Solar radiation effects on unsteady bio convectional flow of viscoelastic nanofluid confined by a wedge

Abstract

Nanofluid suspension comprises a chip-sized rigid particle trapped in the convection fluid. These forms of fluids are known as nanofluids. Nanofluid is more user-friendly in engineering including fuel cells, cooling systems, and a wide range of applications of technologies to improve. The idea of this work is to analyze the Bio convectional stream characteristics and Solutal boundary characteristics of nanofluid flow via a wedge. Furthermore, the consequences of motile bacteria and heat conductivity are considered. When the boundary layer is estimated, the governing equations will be seen. Coupled PDEs are reduced into nonlinear ODEs using the required similarity vector, and the resultant structures are shown using the MATLAB computational tool bvp4c firing (Labotto IIIA formula). The implications of fluid velocity, temperature area, nanoparticles concentration, and microorganism concentration on the induced parameters are shown in graphical and numerical values. The velocity field is booming up for higher fluid parameter values and depressed for larger buoyancy ratio parameter estimations. For increasing levels of both the thermal conductivity parameter and the thermal Biot number, the temperature field rises. The microorganism’s field has dropped for Peclet number values and increased for Microorganisms Biot number values. The concentration field is lowered for the Lewis number while it is increased for the concentration Biot number. The current implications are novel and original for the investigation of flow and heat transfer over a wedge in a viscoelastic nanofluid with thermal conductivity, motile microorganisms, and bioconvection.

Introduction

The heat exchange phenomenon is a vital part of the mass of machine tools, pharmaceutical, food, and manufacturing industry. Any popular applications also include steelmaking processes, heaters, condensers, nuclear reactions, polymer manufacturing, chemical refining, and mechanical and chemical reactions. The scientists and researchers have dedicated their efforts to improving the transport of heat required for different manufacturing applications. In this continuity, different conventional methods have been used, based on the stability of the components and the circumstance. Amongst many others, the use of nonmaterial as a freezing source has been one of the new techniques. The fundamental premise of this driving theory is that metals have a higher thermal conductivity relative to fluids. The Thermo physical properties with the incorporation of certain particles into the fluid have been significantly enhanced. Nanomaterials have also been considered the most appropriate and cheapest coolant for heavy machinery, mechanical, and manufacturing processes. Due to medical uses, nanoparticles were used to identify and cure brain tumors. Choi and Eastman ¹ was the originator of the key proposal on this issue. Buongiorno ² later identified two major properties of nanofluids, namely Brownian motion and thermophoresis consequences for the expansion of nanotechnology heat transmission. Bhatti and Rashidi ³ used the porous sheet to explore the thermo-diffusion the conduct of Williamson nano fluid flow. Muhammad et al. ⁴ to explore the characteristic of a nanofluid turbulent fluid in the continuance of heat and mass fluxes. Elgazery ⁵ ascribed the existence of nanoparticles immersed in a normal fluid with acceptable magnetic field properties to the presence of nanostructures. Hamid and Khan ⁶ investigated the movement of nanoparticles through a stretched sheet in a numerical prolongation. Hsiao ⁷ analyzed the flow of nanofluids with multiple applications such as viscous dispersion and magnetic field. Turkyilmazoglu ⁸ discovered a closed-form connection between two metallic nanoparticles on a curved sheet. Khan and Pop ⁹ examined a conceptual study of nanofluid flow caused by a stretched surface. Gireesha et al. ¹⁰ quantified the influence of heat radiation and chemical reactions in nanofluids across the sheet. Alamri et al. ¹¹ investigated the flow of nanofluid through turbulent flow in the presence of slip. Additional research is being done on these phenomena.^12–18 Bioconvection is the flow of micro hardness fluids caused by microbe bathing. Bioconvection has been used in befoul, fertilizer, and industrial operations. At nanofluid bioconvection, researchers studied density stratification, pattern development caused by micro biomes, nano-materials, and booster internships. Low levels of algae and other ox tactic bacteria. Different types of microorganisms swim in various ways. Geo-tactic microorganisms swim against gravity, whereas gyro-tactic microorganisms swim using a combination of the gravity and viscous torques are addressed by Pedley et al. ¹⁹ and Hill et al. ²⁰ Oxytactic microorganisms typically swim on the way to the upper layer when the upper surface of every substrate opens to the air where the amount of glucose is high. Because microorganisms are heavier than water, increasing their densities on the outer part causes inverted turbulence, which aids in the formation of bioconvection. Most of the consumer and industrial goods may be generated by the introduction of the substances of microorganisms. Sources include biofuels and biodiesel, biomass, fertilizers, micro-microsystems, etc. Kuznetsov ²¹ established the bio-thermal convection of nanoliquids using both microorganisms simultaneously. Alsaedi et al. ²² have projected the relationship of magneto-nanomaterials with gyrotactic microorganisms and permeable surfaces due to overheating. The vertically interacting motion of fluid nanostructures with gyrotactic motile bacteria was shown by Rehman et al. ²³ Recently, Waqas et al. ²⁴ investigated the mechanism of motile microorganisms in a Falkner-Skan flow of metal nanoparticles over a stretched surface in a Falkner-Skan flow of metal nanoparticles across such a stretched sheet. Uddin et al. ²⁵ studied the different slip characteristics in convectional nanofluid flow. Raju and Sandeep ²⁶ discovered the effects of non-Newtonian flow leaked via a revolving plate on the emergence of motile bacteria. Several scholars have recently added to this intriguing phenomenon, as seen in Refs.^27–38

