Erratum: High-order algorithm to simulate thermal convection for a thermo-dependent viscoplastic fluid flow

doi:10.1108/HFF-04-2026-0539

It has come to the attention of the publisher that article Al-Bahrani, B.K. and Al-Muslimawi, A.H. (2025), “High-order algorithm to simulate thermal convection for a thermo-dependent viscoplastic fluid flow”. International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 35 No. 9 pp. 3322–3345, Link to High-order algorithm to simulate thermal convection for a thermo-dependent viscoplastic fluid flowLink to the cited article., had errors in several equations with incorrect brackets/parentheses, overlapping numbers, extra symbol within the mathematical and inline equations only in the PDF version of the article which is now corrected.

The table below provides details of the affected equations which have now been corrected in the PDF:

Equation no.	Incorrect equation	Correct equation
(1)	$\nabla . u = 0, ψ$	∇.u = 0,
(2)	$\frac{\partial u p}{\partial t p} = p \frac{1}{R e p} \nabla . \bar{τ} p - R e p (u . \nabla) u - \nabla, ψ$	$\frac{\partial u}{\partial t} = \frac{1}{R e} (\nabla \cdot \bar{τ} - R e (u \cdot \nabla) u - \nabla p),$
(3)	$\frac{\partial T}{\partial t p} = p \frac{1}{P e p} \nabla^{2} T p - P e p (u . \nabla) T + B r p (\bar{τ} p . \nabla u) . ψ$	$\frac{\partial T}{\partial t} = \frac{1}{P e} (\nabla^{2} T - P e (u . \nabla) T + B r (\bar{τ} : \nabla u)) .$
(4)	$\bar{τ} = μ (\dot{γ}) \bar{γ} \cdot ψ ψ$	$\bar{τ} = μ (\dot{γ}) \bar{\dot{γ}} .$
(5)	$\bar{τ} = {\begin{array}{l} (μ_{p} + \frac{τ_{0}}{\| γ \|} [1 - e p^{(- a \| γ \|)}]) \bar{\dot{γ}} \| \bar{τ} \| > τ_{0}, ψ \\ \bar{\dot{γ}} = 0 \| \bar{τ} \| \leq τ_{0}, ψ \end{array}$	$\bar{τ} = {\begin{array}{l} (μ_{p} + \frac{τ_{0}}{\| \dot{γ} \|} [1 - e^{(- a \| \dot{γ} \|)}]) \bar{\dot{γ}} \| \bar{τ} \| > τ_{0}, \\ \bar{\dot{γ}} = 0 \| \bar{τ} \| \leq τ_{0}, \end{array}$
(6)	$μ = e {\bar{p}}^{b (T - T_{0})} μ (γ) \cdot ψ ψ$	$μ = e^{- b (T - T_{0})} μ (\dot{γ}) .$
(7)	$\frac{\partial}{\partial t p} + p \frac{1}{M a^{2}} (\nabla \cdot u) = 0, ψ ψ$	$\frac{\partial p}{\partial t} + \frac{1}{M a^{2}} (\nabla \cdot u) = 0,$
(8)	$\frac{\partial}{\partial t p} + ξ p + p \frac{1}{M a^{2}} (\nabla \cdot u) = 0, ψ$	$\frac{\partial p}{\partial t} + ξ p + \frac{1}{M a^{2}} (\nabla \cdot u) = 0$
