Effects of Hall Current and Viscous Dissipation on Bioconvection Transport of Nanofluid over a Rotating Disk with Motile Microorganisms

Abstract

The study of rotating-disk heat-flow problems is relevant to computer storage devices, rotating machineries, heat-storage devices, MHD rotators, lubrication, and food-processing devices. Therefore, this study investigated the effects of a Hall current and motile microorganisms on nanofluid flow generated by the spinning of a disk under multiple slip and thermal radiation conditions. The Buongiorno model of a nonhomogeneous nanofluid under Brownian diffusion and thermophoresis was applied. Using the Taylor series, the effect of Resseland radiation was linearized and included in the energy equation. By implementing the appropriate transformations, the governing partial differential equations (PDEs) were simplified into a two-point ordinary boundary value problem. The classical Runge–Kutta dependent shooting method was used to find the numerical solutions, which were validated using the data available in the literature. The velocity, motile microorganism distribution, temperature, and concentration of nanoparticles were plotted and comprehensively analyzed. Moreover, the density number, Sherwood number, shear stresses, and Nusselt number were calculated. The radial and tangential velocity declined with varying values of magnetic numbers, while the concentration of nanoparticles, motile microorganism distribution, and temperature increased. There was a significant reduction in heat transfer, velocities, and motile microorganism distribution under the multiple slip conditions. The Hall current magnified the velocities and reduced the heat transfer. Thermal radiation improved the Nusselt number, while the thermal slip conditions reduced the Nusselt number.

1. Introduction

The heat transport process is imperative in chemical, mechanical, engineering, and biomedical systems. The performance of devices such as heat exchangers, gas turbine blades, combustors, microelectronic boards, and solar panels depends on the rate of heat transport and can be modulated by altering the heat transfer coefficient and improving the thermal exchange properties of the working fluids such as oil, ethylene glycol, and water. The use of nanofluids represents a decisive solution to extemporizing the heat transport process. The applications of nanofluids in numerous disciplines were described by Abu-Nada [1]. Choi and Eastman [2] confirmed the efficacy of this alternative heat transport fluid in various applications by establishing the improved heat transport properties of nanofluids. Thus far, in the field of nanofluidics research, a great number of theoretical studies have been conducted by applied mathematicians, due to the predominant applications such as electronic device cooling, oil recovery systems, solar energy systems, nuclear systems cooling, and renewable energies. Buongiorno [3] established that the non-uniform relative velocity between the nanoparticles and the base fluid plays a role in the advancement of heat transfer in nanofluids through the Brownian diffusion of the nanoparticles. Nield and Kuznetsov [4] examined the effects of nanofluidics on boundary layer transfer with natural convection using a similarity approach. Kuznetsov and Nield [5] re-examined the Cheng–Minkowycz problem related to the Buongiorno model and found that the diffusion generated by the zig-zag movement of nanoparticles causes the development of a heat field. Wakif et al. [6] investigated the Buongiorno hybrid nanofluidic transition carrying solid alumina particles over an extended plate. Rasheed et al. [7] calculated the numerical solution for the 3D flow of nanofluids under rotation and convection conditions. Gumber et al. [8] studied a submerged nanofluid with micropolar particles subjected to Rosseland radiation with transpiration cooling on a vertical plate. Mahanthesh [9] examined the role of nanofluid agglomeration in transport behavior subject to nonlinear radiation and convection conditions. Areekara et al. [10] used the Buongiorno nanofluid model to study the transition of a nanofluid subject to zero-mass flow conditions and highlighted the biomedical applications of the nanofluid. For recent research on nanofluids under various physical conditions, refer to [11,12,13,14,15].

The upward movement of mobile microorganisms generates bioconvection in a fluid. The swimming of mobile microorganisms (e.g., algae and bacteria) in a base fluid makes the fluid denser but prevents the agglomeration of nanoparticles and allows the nanofluid to effectively improve its heat transport characteristics. Bioconvection is useful in a variety of biotechnology applications, including biosensors, biopolymer fabrication, drug delivery, microsystems, and the recovery of microbial oils. Kuznetsov [16] investigated the consequences of oxytactic microorganisms swimming in a nanofluid layer and found that microbes speed up the onset of convection. Sampath et al. [17] analyzed a convective heated and submerged nanofluid with gyrotactic microorganisms and concluded that due to the bioconvection Lewis number, the heat transfer rate increased by 37.3%. Chu et al. [18] examined the effects of gyrotactic microorganisms on a Maxwell nanofluid under Cattaneo–Christov heat flow on a nanomaterial surface. Ayodeji et al. [19] studied a magnetic nanofluid with bioconvection on a stretched plate under the conditions of slip and Brownian motion. They determined that Brownian diffusion and the Peclet number increased the temperature of the nanofluid. Khan et al. [20] predicted the bioconvection of a magnetic nanofluid on a higher paraboloid surface. Shehzad et al. [21] analyzed Cattaneo–Christov heat and mass fluxes in the bioconvection transport of micropolar nanofluid. Khan et al. [22] examined thixotropic nanofluid flow with gyrotactic microorganisms and activation energy. Ferdows et al. [23] reported bioconvection over an exponentially extended plate using magneto-nanofluids. Waqas et al. [24] analyzed the effects of second-order slip conditions, chemical reactions, and thermal radiation on bioconvection in magneto-Carreau–Yasuda nanofluids. Recently, Muhammad et al. [25] observed bioconvection in a thixotropic magneto-nanofluid. However, the phenomenon of bioconvection in rotating geometries under slip conditions has yet to be explored.

