Correction: Sachhin et al. Darcy–Brinkman Model for Ternary Dusty Nanofluid Flow across Stretching/Shrinking Surface with Suction/Injection. Fluids 2024, 9, 94

Sudha Mahanthesh Sachhin; Ulavathi Shettar Mahabaleshwar; David Laroze; Dimitris Drikakis

doi:10.3390/fluids9100241

Figures: In Section 5, we aligned Figures 14–18 by consistently adding all the modelling parameters inside the labels [1]. We also revised the captions for Figures 14–18 to clearly state what they represent [1]. The correct Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 appears below.

Figure 14. Temperature profiles for the dusty and fluid phases versus similarity variable for S = −2.

Figure 15. Temperature profiles for the dusty and fluid phases versus similarity variable for S = 0.

Figure 16. Temperature profiles for the dusty and fluid phases versus similarity variable for S = 2.

Figure 17. Temperature profile versus similarity variable for a shrinking boundary.

Figure 18. Velocity profile versus similarity variable variation in

D a^{- 1}

.

Text Correction: In Section 2, the following text was added: “Similar to previous studies [2]”, “b is a parameter that is b > 0 for heated and b < 0 for cooled plate”. The correct text appears below.

Similar to previous studies [2], here,

u, v, u_{p}

, and

v_{p}

are the velocity components of a fluid and dusty fluid phase along the x- and y-directions, respectively; the dusty and fluid phase temperatures are Tp and T;

μ

is the dynamic viscosity;

ρ

is the effective density;

κ

is the thermal conductivity; b is a parameter that is b > 0 for heated and b < 0 for cooled plate;

σ

is the electrical conductivity;

C_{p}

and

C_{m}

are the specific heat coefficients;

τ_{T}

is the heat equilibrium time;

L_{1}

is the Stokes drag/resistance term;

ν

is the kinematic viscosity of nanoparticles N;

k_{1}

is the flow permeability; and

τ_{v} = \frac{m}{L_{1}}

is a relaxation time parameter, where m denotes the mass of dusty particles [2].

In Section 2, we corrected the typographical error in the definition of the Prandtl number. The correct one is

\Pr = \frac{μ C_{p}}{κ_{f}}

.

In Section 5, we revised the text to avoid ambiguity regarding the results of Figure 14, Figure 15, Figure 16 [1]. The correct text appears below.

Figure 14, Figure 15, Figure 16 show the temperatures for the fluid and dusty phases for different values of S = −2, 0 and 2, respectively. Increasing S value increases the thermal boundary layer thickness of the fluid phase. The dusty phase exhibits an increase in the thermal boundary layer when S increases from −2 to 0, while decreases for S = 2.

Equations: In Equations (35)–(39), there are typographical errors. We revised the subscript thnf to tnf. In Equation (38), we also revised the κ_nf to κ_f. The correct equations appears below:

μ_{t n f} = \frac{1}{{(1 - ϕ_{A g})}^{2.5} {(1 - ϕ_{C u})}^{2.5} {(1 - ϕ_{T i O_{2}})}^{2.5}} .

(35)

\begin{matrix} \frac{ρ_{t n f}}{ρ_{f}} = (1 - ϕ_{A g}) \{(1 - ϕ_{C u}) [\begin{matrix} (1 - ϕ_{T i O_{2}}) \\ + ϕ_{T i O_{2}} \frac{ρ_{T i O_{2}}}{ρ_{f}} \end{matrix}] + ϕ_{C u} \frac{ρ_{C u}}{ρ_{f}}\} \\ + ϕ_{A g} \frac{ρ_{A g}}{ρ_{f}} . \end{matrix}

(36)

\frac{{(ρ C_{p})}_{t n f}}{{(ρ C_{p})}_{f}} = (1 - ϕ_{A g}) \{\begin{matrix} (1 - ϕ_{C u}) \times \\ [\begin{matrix} (1 - ϕ_{T i O_{2}}) \\ + ϕ_{T i O_{2}} \frac{{(ρ C_{p})}_{T i O_{2}}}{{(ρ C_{p})}_{f}} \end{matrix}] \\ + ϕ_{C u} \frac{{(ρ C_{p})}_{C u}}{{(ρ C_{p})}_{f}} \end{matrix}\} + ϕ_{A g} \frac{{(ρ C_{p})}_{A g}}{{(ρ C_{p})}_{f}} .

(37)

\begin{matrix} \frac{κ_{t n f}}{κ_{h n f}} = \frac{κ_{A g} + 2 κ_{h n f} - 2 ϕ_{A g} (κ_{h n f} - κ_{A g})}{κ_{A g} + 2 κ_{h n f} + ϕ_{A g} (κ_{h n f} - κ_{A g})}, \\ \frac{κ_{h n f}}{κ_{n f}} = \frac{κ_{C u} + 2 κ_{n f} - 2 ϕ_{C u} (κ_{n f} - κ_{C u})}{κ_{C u} + 2 κ_{n f} + ϕ_{C u} (κ_{n f} - κ_{C u})}, \\ \frac{κ_{n f}}{κ_{f}} = \frac{κ_{T i O_{2}} + 2 κ_{f} - 2 ϕ_{T i O_{2}} (κ_{f} - κ_{T i O_{2}})}{κ_{T i O_{2}} + 2 κ_{f} + ϕ_{T i O_{2}} (κ_{f} - κ_{T i O_{2}})} . \end{matrix}\}

