Equation no. Incorrect equation Correct...

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)	$\frac{2 R e}{Δ t p} M (p u^{m + 1} - u^{⋆}) = L^{⊤} Δ p^{m + 1}, ψ$	$\frac{2 R e}{Δ t} M (u^{m + 1} - u^{⋆}) = L^{⊤} Δ p^{m + 1},$

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