The traditional problem of the rotating disk was studied by Von Karman [26], who introduced transformations to generate self-similar equations. Flow simulations on a spinning disk provide the information required for various applications, such as spinning machines, computer disk drives, turbine systems, crystal growth processes, spinning viscometers, and food processing. Benton [27] discussed the applications of spinning-disk problems. Turkyilmazoglu [28] considered porous-disk flow with a Hall current and highlighted the consequences and applications of Hall currents. Sheikholeslami [29] investigated a magnetic nanofluid on a rotating disk; similarly, Turkyilmazoglu [30] revealed the heat transfer patterns of the nanofluid under the revolution of the disk. Hayat et al. [31] extended this study [30] to examine the partial slip effects and found that they reduced the velocity and thus accelerated the shear stress on the disk surface. Abdel-Wahed and Akl [32] analyzed the effects of a Hall current on a magneto nanofluid on a rotating disk and reported that the Hall current improved the heat transport on the disk surface. Hayat et al. [33] studied the biphasic flow of a nanofluid on a disk under sliding conditions and described the characteristics of the thermophoresis in the heat field. Rehman et al. [34] extended this study [33] by considering a Casson nanofluid. Recent studies on the Hall current and rotating disks can be found in [35,36,37,38,39,40,41,42].

The literature review revealed that only a few studies have explored the effect of Hall currents and viscous dissipation on nanofluid flow with gyrotactic microorganisms on a spinning disk. The additional features of multiple slip conditions and thermal radiation with viscous heating in the proposed mathematical problem distinguish this novel study from others in the literature. The consideration of Buongiorno’s biphasic model and viscous heating with bioconvection resulted in a highly non-linear model that was solved by the shooting method. The density number, Sherwood number, shear stress, and Nusselt number were calculated and analyzed. The paper is organized as follows: Section 2 deals with the formulation of the model; Section 3 covers the shooting solution of the self-similar equations; the findings are presented in detail in Section 4; and the conclusions are provided in Section 5.

2. Formulation of the Problem

Let us consider the three-dimensional flow of a nanofluid submerged with gyrotactic microorganisms on an infinite revolving disk. The nanofluid of a constant density ($\rho$), thermal conductivity ($k$), specific heat ($\rho C_p$), electrical conductivity ($\sigma$), Brownian motion coefficient ($D_B$), thermophoresis coefficient ($D_T$), microorganism diffusivity ($D_W$), and dynamic viscosity ($\mu$) obeys the non-homogeneous model of Buongiorno. The Hall effect with a strong magnetic field is applied to the flow system. The generalized mathematical problem, which describes the laminar flow, is seen below:

$$\nabla ·\overrightarrow{U}=0$$

(1)

$$\rho \left(\frac{\partial \overrightarrow{U}}{\partial t}+\left(\overrightarrow{U}·\nabla \right)\overrightarrow{U}\right)=-\nabla p+\mu {\nabla }^2\overrightarrow{U}+\overrightarrow{J}\times \overrightarrow{B}$$

(2)

$$\rho C_p\left(\frac{\partial T}{\partial t}+\left(\overrightarrow{U}·\nabla \right)T\right)=k{\nabla }^{2}T+\Phi +{\left(\rho C_p\right)}_{np}\left(D_B\nabla C·\nabla T+\frac{D_T}{T_{\infty }}\nabla T·\nabla T\right)-\nabla q_r,$$

(3)

$$\frac{\partial C}{\partial t}+\left(\overrightarrow{U}·\nabla \right)C=D_B{\nabla }^{2}C+\frac{D_T}{T_{\infty }}{\nabla }^{2}T,$$

(4)

$$\frac{\partial W}{\partial t}+\nabla ·\left(WU+W\gamma -D_W\nabla W\right)=0,$$

(5)

where $\gamma$ is defined as

$$\gamma =\left(\frac{b{W}_c}{C_w-C_{\infty }}\right)\nabla C.$$

(6)

The Ohms law with no electric field to define the Hall current is

$$\overrightarrow{J}=\sigma \left(\overrightarrow{U}\times \overrightarrow{B}-m\left(\overrightarrow{J}\times \overrightarrow{B}\right)\right)$$

(7)

The radiative heat flux $q_r$ is defined as

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\nabla T^4$$

(8)

The operators in the cylindrical system are

$$\left(\nabla ,{\nabla }^2\right)=\left(\frac{\partial }{\partial r}\widehat{e_r}+\frac{1}{r}\frac{\partial }{\partial \phi }\widehat{e_{\phi }}+\frac{\partial }{\partial z}\widehat{e_z}, \frac{{\partial }^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{{\partial }^2}{\partial {\phi }^2}+\frac{{\partial }^2}{\partial z^2}\right),$$

(9)

where $t$ is the time, $\overrightarrow{U}$ is the velocity vector, $\overrightarrow{J}$ is the current density, $\overrightarrow{B}$ is the magnetic field, $p$ is the pressure, $T$ is the temperature, $C$ is the nanoparticle volume fraction, $W$ is the density of microorganisms, ${\left(\rho C_p\right)}_{np}$ is the specific heat of the nanoparticles, $\Phi$ is the viscous dissipation, $T_{\infty }$ is the ambient temperature, $m=1/n_{e}n$ is the Hall factor, $n_e$ is the electron concentration, $n$ is the electron charge, $b$ is the chemotaxis constant, $W_c$ is the extreme speed of cell swimming, $k^{*}$ is the Rosseland mean absorption, ${\sigma }^{*}$ is the Stefan–Boltzmann constant, and $\Delta C$ is the concentration gradient.

