2.1.4 Boundary and initial conditions
Fig. 1 shows the coordinate system for the 2-dimensional (2D) domain. For boundary conditions, we applied constant parabolic velocity to the upstream flow and set zero pressure at the downstream. Considering the 2D domain, for velocity vector \(\mathbf{u}{{=\{u}_{1,}u_{2}\}}^{T}\)upstream, we set:\(\ u_{1}=u_{0}\), \(u_{2}=0\). Along the wall, we consider \(\mathbf{u}=\mathbf{0}\) and \(\tau_{\text{yy}}=0\). The gradient of x-velocity and xx component of extra stress is 0. Thus, the boundary conditions for stress \(\mathbf{\tau}\) would be as following:
\(\tau_{\text{xx}}=2\mu_{b}\lambda{(\frac{\partial u}{\partial y})}^{2}\),\(\ \tau_{\text{xy}}=\mu_{b}\frac{\partial u}{\partial y}\),\(\text{\ \ }\tau_{\text{yy}}=0\),
where \(\ \mathbf{\tau}\ =\par \begin{bmatrix}\end{bmatrix}\). The initial condition for the velocity and extra stress were set as \(\mathbf{u}=\mathbf{0\ }\)and\(\ \mathbf{\tau}\ =\mathbf{0\ }\)in the whole domain.