(9)	$u^{m + \frac{1}{2}} = u^{m} + p \frac{Δ t p}{2 R e p} - R e (u^{m} \cdot \nabla) u^{m} + p \frac{1}{2} \nabla \cdot {\bar{τ}}^{m + \frac{1}{2}} + \nabla \cdot {\bar{τ}}^{m p} - \nabla^{m}, ψ$	$u^{m + \frac{1}{2}} = u^{m} + \frac{Δ t}{2 R e} (- R e (u^{m} \cdot \nabla) u^{m} + \frac{1}{2} (\nabla \cdot {\bar{τ}}^{m + \frac{1}{2}} + \nabla \cdot {\bar{τ}}^{m}) - \nabla p^{m}),$
(10)	$T^{m + \frac{1}{2}} = T^{m} + p \frac{Δ t p}{2 P e p} - P e (u^{m} \cdot \nabla) T^{m} + \frac{1}{2} \nabla^{2} T^{m + \frac{1}{2}} + \nabla^{2} T^{m} + B r ({\bar{r}}^{m p} : p \nabla u^{m}) \leftarrow, ψ$	$T^{m + \frac{1}{2}} = T^{m} + \frac{Δ t}{2 P e} (- P e (u^{m} \cdot \nabla) T^{m} + \frac{1}{2} (\nabla^{2} T^{m + \frac{1}{2}} + \nabla^{2} T^{m}) + B r ({\bar{τ}}^{m} : \nabla u^{m})),$
(11)	$m + \frac{1}{2} = p \frac{m p - \frac{Δ t p}{2 M a^{2}} (\nabla \cdot u^{m + \frac{1}{2}})}{1 + p \frac{ξ Δ t p}{2}}, ψ$	$p^{m + \frac{1}{2}} = \frac{p^{m} - \frac{Δ t}{2 M a^{2}} (\nabla \cdot u^{m + \frac{1}{2}})}{1 + \frac{ξ Δ t}{2}},$
(12)	$u^{m + 1} = u^{m} + p \frac{Δ t p}{R e p} - R e p u^{m + \frac{1}{2}} \cdot \nabla u^{m + \frac{1}{2}} + p^{\frac{1}{2}} (\nabla \cdot {\bar{τ}}^{m + 1} + \nabla \cdot {\bar{τ}}^{m}) - \nabla^{m + \frac{1}{2}}, ψ$	$u^{m + 1} = u^{m} + \frac{Δ t}{R e} (- R e (u^{m + \frac{1}{2}} \cdot \nabla) u^{m + \frac{1}{2}} + \frac{1}{2} (\nabla \cdot {\bar{τ}}^{m + 1} + \nabla \cdot {\bar{τ}}^{m}) - \nabla p^{m + \frac{1}{2}}),$
(13)	$T^{m + 1} = T^{m} + p \frac{Δ t p}{P e p} - P e p u^{m + \frac{1}{2}}, \nabla T^{m + \frac{1}{2}} + \frac{1}{2} (\nabla^{2} T^{m + 1} + \nabla^{2} T^{m}^{p}) + B r p {\bar{t}}^{m + \frac{1}{2}}; p \nabla u^{m + \frac{1}{2}}, ψ$	$T^{m + 1} = T^{m} + \frac{Δ t}{P e} (- P e (u^{m + \frac{1}{2}} \cdot \nabla) T^{m + \frac{1}{2}} + \frac{1}{2} (\nabla^{2} T^{m + 1} + \nabla^{2} T^{m})$ $+ B r ({\bar{τ}}^{m + \frac{1}{2}} : \nabla u^{m + \frac{1}{2}})),$
(14)	$^{m + 1} =^{m p} - Δ t p ξ p^{m + \frac{1}{2}} + p \frac{1}{M a^{2}} (\nabla \cdot u^{m + 1}) . ψ$	$p^{m + 1} = p^{m} - Δ t (ξ p^{m + \frac{1}{2}} + \frac{1}{M a^{2}} (\nabla \cdot u^{m + 1})) .$
(15)	$\frac{2 Re}{Δ t p} M + p \frac{1}{2} S_{u p} Δ u^{m + \frac{1}{2}} = p - S_{u p} + R e N (u^{m}) u^{m p} + L^{⊤}^{m}, ψ$	$[\frac{2 R e}{Δ t} M + \frac{1}{2} S_{u}] Δ u^{m + \frac{1}{2}} = - [S_{u} + R e N (u^{m})] u^{m} + L^{⊤} p^{m},$
(16)	$\frac{2 P e p}{Δ t p} M + p \frac{1}{2} S_{T p} Δ T^{m + \frac{1}{2}} = p - P e N (u^{m}) + S_{T p} T^{m p} + B r Φ (u^{m}), ψ$	$[\frac{2 P e}{Δ t} M + \frac{1}{2} S_{T}] Δ T^{m + \frac{1}{2}} = - [P e N (u^{m}) + S_{T}] T^{m} + B r Φ (u^{m}),$
