Impact of Melting Heat Transfer and Variable Characteristics on an MHD Non-Newtonian Shear-Thinning Fluid Flow with Gyrotactic Microorganisms over a Nonlinear Stretched Surface ()
1. Introduction
The topic of boundary layer flow past a stretching surface is of great curiosity for researchers owing to its numerous applications such as the production of rubber and plastic sheets, aerodynamics, glass blowing, metallurgical processes, and metal extrusion. A good number of investigations may be referred to in this regard highlighting various scenarios. The first step in this direction was taken by Crane [1] who discussed fluid flow past a deformable surface. The behavior of activation energy combined with the convective condition on a magnetohydrodynamic third-grade nanofluid flow over a nonlinear deformable sheet is demonstrated by Hayat et al. [2]. Rasool et al. [3] examined nanofluid flow over a nonlinear elongated surface. The amalgamation of the Soret-Dufour effects on a silver-oxide water-based nanofluid flow on a nonlinear extended surface with slip condition is explored by Bhatti et al. [4]. Khan et al. [5] numerically investigated the impression of variable viscosity amalgamated with inclined Lorentz force on Pseudoplastic nanoliquid flow past a variable deformable surface. Seth et al. [6] discussed slip condition on a MHD nanoliquid flow past a nonlinearly deforming sheet.
In the field of engineering, Reiner-Philippoff (R-P) fluid has a vast utilization. Reiner-Philippoff is a non-Newtonian shear-thinning fluid. The association between shear stress and rate of strain in the R-P fluid model is given in [7] as follows:
$\frac{\partial v}{\partial y}=\frac{\tau}{{\mu}_{\infty}+\frac{\left({\mu}_{0}-{\mu}_{\infty}\right)}{1+{\left(\frac{\tau}{{\tau}_{s}}\right)}^{2}}},$ (1)
where
${\mu}_{0}$ is zero shear viscosity,
${\tau}_{s}$ is the reference shear stress, shear-stress is
$\tau $ and
${\mu}_{\infty}$ is the limiting viscosity. The dimensionless form of Equation (1) is defined by Ahmad [8].
$f\left(\sigma \right)=\frac{\sigma}{1+\left(\raisebox{1ex}{$\lambda -1$}\!\left/ \!\raisebox{-1ex}{$1+{\sigma}^{2}$}\right.\right)},$ (2)
where
$\lambda =\left({\mu}_{0}/{\mu}_{\infty}\right)$ and
$\sigma =\left(\tau /{\tau}_{s}\right)$ . The R-P fluid is a non-Newtonian model which displays all three behaviors i.e., dilatant fluid, pseudoplastic fluid, and Newtonian fluid. The R-P fluid exhibits features of dilatant fluid for
$\lambda <1$ , it behaves as pseudoplastic fluid for
$\lambda >1$ and Newtonian fluid for
$\lambda =1$ . Recent studies featuring R-P fluid flow over varied geometries may be found in [9] [10] [11] [12] [13].
The phenomenon of melting heat transfer over a stretching surface has extensive applications which include fiber technology, the welding process, the freezing treatment of sewage, the preparation of semi conductors, oil extraction, and the melting of permafrost. On a dissipative flow, the effect of melting heat and Joule heating is explored by Hayat et al. [14] across an elongated surface. Mabood et al. [15] numerically discussed the impression of melting heat coalesced with nonlinear thermal radiation on hybrid nanoliquid flow across an elongated sheet. Mabood et al. [16] in another investigation discussed the flow of the Sisko nanoliquid over a deforming surface with melting heat and binary chemical reaction. It is noticed in this study that fluid temperature drops as the heat generation parameter escalates. Hayat et al. [17] numerically illustrated the impact of homogeneous-heterogeneous reactions and melting heat transfer on an extendable surface. Mabood et al. [18] analyzed the impression of melting heat on a radiative Casson fluid flow over a stretching sheet. It is comprehended in this study that by varying the magnetic and permeability parameters, the fluid temperature upsurges.