(38)

\begin{matrix} \frac{σ_{t n f}}{σ_{h n f}} = 1 + \frac{3 (\frac{σ_{A g}}{σ_{h n f}} - 1) ϕ_{A g}}{(\frac{σ_{A g}}{σ_{h n f}} + 2) - (\frac{σ_{A g}}{σ_{h n f}} - 1) ϕ_{A g}}, \\ \frac{σ_{h n f}}{σ_{n f}} = 1 + \frac{3 (\frac{σ_{C u}}{σ_{n f}} - 1) ϕ_{C u}}{(\frac{σ_{C u}}{σ_{n f}} + 2) - (\frac{σ_{C u}}{σ_{n f}} - 1) ϕ_{C u}}, \\ \frac{σ_{n f}}{σ_{f}} = 1 + \frac{3 (\frac{σ_{T i O_{2}}}{σ_{f}} - 1) ϕ_{T i O_{2}}}{(\frac{σ_{T i O_{2}}}{σ_{f}} + 2) - (\frac{σ_{T i O_{2}}}{σ_{f}} - 1) ϕ_{T i O_{2}}} . \end{matrix}\}

(39)

Nomenclature: We added the units that were missing in several parameters and corrected the typographical errors in some of the parameters [1]. The correct Nomenclature appears below.

The authors state that the scientific conclusions are unaffected. These corrections were approved by the Academic Editor. The original publication has also been updated.

Nomenclature

$A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$	Constants
a	Stretching coefficient $(s^{- 1})$
$B_{0}$	Magnetic parameter (Tesla)
$C_{m}, C_{p}$	Specific heat coefficient $({JK}^{- 1} {kg}^{- 1})$
d	Stretching/shrinking parameter
$D a^{- 1}$	Inverse Darcy number
$E c$	Eckert number
$f (η)$	Velocity function fluid phase
$F (η)$	Velocity function dusty phase
$k_{1}$	Permeability of porous medium $(m^{2})$
$l$	Mass number
$L_{1}$	Stokes drag term (kg/s)
$m$	Mass of the dusty particles $(kg)$
M	dimensionless magnetic parameter
$Ni$	Heat source/sink parameter
$Nr$	Thermal radiation parameter
$N$	Quantity of nanoparticles $(m^{- 3})$
$P r$	Prandtl number
p	Pressure $({Nm}^{- 2})$
$q_{r}$	Radiative heat flux $({Wm}^{- 2})$
$Q_{0}$	Heat source/sink $({Wm}^{- 3} K^{- 1})$
S	Dimensionless mass suction/injection parameter
$S > 0$	Mass suction parameter
$S = 0$	No permeability
$T_{p}$	Dusty-phase temperature $(K)$
$T_{w}$	Surface temperature $(K)$
T	Fluid temperature $(K)$
$T_{\infty}$	Ambient temperature $(K)$
$u, v$	x, y-axis velocity of fluid phase $({ms}^{- 1})$
$u_{p}, v_{p}$	x, y-axis velocity of dusty phase $({ms}^{- 1})$
$u_{w}$	Wall velocity $({ms}^{- 1})$
$v_{w}$	Wall mass transfer velocity $({ms}^{- 1})$
x	Coordinate along the plate $(m)$
y	Coordinate normal to the plate $(m)$
Greek symbols
$α$	Stretching speed of dust particles $({ms}^{- 1})$
$β_{T}$	Fluid–particle interaction parameters
$β$	Solution parameters
$λ_{1}$ , $λ_{2}$ , $λ_{3}$	Solution roots
$ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}$	Constants
$η$	Similarity variable
$γ$	Heat coefficient
$Λ$	Brinkman number
$κ$	Thermal conductivity $({Wm}^{- 1} K^{- 1})$
$κ^{*}$	Absorption coefficient $(m^{- 1})$
$μ_{e f f}$	Effective dynamic viscosity $({kg (ms)}^{- 1})$
$μ, μ_{p}$	Dynamic viscosity of the fluid and dusty phase $({kg (ms)}^{- 1})$
$ν, ν_{p}$	Kinematic viscosity of fluid and dusty phase $(m^{2} s^{- 1})$
$ρ$	Fluid density $({kgm}^{- 3})$
$ρ_{p}$	Particle phase density $({kgm}^{- 3})$
$ψ$	Stream function
$σ, σ_{p}$	Electrical conductivity $(S i e m e n s / m = A^{2} s^{3} m^{- 3} {kg}^{- 1})$
$σ^{*}$	Stephen–Boltzmann constant $({Wm}^{- 2} K^{- 4})$
$τ_{T}$	Heat equilibrium time $(s)$
$τ_{v}$	Relaxation time parameter $(s)$
$φ$	Fluid nanoparticle volume fraction ratio
$θ (η)$	Dimensionless temperature of fluid phase
$Φ (η)$	Dimensionless temperature of dusty phase
Abbreviations
HNF	Hybrid nanofluid
ODE	Ordinary differential equation
PDE	Partial differential equation
MHD	Magnetohydrodynamics
BCs	Boundary conditions
TNF	Ternary nanofluid

Reference

Sachhin, S.M.; Mahabaleshwar, U.S.; Laroze, D.; Drikakis, D. Darcy–Brinkman Model for Ternary Dusty Nanofluid Flow across Stretching/Shrinking Surface with Suction/Injection. Fluids 2024, 9, 94. [Google Scholar] [CrossRef]

Figure 14. Temperature profiles for the dusty and fluid phases versus similarity variable for S = −2.

Figure 15. Temperature profiles for the dusty and fluid phases versus similarity variable for S = 0.

Figure 16. Temperature profiles for the dusty and fluid phases versus similarity variable for S = 2.

Figure 17. Temperature profile versus similarity variable for a shrinking boundary.

Figure 18. Velocity profile versus similarity variable variation in

D a^{- 1}

.

Figure 18. Velocity profile versus similarity variable variation in

D a^{- 1}

.

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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

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