Nomenclature
| C Concentration of nanomaterials | |
| ${D}_{B}$ Brownian motion coefficient | |
| ${D}_{m}$ Microorganisms coefficient | |
| ${W}_{c}$ Cell-swimming speed | |
| $K$ Couple-stress number | |
| Nr Buoyancy ratio number | |
| $Rd$ Radiation number | |
| $Pr$ Prandtl parameter | |
| $Nb$ Brownian motion number | |
| $Le$ Lewis parameter | |
| $Pe$ Peclet parameter | |
| N Concentration of microorganisms | |
| ${D}_{T}$ Thermophoresis coefficient | |
| $b$ Chemotaxis constant | |
| $M$ Magnetic number | |
| S Mixed convection number | |
| Nc Bioconvection Rayleigh parameter | |
| $Lb$ Bioconvection Lewis number | |
| ${S}_{1}$ Thermal stratified variable | |
| $Nt$ Thermophoresis number | |
| ${S}_{2}$ Solutal stratified variable | |
| ${S}_{3}$ Microorganism stratified variable | |
| Greek words |
|
| ${{\rm{\Omega }}}_{C}$ Concentration relaxation parameter | |
| $\sigma $ Chemical reaction parameter | |
| ${\gamma }_{1}$ Thermal stratification Biot number | |
| ${\gamma }_{3}$ Microorganisms stratification Biot number | |
| ${C}_{f}$ Skin friction coefficient | |
| $S{h}_{x}$ Local Sherwood number | |
| ${\tau }_{w}$ Wall shear stress | |
| ${q}_{m}$ Mass flux | |
| u,v Components of velocity | |
| ${\sigma }^{* * }$ Stephan Boltzmann constant | |
| ${B}_{0}$ Magnetic field strength | |
| $\nu $ Kinematic viscosity | |
| k Thermal conductivity | |
| ${g}^{* }$ Gravity | |
| ${{\rm{\Omega }}}_{T}$ Thermal relaxation parameter | |
| $\delta $ Temperature difference parameter | |
| ${\gamma }_{2}$ Solutal stratification Biot number | |
| ${\delta }_{1}$ Microorganisms difference parameter | |
| $N{u}_{x}$ Local Nusselt number | |
| $N{n}_{x}$ Local density number of microorganisms | |
| ${q}_{w}$ Heat flux | |
| ${q}_{n}$ Motile density flux | |
| ${\rho }_{f}$ Density of liquid | |
| ${\rho }_{m}$ Density of motile microorganisms | |
| T Temperature of nanoliquids | |
| ${\alpha }_{m}$ Thermal diffusivity | |
| ${\left(\rho c\right)}_{p}$ Heat capacity | |
| ${\rho }_{p}$ Density of nanoparticles |
Nomenclature
1. Introduction
In recent years, scientists have shown remarkable interest to study the heat transport of nanofluid through a stretchable cylinder due to its implications in a wide variety of engineering and industrial applications. Concerning the physical and chemical characteristics of nanofluids, in addition to their superior thermal efficiency, nanofluids can be effectively used in a wide range of potential applications, such as heat exchangers, radiators, domestic refrigerator-freezers, glass blowers, electronic cooling systems (such as flat plates), crystal-growing, solar water heaters and paper products. Nanofluids are suspended nanoparticles with a diameter of <100 nm considered for enhanced heat conductivity. There are many uses of nanofluids in cooling devices, nuclear reactors, micro-electronics and transformer oils.
The main idea of nanofluid is given by Choi [1]. He anticipated that the involvement of metal nanoparticles in the base fluid would improve the thermal conductivity of the regular specific fluid and stimulates the growth of these liquids for heat transfer. Nanofluids have claims for sun powered water warming, upgraded transport, heat move of fridges and coolers, and ideal assimilation of sunlight-based energy. Irfan et al [2] explored the impact of activation energy on Carreau nanofluids because of the contracting/extending layer within the sight of convective conditions, Joule heating and heat source/sink. An investigation to fabricate and comprehend numerical demonstrating for non-Newtonian Williamson liquid for portraying the warm qualities of nanofluid is given by Hashim et al [3]. The objective of Moradi et al [4] was to do an exploratory investigation of the warmth move properties of multi-walled carbon nanotube nanofluids by the utilization of permeable media by a twofold line heat exchanger. Early examinations have made up for the unique consideration paid to improving energy development by working with nanoparticles to determine this perilous issue. On the slip side, within the sight of non-Newtonian nanofluids, the impacts of liquid stream, the warmth move of nanoparticles and the gyrotactic microorganism are discussed by Khan et al [5]. They found that nanofluid stream, heat move, and gyrotactic convergences of microorganisms had utilitarian ramifications for inactively directed nanofluid limit model comparative with effectively controlled nanofluid conditions. Zuhra et al [6] explored the gyrotactic microorganisms and nanoparticles alongside viscoelastic nanofluid and heat move at higher temperatures with the thermophoresis boundary. The latest EMHD nanofluids motion computational technology model on a fixed thickness sheet with fixed fluid characteristics is explored by Irfan et al [7]. The motion of chemically reacting stagnation point Powell–Eyring nanofluids by inclination cylinder with Cattaneo–Christov heat transfer is explored by Reddy et al [8]. The nature of nonlinear heat radiation via the movement of modified 2nd-grade nanoliquid was observed by Khan et al [9]. Alghamdi [10] deliberated magnetized nanoliquid flow by rotatable disk. Action of nanomaterial to diffuse blood by cylindrical tube exploiting a single kernel is explored by Uddin et al [11]. The non-uniform hemodynamic nanoliquid motion in the absence of an existing magnetic field is analyzed by Abbas et al [12]. Several scientists' works on nanofluid are supported via attempts [13–23].
