Steady Thermal Analysis of Two-Dimensional Cylindrical Pin Fin with a Nonconstant Base Temperature

Table 1

Extra point symmetries admitted by (2.11).

𝑠(𝑤)	Symmetries
0	Γ2=−(𝑥𝜕𝑥+𝑟𝜕𝑟), Γ3=𝑤𝜕𝑤−2𝑥𝜕𝑥−2𝑟𝜕𝑟, Γ4=𝑥𝑤𝜕𝑤+(𝑟^2−𝑥^2)𝜕𝑥−2𝑥𝑟𝜕𝑟, Γ5=ℱ(𝑟,𝑥)𝜕𝑤,
	where ℱ𝑟𝑟+(ℱ𝑟/𝑟)+ℱ𝑥𝑥=0.
𝑤	Γ2=−𝑤𝜕𝑤, Γ3=ℱ(𝑟,𝑥)𝜕𝑤, where ℱ𝑟𝑟+ℱ𝑟/𝑟+ℱ𝑥𝑥=0.
𝑤^𝑝, 𝑝≠5	Γ2=1/(𝑝−5)(2𝑤𝜕𝑤+(1−𝑝)𝑥𝜕𝑥+(1−𝑝)𝑟𝜕𝑟)
𝑝=5	Γ3=−𝑥𝑤𝜕𝑤+(𝑥^2−𝑟^2)𝜕𝑥+2𝑥𝑟𝜕𝑟
𝑒^𝑝𝑤	Γ2=(1/𝑝)𝜕𝑤−(𝑥/2)𝜕𝑥−(𝑟/2)𝜕𝑟
(𝑎+𝑏𝑤)^𝑞	Γ2=1/(𝑞−5)((2(𝑎+𝑏𝑤)/𝑛)𝜕𝑤+(1−𝑞/𝑞−5)𝑥𝜕𝑥+((1−𝑞)/(𝑞−5))𝑟𝜕𝑟)
𝑞=5	Γ3=−(𝑟(𝑎+𝑏𝑤)/2𝑛)𝜕𝑤+((𝑥^2−𝑟^2)/2)𝜕𝑥+𝑥𝑟𝜕𝑟