## 3 Numerical Algorithms and Computational Methodologies
### Coupled Simulations for Core-Fluid, Shell-Fluid, and Connecting Wall
#### 3.5.2 Initial and Boundary Conditions for Coupled Simulations--Shell-Fluid  
The initial and inlet boundary conditions for the shell region, velocity, temperature, and pressure, are similar to those for the core fluid. At the outlet, the velocity and temperature are assigned zero-gradient boundary conditions while pressure is set to zero. At the walls, no-slip boundary conditions are used for the velocity, while a zero-gradient boundary condition is applied for the pressure. As with the core-fluid and the connecting wall interface, a coupled temperature-boundary condition is employed at the interface of the shell-fluid and the connecting wall.  
3.5.3. Initial and Boundary Conditions for Coupled Simulations--Connecting Wall  
The coupled temperature-boundary condition is applied at the interfaces on either side of the connecting wall, i.e., to both the connecting wall/core-fluid interface and the connecting wall/shell-fluid interface.  
3.5.4. Discretisation Schemes for Coupled Simulations--Core-Fluid  
The time derivatives are discretised using the implicit Euler scheme. A second-order bounded Gauss linear scheme is used in the discretisation of Laplacian terms. The least squares scheme is applied on the space-gradient terms. The convective terms are discretised via a Universally Bounded Interpolation scheme (Cubista). The solutions of the systems of linear equations for temperature, velocity, and polymer-stresses employ a Preconditioned Bi-Conjugate Gradient (PBiCG) solver in conjunction with a simplified Diagonal Incomplete LU (DILU) pre-conditioner. The pressure equation is solved with a Preconditioned Conjugate Gradient (PCG) solver using Diagonal Incomplete-Cholesky (DIC) pre-conditioners.  
3.5.5. Discretisation Schemes for Coupled Simulations--Shell-Fluid  
The implicit Euler scheme is used to discretise time derivatives and a second-order bounded Gauss linear scheme is utilised for space-gradient terms. The Laplacian terms are treated via Gauss linear orthogonal schemes. For convective terms the Cubista and Gauss linear schemes are employed. The solver for the discretised temperature and velocity equations is the PBiCGStab solver with a DILU pre-conditioner. A Geometric-Algebra Multi-Grid (GAMG) solver is used in conjunction with a DIC pre-conditioner for the solution of the pressure equation.  
3.5.6. Discretisation Schemes for Coupled Simulations--Connecting Wall  
The space-gradient terms are discretised using the Gauss linear scheme, while Laplacian terms employ the second-order bounded linear scheme. The time derivatives are treated via the implicit Euler scheme. The temperature equation for the connecting wall is solved using a PCG solver in conjunction with a DIC pre-conditioner.  
4. Numerical Results and Discussion  
4.1. Mesh Convergence  
The mesh independence (or alternatively mesh convergence) in the present simulations is illustrated in Figure 3. The mesh sizes in Figure 3 correspond to the number of computational cells in Table 1.  
Given the huge computational costs incurred in running the simulations with such high numbers of computational cells, it is prudent to balance the requirements of mesh convergence, on the one hand, and computational costs, on the other. Figure 3 illustrates mesh independence for the three vastly different mesh sizes. For these reasons, all subsequent simulations in this work are conducted via mesh size 1, i.e., with 120,000 computational cells. It should be remarked that similar mesh independence was observed with much fewer than 120,000 computational cells.  
\begin{table}
\begin{tabular}{c c} \hline mesh 1 & 120,000 cells \\ \hline mesh 2 & 640,000 cells \\ \hline mesh 3 & 800,000 cells \\ \hline \end{tabular}
\end{table}
Table 1: Mesh sizes corresponding to Figure 3.