The transient heat/mass transfer properties during static film flash of aqueous NaCl solution though experiments with water film concentration is investigated by Zhang et al [24]. A comprehensive numerical study of heat transfer model including both conduction and radiation under contact and non-contact conditions was anticipated by Wang et al [25]. Jia et al [26] examined the thermally conductive polymer composites with the integration of superior thermal conductivity.
Bioconvection can be categorized as the spectacle of the macroscopic motion of the fluid created by the gradient of density that has been formed by the collective directional swimming of micro-organisms. Such micro-organisms can be categorized as gyrotactic, oxytactic, gravitaxis and chemotaxis depending on the cause of implementation. Such self-propelled micro-organisms want to collect near upper zone of liquid layer so that they pass there, which builds up a thick upper surface and become unstable/destabilized. Then upward swimming induces microorganisms to crumble and macroscopic convection to form. Bioconvection can be used in a wide variety of applications such as biological applications and microsystems, the pharmaceutical industry, biopolymer synthesis, environmentally safe applications, sustainable fuel cell technologies, microbial enhanced oil recovery, biosensors and biotechnology, and continuous modifications in mathematical modeling. Improvements in laboratory and field testing are utilized to develop the design of such structures. Kuznetsov [27] developed the principle of nanofluid bioconvection. Bioconvection is an idea used to clarify the system of incautious model improvement in microorganism liquids, for example, microbes and green growth was explained by Ghorai and Hill [28]. The homogenous miniature polar bioconvective liquid stream with nanoparticles and gyrotactic microorganisms is investigated by Atif et al [29]. Two-dimensional bioconvection blended pressure liquid stream with nanoparticles and gyrotactic motile microorganisms is concentrated by Khan et al [30]. A bioconvection stream of summed up 2nd grade nanofluid is explored by Li et al [31]. The non-warm radiation impacts of the Oldroyd-B nanofluid stream by swimming motile microorganisms past the turning plate have inspected by Waqas et al [32]. Number of researchers in bioconvection has been approved out by investigations [33–42].
Above works investigation stimulates the current study, the foremost impartial to examine the upshot of motile microorganisms, activation energy and Cattaneo Christov double-diffusion on the flow of Newtonian nanofluid over an inclined stretching cylinder. The leading expressions of energy, momentum, volumetric and motile concentrations are initially concentrated to a self-similar arrangement by using appropriate similarities. Then obtained system is cracked numerically by bvp4c scheme, a program in the MATLAB. The graphical results of prominent parameters against all-controlling concentrations are illustrated through plots.
2. Mathematical modeling
Consider the flow of Newtonian nanofluid flow with the consequences of activation energy, magnetic field and motile microorganisms' concentration through the inclined stretching cylinder (see figure 1). Bioconvection mechanism and Buongiorno relation are delighted to illustrate Brownian and thermophoresis diffusions and motility of liquid because of motile microorganisms and nanomaterials. The relation for the Newtonian fluid is1 ) is a mathematical form of Newton's law of viscosity. Here ${\tau }_{yx}$ is shear stress and $\mu $ is called dynamic viscosity. The power-law model is expressed in equation (2 ) which can be reduced to the Newton's law of viscosity by putting $\mu =k$ and $n=1$ where n is flow behavior index. Therefore3 ) is kinematic viscosity. Flow concerning differential equations is given as:1 )–(10 ), the components of velocity across x and r axis are symbolized as $u\,$ and $v$ respectively, ${\rho }_{f}$ fluid density, ${\rho }_{m}$ density of microorganisms, ${\rho }_{f}$ density of base liquid, $\mu $ dynamic viscosity, $v=\tfrac{\mu }{{\rho }_{f}}$ kinematic viscosity, $C$ concentration of nanoparticles, $T$ temperature, ${T}_{w}$ and ${T}_{\infty }$ temperatures at the surface and away from the wall respectively, ${D}_{B},$ ${D}_{T}\,\,$ and ${D}_{m}$ Brownian diffusion, thermophoresis features and microorganisms diffusion coefficient respectively, ${C}_{w}$ nanomaterial concentration due to stretching cylinder, ${C}_{\infty }$ concentration of nanofluid away from the wall. Using the following similarity transformation to reduce the PDEs into ODEs:
