Governing Equations

The equations of the motion of a viscoelastic fluid are:  

$$\nabla \cdot {\tilde{\bf v}}={\bf 0},$$ (148)

 

$$\tilde{\rho} \frac{D \tilde {{\bf v}}}{D\tilde{t}}=-\nabla \tilde{p} +\nabla \cdot \tilde{{\bf T}}.$$ (149)

For the extra stress we use the Oldroyd-B constitutive model given by

$$\tilde{{\bf T}}=2\eta_{s}{\bf D}+\tilde{\mbox{\boldmath$\tau$}},$$ (150)

where $\tilde{\mbox{\boldmath$\tau$}}$, the Maxwell stress, satisfies the equation  

$$\tilde{\mbox{\boldmath$\tau$}} +\lambda ( \frac{D \tilde{\m... ...ilde{\mbox{\boldmath$\tau$}}{\bf L}^{T} )=2\eta_{p} {\bf D}.$$ (151)

Here $\tilde{{\bf v}}$ is the velocity, $\tilde{p}$ is the pressure, $\tilde{\rho}$ is the density, $\lambda$ is the relaxation time, while $\eta_{s}$ and $\eta_{p}$ are solvent and polymer viscosities respectively; L is the velocity gradient tensor (with the convention that $L_{ij}=\partial v_i/\partial x_j$)and D its symmetric part.