The microscopic organisms (microorganisms) can be seen only through an optical microscope. Microorganisms are everywhere around us and these are the root cause of many diseases including citrus canker, cholera, and tuberculosis. The gyrotactic microorganisms live on the still water and their movement is opposite to gravity. The random motion of the microorganisms is translated as bioconvection. Researchers are motivated to discuss the bioconvection phenomenon in numerous circumstances. Waqas et al. [19] studied the rate of mass transfer combined with gyrotactic microorganisms on a non-Newtonian nanofluid flow over an elongated surface. The impacts of the homogeneous-heterogeneous reactions and gyrotactic microorganisms on a Maxwell nanofluid flow past a deformable surface are scrutinized by Sohail et al. [20]. Waqas et al. [21] explored the Jeffery nanofluid flow over an elongated surface with motile microorganisms and stratification. Naz et al. [22] deliberated a numerical solution of the Cross nanofluid flow containing gyrotactic motile microorganisms with irreversibility analysis.
It is inferred from the above-cited literature that abundant studies may be found that discuss the impression of gyrotactic microorganisms embedded with non-Newtonian fluid flows. But very little research can be quoted that analyzed the flow of the Reiner-Philippoff fluid over varied geometries. Nevertheless, on a magnetohydrodynamic Reiner-Philippoff nanoliquid flow, gyrotactic microorganisms amalgamated with zero mass flux at the surface and melting heat transfer conditions over a nonlinear extended surface is still not discussed in the literature. An inbuilt function of MATLAB (bvp4c) software is used to solve the ODEs. The parameters emerging in the model are discussed versus allied profiles through graphical illustrations. The outcomes are quite interesting.
2. Equations
We consider a two-dimensional Reiner-Philippoff nanofluid flow with a non-uniform magnetic field
$B\left(x\right)=\left({B}_{0}/{x}^{0.3}\right)$ (is magnetic field strength) perpendicular to the nonlinear extended surface with stretching velocity where d is a constant (Figure 1). The flow is assisted by the impact of gyrotactic microorganisms. To explore the heat transfer phenomenon, melting heat and zero mass flux conditions are analyzed.
The equations governing the mathematical problem are:
${u}_{x}+{v}_{y}=0,$ (3)
$u{u}_{x}+v{u}_{y}=\frac{1}{\rho}{\tau}_{y}-\frac{\sigma {B}^{2}u}{\rho},$ (4)
$u{T}_{x}+v{T}_{y}-\tau \left[{D}_{B}{C}_{y}{T}_{y}+\frac{{D}_{T}}{{T}_{\infty}}{\left({T}_{{}_{y}}\right)}^{2}\right]=\frac{1}{\rho {C}_{p}}\frac{\partial}{\partial y}\left(\kappa \left(T\right){T}_{y}\right),$ (5)
Figure 1. Schematic diagram of the Reiner-Philippoff nanofluid flow.
$u{C}_{x}+v{C}_{y}=\frac{\partial}{\partial y}\left({D}_{B}\left(C\right){C}_{y}\right)+\frac{{D}_{T}}{{T}_{\infty}}{T}_{yy},$ (6)
$u{n}_{x}+v{n}_{y}+\frac{b{W}_{c}}{{C}_{\infty}}\left[{n}_{y}{C}_{y}+n{C}_{yy}\right]={D}_{n}{n}_{yy}.$ (7)
In Equations (5) and (6)
$\kappa \left(T\right),{D}_{B}\left(C\right)$ depicts temperature-dependent thermal conductivity and dependence of molecular diffusion on concentration. The mathematical form of
$\kappa \left(T\right),{D}_{B}\left(C\right)$ is specified as:
$\kappa \left(T\right)={\kappa}_{\infty}\left(1+{h}_{1}\theta \right),$ (8)
where
${\kappa}_{\infty}$ is thermal conductivity at the free stream and
${h}_{1}$ is the variable thermal conductivity parameter. Similarly,
${D}_{B}\left(C\right)={D}_{{B}_{\infty}}\left(1+{h}_{2}\varphi \right),$ (9)
where
${D}_{{B}_{\infty}}$ is molecular diffusivity at the free stream and
${h}_{2}$ is a variable molecular diffusivity parameter.