As seen in the geometric diagram (Figure 1), the disk rotates with a velocity $\mathsf{\Omega }$ around the $z$-axis. Considering a cylindrical system ($r, \phi ,z$), the $r$-axis is constrained along the radial direction, and the $z$-axis is taken as normal to it. The disk surface has a uniform temperature, nanoparticle concentration, and density of microorganisms, which are constant under ambient conditions. A steady flow is considered, without suction or injection. The disk surface is subjected to multiple slippage conditions. The simplified component equations are as follows (see [30,32,36]):

$$\frac{\partial u}{\partial r}+\frac{v}{r}+\frac{\partial w}{\partial z}=0,$$

(10)

$$\rho \left(u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z }-\frac{v^2}{r}\right)=-\frac{\partial p}{\partial r}+\mu \left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}-\frac{u}{r^2}+\frac{{\partial }^{2}u}{\partial z^2}\right)-\frac{\sigma B_0^2}{1+m^2}\left(u-mv\right),$$

(11)

$$\rho \left(u\frac{\partial v}{\partial r}+w\frac{\partial v}{\partial z }+\frac{uv}{r}\right)=\mu \left(\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\frac{\partial v}{\partial r}-\frac{v}{r^2}+\frac{{\partial }^{2}v}{\partial z^2}\right)-\frac{\sigma B_0^2}{1+m^2}\left(v+mu\right),$$

(12)

$$\rho \left(u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu \left(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial z^2}\right),$$

(13)

$$\rho C_p\left(u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}\right)=\left(k+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\right)\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r }+\frac{{\partial }^{2}T}{\partial z^2}\right)+D_B{\left(\rho C_p\right)}_{np}\left(\frac{\partial T}{\partial z}\frac{\partial C}{\partial z}+\frac{\partial T}{\partial r}\frac{\partial C}{\partial r}\right)$$

$$+\frac{D_T{\left(\rho C_p\right)}_{np}}{T_{\infty }}\left({\left(\frac{\partial T}{\partial z}\right)}^2+{\left(\frac{\partial T}{\partial r}\right)}^2\right)+\mu \left({\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial r}\right)}^2+{\left(\frac{\partial v}{\partial r}-\frac{v}{r}\right)}^2\right)$$

$$+2\mu \left({\left(\frac{\partial u}{\partial r}\right)}^2+{\left(\frac{u}{r}\right)}^2+{\left(\frac{\partial w}{\partial z}\right)}^2\right),$$

(14)

$$u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}=D_B\left(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r }+\frac{{\partial }^{2}C}{\partial z^2}\right)+\frac{D_T}{T_{\infty }}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r }+\frac{{\partial }^{2}T}{\partial z^2}\right),$$

(15)

$$u\frac{\partial W}{\partial r}+w\frac{\partial W}{\partial z}+\frac{b{W}_c}{C_w-C_{\infty }}\left(\frac{\partial W}{\partial z}\frac{\partial C}{\partial z}+W\frac{{\partial }^{2}C}{\partial z^2}\right)=D_W\left(\frac{{\partial }^{2}W}{\partial r^2}+\frac{1}{r}\frac{\partial W}{\partial r }+\frac{{\partial }^{2}W}{\partial z^2}\right),$$

(16)

The apposite boundary conditions are (see [32,36])

$$u=L_1\frac{\partial u}{\partial z}, v=v_w+L_1\frac{\partial v}{\partial z}, w=0, T=T_w+L_2\frac{\partial T}{\partial z},$$

$$C=C_w+L_3\frac{\partial C}{\partial z}, W=W_w+L_4\frac{\partial W}{\partial z} \mathrm{at} z=0$$

(17)

$$u\to 0, v\to 0, T\to T_{\infty },C\to C_{\infty }, W\to W_{\infty } \mathrm{as} z\to \infty ,$$

(18)

where $L_i$ (i = 1,2,3,4) are the slip coefficients; $v_w=r\mathsf{\Omega }$; and subscripts $w$ and $\infty$ denote the disk surface and ambient state, respectively.

Now consider the following transformations ([26,32,42]):

$$\xi =\frac{z}{r}\sqrt{Re}, f^{\prime }\left(\xi \right)=\frac{u}{r\mathsf{\Omega }}, g\left(\xi \right)=\frac{v}{r\mathsf{\Omega }}, f\left(\xi \right)=\frac{w}{-\sqrt{2\mathsf{\Omega }\nu }},$$

$$P\left(\xi \right)=\frac{p_{\infty }-p}{\mathsf{\Omega }\mathsf{\mu }}, \theta \left(\xi \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}, \varphi \left(\xi \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }}, \mathsf{\Psi }\left(\xi \right)=\frac{W-W_{\infty }}{W_w-W_{\infty }}$$

(19)

where $\xi , f^{\prime }\left(\xi \right), g\left(\xi \right), f\left(\xi \right), P,\varphi , \mathsf{\Psi }\left(\xi \right),$ and $\theta$ are the dimensionless similarity variable, radial velocity, tangential velocity, axial velocity, pressure, nanoparticle concentration, microorganism distribution, and temperature, respectively.

