# Computation of Melting Dissipative Magnetohydrodynamic Nanofluid Bioconvection with Second-order Slip and Variable Thermophysical Properties

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## Abstract

**:**

## 1. Introduction

_{3}–KNO

_{3}(60:40 ratio) binary salt selected as phase change material) with nanoparticles using the direct-synthesis method as a possible working medium for concentrating solar plants. Sheikholeslami and Rokni [25] simulated the nanofluid flow over a stretching plate in the presence of a magnetic field, strong radiative heat transfer and melting, computing elevation in both nanofluid velocity and concentration with melting parameter. Mastiani et al. [26] analyzed the melting of paraffin-based copper nanoparticle-doped nanofluids in an annulus using an unstructured finite volume method to track the solid and liquid interface. They noted that heat transfer rate and melting time are respectively increased and reduced with a rise in nanoparticle volume fraction.

## 2. Problem Formulation

- $\mathrm{Pr}=\frac{{\upsilon}_{\infty}}{{\alpha}_{\infty}}$: Prandtl number,
- $Nb=\frac{\tau {D}_{B,\infty}\left({C}_{w}-{C}_{\infty}\right)}{{\alpha}_{\infty}}$: Brownian motion parameter,
- $Nt=\frac{\tau {D}_{T}\left({T}_{\infty}-{T}_{m}\right)}{{\alpha}_{\infty}{T}_{\infty}}$: thermophoresis parameter,
- $Sc=\frac{{\upsilon}_{\infty}}{{D}_{B,\infty}}$: Schmidt number,
- $Pe=\frac{\tilde{b}{W}_{c}}{{D}_{m,\infty}}$: bioconvection Péclet number,
- $Sb=\frac{{\upsilon}_{\infty}}{{D}_{m,\infty}}$: bioconvection Schmidt number,
- $M=\frac{\sigma {B}_{0}^{2}}{a{\rho}_{\infty}}$: magnetic body force parameter,
- $M\text{\hspace{0.17em}}e=\frac{{c}_{f}\left({T}_{\infty}-{T}_{m}\right)}{\lambda +{c}_{s}\left({T}_{m}-{T}_{0}\right)}$: melting (phase change) parameter,
- $Ec=\frac{{u}_{w}^{2}}{{c}_{p}\left({T}_{\infty}-{T}_{m}\right)}$: Eckert number,
- $\delta =A\sqrt{\frac{a}{{\upsilon}_{\infty}}}\left(>0\right)$: first-order hydrodynamic slip parameter,
- $\gamma =B\left(\frac{a}{{\upsilon}_{\infty}}\right)\left(<0\right)$: second-order hydrodynamic slip parameter.

## 3. Numerical Computations with MATLAB BVP4C Code

^{−5}produces every numerical solution in this problem. Mesh selection and error control are based on the residual of the continuous solution. The range of parameters are: $\delta >0,\gamma <0,$ $0<{h}_{2},{h}_{4},{h}_{6},{h}_{8}\le 1,$ $0\le Nb,Nt,Ec,Me,M\le 5,$ and $0<Pe,Sc,Sb\le 10.$ To validate the present solution, comparison has been made with previously published data from the literature for the skin friction in Table 1 and Table 2, and favourable agreement is achieved.

## 4. Further Validation with Variational Iteration Method (VIM)

## 5. Results and Discussion

_{b}) and bioconvection Lewis number (Le

_{b}). Ordinary Péclet number is associated with convective heat transfer processes. In bioconvection, the Péclet number when sufficiently high has been shown to significantly change patterns of the motile micro-organism flow. Bioconvection is due to the internal energy of the microorganisms. With greater swimming speed (higher Péclet number), the micro-organisms move faster, which reduces their concentrations. At lower Péclet numbers the effect is reversed, i.e., motility of the micro-organisms is inhibited and move slower. This then leads to higher and more homogenous concentrations. The second-order slip parameter also arises in the velocity wall boundary condition but induces a lesser effect than first-order slip. It weakly decreases motile micro-organism species boundary layer thickness whereas the bioconvection Péclet number generates a strong reduction.