The goal of this research is to look the aspects of Bioconvectional flow viscoelastic nanofluid and Solutal boundary conditions using a wedge. Here we discussed the significance of three cases of wedge like as (shrinking $\alpha =0.5$ , static $\alpha =0.0$ , and stretching $\alpha =0.5$ ). We use water as a base fluid and metallic nanoparticles in this investigation. The consequences of motile microorganisms and thermal conductivity are also explored. To solve higher order differential equations into first order differential equations by using the shooting technique. Graphical and numerical results are calculated by using the bvp4c built-in function in MATLAB mathematical tool.

Mathematically and physically formulation

Physical depiction of the model

Here, we modeled the bioconvective flow aspects and solutal boundaries features of Viscoelastic fluid flow through a wedge. Furthermore, the effects of motile microorganisms and heat conductivity are considered. The thermophoresis effects and Brownian motion are also discussed. The constant strength is applied normally to the wedge with external velocity $u_e\left(x\right)=a{x}^m$ . Here a&m are the constants and the coordinates are $x-\mathrm{axis}\mathrm{and}y-\mathrm{axis}$ perpendicular to the wedge.

The main assumptions of the flow problem is listed below

• The two-dimensional flow of viscoelastic nanofluid flow is studied.

• Here we use wedge shape geometry (shrinking $\alpha =0.5$ , static $\alpha =0.0$ , and stretching $\alpha =0.5$ ).

• The effects of bioconvection with motile microorganisms are discussed with the effects of thermal radiation.

• The aspects of thermophoresis effects and Brownian motion are analyzed here.

• Here we use Solutal boundary conditions with heat and mass thermal conductivity.

Figure 1 shows the effects of flow problem.

Figure 1.

Physically model and coordinate system of a wedge.

Mathematical formulation

The governing equations for represented problem are:

Equation of mass

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(1)

Equation of momentum

$$\begin{array}{c}u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=u_e\frac{\partial u_e}{\partial x}+v\frac{{\partial }^{2}u}{\partial y^2}+k_0\hfill \\ (u\frac{{\partial }^{3}u}{\partial x\partial y^2}+\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\partial u}{\partial y}\frac{{\partial }^{2}v}{\partial y^2}+v\frac{{\partial }^{3}u}{\partial y^3})\hfill \\ +\frac{1}{{\rho }_f}\left[\begin{array}{c}(1-C_f){\rho }_f{\beta }^{**}g*(T-T_{\infty })-({\rho }_p-{\rho }_f)g^{*}(C-C_{\infty })\hfill \\ \multicolumn{1}{c}{-(N-N_{\infty })g^{*}\gamma ({\rho }_m-{\rho }_f)}\end{array}\right],\hfill \end{array}$$

