Application of boundary-fitted convolutional neural network to simulate non-Newtonian fluid flow behavior in eccentric annulus

Abstract

Mathematical simulation of non-Newtonian fluid flow is an enduring problem with imperative influence on numerous industrial processes such as oil and gas drilling, food processing, etc. The relation between shear rate and shear stress is nonlinear for non-Newtonian fluids, which results in a highly nonlinear governing equation for fluid flow in an irregular geometry. Analytical solution does not exist for such governing equations and is generally solved by algebraic and iterative methods, which is computationally intensive. Convolutional Networks can learn a complex and high-dimensional functional space solution and may have high accuracy but depend significantly on the quality of training data. One of the prominent challenges in using a Convolutional network is the limiting performance, and the proposed solution may become inconclusive in a small data regime over an irregular geometry. A novel algorithm, Boundary Fitted Convolutional Network, is proposed in this research, which can proficiently solve a partial differential equation for an irregular geometry. This research aims to simulate a Power-Law non-Newtonian fluid in an eccentric annulus with a convolutional network without using training data. The governing equations are transformed from physical domain to computational plane using boundary-fitted coordinate system and then solved by minimizing physics-based residuals. Thus, establishing a benchmark investigation in non-Newtonian fluid flow. The Dirichlet and Neumann boundary conditions are applied in a ‘hard’ manner. The simulated results and parametric analysis conclude that the proposed algorithm can decipher the non-Newtonian fluid mechanics from the governing equations. The algorithms also explain the effect of geometric and rheological parameters on the fluid flow attributes. The performance of the algorithm is validated with experimental data available from published studies. The statistical error estimation exhibits an average root mean squared error of 0.228 and mean absolute percentage error of 8.21% for four different samples of Power-Law fluid, with varying eccentricities. A comprehensive discussion to train the unsupervised convolutional network, and the spectrum of hyperparameters considered to expedite convergence is also highlighted.

Appendices

Appendix A

The governing equation used in this study is given by Eq. (A1)

$$\frac{\partial }{\partial x}\left( {\mu \frac{\partial w}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {\mu \frac{\partial w}{{\partial y}}} \right) = \frac{\partial P}{{\partial z}}$$

(A1)

The mathematical relation used to transform the cartesian-based governing equation into boundary fitted coordinate system is given by Eqs. (A2)–(A3).

$$\frac{\partial }{\partial x} = \frac{1}{J}\left( {\frac{\partial }{\partial \xi }y_{\eta } - \frac{\partial }{\partial \eta }y_{\xi } } \right)$$

(A2)

$$\frac{\partial }{\partial y} = \frac{1}{J}\left( {\frac{\partial }{\partial \eta }x_{\xi } - \frac{\partial }{\partial \xi }x_{\eta } } \right)$$

(A3)

where the values of $y_{\eta } , y_{\xi } ,x_{\xi } ,x_{\eta }$ are determined from Eqs. (A4.1)–(A.4.4).

$$y_{\eta } = \frac{\partial y}{{\partial \eta }}$$

(A4.1)

$$y_{\xi } = \frac{\partial y}{{\partial \xi }}$$

(A4.2)

$$x_{\xi } = \frac{\partial x}{{\partial \xi }}$$

(A4.3)

$$x_{\eta } = \frac{\partial x}{{\partial \eta }}$$

(A4.4)

Substituting Eq. (A2) in $\frac{\partial }{\partial x}\left( {\mu \frac{\partial w}{{\partial x}}} \right)$ yields the expression given by Eq. (A5.1).

$$\frac{1}{J}\left\{ {y_{\eta } \frac{\partial }{\partial \xi }\left[ {\mu \frac{1}{J}\left( {y_{\eta } \frac{\partial w}{{\partial \xi }} - y_{\xi } \frac{\partial w}{{\partial \eta }}} \right)} \right] - y_{\xi } \frac{\partial }{\partial \eta }\left[ {\mu \frac{1}{J}\left( {y_{\eta } \frac{\partial w}{{\partial \xi }} - y_{\xi } \frac{\partial w}{{\partial \eta }}} \right)} \right] } \right\}$$

(A5.1)

Substituting Eq. (A3) in $\frac{\partial }{\partial y}\left( {\mu \frac{\partial w}{{\partial y}}} \right)$ yields the expression given by Eq. (A5.2).

$$\frac{1}{J}\left\{ {x_{\xi } \frac{\partial }{\partial \eta }\left[ {\mu \frac{1}{J}\left( {x_{\xi } \frac{\partial w}{{\partial \eta }} - x_{\eta } \frac{\partial w}{{\partial \xi }}} \right)} \right] - x_{\eta } \frac{\partial }{\partial \xi }\left[ {\mu \frac{1}{J}\left( {x_{\xi } \frac{\partial w}{{\partial \eta }} - x_{\eta } \frac{\partial w}{{\partial \xi }}} \right)} \right] } \right\}$$