(17)	$M p^{m + \frac{1}{2}} = p \frac{M^{m p} - \frac{Δ t}{2 M a 2} L u^{m + \frac{1}{2}}}{1 + p \frac{ξ Δ t p}{2}}, ψ$	$M_{p} p^{m + \frac{1}{2}} = \frac{M_{p} p^{m} - \frac{Δ t}{2 M a^{2}} L u^{m + \frac{1}{2}}}{1 + \frac{ξ Δ t}{2}},$
(18)	$\frac{R e p}{Δ t p} M + p \frac{1}{2} S_{u p} Δ u^{m + 1} = p - S_{u} u^{m p} - R e N p u^{m + \frac{1}{2}} u^{m + \frac{1}{2}} + L^{⊤}^{m + \frac{1}{2}}, ψ$	$[\frac{R e}{Δ t} M + \frac{1}{2} S_{u}] Δ u^{m + 1} = - S_{u} u^{m} - R e N (u^{m + \frac{1}{2}}) u^{m + \frac{1}{2}} + L^{⊤} p^{m + \frac{1}{2}},$
(19)	$\frac{P e}{Δ t p} M + p \frac{1}{2} S_{T p} Δ T^{m + 1} = p - P e N p u^{m + \frac{1}{2}} T^{m + \frac{1}{2}} - S_{T} T^{m p} + B r Φ p u^{m + \frac{1}{2}}, ψ$	$[\frac{P e}{Δ t} M + \frac{1}{2} S_{T}] Δ T^{m + 1} = - P e N (u^{m + \frac{1}{2}}) T^{m + \frac{1}{2}} - S_{T} T^{m} + B r Φ (u^{m + \frac{1}{2}}),$
(20)	$M Δ p^{m + 1} = p - Δ t p ξ M p^{m + \frac{1}{2}} - \frac{1}{M a^{2}} L u^{m + 1}, ψ$	$M_{p} Δ p^{m + 1} = - Δ t (ξ M_{p} p^{m + \frac{1}{2}} - \frac{1}{M a^{2}} L u^{m + 1}),$
(21)	$\frac{2 R e p}{Δ t p} M + p \frac{1}{2} S_{u p} Δ u^{m + \frac{1}{2}} = p - S_{u p} + R e N (u^{m}) u^{m p} + L^{⊤}^{m}, ψ$	$[\frac{2 R e}{Δ t} M + \frac{1}{2} S_{u}] Δ u^{m + \frac{1}{2}} = - [S_{u} + R e N (u^{m})] u^{m} + L^{⊤} p^{m},$
(22)	$\frac{2 P e p}{Δ t p} M + p \frac{1}{2} S_{T p} Δ T^{m + \frac{1}{2}} = - P e N (u^{m}) - S_{T p} T^{m} + B r Φ (u^{m}), ψ$	$[\frac{2 P e}{Δ t} M + \frac{1}{2} S_{T}] Δ T^{m + \frac{1}{2}} = [- P e N (u^{m}) - S_{T}] T^{m} + B r Φ (u^{m}),$
(23)	$\frac{R e p}{Δ t p} M + p \frac{1}{2} S_{u p} Δ u^{⋆} = p - S_{u} u^{m p} - R e N p u^{m + \frac{1}{2}} u^{m + \frac{1}{2}} + L^{⊤}^{m p}, ψ$	$[\frac{R e}{Δ t} M + \frac{1}{2} S_{u}] Δ u^{⋆} = - S_{u} u^{m} - R e N (u^{m + \frac{1}{2}}) u^{m + \frac{1}{2}} + L^{⊤} p^{m},$
(24)	$\frac{P e}{Δ t p} M + \frac{1}{2} S_{T p} Δ T^{m + 1} = p - P e N u^{m + \frac{1}{2}} T^{m + \frac{1}{2}} - S_{T} T^{m p} + B r Φ p u^{m + \frac{1}{2}}, ψ$	$[\frac{P e}{Δ t} M + \frac{1}{2} S_{T}] Δ T^{m + 1} = - P e N u^{m + \frac{1}{2}} T^{m + \frac{1}{2}} - S_{T} T^{m} + B r Φ (u^{m + \frac{1}{2}}),$
(25)	$K Δ p^{m + 1} = p - \frac{2 R e}{Δ t p} L u^{⋆}, ψ$	$K Δ p^{m + 1} = - \frac{2 R e}{Δ t} L u^{⋆},$
(26)	$\frac{2 R e}{Δ t p} M (p u^{m + 1} - u^{⋆}) = L^{⊤} Δ p^{m + 1}, ψ$	$\frac{2 R e}{Δ t} M (u^{m + 1} - u^{⋆}) = L^{⊤} Δ p^{m + 1},$