(2)

Equation of species diffusion

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=({\alpha }_f+\frac{16{\sigma }_q{T}_{\infty }^3}{3k^{*}\left(\rho C\right)f})\frac{{\partial }^{2}T}{\partial y^2}}\\ +\frac{1}{\rho c_p}\frac{\partial }{\partial z}\left[k\left(T\right)\frac{\partial T}{\partial z}\right]+\frac{k}{\rho c_p}\frac{{\partial }^{2}T}{\partial y^2},\hfill \end{array}$$

(3)

Here,

$$K\left(T\right)=k_{\infty }(1+{\in }_1\frac{T-T_{\infty }}{\mathrm{\Delta }T})$$

$$\begin{array}{c}u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=\frac{1}{\rho c_p}\frac{\partial }{\partial z}\left[D\left(C\right)\frac{\partial C}{\partial z}\right]+D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2},\hfill \end{array}$$

(4)

Here

$$D\left(C\right)=k_{\infty }(1+{\in }_2\frac{C-C_{\infty }}{\mathrm{\Delta }C})$$

Equation of species concentration

$$u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}=D_m\left(\frac{{\partial }^{2}N}{\partial y^2}\right)-\frac{b{W}_c}{(C_w-C_{\infty })}\left[\frac{\partial }{\partial y}\left(N\frac{\partial C}{\partial y}\right)\right],$$

(5)

Associated boundary conditions

$$\begin{array}{c}u=0,v=0,-k\frac{\partial T}{\partial y}=h_f(T_w-T),-D_B\frac{\partial C}{\partial y}=h_g(C_w-C),\hfill \\ \multicolumn{1}{c}{-D_m\frac{\partial N}{\partial y}=h_n(N_w-N),\mathrm{at}y=0}\\ \multicolumn{1}{c}{u\to u_e\left(x\right),\frac{\partial u}{\partial y}\to 0,T\to T_{\infty },C\to C_{\infty },N\to N_{\infty }\mathrm{as}y\to \infty }\end{array}\},$$

(6)

Where $\rho$ is density, $c_p$ is the specific heat, ${\beta }_T$ is the coefficient of thermal expansion, ${\in }_1\mathrm{\&}{\in }_2$ is the thermal conductivity, $\upsilon$ is the kinematic viscosity, $g$ is the acceleration of gravity, $\mathrm{\Omega }=\beta \pi$ is angle of wedge, $T$ is temperature, $C$ is concentration, $uv$ are velocity components, $\beta$ the wedge angle parameter, $k_0$ is the viscoelasticity parameter and $N$ is the microorganisms.

Transformations analysis and solution

The similarity transformations are

$$\begin{array}{c}\zeta =\sqrt{\frac{u_e\left(x\right)}{\mathrm{ux}}}y,\psi =\sqrt{\mathrm{vx}{u}_e\left(x\right)}f\left(\zeta \right),\hfill \\ \multicolumn{1}{c}{\theta \left(\zeta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi \left(\zeta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\chi \left(\zeta \right)=\frac{N-N_{\infty }}{N_w-N_{\infty }}}\end{array}\},$$

(7)

Dimensionless transformed model

$$\begin{array}{c}m({f\prime }^2-1)-\frac{m+1}{2}\mathrm{ff}″-f″\prime -K_1\hfill \\ [(3m-1)f\prime f″\prime -\frac{(3m-1)}{2}{f″}^2-\frac{(m+1)}{2}f{f}^{\mathrm{iv}}]\hfill \\ +\lambda (\theta -\mathrm{Nr}\varphi -\mathrm{Nc}\chi )=0,\hfill \end{array}$$

(8)