$\begin{eqnarray}{\tau }_{yx}=\mu \displaystyle \frac{{\rm{d}}u}{{\rm{d}}y},\end{eqnarray}$
$\begin{eqnarray}{\tau }_{yx}=k{\left(\displaystyle \frac{{\rm{d}}u}{{\rm{d}}y}\right)}^{n},\end{eqnarray}$
in which equation ( $\begin{eqnarray}\nu =\displaystyle \frac{\mu }{\rho }.\end{eqnarray}$
Equation ( $\begin{eqnarray}\displaystyle \frac{\partial \left(ru\right)}{\partial x}+\displaystyle \frac{\partial \left(rv\right)}{\partial r}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}u\displaystyle \frac{\partial u}{\partial x}+v\displaystyle \frac{\partial u}{\partial r}=\nu \left(\displaystyle \frac{{\partial }^{2}u}{\partial {x}^{2}}+\displaystyle \frac{1}{r}\displaystyle \frac{\partial u}{\partial r}\right)+{u}_{e}\displaystyle \frac{\partial {u}_{e}}{\partial x}-\displaystyle \frac{\sigma {B}_{0}^{2}}{{\rho }_{f}}\\ \,\times \left(u-{u}_{e}\right)+\displaystyle \frac{g* }{{\rho }_{f}}\left[\begin{array}{l}\left(1-{C}_{f}\right){\rho }_{f}{\beta }^{* * }\left(T-{T}_{\infty }\right)\\ -\left({\rho }_{p}-{\rho }_{f}\right)\left(C-{C}_{\infty }\right)\\ -\left(N-{N}_{\infty }\right){\gamma }^{* }\left({\rho }_{m}-{\rho }_{f}\right)\end{array}\right]\\ \,\times \,\cos \left(\gamma \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}u\displaystyle \frac{\partial T}{\partial x}+v\displaystyle \frac{\partial T}{\partial r}+{{\rm{\Gamma }}}_{E}\left\{\begin{array}{l}{u}^{2}\displaystyle \frac{{\partial }^{2}T}{\partial {x}^{2}}+{v}^{2}\displaystyle \frac{{\partial }^{2}T}{\partial {r}^{2}}+2uv\displaystyle \frac{{\partial }^{2}T}{\partial x\partial r}\\ \,+u\displaystyle \frac{\partial u}{\partial x}\displaystyle \frac{\partial T}{\partial x}\\ \,+u\displaystyle \frac{\partial v}{\partial x}\displaystyle \frac{\partial T}{\partial r}+v\displaystyle \frac{\partial u}{\partial r}\displaystyle \frac{\partial T}{\partial x}\\ \,+v\displaystyle \frac{\partial v}{\partial r}\displaystyle \frac{\partial T}{\partial r}\end{array}\right\}\\ \,=\displaystyle \frac{{\alpha }_{m}}{r}\displaystyle \frac{\partial }{\partial r}\left(\displaystyle \frac{\partial T}{\partial r}\right)+\tau \left(\displaystyle \frac{{D}_{T}}{{T}_{\infty }}{\left(\displaystyle \frac{\partial T}{\partial r}\right)}^{2}+{D}_{B}\displaystyle \frac{\partial T}{\partial r}\displaystyle \frac{\partial C}{\partial r}\right)\\ \,-\displaystyle \frac{1}{{\rho }_{f}{C}_{p}r}\left(\displaystyle \frac{4}{3}\right)\displaystyle \frac{{\sigma }^{* * }}{{k}^{* }}\displaystyle \frac{\partial }{\partial x}\left(r\displaystyle \frac{\partial {T}^{4}}{\partial r}\right)+\displaystyle \frac{{Q}_{0}}{{\rho }_{f}{C}_{p}}\left(T-{T}_{\infty }\right)\\ \,+\displaystyle \frac{\sigma {B}_{0}^{2}{u}^{2}}{{\rho }_{f}{C}_{p}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}u\displaystyle \frac{\partial C}{\partial x}+v\displaystyle \frac{\partial C}{\partial r}+{{\rm{\Gamma }}}_{C}\left\{\begin{array}{l}{u}^{2}\displaystyle \frac{{\partial }^{2}C}{\partial {x}^{2}}+{v}^{2}\displaystyle \frac{{\partial }^{2}C}{\partial {r}^{2}}+2uv\displaystyle \frac{{\partial }^{2}C}{\partial x\partial r}\\ \,+u\displaystyle \frac{\partial u}{\partial x}\displaystyle \frac{\partial C}{\partial x}\\ \,+u\displaystyle \frac{\partial v}{\partial x}\displaystyle \frac{\partial C}{\partial r}+v\displaystyle \frac{\partial u}{\partial r}\displaystyle \frac{\partial C}{\partial x}\\ \,+v\displaystyle \frac{\partial v}{\partial r}\displaystyle \frac{\partial C}{\partial r}\end{array}\right\}\\ \,={D}_{B}\left(\displaystyle \frac{1}{r}\displaystyle \frac{\partial C}{\partial r}+\displaystyle \frac{{\partial }^{2}C}{\partial {r}^{2}}\right)+\displaystyle \frac{{D}_{T}}{{T}_{\infty }}\left(\displaystyle \frac{1}{r}\displaystyle \frac{\partial T}{\partial r}+\displaystyle \frac{{\partial }^{2}T}{\partial {r}^{2}}\right)\\ \,-K{r}^{2}\left(C-{C}_{\infty }\right){\left(\displaystyle \frac{T}{{T}_{\infty }}\right)}^{n}\exp \left(\displaystyle \frac{-{E}_{a}}{\kappa T}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}u\displaystyle \frac{\partial N}{\partial x}+v\displaystyle \frac{\partial N}{\partial r}+\left[\displaystyle \frac{N}{r}\displaystyle \frac{\partial C}{\partial r}+\displaystyle \frac{\partial N}{\partial r}\displaystyle \frac{\partial C}{\partial r}+N\displaystyle \frac{{\partial }^{2}C}{\partial {r}^{2}}\right]\\ \,\times \displaystyle \frac{b{W}_{c}}{\left({C}_{w}-{C}_{0}\right)}={D}_{m}\left(\displaystyle \frac{{\partial }^{2}N}{\partial {r}^{2}}+\displaystyle \frac{1}{r}\displaystyle \frac{\partial N}{\partial r}\right),\end{array}\end{eqnarray}$
