## 2 Methodology
### Uncertain Thermal Performance and Flow Behavior Prediction  
Once the material properties of nanofluid and the boundary conditions are determined, the thermal performance and flow behavior can be obtained based on the physical model and numerical method presented in Section 2.2. A numerical calculation under determined parameters is called a deterministic calculation or a sample, which is widely utilized to study the thermal performance and flow behavior of forced convection problems by a series of deterministic calculations. As mentioned before, the forced convection problem is always at the mercy of uncertainties. Therefore, IM is employed to analyze the uncertain forced convection problem. This section aims to build the surrogate model of the uncertain forced convection problem and then predict the bounds of interested variables such as temperature, velocity, and Nusselt number. Firstly, a deterministic calculation is described as a black box of which the input is the physical model, material properties and boundary conditions, and the output is the results of the forced convection problem such as temperature, velocity, and Nusselt number. The function of the black box is defined as:  
\[\mathbf{Y}=f(\xi^{I}) \tag{15}\]  
where \(\mathbf{Y}\) is the vector which consists of the outputs of the black box;\(f\) is the function of the black box and \(\xi^{I}\) is the interval parameter vector which consists of the interval variables of inputs of the black box. The uncertainties are regarded as interval variables. And the interval parameter vector can be defined by:  
\[\mathbf{\xi}^{I}=\Big{\{}\mathbf{\xi}:\mathbf{\xi}<\mathbf{\xi}<\overline{\mathbf{\xi}}\Big{\}}= \big{[}\mathbf{\xi}^{c}-\mathbf{\beta}^{xc}_{\mathbf{\xi}},\mathbf{\xi}^{c}+\mathbf{\beta}^{c} \big{]} \tag{16}\]  
where \(\mathbf{\xi}\) and \(\overline{\mathbf{\xi}}\) are the vectors which consists of the lower and upper bounds of uncertainties; \(\overline{\mathbf{\xi}}^{c}\) is the vector which consists of the nominal values of uncertainties; and \(\mathbf{\beta}\) is the vector of deviation coefficients. SM is a traditional and robust method to predict the uncertain properties of various systems without requiring the probability distributions of the uncertain variables. To carry out SM, the interval variable is divided into equidistant samples and deterministic calculations are conducted at every sample. Then, the bounds of interested properties are obtained based on the results of all samples. However, the requirement of a large amount of samples, limits the application of SM because the number of samples will increase exponentially with the dimension of uncertainties. To overcome the large amount of samples, IM is adopted. With the help of the Chebyshev polynomial approximation, the surrogate model of the black box can be defined as:  
\[Y\cong\widetilde{f}(\xi^{I})=\mathbf{a}^{T}\mathbf{C}(\xi^{I}) \tag{17}\]  
and  
\[\mathbf{a}=\left[ \begin{array}{cccc}a_{0,0,\cdots,0}&a_{1,0,\cdots,0}&\cdots&a_{ m,m,\cdots,m}\end{array} \right] \tag{18}\]  
\[\mathbf{C}=\left[ \begin{array}{cccc}C_{0,0,\cdots,0}(\xi^{I})&C_{1,0,\cdots,0}( \xi^{I})&\cdots&C_{m,m,\cdots,m}(\xi^{I})\end{array} \right] \tag{19}\]  
where \(\mathbf{a}\) and \(\mathbf{C}\) are the vectors of coefficients and basic functions of Chebyshev polynomial approximation; and \(m\) is number of sample points in every dimension. The basic functions of Chebyshev polynomial approximation can be expressed as:  
\[C_{i_{1},i_{2},\cdots,i_{n}}(\xi^{I})=\cos(i_{1}\theta_{1})\cos(i_{2}\theta_{2 })\cdots\cos(i_{n}\theta_{n}) \tag{20}\]  
where  
\[\theta_{i}=\arccos\frac{2\xi_{i}^{I}-(\overline{\xi}_{i}+\underline{\xi}_{i}) }{\overline{\xi}_{i}-\underline{\xi}_{i}}\in\geq[0,\,\pi],\,\xi_{i}^{I}=\left[ \underline{\xi}_{i},\,\overline{\xi}_{i}\right] \tag{21}\]  
where \(\underline{\xi}_{i}^{I}\) is the \(i\)th term of the interval vector \(\xi^{I}\); and \(n\) is the dimension of Chebyshev polynomial which is equal to the dimension of the interval variables.