Plugging Equation (19) into (10)–(18), we obtain the self-similar equations:

$$2f^{‴}+2f{f}^{″}-f^{\prime 2}+g^2-\frac{M}{1+m^2}\left(f^{\prime }-mg\right)=0,$$

(20)

$$2g^{″}+2f{g}^{\prime }-2f^{\prime }g-\frac{M}{1+m^2}\left(g+m{f}^{\prime }\right)=0,$$

(21)

$$P^{\prime }+f{f}^{\prime }+f^{\prime }=0,$$

(22)

$$\frac{3+4R}{3Pr}{\theta }^{″}+f{\theta }^{\prime }+Nb{\theta }^{\prime }{\varphi }^{\prime }+Nt{\theta }^{\prime 2}+Ec\left(f^{″2}+6R{e}^{-1}{f}^{\prime 2}\right)=0,$$

(23)

$${\varphi }^{″}+PrLef{\varphi }^{\prime }+\frac{Nt}{Nb}{\theta }^{″}=0,$$

(24)

$${\mathsf{\Psi }}^{″}+Lbf {\mathsf{\Psi }}^{\prime }-Pe\left({\varphi }^{″}\left(\mathsf{\Psi }+\mathsf{\Xi }\right)+ {\mathsf{\Psi }}^{\prime }{\varphi }^{\prime }\right)=0,$$

(25)

Similarly, the boundary conditions (17) and (18) yield

$$f\left(\xi \right)=0, f^{\prime }\left(\xi \right)={\alpha }_1{f}^{″}\left(\xi \right), g\left(\xi \right)=1+{\alpha }_1{g}^{\prime }\left(\xi \right), \theta \left(\xi \right)=1+{\alpha }_2{\theta }^{\prime }\left(\xi \right),$$

$$\varphi \left(\xi \right)=1+{\alpha }_3{\varphi }^{\prime }\left(\xi \right), \mathsf{\Psi }\left(\xi \right)=1+{\alpha }_4{\mathsf{\Psi }}^{\prime }\left(\xi \right) \mathrm{at} \xi =0,$$

(26)

$$f^{\prime }\left(\xi \right)\to 0, g\left(\xi \right)\to 0, \theta \left(\xi \right)\to 0, \varphi \left(\xi \right)\to 0, \mathsf{\Psi }\left(\xi \right)\to 0 \mathrm{as} \xi \to \infty$$

(27)

where $M=\frac{\sigma B_0^2}{\rho \mathsf{\Omega }}$ denotes the magnetic parameter, $R=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k{k}^{*}}$ denotes the radiation parameter, $Pr=\frac{\mu C_p}{k}$ denotes the Prandtl number, $Nb=\frac{{\left(\rho C_p\right)}_{np}{D}_B\left(C_w-C_{\infty }\right)}{\mu C_p}$ denotes the the Brownian motion parameter, $Le=\frac{k}{\rho C_p{D}_B}$ denotes the Lewis number, $Nt=\frac{{\left(\rho C_p\right)}_{np}{D}_T\left(T_w-T_{\infty }\right)}{T_{\infty }\mu C_p}$ denotes the thermophoresis number, $Lb=\frac{\nu }{D_B}$ denotes the bioconvection Lewis number, $Pe=\frac{b{W}_c}{D_W}$ denotes the Peclet number, $\mathsf{\Xi }=\frac{C_{\infty }}{\left(C_w-C_{\infty }\right)}$ denotes the nanoparticle concentration ratio, $Ec=\frac{v_w^2}{C_p\left(T_w-T_{\infty }\right)}$ denotes the Eckert number, ${\alpha }_1=L_1\sqrt{\frac{2\mathsf{\Omega }}{\nu }}$ denotes the velocity slip parameter, ${\alpha }_2=L_2\sqrt{\frac{2\mathsf{\Omega }}{\nu }}$ denotes the thermal slip parameter, ${\alpha }_3=L_3\sqrt{\frac{2\mathsf{\Omega }}{\nu }}$ denotes the solutal slip parameter, and ${\alpha }_4=L_4\sqrt{\frac{2\mathsf{\Omega }}{\nu }}$ denotes the microorganism slip parameter.

The shear stress on the disk surface along the tangential and radial directions is given by:

$${\tau }_{\phi }={\left[\mu \left(\frac{\partial v}{\partial z }+\frac{1}{z}\frac{\partial w}{\partial \phi }\right)\right]}_{z=0}$$

(28)

$${\tau }_r={\left[\mu \left(\frac{\partial u}{\partial z }+\frac{\partial w}{\partial r}\right)\right]}_{z=0}$$

(29)

The total heat flux, nanoparticle mass flux, and microorganism mass flux on the disk surface are defined as:

$$q_w={\left[q_r-k\frac{\partial T}{\partial z }\right]}_{z=0}$$

(30)

$$q_m={\left[-D_B\frac{\partial C}{\partial z }\right]}_{z=0}$$

(31)

$$q_n={\left[-D_W\frac{\partial W}{\partial z }\right]}_{z=0}$$

(32)

Now, the coefficients of the wall friction along the tangential and radial directions, the Nusselt number, the Sherwood number, and the local motile number are computed using:

$$C_g=\frac{{\tau }_{\phi }}{\frac{1}{2}\rho v_w^2}$$

(33)

$$C_f=\frac{{\tau }_r}{\frac{1}{2}\rho v_w^2},$$

(34)

$$Nu=\frac{r{q}_w}{k\left(T_w-T_{\infty }\right)}$$

(35)

$$Sh=\frac{r{q}_m}{D_B\left(C_w-C_{\infty }\right)},$$

(36)

$$Dn=\frac{r{q}_n}{D_W\left(W_w-W_{\infty }\right)},$$

(37)

The self-similar forms of (33)–(37) are given below:

$$R{e}^{1/2}{C}_g=g^{\prime }\left(0\right), R{e}^{1/2}{C}_f=f^{″}\left(0\right), R{e}^{-1/2}Nu=-\left(1+\frac{4}{3}R\right){\theta }^{\prime }\left(0\right),$$

(38)

$$R{e}^{-1/2}Sh=-{\varphi }^{\prime }\left(0\right), R{e}^{-1/2}Dn=-{\mathsf{\Psi }}^{\prime }\left(0\right),$$

(39)

where $Re=r{v}_w/\mathsf{\Omega }$ is the local Reynolds number.