Boundary constraints are:
$u(x,y)={U}_{w}(x)=d{x}^{1/3}\text{,}\kappa \left(T\right){T}_{y}=\rho \left[\chi +{c}_{s}\left({T}_{m}-{T}_{0}\right)\right]v,\text{}T={T}_{m},$
$\frac{{D}_{T}}{{T}_{\infty}}{T}_{y}+{D}_{B}{C}_{y}=0,\text{}n={n}_{w},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{at}y=0,$ (10)
$u\to \text{0,}T\to {T}_{\infty},C\to {C}_{\infty},n\to {n}_{\infty},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\to \infty .$
Here
$b,{B}_{0},C,\sigma ,{c}_{s},{c}_{p},{W}_{c},{D}_{B},T,{D}_{T},(u,v),{D}_{n},\kappa \left(T\right),\rho ,{T}_{m},{T}_{\infty}\text{,}\chi ,n,{n}_{w},{U}_{w}(x),(x,y)$ and
${n}_{\infty}$ are chemotaxis constant, the magnitude of magnetic field strength, concentration of nanoparticles, the conductivity of electricity, heat capacity of solid surface, specific heat capacitance, maximum cell swimming speed, Brownian motion coefficient, the temperature of boundary layer fluid, thermophoresis coefficient, components of velocity, micro-organism diffusion coefficient, thermal conductivity, density, the temperature of the wall, temperature of the ambient fluid, latent heat, concentration of micro-organisms, the surface concentration of micro-organisms, stretching velocity, coordinate axes and ambient concentration of micro-organisms.
Using local similarity transformation defined in [8]:
$\begin{array}{l}\zeta =\sqrt{\frac{{U}_{w}}{vx}}y,\text{}\Psi =\sqrt{{U}_{w}xv}f\left(\zeta \right),\text{}\tau =\rho \sqrt{{d}^{3}v}m\left(\zeta \right),\text{}C={C}_{\infty}\varphi \left(\zeta \right)+{C}_{\infty},\\ T=\left({T}_{\infty}-{T}_{m}\right)\theta \left(\zeta \right)+{T}_{m},\text{}n=\left({n}_{w}-{n}_{\infty}\right)R\left(\zeta \right)+{n}_{\infty}.\end{array}$ (11)
Equation (3) is satisfied. However, Equations (4)-(6), (9), and (10) are transmuted as:
${m}^{\prime}=\frac{1}{3}{({f}^{\prime})}^{2}-\frac{2}{3}f{f}^{\u2033}+Ha{f}^{\prime},$ where
$m={f}^{\u2033}\frac{{m}^{2}+\lambda {\gamma}^{2}}{{m}^{2}+{\gamma}^{2}},$ (12)
$\left(1+{h}_{1}\theta \right){\theta}^{\u2033}+{h}_{1}{{\theta}^{\prime}}^{2}+{N}_{b}{\theta}^{\prime}{\varphi}^{\prime}+\frac{2}{3}\mathrm{Pr}f{\theta}^{\prime}+{N}_{t}{({\theta}^{\prime})}^{2}=0,$ (13)
$\left(1+{h}_{2}\varphi \right){\varphi}^{\u2033}+{h}_{2}{{\varphi}^{\prime}}^{2}+\frac{2}{3}{S}_{c}f{\varphi}^{\prime}+{N}_{t}\frac{{\theta}^{\u2033}}{{N}_{b}}=0,$ (14)
