TimeDependent Flow of WaterBased CoFe_{2}O_{4}MnZnFe_{2}O_{4} Nanoparticles over a Shrinking Sheet with Mass Transfer Effect in Porous Media
Abstract
:1. Introduction
2. Description and Framework of the Mathematical Model
3. Time Stability Analysis
4. Analysis of Results and Discussion
5. Conclusions
 The mass transfer rate and the friction factor accelerate for the first branch solution, but decelerate for the second branch solution for high values of nanoparticle volume fraction ${\phi}_{hnf}$, while the heat transfer rate abruptly diminished in both branch solutions.
 The heat and mass transfer rates, as well as the friction factor, enrich for the first solution for larger values of $K$, while the opposite is seen for the second solution.
 The magnitude of the critical values augments with larger values of ${\phi}_{hnf}$ and $K$, which ultimately causes the separations in the boundary layer to diminish.
 The first branch solution shows an inertial decay of disturbance, which is thus physically reliable as time passes, whereas the second branch solution shows the opposite response.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
$a,\alpha $  constant 
C  concentration 
${C}_{\infty}$  ambient concentration 
${C}_{w}$  surface concentration 
${C}_{f}$  skin friction coefficient 
${C}_{p}$  specific heat at constant pressure (J kg^{−1} K^{−1}) 
${D}_{m}$  mass diffusivity coefficient 
$f$  dimensionless stream function 
$F,G,H$  arbitrary functions 
${K}_{0}$  permeability of the porous media 
$K$  porous medium parameter 
$k$  thermal conductivity of the fluid (W m^{−1} K^{−1}) 
$N{u}_{x}$  local Nusselt number 
$\mathrm{Pr}$  Prandtl number 
${Q}_{0}$  heat sink or source coefficient 
$Q$  heat sink or source parameter 
${\mathrm{Re}}_{x}$  local Reynolds number 
$S$  mass flux velocity parameter 
$Sc$  Schmidt number 
$S{h}_{x}$  local Sherwood number 
$t$  time (s) 
$T$  fluid temperature (K) 
${T}_{\infty}$  ambient temperature (K) 
${T}_{w}$  surface temperature (K) 
$u,v$  velocity component in the x and y directions (m s^{−1}) 
${u}_{w}$  velocity of the surface (m s^{−1}) 
${v}_{w}$  velocity of the mass flux (m s^{−1}) 
$x,y$  Cartesian coordinates (m) 
Greek symbols  
$\alpha $  constant 
$\beta $  unsteady parameter 
$\mathsf{\Gamma}$  dimensionless time variable 
$\gamma $  eigenvalue 
$\eta $  similarity variable 
$\theta $  dimensionless temperature 
$\lambda $  stretching/shrinking parameter 
$\mu $  dynamic viscosity (kg m^{−1} s^{−1}) 
$\nu $  kinematic viscosity of the fluid (m^{2} s^{−1}) 
$\rho $  density of the fluid (kg m^{−3}) 
${\phi}_{1}$  nanoparticle volume fractions for CoFe_{2}O_{4} (cobalt ferrite) 
${\phi}_{2}$  nanoparticle volume fractions for MnZnFe_{2}O_{4} (manganesezinc ferrite) 
${\phi}_{hnf}$  hybrid nanoparticle volume fractions 
$\chi $  dimensionless concentration 
$\psi $  stream function 
Subscripts  
$f$  base fluid 
$hnf$  hybrid nanofluid 
$n1$  solid component for CoFe_{2}O_{4,} (cobalt ferrite) 
$n2$  solid component for MnZnFe_{2}O_{4} (manganesezinc ferrite) 
Superscript  
$\prime $  differentiation with respect to $\eta $ 
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Properties  ρ(kg/m^{3})  C_{p}(J/kg K)  k(W/mk)  Pr 

Water  997.1  4179  0.613  6.2 
CoFe_{2}O_{4}  4907  700  3.7  
$\mathrm{Mn}{\mathrm{ZnF}}_{2}{\mathrm{O}}_{4}$  4900  800  5 
$\mathit{K}$  Present Results  Kameswaran et al. [47] 

0.5  1.22474487  1.22474487 
1.0  1.41421356  1.41421356 
1.5  1.58113883  1.58113883 
2.0  1.73205081  1.73205081 
5.0  2.44948974  2.44948974 
$${\mathit{\phi}}_{\mathit{h}\mathit{n}\mathit{f}}$$

$$\mathit{K}$$

$$\mathit{S}$$

$$\mathit{\beta}$$

$$\mathit{Q}$$

$${\mathit{f}}^{\u2033}(0)$$

$${\mathit{\theta}}^{\prime}(0)$$

$${\mathit{\chi}}^{\prime}(0)$$


0.00  0.1  2.1  −1.0  0.5  1.0326  12.5602  2.0355 
0.01  1.0759  12.2887  2.0389  
0.02  1.1129  12.0244  2.0418  
0.02  0.0  0.9167  12.0170  2.0277  
0.2  1.2512  12.0295  2.0511  
0.3  1.3628  12.0335  2.0582  
0.1  2.15  1.2340  12.3364  2.0990  
2.2  1.3409  12.6469  2.1546  
2.3  1.5297  13.2649  2.2630  
2.1  −2.0  0.6321  12.2673  2.2218  
−3.0  0.1120  12.4926  2.3703  
−4.0  −0.4643  12.7034  2.4991  
−1.0  0.0  1.1129  12.2791  2.0418  
1.5  1.1129  11.4768  2.0418  
2.0  1.1129  11.1796  2.0418 
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Waini, I.; Khan, U.; Zaib, A.; Ishak, A.; Pop, I.; Akkurt, N. TimeDependent Flow of WaterBased CoFe_{2}O_{4}MnZnFe_{2}O_{4} Nanoparticles over a Shrinking Sheet with Mass Transfer Effect in Porous Media. Nanomaterials 2022, 12, 4102. https://doi.org/10.3390/nano12224102
Waini I, Khan U, Zaib A, Ishak A, Pop I, Akkurt N. TimeDependent Flow of WaterBased CoFe_{2}O_{4}MnZnFe_{2}O_{4} Nanoparticles over a Shrinking Sheet with Mass Transfer Effect in Porous Media. Nanomaterials. 2022; 12(22):4102. https://doi.org/10.3390/nano12224102
Chicago/Turabian StyleWaini, Iskandar, Umair Khan, Aurang Zaib, Anuar Ishak, Ioan Pop, and Nevzat Akkurt. 2022. "TimeDependent Flow of WaterBased CoFe_{2}O_{4}MnZnFe_{2}O_{4} Nanoparticles over a Shrinking Sheet with Mass Transfer Effect in Porous Media" Nanomaterials 12, no. 22: 4102. https://doi.org/10.3390/nano12224102