## 6. Conclusions

- Increasing magnetic body force parameter reduces velocity, nanoparticle concentration and motile microorganism species number density, whereas hydrodynamic boundary layer thickness is increased as are temperatures.
- Increasing Brownian motion and thermophoresis effect lead to an enhancement in temperatures. Increasing melting effect accelerates the flow (via momentum diffusion assistance) whereas it decreases temperatures. Increasing first-order slip and temperature-dependent thermal conductivity parameters significantly cool the boundary layer regime.
- Nanoparticle volume fraction and motile microorganism density number are both elevated with increasing mass diffusivity parameter whereas they are depressed with greater Schmidt number.
- Motile microorganism density number is boosted with increasing microorganism diffusivity parameter, whereas the converse response is induced with increasing bioconvection Schmidt number.
- Increasing temperature-dependent viscous parameter and Eckert number increases temperatures but reduce nanoparticle volume fraction.
- Local Sherwood number is depressed with Brownian motion parameter and Schmidt number, whereas it is enhanced with thermophoresis parameter.
- Increasing nanomaterial viscosity, viscous dissipation and melting heat transfer reduce local Nusselt number at the sheet surface.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$a$ | dimensional positive constant $\left(\frac{1}{s}\right)$ |

$A$ | dimensional positive constant $\left(m\right)$ |

$\tilde{b}$ | chemotaxis constant $(m)$ |

$B$ | positive constant $\left({m}^{2}\right)$ |

${B}_{0}$ | magnetic field and strength $\left(\frac{kg}{A\xb7{s}^{2}}\right)=Tesla$ |

${c}_{p}$ | specific heat at constant pressure $\left(\frac{J}{kg\xb7K}\right)$ |

${c}_{s}$ | specific heat of the solid particle at constant pressure $\left(\frac{J}{kg\xb7K}\right)$ |

${c}_{f}$ | specific heat of the base fluid at constant pressure $\left(\frac{J}{kg\xb7K}\right)$ |

${C}_{w}$ | wall nanoparticle volume fraction $(-)$ |

${C}_{\infty}$ | ambient nanoparticle volume fraction $(-)$ |

${C}_{{f}_{x}}$ | skin friction coefficient $(-)$ |

${D}_{B}\left(C\right)$ | variable Brownian diffusion coefficient $\left(\frac{{m}^{2}}{s}\right)$ |

${D}_{B,\infty}$ | constant Brownian diffusion coefficient $\left(\frac{{m}^{2}}{s}\right)$ |

${D}_{m}\left(C\right)$ | variable diffusivity of microorganisms $\left(\frac{{m}^{2}}{s}\right)$ |

${D}_{m,\infty}$ | constant diffusivity of microorganisms $\left(\frac{{m}^{2}}{s}\right)$ |

${f}^{\prime}\text{\hspace{0.17em}}(\eta )$ | dimensionless velocity $(-)$ |

${h}_{1}$ | dimensional positive constant $\left(\frac{1}{K}\right)$ |

${h}_{2}$ | temperature dependent viscous parameter $(-)$ |

${h}_{3}$ | dimensional positive constant $\left(\frac{1}{K}\right)$ |

${h}_{4}$ | temperature dependent thermal conductive parameter $(-)$ |

${h}_{5}$ | positive constant $(-)$ |

${h}_{6}$ | mass diffusivity parameter $(-)$ |

${h}_{8}$ | microorganism diffusivity parameter $(-)$ |

$k\left(T\right)$ | variable thermal conductivity of nanofluid $\left(\frac{W}{m\xb7K}\right)$ |

${k}_{\infty}$ | constant thermal conductivity of nanofluid $\left(\frac{W}{m\xb7K}\right)$ |