Here $K_1(=\frac{k_{0}a{x}^{m-1}}{v})$ is the fluid parameter (Viscoelastic fluid), the mixed convection parameter is $\lambda (=\frac{{\beta }^{*}g(1-C_{\infty })(T_w-T_{\infty })}{(m+1)u_e^2})$ , $\mathrm{Nc}(=\frac{{\gamma }^{**}({\rho }_m-{\rho }_f)(N_w-N_{\infty })}{(1-C_{\infty })(T_w-{\tilde{T}}_{\infty })\beta })$ is the bioconvection Rayleigh number $\mathrm{Nr}(=\frac{({\rho }_p-{\rho }_f)(C_w-C_{\infty })}{(1-C_{\infty })(T_w-T_{\infty }){\beta }^{*}})$ is the buoyancy ratio parameter,

$$\begin{array}{c}(1+{\in }_1\mathrm{Rd}(1+({\theta }_f-1){\theta }^3))\theta ″+{\in }_1\theta {\prime }^2\hfill \\ +\mathrm{Pr}(\frac{m+1}{2}f\theta \prime -\mathrm{mf}\prime \theta )=0,\hfill \end{array}$$

(9)

Here $\mathrm{Pr}(=\frac{v}{\alpha })$ is the Prandtl number, thermophoresis parameter $\mathrm{Nt}(=\frac{E{D}_T(T_w-T_{\infty })}{T_{\infty }\alpha })$ , Brownian motion parameter $\mathrm{Nb}(=\frac{E{D}_B(C_w-C_{\infty })}{\alpha })$ , $\mathrm{Rd}(=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k{k}^{*}})$ is the radiation parameter, temperature ratio parameter ${\theta }_f(=\frac{T_w}{T_{\infty }})$ ,

$$(1+{\in }_2)\varphi ″+{\in }_2\varphi {\prime }^2+\mathrm{Le}\mathrm{Pr}f\varphi \prime +\frac{\mathrm{Nt}}{\mathrm{Nb}}\theta ″=0,$$

(10)

Here, $\mathrm{Le}(=\frac{\alpha }{D_B})$ is Lewis number.

$$\chi ″+\mathrm{Lbf}\chi \prime -\mathrm{Pe}(\varphi ″(\chi +\mathrm{\Omega })+\chi \prime \varphi \prime )=0,$$

(11)

Here $\mathrm{\Omega }(=\frac{N_{\infty }}{N_w-N_{\infty }})$ is the microorganism difference parameter, $\mathrm{Lb}(=\frac{\nu }{D_m})$ is bioconvection Lewis number, $\mathrm{Pe}(=\frac{b{W}_c}{D_m})$ is Peclet number.

Modified boundary conditions

$$\begin{array}{c}f\left(0\right)=0,f\prime \left(0\right)=1,\theta \prime \left(0\right)=-A_1(1-\theta \left(0\right)),\hfill \\ \multicolumn{1}{c}{\varphi \prime \left(0\right)=-A_2(1-\varphi \left(0\right)),\chi \prime \left(0\right)=-A_3(1-\chi \left(0\right)),\mathrm{at}\zeta =0}\\ \multicolumn{1}{c}{f\prime \left(\zeta \right)=1,f″\left(\zeta \right)=0,\theta \left(\zeta \right)=0,\varphi \left(\zeta \right)=0,\chi \left(\zeta \right)=0\mathrm{as}\zeta \to \infty }\end{array}\},$$

(12)

Here $A_1(=\frac{h_f}{k}\sqrt{\frac{\nu }{a}})$ is thermal stratification Biot number, $A_2(=\frac{h_g}{D_B}\sqrt{\frac{\nu }{a}})$ is mass/Solutal stratification Biot number, $A_3(=\frac{h_k}{D_m}\sqrt{\frac{\nu }{a}})$ the microorganisms stratification Biot number.

$${\mathrm{Re}}_x(=\frac{u_{e}x}{v})(=\frac{a{x}^{m+1}}{v}),$$

(13)

Here ${\mathrm{Re}}_x$ is the Reynolds number.