with boundary conditions $\begin{eqnarray}\begin{array}{l}u={u}_{w}\left(x\right)=cx,v=0,-k\displaystyle \frac{\partial T}{\partial r}={h}_{f}\left({T}_{w}-T\right),\\ \,-{D}_{B}\displaystyle \frac{\partial C}{\partial r}={h}_{g}\left({C}_{w}-C\right)\,,\\ \,-{D}_{m}\displaystyle \frac{\partial N}{\partial r}={h}_{m}\left({N}_{w}-N\right)\,,\,at\,r=R,\end{array}\end{eqnarray}$
$\begin{eqnarray}u\to 0,T\to {T}_{\infty },C\to {C}_{\infty },N\to {N}_{\infty }\,as\,r\to \infty .\end{eqnarray}$
In equations ( $\begin{eqnarray}\begin{array}{l}u=\displaystyle \frac{{U}_{0}x}{L}f^{\prime} \left(\zeta \right),v=\displaystyle \frac{R}{r}\sqrt{\displaystyle \frac{{U}_{0}\nu }{L}}f\left(\zeta \right),\\ \zeta =\displaystyle \frac{{r}^{2}-{R}^{2}}{2R}{\left(\displaystyle \frac{{U}_{0}}{\nu L}\right)}^{\displaystyle \frac{1}{2}},\\ \psi =\left(\displaystyle \frac{{U}_{0}{x}^{2}}{L}\right)C\left(\zeta \right)=\displaystyle \frac{C-{C}_{\infty }}{{C}_{w}-{C}_{0}},\\ T\left(\zeta \right)=\displaystyle \frac{T-{T}_{\infty }}{{T}_{w}-{T}_{0}},\\ \chi \left(\zeta \right)=\displaystyle \frac{N-{N}_{\infty }}{{T}_{w}-{T}_{0}}.\end{array}\end{eqnarray}$
$\begin{eqnarray}u=\displaystyle \frac{1}{r}\left(\displaystyle \frac{\partial \psi }{\partial r}\right),\,v=-\displaystyle \frac{1}{r}\left(\displaystyle \frac{\partial \psi }{\partial x}\right),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}\left(1+2\alpha \eta \right)f^{\prime\prime\prime} +2\alpha f^{\prime\prime} +ff^{\prime\prime} -{f^{\prime} }^{2}\\ \,-{\gamma }^{2}\left(f^{\prime} -A\right)+{A}^{2}\\ \,+S\left(\theta -Nr\phi -Nc\chi \right)\cos \,\gamma =0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left[\left\{1+Rd{\left(1+\left({\theta }_{w}-1\right)\theta \right)}^{3}\right\}\left(1+2\alpha \zeta \right)\theta ^{\prime} \right]\theta ^{\prime\prime} \\ \,+2\alpha \theta ^{\prime} +PrNb\left(1+2\alpha \zeta \right)\left(\theta ^{\prime} \phi ^{\prime} \right)\\ \,+Pr\left(f\theta ^{\prime} +Q\theta -f^{\prime} \theta +Ec{\gamma }^{2}{f}^{2}\right)\\ \,+PrNb\left(1+2\alpha \zeta \right)\left(\displaystyle \frac{Nt}{Nb}{\theta }^{2}\right)\\ \,-Pr{{\rm{\Omega }}}_{T}\left[{f}^{2}\theta ^{\prime\prime} -ff^{\prime} \theta ^{\prime} +\theta {f^{\prime} }^{2}-\theta ff^{\prime\prime} \right]=0\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left(1+2\alpha \zeta \right)\left(\phi ^{\prime\prime} +\displaystyle \frac{Nt}{Nb}{\theta }^{2}\right)+PrLe\left(f\phi -f^{\prime} \phi \right)\\ \,+2\alpha \left(\phi ^{\prime} +\displaystyle \frac{Nt}{Nb}\theta ^{\prime} \right)\\ \,-PrLe{{\rm{\Omega }}}_{C}\left[{f}^{2}\phi ^{\prime\prime} -ff^{\prime} \phi ^{\prime} +{f^{\prime} }^{2}\phi -\phi ff^{\prime\prime} \right]\\ \,-LePr\sigma {\left(1+{\delta }_{0}\theta \right)}^{n}\phi \exp \left(\displaystyle \frac{-E}{1+{\delta }_{0}\theta }\right)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left(1+2\alpha \zeta \right)\chi ^{\prime\prime} +2\alpha \chi ^{\prime} +Lb\chi ^{\prime} f\\ \,-Pe\left(\phi ^{\prime\prime} \left(\chi +\omega \right)+\chi ^{\prime} \phi ^{\prime} \right)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}f\left(0\right)=0,f^{\prime} \left(0\right)=1,\,f^{\prime} \left(\infty \right)\to 0,\\ \theta ^{\prime} =-{\lambda }_{1}\left(1-\theta \left(0\right)\right),\,\theta \left(\infty \right)\to 0,\\ \phi ^{\prime} =-{\lambda }_{2}\left(1-\phi \left(0\right)\right),\,\phi \left(\infty \right)\to 0,\\ \chi ^{\prime} =-{\lambda }_{3}\left(1-\chi \left(0\right)\right),\,\chi \left(\infty \right)\to 0.\end{array}\end{eqnarray}$