Equation no.	Incorrect equation	Correct equation
(1)	$\nabla . u = 0, ψ$	∇.u = 0,
(2)	$\frac{\partial u p}{\partial t p} = p \frac{1}{R e p} \nabla . \bar{τ} p - R e p (u . \nabla) u - \nabla, ψ$	$\frac{\partial u}{\partial t} = \frac{1}{R e} (\nabla \cdot \bar{τ} - R e (u \cdot \nabla) u - \nabla p),$
(3)	$\frac{\partial T}{\partial t p} = p \frac{1}{P e p} \nabla^{2} T p - P e p (u . \nabla) T + B r p (\bar{τ} p . \nabla u) . ψ$	$\frac{\partial T}{\partial t} = \frac{1}{P e} (\nabla^{2} T - P e (u . \nabla) T + B r (\bar{τ} : \nabla u)) .$
(4)	$\bar{τ} = μ (\dot{γ}) \bar{γ} \cdot ψ ψ$	$\bar{τ} = μ (\dot{γ}) \bar{\dot{γ}} .$
(5)	$\bar{τ} = {\begin{array}{l} (μ_{p} + \frac{τ_{0}}{\| γ \|} [1 - e p^{(- a \| γ \|)}]) \bar{\dot{γ}} \| \bar{τ} \| > τ_{0}, ψ \\ \bar{\dot{γ}} = 0 \| \bar{τ} \| \leq τ_{0}, ψ \end{array}$	$\bar{τ} = {\begin{array}{l} (μ_{p} + \frac{τ_{0}}{\| \dot{γ} \|} [1 - e^{(- a \| \dot{γ} \|)}]) \bar{\dot{γ}} \| \bar{τ} \| > τ_{0}, \\ \bar{\dot{γ}} = 0 \| \bar{τ} \| \leq τ_{0}, \end{array}$
(6)	$μ = e {\bar{p}}^{b (T - T_{0})} μ (γ) \cdot ψ ψ$	$μ = e^{- b (T - T_{0})} μ (\dot{γ}) .$
(7)	$\frac{\partial}{\partial t p} + p \frac{1}{M a^{2}} (\nabla \cdot u) = 0, ψ ψ$	$\frac{\partial p}{\partial t} + \frac{1}{M a^{2}} (\nabla \cdot u) = 0,$
(8)	$\frac{\partial}{\partial t p} + ξ p + p \frac{1}{M a^{2}} (\nabla \cdot u) = 0, ψ$	$\frac{\partial p}{\partial t} + ξ p + \frac{1}{M a^{2}} (\nabla \cdot u) = 0$
(9)	$u^{m + \frac{1}{2}} = u^{m} + p \frac{Δ t p}{2 R e p} - R e (u^{m} \cdot \nabla) u^{m} + p \frac{1}{2} \nabla \cdot {\bar{τ}}^{m + \frac{1}{2}} + \nabla \cdot {\bar{τ}}^{m p} - \nabla^{m}, ψ$	$u^{m + \frac{1}{2}} = u^{m} + \frac{Δ t}{2 R e} (- R e (u^{m} \cdot \nabla) u^{m} + \frac{1}{2} (\nabla \cdot {\bar{τ}}^{m + \frac{1}{2}} + \nabla \cdot {\bar{τ}}^{m}) - \nabla p^{m}),$
(10)	$T^{m + \frac{1}{2}} = T^{m} + p \frac{Δ t p}{2 P e p} - P e (u^{m} \cdot \nabla) T^{m} + \frac{1}{2} \nabla^{2} T^{m + \frac{1}{2}} + \nabla^{2} T^{m} + B r ({\bar{r}}^{m p} : p \nabla u^{m}) \leftarrow, ψ$	$T^{m + \frac{1}{2}} = T^{m} + \frac{Δ t}{2 P e} (- P e (u^{m} \cdot \nabla) T^{m} + \frac{1}{2} (\nabla^{2} T^{m + \frac{1}{2}} + \nabla^{2} T^{m}) + B r ({\bar{τ}}^{m} : \nabla u^{m})),$
(11)	$m + \frac{1}{2} = p \frac{m p - \frac{Δ t p}{2 M a^{2}} (\nabla \cdot u^{m + \frac{1}{2}})}{1 + p \frac{ξ Δ t p}{2}}, ψ$	$p^{m + \frac{1}{2}} = \frac{p^{m} - \frac{Δ t}{2 M a^{2}} (\nabla \cdot u^{m + \frac{1}{2}})}{1 + \frac{ξ Δ t}{2}},$
(12)	$u^{m + 1} = u^{m} + p \frac{Δ t p}{R e p} - R e p u^{m + \frac{1}{2}} \cdot \nabla u^{m + \frac{1}{2}} + p^{\frac{1}{2}} (\nabla \cdot {\bar{τ}}^{m + 1} + \nabla \cdot {\bar{τ}}^{m}) - \nabla^{m + \frac{1}{2}}, ψ$	$u^{m + 1} = u^{m} + \frac{Δ t}{R e} (- R e (u^{m + \frac{1}{2}} \cdot \nabla) u^{m + \frac{1}{2}} + \frac{1}{2} (\nabla \cdot {\bar{τ}}^{m + 1} + \nabla \cdot {\bar{τ}}^{m}) - \nabla p^{m + \frac{1}{2}}),$
(13)	$T^{m + 1} = T^{m} + p \frac{Δ t p}{P e p} - P e p u^{m + \frac{1}{2}}, \nabla T^{m + \frac{1}{2}} + \frac{1}{2} (\nabla^{2} T^{m + 1} + \nabla^{2} T^{m}^{p}) + B r p {\bar{t}}^{m + \frac{1}{2}}; p \nabla u^{m + \frac{1}{2}}, ψ$	$T^{m + 1} = T^{m} + \frac{Δ t}{P e} (- P e (u^{m + \frac{1}{2}} \cdot \nabla) T^{m + \frac{1}{2}} + \frac{1}{2} (\nabla^{2} T^{m + 1} + \nabla^{2} T^{m})$ $+ B r ({\bar{τ}}^{m + \frac{1}{2}} : \nabla u^{m + \frac{1}{2}})),$