${R}^{\u2033}+\frac{2}{3}f{R}^{\prime}Lb-Pe\left[{\varphi}^{\prime}{R}^{\prime}+\left(R+\omega \right){\varphi}^{\u2033}\right]=0.$ (15)
boundary constraints
$\begin{array}{l}{f}^{\prime}(\zeta )=1,\text{}M\left(1+{h}_{1}\theta \right){\theta}^{\prime}(\zeta )+\mathrm{Pr}f(\zeta )=0,\text{}\theta (\zeta )=0,\text{}{N}_{t}{\theta}^{\prime}(\zeta )+{N}_{b}{\varphi}^{\prime}(\zeta )=0,\\ R(\zeta )=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{at}\zeta =0,\\ f(\zeta )\to 0,\text{}\theta (\zeta )\to 1,\text{}\varphi (\zeta )\to 0,\text{}R(\zeta )\to 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{as}\zeta \to \infty .\end{array}$ (16)
Here,
$Ha,M,{P}_{e},\lambda ,{N}_{b},\gamma ,Pr,\omega ,{N}_{t},{L}_{b}$ and
${S}_{c}$ denote the Hartmann number, melting parameter, Peclet number, fluid parameter, Brownian motion parameter, Bingham number, Prandtl number, bioconvection concentration difference parameter, Thermophoresis parameter, Lewis number, and Schmidt number. These are demarcated as under:
$\begin{array}{l}Ha=\frac{\sigma {B}_{0}^{2}}{\rho {U}_{0}},M=\frac{{c}_{f}\left({T}_{\infty}-{T}_{m}\right)}{\chi +{c}_{s}\left({T}_{m}-{T}_{0}\right)},{P}_{e}=\frac{b{W}_{c}}{\nu},\lambda =\frac{{\mu}_{0}}{{\mu}_{\infty}},{N}_{b}=\frac{\tau {D}_{B}{C}_{\infty}}{\nu},\gamma =\frac{{\tau}_{s}}{\rho \sqrt{{d}^{3}\nu}},\\ \mathrm{Pr}=\frac{\mu {c}_{p}}{k},\omega =\frac{{n}_{\infty}}{{n}_{w}-{n}_{\infty}},{N}_{t}=\frac{\tau {D}_{B}\left({T}_{\infty}-{T}_{m}\right)}{{T}_{\infty}\nu},{L}_{b}=\frac{\nu}{{D}_{n}},{S}_{c}=\frac{\nu}{{D}_{B}}.\end{array}$ (17)
The mathematical forms of heat and mass flux are defined as:
$N{u}_{x}=\frac{x{Q}_{w}}{{\kappa}_{\infty}\left({T}_{\infty}-{T}_{m}\right)}\text{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{Q}_{w}={-\kappa \left(T\right){T}_{y}|}_{y=0},$ (18)
$S{h}_{x}=\frac{x{Q}_{m}}{{D}_{{B}_{\infty}}{C}_{\infty}}\text{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{Q}_{m}={-{D}_{B}\left(C\right){C}_{y}|}_{y=0}.$ (19)
The non-dimensional form of (18) and (19) are given as:
$N{u}_{x}{\mathrm{Re}}_{x}^{-0.5}=-\left(1+{h}_{1}\theta \left(0\right)\right){\theta}^{\prime}(0),$ (20)
$S{h}_{x}{\mathrm{Re}}_{x}^{-0.5}=-\left(1+{h}_{2}\varphi \left(0\right)\right){\varphi}^{\prime}(0).$ (21)
The rate of heat transfer for numeric values of Pr is presented in Table 1. A fabulous association is observed between the numeric outcomes of the present study with the published work of Sajid et al. [9] and Reddy et al. [11].
