Write the possible solution steady state of two dimensional heat flow equation
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The two-dimensional heat balance equation is given by (see, e.g., [16, 17])1()1+1()1=().(2.1)The imposed boundary conditions are()1=−()0,1−∞,1=0,,1=−∞1,1=,1=0,1=0,()1=−()1,2−∞,1=2.(2.2)
Here, is the dimensionless temperature, is the fin base temperature, is the heat transfer coefficient, 1 is the longitudinal coordinate, 1 is the transverse coordinate, is the internal heat generation function, and is the thermal conductivity. Several authors have considered the two-dimensional problem with =0 and thermal conductivity being a constant (see, e.g., [19, 20]) and the case =0 with a temperature-dependent thermal conductivity [21].
Introducing the dimensionless variables=−∞−∞,=1,=1/2,()=(),ℎ()=()ℎ,2=/22,()=()(/2)2−∞,(2.3)
we obtain2()+()=2().(2.4)
The corresponding dimensionless boundary conditions are()=−Biℎ(),=0,(1,)=(),=1,=0,=0,()=−Biℎ(),=1,(2.5)where is the fin extension factor (purely geometric parameter), and Bi=ℎ/ and Bi=ℎ(/2)/ are the Biot numbers. is the reciprocal to aspect ratio (see, e.g., [16]). ℎ and are the heat transfer at the base and thermal conductivity of the fin at the ambient temperature, respectively.
Here, is the dimensionless temperature, is the fin base temperature, is the heat transfer coefficient, 1 is the longitudinal coordinate, 1 is the transverse coordinate, is the internal heat generation function, and is the thermal conductivity. Several authors have considered the two-dimensional problem with =0 and thermal conductivity being a constant (see, e.g., [19, 20]) and the case =0 with a temperature-dependent thermal conductivity [21].
Introducing the dimensionless variables=−∞−∞,=1,=1/2,()=(),ℎ()=()ℎ,2=/22,()=()(/2)2−∞,(2.3)
we obtain2()+()=2().(2.4)
The corresponding dimensionless boundary conditions are()=−Biℎ(),=0,(1,)=(),=1,=0,=0,()=−Biℎ(),=1,(2.5)where is the fin extension factor (purely geometric parameter), and Bi=ℎ/ and Bi=ℎ(/2)/ are the Biot numbers. is the reciprocal to aspect ratio (see, e.g., [16]). ℎ and are the heat transfer at the base and thermal conductivity of the fin at the ambient temperature, respectively.
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Give three possible solutions of two dimensional steady state heat flow equation
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