3. Numerical Approach

The nonlinear parametrized and normalized governing Equations (20)–(25) subject to (26) and (27) were coupled, but the solutions in closed form were impractical; therefore, they were solved using the shooting method combined with the Runge–Kutta method (see [43] for more details). As a first step, let us introduce

$$\left({\mathcal{Z}}_1, {\mathcal{Z}}_2, {\mathcal{Z}}_3, {\mathcal{Z}}_4,{\mathcal{Z}}_5, {\mathcal{Z}}_6, {\mathcal{Z}}_7, {\mathcal{Z}}_8,{\mathcal{Z}}_9,{\mathcal{Z}}_{10},{\mathcal{Z}}_{11}\right)=\left(f,f^{\prime },f^{″},g,g^{\prime },\theta , {\theta }^{\prime },\varphi ,{\varphi }^{\prime },\mathsf{\Psi },{\mathsf{\Psi }}^{\prime }\right),$$

to obtain the initial value problem

$${\mathcal{Z}}_1^{\prime }={\mathcal{Z}}_2,\to {\mathcal{Z}}_1\left(0\right)=0,$$

(40)

$${\mathcal{Z}}_2^{\prime }={\mathcal{Z}}_3,\to {\mathcal{Z}}_2\left(0\right)={\alpha }_1{s}_1,$$

(41)

$${\mathcal{Z}}_3^{\prime }=\left(-2{\mathcal{Z}}_1{\mathcal{Z}}_3+{\mathcal{Z}}_2^2-{\mathcal{Z}}_4^2+\frac{M}{1+m^2}\left({\mathcal{Z}}_2-m{\mathcal{Z}}_4\right)\right)/2, {\mathcal{Z}}_3\left(0\right)=s_1$$

(42)

$${\mathcal{Z}}_4^{\prime }={\mathcal{Z}}_5,\to {\mathcal{Z}}_4\left(0\right)=1+{\alpha }_1{s}_2,$$

(43)

$${\mathcal{Z}}_5^{\prime }=\left(-2{\mathcal{Z}}_1{\mathcal{Z}}_5+2{\mathcal{Z}}_2{\mathcal{Z}}_4+\frac{M}{1+m^2}\left({\mathcal{Z}}_4+m{\mathcal{Z}}_2\right)\right)/2,{\mathcal{Z}}_5\left(0\right)=s_2$$

(44)

$${\mathcal{Z}}_6^{\prime }={\mathcal{Z}}_7\to {\mathcal{Z}}_6\left(0\right)=1+{\alpha }_2{s}_3,$$

(45)

$${\mathcal{Z}}_7^{\prime }=\left(\frac{-3Pr}{3+4R}\right)\left({\mathcal{Z}}_1{\mathcal{Z}}_7+Nb{\mathcal{Z}}_7{\mathcal{Z}}_9+Nt{\mathcal{Z}}_7^2+Ec\left({\mathcal{Z}}_3^2+6R{e}^{-1}{\mathcal{Z}}_2^2\right)\right),\to {\mathcal{Z}}_7\left(0\right)=s_3,$$

(46)

$${\mathcal{Z}}_8^{\prime }={\mathcal{Z}}_9, {\mathcal{Z}}_8\left(0\right)=1+{\alpha }_3{s}_4,$$

(47)

$${\mathcal{Z}}_9^{\prime }=-PrLe{\mathcal{Z}}_1{\mathcal{Z}}_9-\frac{Nt}{Nb}{\mathcal{Z}}_7\prime \to {\mathcal{Z}}_9\left(0\right)=s_4,$$

(48)

$${\mathcal{Z}}_{10}^{\prime }={\mathcal{Z}}_{11},\to {\mathcal{Z}}_{10}\left(0\right)=1+{\alpha }_4{s}_5,$$

(49)

$${\mathcal{Z}}_{11}^{\prime }=-Lb{\mathcal{Z}}_1{\mathcal{Z}}_{11}+Pe\left({\mathcal{Z}}_9^{\prime }\left({\mathcal{Z}}_{10}+\mathsf{\Xi }\right)+{\mathcal{Z}}_9{\mathcal{Z}}_{11}\right),\to {\mathcal{Z}}_{11}\left(0\right)=s_5,$$

(50)

where $s_i$ ($i=1,2,3,4,5$) are estimated unknowns. The classical Runge–Kutta method was used to solve the above problem with a suitable guess for $s_i$’s. Using the Newton–Raphson method, the guesses were refined until the solution satisfied the required accuracy of ${10}^{-6}$ (see [44,45] for more details). For a validation of the obtained results, we drew a comparison with the available data (Rehman et al. [34] and Hayat et al. [33]); the compared data are presented in

Table 1. The values of $R{e}^{-1/2}Nu$ for different values of $M$ and ${\alpha }_1$ when $Rd=m={\alpha }_2={\alpha }_3=Ec=0$ in the absence of microorganisms.

$\mathit{M}$	${\mathit{\alpha }}_1$	Hayat et al. [33]	Rehman et al. [34]	Our Results
0	0.7	0.30494	0.3050	0.304948
0.7	0.7	0.24421	0.2443	0.244214
1.4	0.7	0.17566	0.1757	0.175661
0.3	0.2	0.32655	0.3266	0.326552
0.3	0.5	0.30360	0.3036	0.303607
0.3	0.8	0.28715	0.2872	0.287155

. Our results coincided exactly with those of [33] and [34].