Here, curvature parameter is $\alpha =\displaystyle \frac{1}{R}\sqrt{\tfrac{\nu L}{{U}_{0}},}$ magnetic parameter is $M=\sqrt{\tfrac{\sigma {B}_{0}^{2}}{{\rho }_{f}a},}$ thermal radiation parameter is $Rd=\tfrac{16{\sigma }^{* * }{T}_{\infty }^{3}}{3{k}^{* }k},$ Prandtl number is $Pr=\tfrac{\nu }{{\alpha }_{m}},$ Brownian motion parameter is $Nb=\tfrac{\tau {D}_{B}\left({C}_{w}-{C}_{\infty }\right)}{\nu },$ temperature ratio parameter is ${\theta }_{w}=\tfrac{{T}_{w}}{{T}_{\infty }},$ thermal relaxation parameter is ${{\rm{\Omega }}}_{T}={{\rm{\Gamma }}}_{E}\tfrac{{U}_{0}}{L},$ thermophoresis parameter is $Nt=\tfrac{\tau {D}_{T}\left({T}_{w}-{T}_{\infty }\right)}{\nu {T}_{\infty }},$ heat generation parameter is $Q=\tfrac{L{Q}_{0}}{{U}_{0}\rho {C}_{p}},$ concentration relaxation parameter is ${{\rm{\Omega }}}_{C}={{\rm{\Gamma }}}_{C}\tfrac{{U}_{0}}{L},$ activation energy parameter is $E\left(=\tfrac{{E}_{a}}{\kappa {T}_{\infty }}\right),$ Lewis number is $Le=\tfrac{{\alpha }_{m}}{{D}_{B}},$ buoyancy ratio parameter is $Nr=\tfrac{({\rho }_{p}-{\rho }_{f})({C}_{w}-{C}_{\infty })}{{\beta }^{* * }{\rho }_{f}(1-{C}_{\infty })\left({T}_{w}-{T}_{\infty }\right)},$ mixed convection parameter is $S=\left(\tfrac{\beta * * g* (1-{C}_{\infty })({T}_{w}-{T}_{\infty })}{\nu {U}_{0}^{2}}\right),$ bioconvection Rayleigh number is $Nc=\tfrac{\gamma ({\rho }_{m}-{\rho }_{f})({N}_{w}-{N}_{\infty })}{{\beta }^{* * }{\rho }_{f}(1-{C}_{\infty })\left({T}_{w}-{T}_{\infty }\right)},$ bioconvection Lewis number, Peclet number and microorganisms difference number are $Lb=\tfrac{\nu }{{D}_{m}},Pe=\tfrac{b{W}_{c}}{{D}_{m}},\omega =\tfrac{{N}_{\infty }}{{N}_{w}-{N}_{0}}$ respectively, thermal Biot number is ${\lambda }_{1}\,=\left(\tfrac{{h}_{f}}{k}\right)\sqrt{\nu L/{U}_{0}},$ mass Biot number is ${\lambda }_{2}=\left(\tfrac{{h}_{g}}{{D}_{B}}\right)\sqrt{\nu L/{U}_{0}}$ and microorganism Biot number is ${\lambda }_{3}=\left(\tfrac{{h}_{m}}{{D}_{m}}\right)\sqrt{\nu L/{U}_{0}}.$
Figure 1. Flow field and coordinate system. |
The skin friction is specified as:18 ) in dimensionless form is
$\begin{eqnarray}{C}_{f}=\displaystyle \frac{{\tau }_{w}}{{\rho }^{\displaystyle \frac{{U}^{2}}{2}}},\,{\tau }_{w}=\mu \left(1+\displaystyle \frac{1}{\beta }\right){\left(\displaystyle \frac{\partial u}{\partial r}\right)}_{r=R}.\end{eqnarray}$
Equation ( $\begin{eqnarray}0.5{C}_{f}\sqrt{R{e}_{x}}=f^{\prime\prime} \left(0\right).\end{eqnarray}$
The aspects for the local Nusselt, Sherwood and motile microorganism numbers are $\begin{eqnarray}\begin{array}{l}N{u}_{x}=\displaystyle \frac{x{q}_{w}}{k\left({T}_{w}-{T}_{0}\right)},\,{q}_{w}=-k{\left(\displaystyle \frac{\partial T}{\partial r}\right)}_{r=R}+{\left(qr\right)}_{r=R},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}Sh=\displaystyle \frac{x{j}_{w}}{D\left({C}_{w}-{C}_{0}\right)},\,{j}_{w}=-D{\left(\displaystyle \frac{\partial C}{\partial r}\right)}_{r=R},\,\\ Sn=\displaystyle \frac{x{q}_{n}}{{D}_{m}\left({N}_{w}-{N}_{0}\right)},\,{q}_{n}=-{D}_{m}{\left(\displaystyle \frac{\partial N}{\partial r}\right)}_{r=R}.\end{array}\end{eqnarray}$
These expressions are pre-arranged for the dimensionless form as $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{N{u}_{x}}{\sqrt{R{e}_{x}}}=-\left(1+\displaystyle \frac{4}{3}{R}_{d}\right)\theta ^{\prime} \left(\zeta \right),\,as\,\zeta \to 0,\\ \displaystyle \frac{S{h}_{x}}{\sqrt{R{e}_{x}}}=-\phi ^{\prime} \left(\zeta \right),\,as\,\zeta \to 0,\\ \displaystyle \frac{S{n}_{x}}{\sqrt{R{e}_{x}}}=-\chi ^{\prime} \left(\zeta \right),\,as\,\zeta \to 0.\end{array}\end{eqnarray}$
3. Numerical scheme
In this segment, the mathematical simulation of the flow model, equations (13 )–(16 ) and boundary conditions (17 ) are transformed from PDEs to ordinary differential equations. For obtained numerical results, the bvp4c tool in MATLAB computational software is used. The governing equations are renewed from higher to 1st order as given below:
$\begin{eqnarray}\begin{array}{l}f={z}_{1},f^{\prime} ={z}_{2},f^{\prime\prime} ={z}_{3},f^{\prime\prime\prime} ={z^{\prime} }_{3},\theta ={z}_{4},\theta ^{\prime} ={z}_{5},\\ \theta ^{\prime\prime} ={z^{\prime} }_{5},\phi ={z}_{6},\phi ^{\prime} ={z}_{7},\phi ^{\prime\prime} ={z^{\prime} }_{7},\chi ={z}_{8},\chi ^{\prime} ={z}_{9},\\ \chi ^{\prime\prime} ={z^{\prime} }_{9}\end{array}\end{eqnarray}$
$\begin{eqnarray}{z^{\prime} }_{3}=\displaystyle \frac{{z}_{2}^{2}-2\alpha {z}_{3}-{z}_{1}{z}_{3}+{\gamma }^{2}\left({z}_{2}-A\right)-{A}^{2}-S\left({z}_{4}-Nr{z}_{6}-Nc{z}_{8}\right)\cos \,\gamma }{\left(1+2\alpha \zeta \right)}\end{eqnarray}$
