 2: Repercussions of λ values on ( )f  Figure 3: Repercussions of Nb values on ( )  Figure 4: Repercussions of Nt values on ( )  As seen in Figure 2, a rise in the mixed convection parameter λ results in an augmented in flow velocity ( )f  . With respect to the density variation and temp. gradient, the flow has a stronger buoyancy effect by the mixed convection. The fluid flow was improved by this phenomenon. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2206 https://internationalpubls.com With increased values of the thermophoresis parameter Nt and the Brownian motion parameter Nb, Figures 3 and 4 show a notable increasing behaviour of ( )  . The increased heat transfer to rise ( )  is resulting from the quick, arbitrary movement of nanoparticles, that are identified by higher Nb. Comparably, an increase in Nt indicates a stronger thermophoretic impact, which shifts the hotter regime of the nanoparticles to the colder one and widens the thermal distribution. Figure 5: Repercussions of E values on  Figure 6: Repercussions of Cr values on  Figure 7: Repercussions of Rb values on ( )  Figure 8: Repercussions of Lb values on ( )  The concentration profile is influenced by A.E., as seen in Figure 5. There is an increase in E. In the end, the circumstances favor fluid substances that produce higher concentrations. Therefore, as the A.E. is increased, the density gradient rises. The decrease in ϕ(η) that results from an increased value of the Cr is seen in Figure 6, where the chemical reaction speeds up to reduce the nanoparticles’ concentration. Figure 9: Repercussions of Pe values on ( )  Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2207 https://internationalpubls.com Figure 7 illustrates how the bioconvection Rayleigh number Rb is directly responsible for providing an increment of ( )  . As Rb rises, so does the density of the motile microbes. The Bio. LN (Lb) of mobile bacteria is the product of their mass diffusivity with heat diffusivity. In biology or medicine, this gauge characterizes the heat transfer brought on by microorganisms. This feature is reversed near a free stream, where the motile microbe density decreases and the Lb value rises shown in Figure 8. Due to their ability to move independently, microorganisms induce a change in profile by reducing the amount of surface bioconvection. Figure 9 illustrates how the dimensionless density of microorganisms decreases as the Peclet number (Pe) grows. A higher Pe value indicates a stronger pattern of microbial movement, which decreases the microorganism profile. This number quantifies the intensity of directed and random swimming in motile microorganisms. Table 1 shows how several factors, including the Williamson fluid parameter We, magnetic field parameter M, suction/injection parameter s, and affect the skin friction coefficient. It is demonstrated that the xx fRe C is decreased of the variations in the We. More fluid motion resistance can be produced by longer relaxation times when the Williamson fluid value is higher. Consequently, the coefficient of skin friction decreases. Table 1: Impact of xx fRe C We s M xx fRe C P. Priyadharshini et al [39] Present calculations 0.1 0.2 2.0 1.7543 1.7537 0.2 1.6830 1.6765 0.3 1.6200 1.6198 0.1 1.7976 1.7989 0.2 1.7543 1.7537 0.3 1.7123 1.7118 0.1 1.2035 1.2018 0.2 1.2383 1.2342 0.3 1.2722 1.1699 It has been shown that a lower xx fRe C is obtained with increased suction/injection parameter s. This suggests that higher fluid flow resistance is caused by an increase in the porosity of the stretched sheet. According to our study, xx fRe C increases with rising the values of M. The speed at which fluid particles tend to flow is determined by the Lorentz force. At the surface level, there is an increase in xx fRe C . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2208 https://internationalpubls.com Table 2: Impact of xNu Pr K Nt Nb (0)xNu  = − P. Priyadharshini et al 391] Present calculations 1.3 0.2 0.1 0.1 0.9475 0.9468 1.4 0.9881 0.9672 1.5 1.0271 0.9998 0.2 0.9475 0.9468 0.6 0.7731 0.7939 1 0.6631 0.6781 0.1 0.9475 0.9468 0.4 0.8551 0.8677 0.7 0.7617 0.7761 0.1 0.9475 0.9468 0.4 0.8435 0.8528 0.7 0.7466 0.753 Table 2 illustrates how the radiation parameter R affects fluid temperature. The thermal boundary layer’s width as well as the dispersion of rise as the values of the R do. The Rosseland radiative absorption is decreasing, which causes the radiative heat change to diverge more. As a consequence, the increase in the radiative heat transmission rate causes the fluid's temperature to rise. As an outcome, the xNu falls. Diffusion thermophoresis factor causes xNu to decrease. The hot zone's high energy levels and fluid molecular movement, which force the nanoparticles out of the area, are responsible for this phenomena. Heat transfer occurs more quickly when heated particles pass from the region of high temperature to the cold zone. The fluid's energy increases as the Brownian motion factor Nb grows. This process illustrates the Brownian motion Nb increase, which describes the zigzag movement fluid particles. When there is a boost in Brownian motion, the value of xNu decreases substantially because there are more fluid particle collisions. Table 3: Impact of xSh m E Nt Nb (0)xSh = − P. Priyadharshini et al [39] Present calculations 0.3 0.2 0.1 0.1 0.7528 0.7532 0.5 0.8807 0.8821 0.7 0.9923 0.9967 0.2 0.7528 0.7532 0.7 0.7338 0.7299 1.3 0.7127 0.710 0.1 0.7528 0.7532 0.4 1.0846 1.0989 0.7 1.1316 1.1441 0.1 0.7528 0.7532 0.2 0.3566 0.3788 0.3 −0.0182 0.0005 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2209 https://internationalpubls.com