4. Results and Discussion

In this section, we analyze the physical effects of the magnetic field ($M$), Hall current ($m$), velocity slip (${\alpha }_1$), viscous dissipation ($Ec$), Rosseland radiation ($R$), Brownian motion ($Nb$), thermophoresis ($Nt$), Reynolds number ($Re$), thermal slip (${\alpha }_2$), nanoparticle solute slip (${\alpha }_3$), Lewis number ($Le$), motile microorganism slip (${\alpha }_4$), and bioconvection Lewis number ($Lb$) on the radial velocity ($f\prime \left(\xi \right)$), tangential velocity ($g\left(\xi \right)$), temperature ($\theta \left(\xi \right)$), nanoparticle concentration ($\varphi \left(\xi \right)$), and motile microorganism distribution ($\mathsf{\Psi }\left(\xi \right)$) through the graphical representations presented in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25. Furthermore, the radial wall-friction coefficient (${\mathrm{Re}}^{1/2}{C}_f$), tangential wall-friction coefficient (${\mathrm{Re}}^{1/2}{C}_g$), Nusselt number (${\mathrm{Re}}^{-1/2}Nu$), Sherwood number (${\mathrm{Re}}^{-1/2}Sh$), and motile microorganism number (${\mathrm{Re}}^{-1/2}Dn$) are computed and presented in Figure 26, Figure 27, Figure 28, Figure 29, Figure 30 and Figure 31 and

Table 2. The values of ${\mathrm{Re}}^{1/2}{C}_f$, ${\mathrm{Re}}^{1/2}{C}_g$, ${\mathrm{Re}}^{-1/2}Nu$, ${\mathrm{Re}}^{-1/2}Sh$, and ${\mathrm{Re}}^{-1/2}Dn$ for different values of $m, {\alpha }_1$, and $M$ when $Pr=6, R={\alpha }_2={\alpha }_3={\alpha }_4=Nb=0.5, Ec=Nt=Pe=0.1,Re=1, Le=Lb=3$, and $\mathsf{\Xi }=0.2$.

${\mathit{\alpha }}_1$	$\mathit{m}$	$\mathit{M}$	$\mathbf{R}{\mathbf{e}}^{1/2}{\mathit{C}}_{\mathit{f}}$	$\mathbf{R}{\mathbf{e}}^{1/2}{\mathit{C}}_{\mathit{g}}$	$\mathbf{R}{\mathbf{e}}^{-1/2}\mathit{N}\mathit{u}$	$\mathbf{R}{\mathbf{e}}^{-1/2}\mathit{S}\mathit{h}$	$\mathbf{R}{\mathbf{e}}^{-1/2}\mathit{D}\mathit{n}$
0	0	0.5	0.27223285	−0.60016174	0.18411756	0.61217292	0.35891396
	0.5		0.34804091	−0.60760591	0.20678466	0.66211221	0.40193257
	1		0.38657686	−0.57472358	0.21535335	0.68590441	0.42344578
	0.5	0	0.36000722	−0.43512123	0.21557962	0.68362256	0.42656590
		1	0.35017124	−0.75136693	0.19808066	0.64951373	0.38490344
		2	0.34939527	−0.98614427	0.18346595	0.63623923	0.36227622
1	0	0.5	0.06907399	−0.37685871	0.19043449	0.65499791	0.35472070
	0.5		0.09729511	−0.38822687	0.23499319	0.73620866	0.41653888
	1		0.11529524	−0.38145137	0.25654230	0.77752075	0.44911090
	0.5	0	0.12206753	−0.31990335	0.26542362	0.79444265	0.46550481
		1	0.08479902	−0.43742667	0.21112374	0.69834925	0.38281983
		2	0.07294442	−0.50408669	0.17797708	0.65135755	0.33905929

and

Table 3. The values of ${\mathrm{Re}}^{-1/2}Nu$, ${\mathrm{Re}}^{-1/2}Sh$, and ${\mathrm{Re}}^{-1/2}Dn$ for different values of $R, {\alpha }_2,{\alpha }_3$, and ${\alpha }_4$ when $Pr=6, {\alpha }_1=R=Ha=m=Nb=0.5, Ec=Nt=Pe=0.1,Re=1, Le=Lb=3$, and $\mathsf{\Xi }=0.2$.

$\mathit{R}$	${\mathit{\alpha }}_2$	${\mathit{\alpha }}_3$	${\mathit{\alpha }}_4$	$\mathbf{R}{\mathbf{e}}^{-1/2}\mathit{N}\mathit{u}$	$\mathbf{R}{\mathbf{e}}^{-1/2}\mathit{S}\mathit{h}$	$\mathbf{R}{\mathbf{e}}^{-1/2}\mathit{D}\mathit{n}$
0	0.5	0.5	0.5	0.09536398	0.74735703	0.42765668
1				0.37593681	0.73549955	0.42662967
2				0.61025676	0.73202599	0.42630997
0.5	0	0.5	0.5	0.25909380	0.74013993	0.42704913
	1			0.22314167	0.73889796	0.42692132
	2			0.19518753	0.73798530	0.42682639
0.5	0.5	0	0.5	0.13514058	1.17259288	0.45258266
		1		0.30584709	0.53909050	0.41493130
		2		0.37956428	0.34920648	0.40338675
0.5	0.5	0.5	0	0.23995186	0.73946890	0.53998522
			1	0.23995170	0.73946891	0.35308806
			2	0.23995170	0.73946888	0.26230148

. In the simulations, unless indicated otherwise, we set $Pr=6, Ha=m=R={\alpha }_1={\alpha }_2={\alpha }_3={\alpha }_4=Nb=0.5, Ec=Nt=Pe=0.1,Re=1, Le=Lb=3$, and $\mathsf{\Xi }=0.2$.