$\begin{eqnarray}{z^{\prime} }_{5}=\displaystyle \frac{\begin{array}{l}-PrNb\left(1+2\alpha \zeta \right)\left({z}_{5}{z}_{7}\right)-Pr\left({z}_{1}{z}_{5}+Q{z}_{4}-{z}_{2}{z}_{4}+Ec{\gamma }^{2}{z}_{1}^{2}\right)\\ +Pr{{\rm{\Omega }}}_{T}\left[-{z}_{1}{z}_{2}{z}_{5}+{z}_{4}{z}_{2}^{2}-{z}_{4}{z}_{1}{z}_{3}\right]-PrNb\left(1+2\alpha \zeta \right)\left(\displaystyle \frac{Nt}{Nb}{z}_{4}^{2}\right)\end{array}}{\left[\left\{1+Rd{\left(1+\left({\theta }_{w}-1\right)\theta \right)}^{3}\right\}\left(1+2\alpha \zeta \right)\theta ^{\prime} \right]-Pr{{\rm{\Omega }}}_{T}{z}_{1}^{2}},\end{eqnarray}$
$\begin{eqnarray}{z^{\prime} }_{7}=\displaystyle \frac{\begin{array}{l}LePr\sigma {\left(1+{\delta }_{0}{z}_{4}\right)}^{n}{z}_{6}exp\left(\displaystyle \frac{-E}{1+{\delta }_{0}{z}_{4}}\right)-\left(1+2\alpha \zeta \right)\displaystyle \frac{Nt}{Nb}{z}_{4}^{2}-PrLe\left({z}_{1}{z}_{6}-{z}_{2}{z}_{6}\right)\\ -2\alpha \left({z}_{7}+\displaystyle \frac{Nt}{Nb}{z}_{5}\right)+PrLe{{\rm{\Omega }}}_{C}\left[-{z}_{1}{z}_{2}{z}_{7}+{z}_{2}^{2}{z}_{6}-{z}_{6}{z}_{1}{z}_{3}\right]\end{array}}{\left(1+2\alpha \zeta \right)-PrLe{{\rm{\Omega }}}_{C}{z}_{1}^{2}},\end{eqnarray}$
$\begin{eqnarray}{z^{\prime} }_{9}=\displaystyle \frac{-2\alpha {z}_{9}-Lb{z}_{1}{z}_{9}+Pe\left({z^{\prime} }_{7}\left({z}_{8}+\omega \right)+{z}_{9}{z}_{7}\right)}{\left(1+2\alpha \zeta \right)},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{z}_{1}\left(0\right)=0,{z}_{2}\left(0\right)=1,\,{z}_{2}\left(\infty \right)\to 0,\\ {z}_{5}=-{\lambda }_{1}\left(1-{z}_{4}\left(0\right)\right),\,{z}_{4}\left(\infty \right)\to 0,\\ {z}_{7}=-{\lambda }_{2}\left(1-{z}_{6}\left(0\right)\right),\,{z}_{6}\left(\infty \right)\to 0,\\ {z}_{9}=-{\lambda }_{3}\left(1-{z}_{8}\left(0\right)\right),\,{z}_{8}\left(\infty \right)\to 0.\end{array}\end{eqnarray}$
4. Results and discussion
The consequences of magnetic parameter $M$ versus velocity $f^{\prime} $ are illuminated in the figure 2. Velocity of nanoliquid $f^{\prime} $ decays for growing data of magnetic parameter for both values (α = 0 and 0.3). Significantly, this shows that growth in magnetic parameter values support the retarding force which promptly mark the fluid dynamic. The significance of the buoyancy ratio parameter $Nr$ for the velocity profile $f^{\prime} $ is depicted in figure 3. Buoyancy-ratio number reduced the $f^{\prime} $ for both estimations (α = 0 and 0.3). In figure 4, the effects of inclination for velocity $f^{\prime} $ are observed. The increasing inclination falls the velocity of the fluid for both cases (α = 0 and 0.3). Figure 5 shows velocity $f^{\prime} $ for the various estimations of bio-convective Rayleigh parameter $Nc.$ Increasing behavior of bio-convective Rayleigh parameter declines velocity distribution for both situations (α = 0 and 0.3). Figure 6 delineates the nature of the mixed convection parameter for velocity $f^{\prime} .$ The velocity of fluid magnifies for mixed convection parameter for both cases of plate and cylinder. Figure 7 reveals the effects of thermal relaxation parameter ${{\rm{\Omega }}}_{T}$ on the temperature field $\theta .$ From the curves, it is noted that the increasing thermal relaxation parameter reduces the temperature of the fluid. In figure 8, the nature of $Pr$ on $\theta $ is explained. Temperature of liquid $\theta $ depreciated by Prandtl number for both values (α = 0 and 0.3). This is due to the rising Prandtl number, which increases the heat transfer rate that cools the process and decreases the temperature within the boundary layer. Fluid with a progressive Prandtl number keeps low thermal conductivity, which diminishes the temperature and thermal boundary-layer thickness. Figure 9 elucidates the outcome of ${\lambda }_{1}$ on temperature $\theta .$ Temperature of the fluid $\theta $ augmented by swelling values of thermal stratification for both cases i.e. plate/cylinder. The impact of ${\theta }_{w}$ against temperature profile $\theta $ for both cases (α = 0 and 0.3) is explored in figure 10. It is regarded that the temperature of fluid $f^{\prime} $ boosted up by higher the variation of temperature ratio parameter. Figure 11 is reflecting the outcome of $Nb$ for the concentration field $\phi .$ It is logical that a rise in Brownian motion parameter decay the concentration of the nanoparticles for both values (α = 0 and 0.3). In the nanofluid structure, because of the existence of nanoparticles, the Brownian motion occurs and with the intensification in Nb the Brownian motion is exaggerated and hence the nanoparticles concentration and boundary layer thickness reduce. Higher estimations of Brownian parameter fall boundary-layer thicknesses, as results the concentration reduces. To distinguish the consequences of $Nt$ versus volumetric concentration $\phi ,$ figure 12 is portrayed. The thermophoresis parameter marks an increasing impact on the volumetric concentration of nanoparticles $\phi .