Numerical approach

The non-linear dimensionless ODEs (08–11) with linked boundary constraints (12) are solved graphically and numerically by built-in function bvp4c solver which implements the Lobatto-IIIa formula in mathematical software MATLAB.

Let

$$\begin{array}{c}f=p_1,f\prime =p_2,f″=p_3,f″\prime =p_4,f^{\mathrm{iv}}={p\prime }_4\hfill \\ \multicolumn{1}{c}{\theta =p_5,\theta \prime =p_6,\theta ″={p\prime }_6}\\ \multicolumn{1}{c}{\varphi =p_7,\varphi \prime =p_8,\varphi ″={p\prime }_8}\\ \multicolumn{1}{c}{\chi =p_9,\chi \prime =p_{10},\chi ″={p\prime }_{10}}\end{array}\},$$

(14)

$$\begin{array}{c}\multicolumn{1}{c}{p{\prime }_4}\end{array}=\frac{-m({p_2}^2-1)+\frac{m+1}{2}{p}_1{p}_3-p_4+K_1[(3m-1)p_2{p}_4-\frac{(3m-1)}{2}{p_3}^2]-\lambda \left(\begin{array}{c}{p}_5-\mathrm{Nr}{p}_7\hfill \\ \multicolumn{1}{c}{-\mathrm{Nc}{p}_9}\end{array}\right)}{K_1\frac{(m+1)}{2}{p}_1},$$

(15)

$$p{\prime }_6=\frac{-{\in }_1{p_6}^2-\mathrm{Pr}(\frac{m+1}{2}{p}_1{p}_6-m{p}_2{p}_5)}{(1+{\in }_1\mathrm{Rd}(1+({\theta }_f-1){p_5}^3))},$$

(16)

$$p{\prime }_8=\frac{-{\in }_2{p_8}^2-\mathrm{Le}\mathrm{Pr}{p}_1{p}_8-\frac{\mathrm{Nt}}{\mathrm{Nb}}{p\prime }_6}{(1+{\in }_2)},$$

(17)

$$p{\prime }_{10}=-\mathrm{Lb}{p}_1{p}_{10}+\mathrm{Pe}(={p\prime }_8(p_9+{\delta }_1)+p_{10}{p}_8),$$

(18)

With

$$\begin{array}{c}{p}_1\left(0\right)=0,p_2=1,p_6\left(0\right)=-A_1(1-p_5\left(0\right)),\hfill \\ \multicolumn{1}{c}{p_8\left(0\right)=-A_2(1-p_7\left(0\right)),p_{10}\left(0\right)=-A_3(1-p_9\left(0\right)),\mathrm{at}\zeta =0}\\ \multicolumn{1}{c}{p_2\left(\zeta \right)=1,p_3\left(\zeta \right)=0,p_5\left(\zeta \right)=0,p_7\left(\zeta \right)=0,p_9\left(\zeta \right)=0\mathrm{as}\zeta \to \infty }\end{array}\},$$

(19)