4.1. Velocity Profiles ($f^{\prime }\left(\xi \right),g\left(\xi \right)$)

Figure 2 and Figure 3 show the effect of the magnetic number $M$ on the radial velocity ($f^{\prime }\left(\xi \right)$) and tangential velocity ($g\left(\xi \right)$), respectively. Both velocities were reduced under a stronger magnetic field, because the higher the $M$ value, the greater the Lorentz force (typically a repellent force). The Lorentz force opposes fluid motion on the surface of a disk, and hence the fluid motion slowed down in both the radial and tangential directions. Therefore, both $f^{\prime }\left(\xi \right)$ and $g\left(\xi \right)$ decreased as $M$ increased. Here, $M=0$ corresponds to the absence of a Lorentz force, at which point the velocities were highest over the entire surface of the disk. The effect of $m$ on $f^{\prime }\left(\xi \right)$ and $g\left(\xi \right)$ is depicted in Figure 4 and Figure 5, respectively. Note that a stronger Hall current facilitates the movement of the fluid, so the velocities increased with $m$. The Hall current had an increased effect on the radial velocity compared to the velocity in other directions. Figure 6 and Figure 7 present the impact of ${\alpha }_1$ on $f^{\prime }\left(\xi \right)$ and $g\left(\xi \right)$. Here, ${\alpha }_1=0$ corresponds to the no-slip velocity conditions. The radial velocity on the disk surface was greater under the slip velocity conditions, whereas the tangential velocity component on the disk surface was smaller under the slip conditions. Both velocity components ($f^{\prime }\left(\xi \right)$ and $g\left(\xi \right)$) diminished with the increase in the values of ${\alpha }_1$. Physically, the slip conditions on the disk serve as a repellent agent that reduces the velocity.

4.2. Temperature Profile ($\theta \left(\xi \right)$)

Figure 8 and Figure 9 illustrate the effect of $M$ and $m$ on $\theta \left(\xi \right)$. In the absence of a magnetic field, the temperature $\theta \left(\xi \right)$ on the disk surface was lower. Increasing the $M$ value led to a higher temperature $\theta \left(\xi \right)$, while the opposite trend was observed for increasing the $m$ value. Figure 10 shows that the temperature $\theta \left(\xi \right)$ increased steadily as the Eckert numbers increased. Physically, the kinetic energy resulting from viscous heating acts as a heat-generating agent that improves the temperature $\theta \left(\xi \right)$. Figure 11 shows that increasing the radiation number led to a significant improvement in the temperature $\theta \left(\xi \right)$ due to the electromagnetic waves exerted by the solar radiation. Figure 11 demonstrates that the solution converged when $R=0$. The impacts of $Nb$ and $Nt$ on $\theta \left(\xi \right)$ are depicted in Figure 12 and Figure 13. Brownian motion creates additional thermal energy due to the accidental motion of nanoparticles, which improves the thermal distribution $\theta \left(\xi \right)$. Likewise, the heat migration of the nanoparticles acts as a heat-generating agent, causing an increase in $\theta \left(\xi \right)$. The $Nt$ had more of a pronounced effect than the $Nb$ on the $\theta \left(\xi \right)$. Larger Reynolds numbers ($Re$) correspond to a weaker viscous force, and such a weaker viscous force reduced the $\theta \left(\xi \right)$. In Figure 15, ${\alpha }_2=0$. represents conditions without thermal slip, where $\theta \left(\xi \right)$ had the value 1, corresponding to isothermal conditions. For a larger ${\alpha }_2$, the temperature $\theta \left(\xi \right)$ dropped on the disk surface and decreased throughout the fluid region as ${\alpha }_2$ increased.

4.3. Nanoparticle Concentration Profile ($\varphi \left(\xi \right)$)

The effects of the Hall current and magnetic field on $\varphi \left(\xi \right)$ were contradictory, as shown in Figure 16 and Figure 17. That is, $\varphi \left(\xi \right)$ rose at the $M$ values decreased and the values of $m \mathrm{increased}$. The increase in the Brownian number reduced $\varphi \left(\xi \right),$ as shown in Figure 18. The greater the strength of Brownian motion, the lower the concentration gradient of the nanoparticles; therefore, $\varphi \left(\xi \right)$ was reduced. The impact of $Nt$ on $\varphi \left(\xi \right)$ is reported in Figure 19, which shows that the $\varphi \left(\xi \right)$ increased pointedly with rising $Nt$ values. The higher the $Nt$ value, the higher the nanoparticle concentration gradient, which caused $\varphi \left(\xi \right)$ to increase unevenly. In Figure 20, ${\alpha }_3=0$ does not imply slip conditions in the solute, and therefore the $\varphi \left(\xi \right)$ increased on the disk surface. Furthermore, increasing the ${\alpha }_3$ values pointedly diminished the nanoparticle concentration profile. Figure 21 shows that $\varphi \left(\xi \right)$ declined with an increase in the Lewis number. Physically, the larger the $Le$, the weaker the diffusivity of the solute, thus reducing the $\varphi \left(\xi \right) \mathrm{value}$.

