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

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 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:On page 3325, the incorrect equation Re=pρU2−nDnpμ0,Pe=pρCpUlpkp,Br=pμ0U2kΔTp, has been corrected to Re=ρU2−nDnμ0,Pe=ρCpUlk,Br=μ0U2kΔT,.On page 3326, the incorrect equation τ¯p, γ¯p, |γ|=12Πγ=(12γ¯:γ¯), f=pτwp12ρue2=pDΔp2lρuep2,Nu=pHxDk. has been corrected to τ¯, γ¯. |γ˙|=12Πγ˙=(12γ˙¯:γ˙¯), fp=τw12ρue2=DΔp2lρue2,Nu=HxDk. 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:Mij=p∫ΩψiψjrdΩ,Mpij=p∫Ωp(ϕiϕjrdΩ,Nij(u)=ρ∫Ωp(ψipuklψlp∂ψj∂xkrdΩ,ψψ(Lk)ij=p∫Ωϕi∇kψjrdΩ,(Su)ij=(Slq)ijp+(Vlqp)ij,ψ(STp)ij=ρ∫Ω(∇ψi⋅∇ψjp+ψiprp∂ψ∂rprdΩ,Φi=p12∫Ωpμ(γ)ip∂ψqp∂xkpulqp+p∂ψq∂xlukpqp2+2ψqrurpqp2rdΩ,k,l=1,2.∇ψip=∂ψip∂rp,∂ψi∂zp⊤,(Slq)ij=p∫Ωpμ(γ) χlkp∂ψip∂xkp∂ψjp∂xkprdΩ,ψifpl=q,(Slqb)ij=p∫Ωpμ(γ)←∂ψip∂xqp∂ψj∂xl)rdΩ,ψifpl ≠ q,χlk=2ifpl =k and 1 ifpl ≠ k,ψ(Vlqb)ijp=p2ψiψjr2, ψifpl =q=1,(Vlq)ij=0,ψifpl≠q, suchthat l,k,q=1,2On page 3330, the below incorrect inline equations have now been corrected in the PDF:Δum+12=um+12−um,Δum+1=um+1−um,Δpm+1=m+1−m,ψΔTm+12=Tm+12−Tm,ΔTm+1=Tm+1−Tm⋅ψKij=p∫Ωp(∇ϕi⋅∇ϕjrdΩ⋅ψOn page 3331, the incorrect inline equations ur=0,uz=ufully−developed (r)⋅ψ, usp=βsτrz|wall⋅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.