Tabular values

Results and discussion

In this part, we investigate the graphical results of the following physical parameters (fluid parameter, thermal stratification Biot number, bioconvection Lewis number, mixed convection parameter, thermophoresis parameter, buoyancy ratio parameter, mass/Solutal stratification Biot number, Brownian motion parameter, Prandtl number, temperature ratio parameter, Lewis number, Peclet number, microorganisms stratification Biot number) on the Bioconvectional flow aspects and Solutal boundary features of nanofluid flow over a wedge. In addition, the results of microorganisms and thermal conductivity are also taken into account. Figure 2 analyzes the alteration of $K_1$ through $f\prime \left(\zeta \right)$ . In generally analyzing the reported results; it can conclude that the velocity distribution $f\prime \left(\zeta \right)$ boomed for the higher values fluid relaxation parameter ${\lambda }_n$ for stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. Figure 3 interprets the consequences of $\mathrm{Nr}$ against $f\prime \left(\zeta \right)$ . It is examined that flow of fluid $f\prime \left(\zeta \right)$ decays for enlarging estimation of buoyancy ratio parameter $\mathrm{Nr}$ for stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. Figure 4 is intended to investigate the estimation of velocity distribution $f\prime \left(\zeta \right)$ for mixed convection parameters $\lambda$ . Physically interpretation $f\prime \left(\zeta \right)$ is always intensified by enlarging the consequence $\lambda$ . The rate of velocity distribution $f\prime \left(\zeta \right)$ boosted due to uplift the $\lambda$ . Figure 5 divulged the consequence of $\mathrm{Nc}$ via $f\prime$ . Investigation of this Figure, represented that by expanding the $\mathrm{Nc}$ via $f\prime \left(\zeta \right)$ is reduced for stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. Figure 6 discloses the effect of $M$ over the velocity distribution $f\prime \left(\zeta \right)$ . It is understood from this figure that the growing value of the $M$ cases declined $f\prime \left(\zeta \right)$ . Lorentz forces are created physically by a magnetic field that resists flow motion and causes velocity to decrease. To envision the model of temperature distribution $\theta$ for amplified variation $A_1$ in Figure 7 is sketched for stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. This representation scrutinized that temperature distribution $\theta$ builds up owing to enhancing the 7 magnitude $A_1$ . Figure 8 is purposed to resolve the visualization of ${\in }_1$ the over-temperature field $\theta$ . Generally, it has been observed that enhancing the value of a thermal field $\theta$ by expanding the importance ${\in }_1$ . Figure 9 displays the impression of Prandtl number $\mathrm{Pr}$ versus $\theta$ . Since these conspiracies, we experimented that temperature distribution $\theta$ decreasing aggravated the variation of Prandtl number $\mathrm{Pr}$ for stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. Figure 10 illustrates the significance of ${\theta }_f$ the temperature distribution $\theta$ . The temperature distribution $\theta$ uplifted passing through higher assessment ${\theta }_f$ . Figure 11 demonstrates the property of thermophoresis parameter $\mathrm{Nt}$ against volumetric nanoparticles concentration $\varphi$ . It is examined that the concentration distribution $\varphi$ increases in intensity the thermophoresis parameter $\mathrm{Nt}$ . Physical thermophoresis forces were discovered, which improved the colloidal motion of nanoparticles and increased the heat transfer rate. Figure 12 clarifies the significance of $Le$ for $\varphi$ . The increasing valuation of $\mathrm{Le}$ diminishes $\varphi$ for stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. The performance of fluid concentration $\varphi$ is shown against the mass Biot number $A_2$ in Figure 13. It is witnessed that $A_2$ intensifies $\varphi$ . Outstanding features of $\mathrm{Nb}$ for $\varphi$ are delineated in Figure 14. It is animated that the concentration field $\varphi$ diminishes by elevation in the magnitude of $\mathrm{Nb}$ both estimations of stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. Physically, the nanoparticles travel from a warm to a cold surface as a result of bigger Brownian diffusions. The mass transfer $\varphi$ with some of Arrhenius activation energy $E$ is shown in Figure 15. It can be similar to the ${\in }_2$ mass transfer $\varphi$ increases. We can see this by nanoparticles concentration $\varphi$ and Solutal boundary layer thickness intensifies for enlarging amounts of ${\in }_2$ . Figure 16 demonstrates the estimation of the microorganism’s concentration $\chi$ concerning the Peclet number $\mathrm{Pe}$ . From this scenario, we noted that microorganism distribution $\chi$ diminishes for enlarger numbers $\mathrm{Pe}$ . Figure 17 elaborates the impact of $A_3$ on $\chi$ . From this picture, we analyzed that $\chi$ enhanced for a larger magnitude of $A_3$ stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. Figure 18 describes the character of microorganisms profile $\chi$ for different variations $\mathrm{Lb}$ . It can be shown that higher estimation of $\mathrm{Lb}$ causes $\chi$ to decline for stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ wedge. This occurs because increasing the values of the bioconvection Lewis number raises the viscous diffusion rate, causing the density profiles in the boundary layer to decrease. Figures 19 and 20 describes the statistical performance of the magnetic field and buoyancy ratio parameter for velocity profile. It can be shown that greater variations of magnetic field and buoyancy ratio parameter causes velocity profile to boost up for stretching $\alpha =0.5$ , static $\alpha =0.0$ , and shrinking $\alpha =0.5$ . Figures 21 to 23 describe the statistical performance of Prandtl number, Lewis number, and Bioconvection Lewis number for thermal, concentration, and microorganism’s profiles. The higher values of parameters declined the thermal, concentration, and microorganisms profile. Table 1 is computed to explore the pattern $-f″\left(0\right)$ via parameters. The $-f″\left(0\right)$ increased via magnetic parameter while declining for mixed convection parameter. Table 2 is investigated in order to examine the elements of $-\theta \prime \left(0\right)$ flowing parameters. Calculated with the following data analysis, it is observed that $-\theta \prime \left(0\right)$ boosts with the enhancement of thermal Biot number while reducing the temperature ratio parameter. Table 3 shows the values $\varphi \prime \left(0\right)$ Better estimates of several parameters. This table revealed that the local Sherwood number $\varphi \prime \left(0\right)$ augmented for $\mathrm{Pr}{A}_2$ . The exact solutions of local microorganism numbers $-\chi \prime \left(0\right)$ using flow parameters are shown in Table 4. Here local density number of microorganisms $-\chi \prime \left(0\right)$ improved for greater variety $A_3$ while reducing the fluid parameter.