4.4. Motile Microorganism Profile ($\Psi \left(\xi \right)$)

Figure 22 and Figure 23 illustrate the motile microorganism distribution $\mathsf{\Psi }\left(\xi \right)$ with respect to the magnetic number $\left(M\right)$ and the Hall number $\left(m\right)$, respectively. The $\mathsf{\Psi }\left(\xi \right) \mathrm{profile}$ decreased with an increase in the Hall factor $\left(m\right)$, while it improved with a rise in the $M$ value. Physically, the Lorentz force facilitates the increased diffusivity of microorganisms and, thereby, amplifies the $\mathsf{\Psi }\left(\xi \right) \mathrm{profile}$. The higher the bioconvection Lewis number, the weaker the diffusivity of the bioconvection; therefore, for larger $Lb$ values, the $\mathsf{\Psi }\left(\xi \right)$ profile became smaller, as seen in Figure 24 and Figure 25, which display the effects of ${\alpha }_4$ on the $\mathsf{\Psi }\left(\xi \right)$ profile. The microorganism profile decreased under slip conditions.

4.5. Physical Quantities ($R{e}^{1/2}{C}_f$, $R{e}^{1/2}{C}_g$, $R{e}^{-1/2}Nu$, $R{e}^{-1/2}Sh$, and $R{e}^{-1/2}Dn$)

Figure 26 shows the contours of ${\mathrm{Re}}^{1/2}{C}_f$ plotted against $m$ and $M$. Increasing the $m$ value reduced the ${\mathrm{Re}}^{1/2}{C}_f$, while the effect of $M$ was almost invariant for different values of $m$. A lower value of $M$ and a higher value of $m$ produced the lowest radial shear stress on the disk wall. Figure 27 demonstrates that ${\mathrm{Re}}^{1/2}{C}_g$ did not vary with respect to $M$ for different values of $m$. Lower values of $Ec$ and $Nb$ achieved the maximum ${\mathrm{Re}}^{-1/2}Nu$ value (see Figure 28). Figure 29 shows that ${\mathrm{Re}}^{-1/2}Nu$ decreased with increased radiation, while it rose in conjunction with $Nt$. Therefore, to achieve a maximum ${\mathrm{Re}}^{-1/2}Nu$, $\mathrm{the} Nt$ value must be as high as possible and the $R$ value must be low. The effects of ${\alpha }_3$ and $Le$ on ${\mathrm{Re}}^{-1/2}Sh$ conflicted (see Figure 30). The non-linear effects of $Nb$ and $Nt$ on ${\mathrm{Re}}^{-1/2}Sh$ are illustrated in Figure 31.

Table 2 presents the behavior of ${\mathrm{Re}}^{1/2}{C}_f$, ${\mathrm{Re}}^{1/2}{C}_g$, ${\mathrm{Re}}^{-1/2}Nu$, ${\mathrm{Re}}^{-1/2}Sh$, and ${\mathrm{Re}}^{-1/2}Dn$ for different values of $m, {\alpha }_1$, and $M$. Under no-slip velocity conditions, ${\mathrm{Re}}^{1/2}{C}_f$, ${\mathrm{Re}}^{1/2}{C}_g$, ${\mathrm{Re}}^{-1/2}Nu$, and ${\mathrm{Re}}^{-1/2}Sh$ were higher than under velocity slip conditions, while this result was reversed for ${\mathrm{Re}}^{-1/2}Dn$. Both ${\mathrm{Re}}^{1/2}{C}_f$ and ${\mathrm{Re}}^{1/2}{C}_g$ increased in conjunction with $m$, while the opposite was true for $M$. ${\mathrm{Re}}^{-1/2}Nu$, ${\mathrm{Re}}^{-1/2}Sh$, and ${\mathrm{Re}}^{-1/2}Dn$ increased with $m$, while the opposite was true for $M$. Table 3 presents the behavior of ${\mathrm{Re}}^{-1/2}Nu$, ${\mathrm{Re}}^{-1/2}Sh$, and ${\mathrm{Re}}^{-1/2}Dn$ for different values of $R, {\alpha }_2,{\alpha }_3,$ and ${\alpha }_4$.${\mathrm{Re}}^{-1/2}Nu$ increased as the values of $R$ and ${\alpha }_3$ increased, while this outcome was reversed for ${\alpha }_2$ and ${\alpha }_4$. ${\mathrm{Re}}^{-1/2}Sh$ and ${\mathrm{Re}}^{-1/2}Dn$ diminished as $R, {\alpha }_2,{\alpha }_3$, and ${\alpha }_4 \mathrm{increased}$.

5. Concluding Remarks

Three-dimensional bioconvection viscous dissipating nanofluid flow on a spinning disk was investigated, along with the Hall current, magnetic field, radiative heat transfer, and multiple slip effects. The framework of the Buongiorno nanofluid model was employed. The system of non-linear PDEs was simplified into dimensionless ODEs through a similarity scheme, then solved using the shooting technique (see [46,47,48,49,50]). The main results of the above analysis were:

• The radial velocity diminished with an increase in the magnetic field and rose with an increase in the Hall current.
• The velocities declined as a result of higher velocity slip parameters.
• The temperature field was improved under higher magnetic number, Eckert number, and radiation parameter values.
• The thermophoresis number had a greater impact on the heat field compared to the Brownian number.
• Multiple slip conditions reduced the transport fields.
• The frictional coefficient of the wall in the radial direction was reduced by an increase the Hall current.
• The heat transfer rate was reduced by an increase in the Brownian motion number, while an increase in thermal radiation elevated the heat transfer rate.
• An increase in Brownian motion and thermophoresis reduced the rate of heat transfer.