Table 3 indicates that the xSh (Sherwood number) is a numerical depiction using several parameters, including m , E, Nt and Nb. The is evident the value of xSh increased when m rises. Actually, the rate of reaction approaches the Burke-Schumann limit. The Sherwood number increases due to the lower species' diffusion coefficient. It is seen that the value of xSh down when the dimensionless A.E. values are raises. This suggests that the rate of diffusion surpasses the rate of mass transfer as the Reynolds number increases. There is higher A.E. as the the lower xSh . A higher A.E. indicates that a more successful collision between the particles will need more energy. An increase in the Nb results in an enhancement of values of xSh . Higher values of Nb are seen when the surface roughens due to an enhance in the mobile microorganisms’ density and the xSh . The value of xSh falls down as the values of the Nt increased. This is proven by the fact that, as the fractional derivative parameter increases, the temperature rises because the thermal resistance decreases. In terms of mechanics, the heat flux vector's phase lag and temperature gradients exhibit opposing trends. Table 4: Impact of xNn Pe Lb 1 (0)xNn  = − P. Priyadharshini et al [39] Present calculations 0.1 1.1 0.1 1.1183 1.1187 0.4 1.2968 1.2979 0.7 1.4806 1.4837 1.1 1.1183 1.1187 1.4 1.2857 1.2875 2.0 1.5740 1.5781 0.1 1.1183 1.1187 0.3 1.1270 1.1265 0.5 1.1357 1.1342 The effects of xNn (mobile microorganisms) under various parameters, Pe, Lb and 1 , are shown in Table 4. We may investigate if a larger concentration of motile microorganisms with a higher Peclet Number (Pe) could be advantageous. It is obvious that the interaction of rotational and magnetic fields significantly increases microbial movement. This increase in velocity and concentration fields can be seen as a stretching influence that forms on the flow of plasma nanofluid containing bacteria. The motile microbes improved as the Bio. LNo. values (Lb) enhanced. These results show that as the Lb rises, so does the motile bacterial density and that associated stress lowers as microorganisms move from the sheet over the boundary layer. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2210 https://internationalpubls.com The quantity of motile microbes rises as the 1 (bio-convection difference parameter) varies more quickly. It is discovered that the motile bacteria’s density minimizes as the Pe rises. Temperature and concentration are known to be primarily transmitted by changes in mass and heat, respectively 5. Conclusion In the current study, MHD Williamson fluid flow has been analysed for an exponential stretching sheet by employing numerical approaches. The physical fields of temperature, velocity, and microbe distribution are the ones on which the results of the parameters are listed. The following significant results are summarized: • The velocity of fluid gets enhanced with the mixed convection parameter (λ) and decreases with the magnetic parameter (M). This happens because Lorentz's force causes a high viscosity, which raises the Williamson fluid parameter and rises the resistance to fluid motion and the velocity profile. • The Brownian motion (Nb), and the thermophoresis diffusion (Nt) resulted in a larger temperature distribution due to the rapid and random motion of the nanoparticles. • With chemical reaction parameter (Cr), concentration recurs, and with A.E. parameter (E), it is enhanced. • An increased Bio. LNo. and Peclet number resulted in a reduction in the microorganism profile. These indicates physical measurement of the relative strength and random motion in motile bacteria. Raising the Rayleigh number (bioconvection) Rb results in an enhanced motile microbe profile. Nomenclature Magnetic field parameter: 2 02 = w M l M U   non-dimensional Williamson fluid parameter: 3 wU We l   = suction/injection (s > 0/ s < 0) parameter: 2 0 0 2 V l s a  = Mixed convection: 2 2 (1 )( ) = w w l C T T U    − − Brownian motion factor: ( ) B wD C C Nb   − = Buoyancy ratio factor: ( )( ) (1 )( ) p f w w C C Nr C T T       − = − − − Rayleigh number of bio-convection: ( )( ) (1 )( ) w w m f C C Rb C T T        − − − − = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2211 https://internationalpubls.com Thermophoresis diffusion: ( ) wT T TD Nt T    − = Rate of the reaction (Dimensionless): 22 r m w K l U  = Non-dimensional activation energy(A.E.): Ea E kT = Bio-convectional difference parameter: 1 w N N N    = − Distinct parameter of temperature: wT T T    − = Schmidt number: Sc D  = Peclet number: n dWc Pe D = Bioconvection Lewis Number (Bio. LNo.): n Lb D  = Prandtl number: Pr =   Radiation Parameter: * 34 f T R k k  =  0M Magnetic field coefficient C Concentration T Temperature N Concentration of microorganisms Re Reynold’s number wT Wall temperature T Temperature far away from the plate ,U V Velocity components along x and y -axes  Dynamic viscosity  Kinematic viscosity k Conductivity of heat TD Thermophoretic diffusion coefficient Φ Concentration (Dimensionless) rq Radiative heat flux ρ Density α Thermal diffusivity xf C Skin friction coefficient xNu Nusselt Number Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2212 https://internationalpubls.com xSh Sherwood number xNn Motile density number * Stefan-Boltzmann constant 1k Coefficient of Mean absorption Refrences [1] S.U. Choi and J.A. Eastman (1995), Enhancing thermal conductivity of fluids with nanoparticles, Argonne National Lab., IL (United States), Tech. Rep. [2] Shahid, A.K.; Yufeng, N.; Bagh, A. 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