1 Introduction

We consider a rectangular longitudinal one-dimensional fin, which is attached to a fixed base surface of temperature \(T_\mathrm{b} \) and extends into a fluid of temperature \(T_\mathrm{b} \). The dimensionless governing equation is [1]

$$\begin{aligned} \frac{\mathrm{d}^2\theta }{\mathrm{d}x^2}+\beta \theta \frac{\mathrm{d}^2\theta }{\mathrm{d}x^2}+\beta \left( \frac{\mathrm{d}\theta }{\mathrm{d}x}\right) ^2-M^2(1+\beta \theta )^n=0, \quad \theta >0 \end{aligned}$$
(1.1)

with boundary conditions

$$\begin{aligned} {\theta }'(0)=0 \quad \hbox { and } \quad \theta (1)=1 \end{aligned}$$
(1.2)

where \(\theta \) is the dimensionless temperature, \(\beta ={\uplambda } (T_\mathrm{b} -T_\mathrm{a} )\) is the gradient of thermal conductivity, \(M\) is the thermo-geometric parameter. The exponent,\( n\), represents laminar film boiling or condensation when \(n=-1/4\), laminar natural convection when \(n=1/4\), turbulent natural convection when \(n=1/3\), nucleate boiling when \(n=2\), radiation when \(n=3\), and \(n\) vanishes for a constant heat transfer coefficient.

It is very important to study the effect of \(M\) on the heat transfer. It is obvious that the increase of \(M\) might result in negative \(\theta \) at \(x=0\), contradicting the assumption. It is, therefore, important to identify the maximal value for the thermo-geometric parameter.

The detailed derivation of Eq. (1) was given in [1], and the thermal characteristics were elucidated in [2]. Harley and Moitsheki [1] gave a numerical investigation, and obtained the maximal values for various \(n\). Some effective analytical methods were successfully applied to the problem [3–5]; there are alternative numerical/analytical methods, such as the three-point implicit block multistep method [6], the variational iteration method [7–12], reproducing kernel method [13, 14], and a complete review on various analytical methods is available in [15]. In this paper, we will suggest a simple analytical approach to identification of the maximal value of the thermo-geometric parameter.

2 Maximal Thermo-geometric Parameter

In this study, neither an exact solution nor an approximate solution is searched for, only the maximal \(M\) in Eq. (1.1) is considered. For this end, we choose a very simple trial function in the form

$$\begin{aligned} \theta (x)=a_0 +a_1 x+a_2 x^2 \end{aligned}$$
(2.1)

By the boundary conditions, Eq. (1.2), we have

$$\begin{aligned}&\displaystyle a_1 =0 \end{aligned}$$
(2.2)
$$\begin{aligned}&\displaystyle a_0 +a_1 +a_2 =1 \end{aligned}$$
(2.3)

By Eqs. (1.1) and (1.2), we obtain

$$\begin{aligned} {\theta }''(0)=M^2(1+\beta \theta (0))^{n-1}=M^2(1+\beta a_0 )^n \end{aligned}$$
(2.4)

Eq. (2.4) means

$$\begin{aligned} 2a_2 =M^2(1+\beta a_0 )^n \end{aligned}$$
(2.5)

Submitting Eqs. (2.2) and (2.5) into Eq. (2.3) results in

$$\begin{aligned} a_0 +\frac{M^2}{2}(1+\beta a_0 )^n=1 \end{aligned}$$
(2.6)

Setting \(a_0 =\theta _{\min } (0)=0\), we obtain maximal value for \(M\), which is

$$\begin{aligned} M_{\max } =\sqrt{2} =1.414 \end{aligned}$$
(2.7)

Comparison of Eq. (2.7) with the numerical results given in [1] reveals that the maximal error is 16.5 % for \(-4<n<3\). The accuracy is 3 and 5.4 % for \(n=2\) and \(n=3\), respectively.

When the thermo-geometric parameter reaches its maximal value, thermal instability occurs [2], so in practical applications, we should follow \(M\ll M_{\max } \), and the 16.5 % error is acceptable.

If a higher accurate prediction is needed, the variational iteration algorithm [7, 15] is recommended.

According to the variational iteration method [7, 15], the following iteration formulation (variational iteration algorithm-II [15]) can be constructed

$$\begin{aligned} \theta _{p+1} (x)\!=\!\theta _0 (x)\!+\!\!\int _0^x \!\!{(x\!-\!s)\!\left\{ {\beta \theta _p (s)\frac{\mathrm{d}^2\theta _p (s)}{\mathrm{d}s^2}\!+\!\beta \left( \frac{\mathrm{d}\theta _p (s)}{\mathrm{d}s}\right) ^2\!\!\!-\!\!M^2\left[ {1\!+\!\beta \theta _p (s)} \right] ^n} \right\} \!\mathrm{d}s}\nonumber \\ \end{aligned}$$
(2.8)

We begin with \(\theta _0 (x)=\theta (0)=a_0 \), by Eq. (2.8), we have

$$\begin{aligned} \theta _1 (x)=a_0 +\int _0^x {(x-s)\left\{ {-M^2(1+\beta a_0 )^n} \right\} \mathrm{ds}} =a_0 +\frac{1}{2}M^2(1+\beta a_0 )^nx^2\nonumber \\ \end{aligned}$$
(2.9)

If the first-order approximate solution is enough, then by the boundary condition, \(\theta (1)=1\), the following result is obtained.

$$\begin{aligned} \theta _1 (1)=a_0 +\frac{1}{2}M^2(1+\beta a_0 )^n=1 \end{aligned}$$
(2.10)

which is exactly same with Eq. (2.6).

The solution process can continue without any difficulty by using some mathematical software, and a higher accurate result can be obtained.

3 Conclusion

In practical applications, we need neither an exact solution nor an approximate solution, but a criterion for some parameters in the studied equation, for example, the condition of resonance for a nonlinear oscillator. In this paper, we suggest a simple but effective approach to identification of the maximal thermo-geometric parameter in Eq. (1), the 16.5 % accuracy of the first-order prediction is acceptable considering it should follow \(M\ll M_{\max } \) though a higher accuracy can be obtained by the variational iteration method.