(14)	$^{m + 1} =^{m p} - Δ t p ξ p^{m + \frac{1}{2}} + p \frac{1}{M a^{2}} (\nabla \cdot u^{m + 1}) . ψ$	$p^{m + 1} = p^{m} - Δ t (ξ p^{m + \frac{1}{2}} + \frac{1}{M a^{2}} (\nabla \cdot u^{m + 1})) .$
(15)	$\frac{2 Re}{Δ t p} M + p \frac{1}{2} S_{u p} Δ u^{m + \frac{1}{2}} = p - S_{u p} + R e N (u^{m}) u^{m p} + L^{⊤}^{m}, ψ$	$[\frac{2 R e}{Δ t} M + \frac{1}{2} S_{u}] Δ u^{m + \frac{1}{2}} = - [S_{u} + R e N (u^{m})] u^{m} + L^{⊤} p^{m},$
(16)	$\frac{2 P e p}{Δ t p} M + p \frac{1}{2} S_{T p} Δ T^{m + \frac{1}{2}} = p - P e N (u^{m}) + S_{T p} T^{m p} + B r Φ (u^{m}), ψ$	$[\frac{2 P e}{Δ t} M + \frac{1}{2} S_{T}] Δ T^{m + \frac{1}{2}} = - [P e N (u^{m}) + S_{T}] T^{m} + B r Φ (u^{m}),$
(17)	$M p^{m + \frac{1}{2}} = p \frac{M^{m p} - \frac{Δ t}{2 M a 2} L u^{m + \frac{1}{2}}}{1 + p \frac{ξ Δ t p}{2}}, ψ$	$M_{p} p^{m + \frac{1}{2}} = \frac{M_{p} p^{m} - \frac{Δ t}{2 M a^{2}} L u^{m + \frac{1}{2}}}{1 + \frac{ξ Δ t}{2}},$
(18)	$\frac{R e p}{Δ t p} M + p \frac{1}{2} S_{u p} Δ u^{m + 1} = p - S_{u} u^{m p} - R e N p u^{m + \frac{1}{2}} u^{m + \frac{1}{2}} + L^{⊤}^{m + \frac{1}{2}}, ψ$	$[\frac{R e}{Δ t} M + \frac{1}{2} S_{u}] Δ u^{m + 1} = - S_{u} u^{m} - R e N (u^{m + \frac{1}{2}}) u^{m + \frac{1}{2}} + L^{⊤} p^{m + \frac{1}{2}},$
(19)	$\frac{P e}{Δ t p} M + p \frac{1}{2} S_{T p} Δ T^{m + 1} = p - P e N p u^{m + \frac{1}{2}} T^{m + \frac{1}{2}} - S_{T} T^{m p} + B r Φ p u^{m + \frac{1}{2}}, ψ$	$[\frac{P e}{Δ t} M + \frac{1}{2} S_{T}] Δ T^{m + 1} = - P e N (u^{m + \frac{1}{2}}) T^{m + \frac{1}{2}} - S_{T} T^{m} + B r Φ (u^{m + \frac{1}{2}}),$
(20)	$M Δ p^{m + 1} = p - Δ t p ξ M p^{m + \frac{1}{2}} - \frac{1}{M a^{2}} L u^{m + 1}, ψ$	$M_{p} Δ p^{m + 1} = - Δ t (ξ M_{p} p^{m + \frac{1}{2}} - \frac{1}{M a^{2}} L u^{m + 1}),$
(21)	$\frac{2 R e p}{Δ t p} M + p \frac{1}{2} S_{u p} Δ u^{m + \frac{1}{2}} = p - S_{u p} + R e N (u^{m}) u^{m p} + L^{⊤}^{m}, ψ$	$[\frac{2 R e}{Δ t} M + \frac{1}{2} S_{u}] Δ u^{m + \frac{1}{2}} = - [S_{u} + R e N (u^{m})] u^{m} + L^{⊤} p^{m},$
(22)	$\frac{2 P e p}{Δ t p} M + p \frac{1}{2} S_{T p} Δ T^{m + \frac{1}{2}} = - P e N (u^{m}) - S_{T p} T^{m} + B r Φ (u^{m}), ψ$	$[\frac{2 P e}{Δ t} M + \frac{1}{2} S_{T}] Δ T^{m + \frac{1}{2}} = [- P e N (u^{m}) - S_{T}] T^{m} + B r Φ (u^{m}),$
(23)	$\frac{R e p}{Δ t p} M + p \frac{1}{2} S_{u p} Δ u^{⋆} = p - S_{u} u^{m p} - R e N p u^{m + \frac{1}{2}} u^{m + \frac{1}{2}} + L^{⊤}^{m p}, ψ$	$[\frac{R e}{Δ t} M + \frac{1}{2} S_{u}] Δ u^{⋆} = - S_{u} u^{m} - R e N (u^{m + \frac{1}{2}}) u^{m + \frac{1}{2}} + L^{⊤} p^{m},$
(24)	$\frac{P e}{Δ t p} M + \frac{1}{2} S_{T p} Δ T^{m + 1} = p - P e N u^{m + \frac{1}{2}} T^{m + \frac{1}{2}} - S_{T} T^{m p} + B r Φ p u^{m + \frac{1}{2}}, ψ$	$[\frac{P e}{Δ t} M + \frac{1}{2} S_{T}] Δ T^{m + 1} = - P e N u^{m + \frac{1}{2}} T^{m + \frac{1}{2}} - S_{T} T^{m} + B r Φ (u^{m + \frac{1}{2}}),$
(25)	$K Δ p^{m + 1} = p - \frac{2 R e}{Δ t p} L u^{⋆}, ψ$	$K Δ p^{m + 1} = - \frac{2 R e}{Δ t} L u^{⋆},$
(26)	$\frac{2 R e}{Δ t p} M (p u^{m + 1} - u^{⋆}) = L^{⊤} Δ p^{m + 1}, ψ$	$\frac{2 R e}{Δ t} M (u^{m + 1} - u^{⋆}) = L^{⊤} Δ p^{m + 1},$