Figure 2.

Action on $f\prime$ with $K_1$ .

Figure 3.

Action on $f\prime$ with $\mathrm{Nr}$ .

Figure 4.

Action on $f\prime$ with $\lambda$ .

Figure 5.

Action on $f\prime$ with $\mathrm{Nc}$ .

Figure 6.

Action on $f\prime$ with $M$ .

Figure 7.

Action on $\theta$ with $A_1$ .

Figure 8.

Action on $\theta$ with ${\in }_1$ .

Figure 9.

Action on $\theta$ with $\mathrm{Pr}$ .

Figure 10.

Action on $\theta$ with ${\theta }_f$ .

Figure 11.

Action on $\varphi$ with $\mathrm{Nt}$ .

Figure 12.

Action on $\varphi$ with $\mathrm{Le}$ .

Figure 13.

Action on $\varphi$ with $A_2$ .

Figure 14.

Action on $\varphi$ with $\mathrm{Nb}$ .

Figure 15.

Action on $\varphi$ with ${\in }_2$ .

Figure 16.

Action on $\chi$ with $\mathrm{Pe}$ .

Figure 17.

Action on $\chi$ with $A_3$ .

Figure 18.

Action on $\chi$ with $\mathrm{Lb}$ .

Figure 19.

Action on velocity profile with $M$ .

Figure 20.

Action on velocity profile with $\mathrm{Nr}$ .

Figure 21.

Action on thermal profile with $\mathrm{Pr}$ .

Figure 22.

Action on concentration profile with $\mathrm{Le}$ .

Figure 23.

Action on microorganisms profile with $\mathrm{Lb}$ .

Table 1.

Aspects of $-f″\left(0\right)$ different flow parameters.