$ Nanoparticles wander from hot sector to cold sector owed to progressive thermophoresis evaluation. The nanofluid molecules kinetic energy is enhanced by swelling evaluation of thermophoresis parameter as a result of the concentration upsurges. Figure 13 is fixed to inspect the behavior of Prandtl number $Pr$ for concentration $\phi .$ Rising estimations of Prandtl number decay concentration profile. Physically, the thermal boundary-layer drops with a higher variation of Prandtl and therefore, the concentration profile is dropped. Figure 14 elucidates the significance of $Le$ for $\phi .$ Concentration field $\phi $ decays for the great evaluation of $Le$ for both cases. Figure 15 lights the consequences of $E$ for $\phi .$ The concentration field is enhanced for increasing valuation of $E$ for both values (α = 0 and 0.3). Figure 16 demonstrates influence of ${\lambda }_{2}$ on $\phi $ for both cases (α = 0 and 0.3). Predictably, enhancing values of ${\lambda }_{2}$ amplified concentration of nanoparticles. Figure 17 is captured to observe the concentration relaxation parameter effects ${{\rm{\Omega }}}_{c}$ on $\phi .$ Growing relaxation parameter falls volumetric concentration profile of nanoparticles of the fluid. The upshot of $Pe$ versus microorganism $\chi $ is deliberated in figure 18. Microorganism field $\chi $ is deteriorated for the advanced magnitude of $Pe.$ Figure 19 explores nature of $Lb$ versus microorganism field $\chi .$ Microorganism $\chi $ decays due to higher magnitude of $Lb$ for both cases (α = 0 and 0.3). The upshot of ${\lambda }_{3}$ for $\chi $ is outlined in figure 20. The microorganism's field $\chi $ is boosted for higher variations of microorganism stratification Biot number. Table 1 illustrates that skin friction is boosted for magnetic parameter while it decreases for mixed convection parameter. Table 2 illustrates that Nusselt number is boosted for Prandtl number while decays for temperature ratio number. Table 3 illustrates that Sherwood number is upgraded for increasing values of Lewis number. Table 4 illustrates that density number of motile microorganism is boosted for bioconvection Lewis and Peclet numbers.
Figure 2. Significance of $M$ on $f^{\prime} .$ |
Figure 3. Significance of $Nr$ on $f^{\prime} .$ |
Figure 4. Significance of $\gamma $ on $f^{\prime} .$ |
Figure 5. Significance of $Nc$ on $f^{\prime} .$ |
Figure 6. Significance of $S$ on $f^{\prime} .$ |
Figure 7. Significance of ${{\rm{\Omega }}}_{T}$ on $\theta .$ |
Figure 8. Significance of $Pr$ on $\theta .$ |
Figure 9. Significance of ${\lambda }_{1}$ on $\theta .$ |
Figure 10. Significance of ${\theta }_{w}$ on $\theta .$ |
Figure 11. Significance of $Nb$ on $\phi .$ |
Figure 12. Significance of $Nt$ on $\phi .$ |
Figure 13. Significance of $Pr$ on $\phi .$ |
Figure 14. Significance of $Le$ on $\phi .$ |
Figure 15. Significance of $E$ on $\phi .$ |
Figure 16. Significance of ${\lambda }_{2}$ on $\phi .$ |
Figure 17. Significance of ${{\rm{\Omega }}}_{c}$ on $\phi .$ |
Figure 18. Significance of $Pe$ on $\chi .$ |
Figure 19. Significance of $Lb$ on $\chi .$ |
Figure 20. Significance of ${\lambda }_{3}$ on $\chi .$ |
Table 1. Outcomes of $-f^{\prime\prime} \left(0\right)$ for varying $M,S,Nr\,$ and Nc. |
| $M$ | $S$ | $Nr$ | $Nc$ | $-f^{\prime\prime} \left(0\right)$ |
|---|---|---|---|---|
| 0.1 | 0.3 | 0.1 | 0.1 | 1.0416 |
| 0.5 | 1.0919 | |||
| 0.8 | 1.1250 | |||
| 0.2 | 0.1 | 0.1 | 0.1 | 1.2912 |
| 0.8 | 1.2348 | |||
| 1.6 | 1.1956 | |||
| 0.2 | 0.3 | 0.1 | 0.1 | 1.2746 |
| 0.8 | 1.2814 | |||
| 1.6 | 1.2905 | |||
| 0.2 | 0.3 | 0.1 | 0.1 | 1.2758 |
| 0.8 | 1.2829 | |||
| 1.6 | 1.2924 |
Table 2. Outcomes of $-\theta ^{\prime} \left(0\right)$ for varying $Pr,Nb,Nt,S,Nr,Nc,$ ${\lambda }_{1},{\theta }_{w} \ \&\,{{\rm{\Omega }}}_{T}.$ |
| $Pr$ | $Nb$ | $Nt$ | ${\lambda }_{1}$ | ${\theta }_{w}$ | ${{\rm{\Omega }}}_{T}$ | $Nr$ | $Nc$ | $S$ | $-\theta ^{\prime} \left(0\right)$ |
|---|---|---|---|---|---|---|---|---|---|
| 3.0 | 0.2 | 0.3 | 0.3 | 0.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2198 |
| 4.0 | 0.2294 | ||||||||
| 5.0 | 0.2359 | ||||||||