Continuity equation	Momentum equation	Energy equation
$^{m + \frac{1}{2}} = p^{m p} - \frac{Δ t p}{2} ξ p^{m p} + \frac{1}{M a^{2}} (\nabla \cdot u^{m}) \leftarrow,^{m + 1} =^{m p} - Δ t p ξ p^{m + \frac{1}{2}} + p \frac{1}{M a^{2}} (\nabla \cdot u^{m}) \leftarrow$	$u^{m + \frac{1}{2}} = u^{m} + p \frac{Δ t p}{2 R e p} - Re (u^{m} \cdot \nabla) u^{m} + \nabla \cdot {\bar{t}}^{m p} - \nabla^{m}, u^{m + 1} = u^{m} + p \frac{Δ t p}{R e p} ((- R e p u^{m + \frac{1}{2}} \nabla u^{m + \frac{1}{2}} + \nabla \cdot {\bar{τ}}^{m + \frac{1}{2}} - \nabla^{m})) \cdot$	$T^{m + \frac{1}{2}} = T^{m} + p \frac{Δ t p}{2 P e p} - P e (u^{m} \cdot \nabla) T^{m} + \nabla^{2} T^{m} + B r ({\bar{τ}}^{m p} : p \nabla u^{m}), T^{m + 1} = T^{m} + p \frac{Δ t p}{P e p} - P e (u^{m} \cdot \nabla) T^{m + \frac{1}{2}} + \nabla^{2} T^{m + \frac{1}{2}} + B r ({\bar{τ}}^{m p} : p \nabla u^{m}) .$