$M$	$\lambda$	$\mathrm{Nr}$	$\mathrm{Nc}$	$K_1$	Shrinking $\alpha =0.5$	Static $\alpha =0.0$	Stretching $\alpha =0.5$
0.2	0.1	0.2	0.1	0.3	2.7016	1.6245	0.6921
0.6					3.1086	1.8749	0.8409
1.2					3.6042	2.1855	0.9878
0.1	0.3	0.2	0.1	0.3	2.6016	1.6446	0.6425
	0.7				2.5088	2.8947	0.7839
	1.0				1.3049	2.2959	0.8898
0.1	0.1	0.3	0.1	0.3	1.2038	1.1145	0.6821
		0.7			2.5083	1.5445	0.8803
		1.0			3.7042	2.2859	0.9971
0.1	0.1	0.2	0.3	0.3	1.3016	1.3221	0.6632
			0.7		2.5146	1.4712	0.8931
			1.0		3.7282	2.0803	0.9852
0.1	0.1	0.2	0.1	0.2	2.0016	1.4678	0.6651
				0.5	3.1092	1.9467	0.8409
				0.7	3.9143	2.1468	0.9798

Table 2.

Aspects of $-\theta \prime \left(0\right)$ different flow parameters.

$\mathrm{Pr}$	$\mathrm{Nb}$	$\mathrm{Nt}$	$\mathrm{Le}$	$A_1$	${\theta }_w$	$-\theta \prime \left(0\right)$
3.0	0.2	0.3	2.0	0.5	1.5	0.3963
5.0						0.4124
7.0						0.4216
2.0	0.1	0.3	2.0	0.5	1.5	0.3827
	0.6					0.3787
	1.2					0.3734
2.0	0.2	0.1	2.0	0.5	1.5	0.3839
		0.6				0.3784
		1.2				0.3715
2.0	0.2	0.3	1.2	0.5	1.5	0.3816
			3.0			0.3819
			5.0			0.3821
2.0	0.2	0.5	2.0	0.1	1.5	0.0943
				0.6		0.4367
				1.2		0.6787
2.0	0.2	0.3	2.0	0.5	1.6	0.3935
					1.7	0.3743
					1.8	0.3583

Table 3.

Aspects of $-\varphi \prime \left(0\right)$ different flow parameters.

$\mathrm{Pr}$	$\mathrm{Nb}$	$\mathrm{Nt}$	$\mathrm{Le}$	$A_2$	$-\varphi \prime \left(0\right)$
3.0				0.5	0.4939
5.0					0.5120
7.0	0.2	0.3	2.0		0.5228
2.0	0.1	0.3	2.0	0.5	0.4462
	0.6				0.4975
	1.2				0.5039
2.0	0.2	0.1	2.0	0.5	0.4986
		0.6			0.4501
		1.2			0.4024
2.0	0.2	0.3	1.2	0.5	0.4419
			3.0		0.4978
			5.0		0.5124
2.0	0.2	0.5	2.0	0.1	0.0923
				0.6	0.4143
				1.2	0.7374

Table 4.

Aspects of $-\chi \prime \left(0\right)$ different flow parameters.

$A_3$	$\mathrm{Lb}$	$\mathrm{Pe}$	$K_1$	$-\chi \prime \left(0\right)$
0.1	2.0	0.1	0.3	0.0911
0.6				0.3321
1.2				0.4965
0.5	1.2	0.1	0.3	0.3417
	1.8			0.3677
	2.6			0.3938
0.5	1.0	0.2	0.3	0.2358
		0.8		0.3369
		1.6		0.3380
0.5	1.0	0.1	0.2	0.4392
			0.5	0.3739
			0.7	0.3892

Final remarks

The aim of current research work is to investigate the aspects of Bioconvectional viscoelastic nanofluid flow with Solutal boundary conditions via a wedge. Furthermore, the effects of motile bacteria and heat conductivity are considered. The primary goals are as follows:

• ❖ The velocity profile field is increased for higher estimations fluid parameter while depressed for larger buoyancy ratio parameter.

• ❖ The velocity profile field is boosted up for mixed convection parameter while decreasing the magnetic parameter.

• ❖ The temperature profile is boomed for the higher values of both thermal conductivity parameter and the thermal Biot number.

• ❖ The temperature field is declined for the growing values of Prandtl number and temperature ratio parameter.

• ❖ The concentration field is decreased for Lewis number while increased for concentration Biot number.

• ❖ The microorganism’s field depressed for Peclet number and increased for the increasing values of microorganisms Biot number.