$Me$ | melting (phase change) parameter $(-)$ |

$n$ | density of motile microorganism $(-)$ |

$Nb$ | Brownian motion parameter $(-)$ |

$N{n}_{\overline{x}}$ | local density number of motile microorganisms $(-)$ |

$Nt$ | thermophoresis parameter $(-)$ |

$N{u}_{\overline{x}}$ | local Nusselt number $(-)$ |

$Pe$ | bioconvection Péclet number $(-)$ |

Pr | Prandtl number $(-)$ |

${q}_{m}$ | surface mass flux $(-)$ |

${q}_{n}$ | surface motile microorganism flux $(-)$ |

${q}_{w}$ | Surface heat flux $\left(\frac{W}{{m}^{2}}\right)$ |

Sb | bioconvection Schmidt number $(-)$ |

$Sc$ | Schmidt number $(-)$ |

$S{h}_{\overline{x}}$ | local Sherwood number $(-)$ |

$S{t}_{f},S{t}_{s}$ | Stefan numbers $(-)$ |

$T$ | nanofluid temperature $(K)$ |

${T}_{0}$ | solid surface temperature $(K)$ |

${T}_{m}$ | solid surface temperature $(K)$ |

${\overline{u}}_{slip}$ | velocity slip $\left(\frac{m}{s}\right)$ |

${\overline{u}}_{w}$ | velocity along stretching surface $\left(\frac{m}{s}\right)$ |

$\overline{u}$ | velocity component along the $\overline{x}-$direction $\left(\frac{m}{s}\right)$ |

$u$ | non-dimensional velocity component along the $\overline{x}-$direction $(-)$ |

$\overline{v}$ | velocity components along the $\overline{y}-$direction $\left(\frac{m}{s}\right)$ |

$v$ | non-dimensional velocity component along the $y-$direction $(-)$ |

${W}_{c}$ | maximum cell swimming speed $\left(\frac{m}{s}\right)$ |

$\overline{x}$ | coordinate along the surface $(m)$ |

$x$ | non-dimensional coordinate along the surface $(-)$ |

$\overline{y}$ | coordinate along the surface $(m)$ |

$y$ | non-dimensional coordinate along the surface $(-)$ |

Greek Letters | |

${\alpha}_{\infty}$ | thermal diffusivity $\left(\frac{{m}^{2}}{s}\right)$ |

$\gamma $ | second-order slip parameter $(-)$ |

$\delta $ | first-order slip parameter $(-)$ |

$\eta $ | similarity independent variable $(-)$ |

$\theta (\eta )$ | dimensionless temperature $(-)$ |

$\lambda $ | latent heat of diffusion $\left(\frac{J}{Kg}\right)$ |

$\mu \left(T\right)$ | variable temperature dependent viscosity $\left(\frac{kg}{ms}\right)$ |

${\mu}_{\infty}$ | constant temperature dependent viscosity $\left(\frac{kg}{ms}\right)$ |

${\rho}_{\infty}$ | density of the base fluid $\left(\frac{kg}{{m}^{3}}\right)$ |

$\tau $ | ratio of effective heat capacity of the nanoparticle material to the fluid heat capacity $(-)$ |

${\tau}_{w}$ | wall skin friction in $\overline{x}$ $(Pa)$ |

${\upsilon}_{\infty}$ | constant kinematic viscosity of nanofluid $\left(\frac{{m}^{2}}{s}\right)$ |

$\varphi (\eta )$ | rescaled nanoparticle volume fraction $(-)$ |

$\chi (\eta )$ | rescaled number of motile microorganisms $(-)$ |

Subscripts/Superscripts | |

$w$ | condition at the wall |

$\infty $ | free stream condition |

$\left({}^{\u2019}\right)$ | differentiation with respect to $\eta $ |

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**Figure 2.**Influences of ${h}_{2}$ and $Ec$ on the dimensionless: (