On page 3325, the incorrect equation $R e = p \frac{ρ U^{2 - n} D^{n p}}{μ_{0}}, P e = p \frac{ρ C p U l p}{k p}, B r = p \frac{μ_{0} U^{2}}{k Δ T p},$ has been corrected to $R e = \frac{ρ U^{2 - n} D^{n}}{μ_{0}}, P e = \frac{ρ C_{p} U l}{k}, B r = \frac{μ_{0} U^{2}}{k Δ T},$ ⁠.

On page 3326, the incorrect equation $\bar{τ} p$ ⁠, $\bar{γ} p$ ⁠, $| γ | = \sqrt{\frac{1}{2} Π_{γ}} = (\sqrt{\frac{1}{2} \bar{γ} : \bar{γ}})$ ⁠, $f = p \frac{τ_{w p}}{\frac{1}{2} ρ u_{e}^{2}} = p \frac{D Δ p}{2 l ρ u_{e p}^{2}}, N u = p \frac{H_{x} D}{k} .$ has been corrected to $\bar{τ}$ ⁠, $\bar{γ}$ ⁠. $| \dot{γ} | = \sqrt{\frac{1}{2} Π_{\dot{γ}}} = (\sqrt{\frac{1}{2} \bar{\dot{γ}} : \bar{\dot{γ}}})$ ⁠, $f p = \frac{τ_{w}}{\frac{1}{2} ρ u_{e}^{2}} = \frac{D Δ p}{2 l ρ u_{e}^{2}}, N u = \frac{H_{x} D}{k} .$ respectively in the PDF.