3. Figures
The outcomes of dimensionless parameters are graphically inspected through Figures 2-9. Figure 2 illustrates the upshot of the Bingham number
$\gamma $ versus the fluid velocity profile
${f}^{\prime}(\zeta )$ . It is comprehended that for growing values of
$\gamma $ , fluid velocity deteriorates. For higher values of shear rate, apparent viscosity increases which result in depreciation in
${f}^{\prime}(\zeta )$ . Figure 3 portrays the outcome of varying the Reiner Philippoff fluid parameter
$\lambda $ on
${f}^{\prime}(\zeta )$ . The velocity is enhanced for numerous values of
$\lambda $ As
$\lambda $ is the quotient of shear viscosity to
Table 1. Comparison of rate of heat transfer for numeric values of Pr by considering
$Ha=1,\lambda =1,\gamma =0.7$ and
${N}_{t}={S}_{c}={P}_{e}=\omega ={L}_{b}=0$ with Sajid et al. [9] and Reddy et al. [11].
Figure 2. Upshot of Bingham number on velocity profile.
Figure 3. Upshot of Reiner Philippoff fluid parameter on velocity profile.
Figure 4. Upshot of varying thermal conductivity parameter on temperature profile.
Figure 5. Upshot of melting parameter on temperature profile.
Figure 6. Upshot of varying molecular diffusivity on solutal profile.
Figure 7. Upshot of Peclet number on gyrotactic microorganisms profile.
Figure 8. Upshot of Prandtl number and thermophoresis parameter on heat transfer rate.
Figure 9. Upshot of Brownian motion parameter and Schmidt number on rate of mass transfer.
the limiting viscosity. Higher values of
$\lambda $ result in a decrement in the viscosity due to which the fluid moves freely, and
${f}^{\prime}(\zeta )$ augments.
Figure 4 portrays the impression of varying thermal conductivity parameter
${h}_{1}$ on the thermal field
$\theta (\zeta )$ . An upsurge is seen for higher values of
${h}_{1}$ as collision among the fluid particles escalates. Thus, more heat is transmitted through the fluid. Hence,
$\theta (\zeta )$ elevates. Figure 5 depicts the behavior of the augmenting M versus fluid temperature
$\theta (\zeta )$ . The heat is transferred more promptly due to the difference in temperature between the surface and the ambient fluid. Hence,
$\theta (\zeta )$ deteriorates. Figure 6 illustrates how varying molecular diffusivity parameter
${h}_{2}$ affects the solutal field
$\varphi (\zeta )$ . On amplifying
${h}_{2}$ , an upsurge is noticed in
$\varphi (\zeta )$ . The upshot of the bioconvection Peclet number
${P}_{e}$ on the gyrotactic micro-organisms profile
$R(\zeta )$ is inspected in Figure 7. A decline in
$R(\zeta )$ is witnessed. On escalating
${P}_{e}$ , it is observed that the diffusion of microorganisms reduces. Consequently,
$R(\zeta )$ declines. The behavior of Pr and
${N}_{t}$ on heat transfer rate is depicted in Figure 8. On augmenting Pr and
${N}_{t}$ heat flux upsurges. The influence of
${N}_{b}$ and
${S}_{c}$ on the rate of mass transfer
$S{h}_{x}{\mathrm{Re}}_{x}^{-0.5}$ is revealed in Figure 9. Here,
$S{h}_{x}{\mathrm{Re}}_{x}^{-0.5}$ is a decreasing function of
${N}_{b}$ .
4. Conclusions
Magnetohydrodynamic Reiner-Philippoff nanoliquid flow is inspected over a nonlinear extended surface with the impression of gyrotactic microorganisms. The uniqueness of the envisioned model is upgraded considering the zero-mass flux and melting heat condition. The problem is handled numerically. The following are the most notable findings of this investigation:
・ The velocity profile is elevated by augmenting the fluid parameter, whereas, an opposite behavior is depicted by the Bingham number.
・ An inverse behavior is witnessed in velocity and fluid temperature for the melting heat parameter.
・ Gyrotactic microorganisms profile diminishes on augmenting the Peclet number.
・ The rate of heat and mass transfer depicts an opposite behavior for Brownian motion and thermophoresis parameter.
Acknowledgments
The study is supported by Bahria University, Islamabad, Pakistan.