| 2 | 0.1 | 0.3 | 0.3 | 0.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2052 |
| 0.6 | 0.1976 | ||||||||
| 1.2 | 0.1880 | ||||||||
| 2 | 0.2 | 0.1 | 0.3 | 0.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2067 |
| 0.6 | 0.1988 | ||||||||
| 1.2 | 0.1881 | ||||||||
| 2 | 0.2 | 0.3 | 0.1 | 0.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.0895 |
| 0.8 | 0.2438 | ||||||||
| 1.6 | 0.2659 | ||||||||
| 2 | 0.2 | 0.3 | 0.3 | 1.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2134 |
| 1.6 | 0.2007 | ||||||||
| 1.7 | 0.1876 | ||||||||
| 2 | 0.2 | 0.3 | 0.3 | 0.4 | 1.0 | 0.1 | 0.1 | 0.3 | 0.2017 |
| 2.0 | 0.2127 | ||||||||
| 3.0 | 0.2315 | ||||||||
| 2 | 0.2 | 0.3 | 0.3 | 0.4 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2043 |
| 0.8 | 0.2042 | ||||||||
| 1.6 | 0.2039 | ||||||||
| 2 | 0.2 | 0.3 | 0.3 | 0.4 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2043 |
| 0.8 | 0.2041 | ||||||||
| 1.6 | 0.2039 | ||||||||
| 2 | 0.2 | 0.3 | 0.3 | 0.4 | 1.5 | 0.1 | 0.1 | 0.1 | 0.2039 |
| 0.8 | 0.2053 | ||||||||
| 1.6 | 0.2064 |
Table 3. Outcomes of $-\phi ^{\prime} \left(0\right)$ for varying $Pr,Nb,Nt,S,Nr,Nc,$ ${\lambda }_{2},Le\,\&\,{{\rm{\Omega }}}_{c}.$ |
| $Pr$ | $Nb$ | $Nt$ | ${\lambda }_{2}$ | $Le$ | ${{\rm{\Omega }}}_{c}$ | $Nr$ | $Nc$ | $S$ | $-\phi ^{\prime} \left(0\right)$ |
|---|---|---|---|---|---|---|---|---|---|
| 3 | 0.2 | 0.3 | 0.2 | 0.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.3037 |
| 3.5 | 0.3160 | ||||||||
| 4.0 | 0.3247 | ||||||||
| 2 | 0.1 | 0.3 | 0.2 | 0.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2541 |
| 0.6 | 0.3042 | ||||||||
| 1.2 | 0.3092 | ||||||||
| 2 | 0.2 | 0.1 | 0.2 | 0.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.3028 |
| 0.6 | 0.2467 | ||||||||
| 1.2 | 0.2220 | ||||||||
| 2 | 0.2 | 0.3 | 0.1 | 0.5 | 1.5 | 0.1 | 0.1 | 0.3 | 0.0847 |
| 0.5 | 0.3375 | ||||||||
| 1.0 | 0.5383 | ||||||||
| 2 | 0.2 | 0.3 | 0.2 | 0.1 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2325 |
| 1.6 | 0.2693 | ||||||||
| 2.2 | 0.2900 | ||||||||
| 2 | 0.2 | 0.3 | 0.2 | 0.4 | 1.0 | 0.1 | 0.1 | 0.3 | 0.2840 |
| 2.0 | 0.2956 | ||||||||
| 3.0 | 0.3128 | ||||||||
| 2 | 0.2 | 0.3 | 0.2 | 0.4 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2845 |
| 0.8 | 0.2844 | ||||||||
| 1.6 | 0.2843 | ||||||||
| 0.2 | 0.3 | 0.2 | 0.4 | 1.5 | 0.1 | 0.1 | 0.3 | 0.2845 | |
| 0.8 | 0.2844 | ||||||||
| 1.6 | 0.2843 | ||||||||
| 0.2 | 0.4 | 1.5 | 0.1 | 0.1 | 0.1 | 0.5855 | |||
| 0.8 | 0.5909 | ||||||||
| 1.6 | 0.5974 |
Table 4. Outcomes of $-\chi ^{\prime} \left(0\right)$ for varying $S,$ $Pe,$ ${\lambda }_{3},$ $Lb,$ $Nr$ and $Nc.$ |
| $S$ | $Pe$ | ${\lambda }_{3}$ | $Lb$ | $Nr$ | $Nc$ | $-\chi ^{\prime} \left(0\right)$ |
|---|---|---|---|---|---|---|
| 0.1 | 0.1 | 0.3 | 2 | 0.1 | 0.1 | 0.2744 |
| 0.8 | 0.2756 | |||||
| 1.6 | 0.2766 | |||||
| 0.3 | 0.2 | 0.3 | 2 | 0.1 | 0.1 | 0.2770 |
| 1.0 | 0.2966 | |||||
| 2.0 | 0.3157 | |||||
| 0.3 | 0.1 | 0.1 | 2 | 0.1 | 0.1 | 0.0899 |
| 0.5 | 0.3184 | |||||
| 1.0 | 0.4665 | |||||
| 0.3 | 0.1 | 0.3 | 1.0 | 0.1 | 0.1 | 0.2434 |
| 3.0 | 0.2742 | |||||
| 5.0 | 0.2950 | |||||
| 0.3 | 0.1 | 0.3 | 2 | 0.1 | 0.1 | 0.2803 |
| 0.8 | 0.2798 | |||||
| 1.6 | 0.2794 | |||||
| 0.3 | 0.3 | 2 | 0.1 | 0.1 | 0.2746 | |
| 0.8 | 0.2745 | |||||
| 1.6 | 0.2741 |
5. Conclusions
The flow of Newtonian nanoliquid by an inclined stretchable cylinder with the effects of motile microorganism and activation energy is discussed. The consequences of Cattaneo–Christov model are also considered. The velocity profile is declined by the mounting evaluation in the buoyancy ratio parameter and inclination parameter. The temperature field is reduced by intensifying values of thermal relaxation parameter and it shows growing nature for both thermal Biot number and temperature ratio parameter. The higher valuations of magnetic parameter and mixed convection parameter have reverse nature for velocity profile. Both temperature distribution and volumetric concentration fall for increasing values of the Prandtl number. The volumetric concentration field is boosted for swelling values of activation energy and thermophoresis parameter while show opposite behavior for Lewis number and concentration relaxation parameter. The growing evaluation of microorganism stratified Biot number boosted the motile microorganism's concentration of nanoparticles. Bio-convection Lewis and Peclet numbers reduced the motile microorganism's concentration of nanoparticles.