On page 3327, the below equations in the steps 1 and 2 for Continuity equation, Momentum equation, Energy equation has been corrected in the PDF:

On page 3329, the below incorrect inline equations have now been corrected in the PDF:

M_{i j} = p \int_{Ω} ψ_{i} ψ_{j} r d Ω, M_{p i j} = p \int_{Ω p} (ϕ_{i} ϕ_{j} r d Ω, N_{i j} (u) = ρ \int_{Ω p} (ψ_{i p} u_{k}^{l} ψ_{l p} \frac{\partial ψ_{j}}{\partial x_{k}} r d Ω, ψ ψ {(L_{k})}_{i j} = p \int_{Ω} ϕ_{i} \nabla_{k} ψ_{j} r d Ω, {(S_{u})}_{i j} = {(S_{l q})}_{i j p} + {(V_{l q p})}_{_{i j}}, ψ {(S_{T p})}_{i j} = ρ \int_{Ω} (\nabla ψ_{i} \cdot \nabla ψ_{j p} + \frac{ψ_{i p}}{r p} \frac{\partial ψ}{\partial r p} r d Ω, Φ_{i} = p \frac{1}{2} \int_{Ω p} μ {(γ)}_{i p} [\frac{\partial ψ_{q p}}{\partial x_{k p}} u_{l}^{q p} + p \frac{\partial ψ_{q}}{\partial x_{l}} u_{k p}^{q p^{2}} + 2 \frac{ψ_{q}}{r} u_{r p}^{q p}^{2}] r d Ω, k, l = 1, 2.

\begin{array}{l} \nabla ψ_{i p} = \frac{\partial ψ_{i p}}{\partial r p}, {\frac{\partial ψ_{i}}{\partial z p}}^{⊤}, \\ {(S_{l q})}_{i j} = p \int_{Ω p} μ (γ) χ_{l k p} \frac{\partial ψ_{i p}}{\partial x_{k p}} \frac{\partial ψ_{j p}}{\partial x_{k p}} r d Ω, ψ i f p l = q, {(S_{l q b})}_{i j} = p \int_{Ω p} μ (γ) \leftarrow \frac{\partial ψ_{i p}}{\partial x_{q p}} \frac{\partial ψ_{j}}{\partial x_{l}}) r d Ω, ψ i f p l \neq q, \\ χ_{l k} = 2 i f p l = k a n d 1 i f p l \neq k, ψ \\ {(V_{l q b})}_{i j p} = p \frac{2 ψ_{i} ψ_{j}}{r^{2}}, ψ i f p l = q = 1, {(V_{l q})}_{i j} = 0, ψ i f p l \neq q, s u c h t h a t l, k, q = 1, 2 \end{array}

On page 3330, the below incorrect inline equations have now been corrected in the PDF:

\begin{array}{l} Δ u^{m + \frac{1}{2}} = u^{m + \frac{1}{2}} - u^{m}, Δ u^{m + 1} = u^{m + 1} - u^{m}, Δ p^{m + 1} =^{m + 1} -^{m}, ψ \\ Δ T^{m + \frac{1}{2}} = T^{m + \frac{1}{2}} - T^{m}, Δ T^{m + 1} = T^{m + 1} - T^{m} \cdot ψ \end{array}

K_{i j} = p \int_{Ω p} (\nabla ϕ_{i} \cdot \nabla ϕ_{j} r d Ω \cdot ψ

On page 3331, the incorrect inline equations $u_{r} = 0, u_{z} = u_{f u l l y - d e v e l o p e d} (r) \cdot ψ$ ⁠, $u_{s p} = {β_{s} τ_{r z} |}_{w a l l} \cdot ψ$ have been corrected in the PDF.

Additionally, on the first line of page 3332, “620 triangular” is corrected to “1620 triangular”.

This error was introduced during the article publication process, for which the publisher apologises.

2026

Emerald Publishing Limited

Licensed re-use rights only

Erratum: High-order algorithm to simulate thermal convection for a thermo-dependent viscoplastic fluid flow

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Erratum: High-order algorithm to simulate thermal convection for a thermo-dependent viscoplastic fluid flow

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