(A5.2)

Combining Eqs. (A5.1) and (A5.2) into Eq. (A6)

$$\begin{aligned} &\frac{1}{J}\left\{ \frac{\partial }{\partial \xi }\left[ {\frac{\mu }{J}\left( {\left( {y_{\eta }^{2} + x_{\eta }^{2} } \right)\frac{\partial w}{{\partial \xi }} - \left( {y_{\eta } y_{\xi } + x_{\xi } x_{\eta } } \right)\frac{\partial w}{{\partial \eta }}} \right)} \right] \right.\\ &\left.\quad + \frac{\partial }{\partial \eta }\left[ {\frac{\mu }{J}\left( {\left( {y_{\xi }^{2} + x_{\xi }^{2} } \right)\frac{\partial w}{{\partial \eta }} - \left( {y_{\eta } y_{\xi } + x_{\xi } x_{\eta } } \right)\frac{\partial w}{{\partial \xi }}} \right)} \right] \right\} = \frac{\partial P}{{\partial z}} \end{aligned}$$

(A6)

Appendix B

The central nodes are discretized for the first derivative using fourth-order accuracy given by Eqs. (B1.1) and (B1.2).

$$\frac{\partial w}{{\partial \xi }} \approx \frac{{ - w_{\xi + 2\delta \xi ,\eta } + 8w_{\xi + \delta \xi ,\eta } - 8w_{\xi - \delta \xi ,\eta } + w_{\xi - 2\delta \xi ,\eta } }}{12\delta \xi } + O\left( {\left( {\delta \xi } \right)^{4} } \right)$$

(B1.1)

$$\frac{\partial w}{{\partial \eta }} \approx \frac{{ - w_{\xi ,\eta + 2\delta \eta } + 8w_{\xi ,\eta + \delta \eta } - 8w_{\xi ,\eta - \delta \eta } + w_{\xi ,\eta - 2\delta \eta } }}{12\delta \eta } + O\left( {\left( {\delta \eta } \right)^{4} } \right)$$

(B1.2)

The convolutional filter for the central discretization defined in Eqs. (B1.1) and (B1.2) is given by Eqs. (B2.1) and (B2.2)

$$\frac{\partial w}{{\partial \xi }} \approx \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & { - 8} & 0 & 8 & { - 1} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \times \frac{1}{12\delta \xi }$$

(B2.1)

$$\frac{\partial w}{{\partial \eta }} \approx \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & { - 8} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 8 & 0 & 0 \\ 0 & 0 & { - 1} & 0 & 0 \\ \end{array} } \right] \times \frac{1}{12\delta \eta }$$

(B2.2)

Similarly, the discretization for forward and backward schemes used on the boundary is approximated by third-order accuracy. The forward scheme is given by Eqs. (B3.1) and (B3.2)

$$\frac{\partial w}{{\partial \xi }} \approx \frac{{ - 11w_{\xi ,\eta } + 18w_{\xi + \delta \xi ,\eta } - 9w_{\xi + 2\delta \xi ,\eta } + 2w_{\xi + 3\delta \xi ,\eta } }}{6\delta \xi } + O\left( {\left( {\delta \xi } \right)^{3} } \right)$$

(B3.1)

$$\frac{\partial w}{{\partial \eta }} \approx \frac{{ - 11w_{\xi ,\eta } + 18w_{\xi ,\eta + \delta \eta } - 9w_{\xi ,\eta + 2\delta \eta } + 2w_{\xi ,\eta + 3\delta \eta } }}{6\delta \eta } + O\left( {\left( {\delta \eta } \right)^{3} } \right)$$

(B3.2)

The backward scheme is given by Eqs. (B4.1) and (B4.2)

$$\frac{\partial w}{{\partial \xi }} \approx \frac{{11w_{\xi ,\eta } - 18w_{\xi - \delta \xi ,\eta } + 9w_{\xi - 2\delta \xi ,\eta } - 2w_{\xi - 3\delta \xi ,\eta } }}{6\delta \xi } + O\left( {\left( {\delta \xi } \right)^{3} } \right)$$

(B4.1)

$$\frac{\partial w}{{\partial \eta }} \approx \frac{{11w_{\xi ,\eta } - 18w_{\xi ,\eta - \delta \eta } + 9w_{\xi ,\eta - 2\delta \eta } - 2w_{\xi ,\eta - 3\delta \eta } }}{6\delta \eta } + O\left( {\left( {\delta \eta } \right)^{3} } \right)$$

(B4.2)