**a**) velocity, (

**b**) temperature, (

**c**) nanoparticle volume fraction, (

**d**) micro-organism density number function.

**Figure 4.**Effects ${h}_{6}$ and $Sc$ on the dimensionless (

**a**) nanoparticle volume fraction and (

**b**) micro-organism density number function.

**Figure 6.**Effects M and Me on the dimensionless (

**a**) velocity, (

**b**) temperature, (

**c**) nanoparticle volume fraction and (

**d**) micro-organism density number function.

**Figure 7.**Effects of Nb and Nt on the dimensionless (

**a**) temperature, (

**b**) nanoparticle volume fraction and (

**c**) micro-organism density number function.

**Figure 9.**Local Nusselt number, $-{\theta}^{\prime}(0)$ versus ${h}_{2}$ and Me for different values of Ec.

**Figure 10.**Local Sherwood number, $-{\varphi}^{\prime}\left(0\right)$ versus Sc and Nb for different values of Nt.

**Table 1.**Comparison of skin friction ${f}^{\u2033}(0)$ for different values of $M$ when $\mathrm{Pr}=1,Me=\delta =\gamma =Nb=Nt=Ec=Sc=Pe=Sb={h}_{2}={h}_{4}={h}_{6}={h}_{8}=0.$

$\mathit{M}$ | Hayat et al. [53] (Modified Adomian Decomposition) | Mabood and Mastroberardino [48] (RKF45) | Present Results (BVP4C) | Present Results (VIM) |
---|---|---|---|---|

0 | 1.00000 | 1.000008 | 1.0013962 | 1.0000002 |

1 | −1.41421 | −1.4142135 | −1.4142375 | −1.41422211 |

5 | −2.44948 | −2.4494897 | −2.4494897 | −2.4494901 |

10 | −3.31662 | −3.3166247 | −3.3166248 | −3.3166229 |

50 | −7.14142 | −7.1414284 | −7.1414284 | −7.1414279 |

100 | −10.04987 | −10.049875 | −10.049876 | −10.049868 |

500 | −22.38302 | −22.383029 | −22.383029 | −22.383031 |

1000 | −31.63858 | −31.638584 | −31.638584 | −31.638578 |

**Table 2.**Comparison of skin friction ${f}^{\u2033}(0)$ for $\delta $ when $Me=M={h}_{2}={h}_{4}={h}_{6}={h}_{8}=\gamma =Nb=Nt=Ec=Sc=Pe=Sb=0,\mathrm{Pr}=1.$

$\mathit{\delta}$ | Andersson [54] | Hamad et al. [55] | Present Results (BVP4C) |
---|---|---|---|

0 | 1.0000 | 1.00000000 | 1.00000000 |

0.1 | 0.8721 | 0.87208247 | 0.87204247 |

0.2 | 0.7764 | 0.77637707 | 0.77593307 |

0.5 | 0.5912 | 0.59119548 | 0.59119589 |

1.0 | 0.4302 | 0.43015970 | 0.43016000 |

2.0 | 0.2840 | 0.28397959 | 0.28398932 |

5.0 | 0.1448 | 0.14484019 | 0.14464015 |

10.0 | 0.0812 | 0.08124198 | 0.08124091 |

20.0 | 0.0438 | 0.04378834 | 0.04378790 |

50.0 | 0.0186 | 0.01859623 | 0.01857868 |

100.0 | 0.0095 | 0.00954997 | 0.00954677 |

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Amirsom, N.A.; Uddin, M.J.; Md Basir, M.F.; Kadir, A.; Bég, O.A.; Md. Ismail, A.I. Computation of Melting Dissipative Magnetohydrodynamic Nanofluid Bioconvection with Second-order Slip and Variable Thermophysical Properties. *Appl. Sci.* **2019**, *9*, 2493.
https://doi.org/10.3390/app9122493

**AMA Style**

Amirsom NA, Uddin MJ, Md Basir MF, Kadir A, Bég OA, Md. Ismail AI. Computation of Melting Dissipative Magnetohydrodynamic Nanofluid Bioconvection with Second-order Slip and Variable Thermophysical Properties. *Applied Sciences*. 2019; 9(12):2493.
https://doi.org/10.3390/app9122493

**Chicago/Turabian Style**

Amirsom, Nur Ardiana, Md. Jashim Uddin, Md Faisal Md Basir, Ali Kadir, O. Anwar Bég, and Ahmad Izani Md. Ismail. 2019. "Computation of Melting Dissipative Magnetohydrodynamic Nanofluid Bioconvection with Second-order Slip and Variable Thermophysical Properties" *Applied Sciences* 9, no. 12: 2493.
https://doi.org/10.3390/app9122493