Boundary conditions for fluids with internal orientational degrees of freedom: Apparent velocity slip associated with the molecular alignment

Boundary effects are investigated for fluids with internal orientational degrees of freedom such as molecular liquids, thermotropic and lyotropic liquid crystals, and polymeric fluids. The orientational degrees of freedom are described by the second rank alignment tensor which is related to the birefringence. We use a standard model to describe the orientational dynamics in the presence of flow, the momentum balance equations, and a constitutive law for the pressure tensor to describe our system. In the spirit of irreversible thermodynamics, boundary conditions are formulated for the mechanical slip velocity and the flux of the alignment. They are set up such that the entropy production at the wall inferred from the entropy flux is positive definite. Even in the absence of a true mechanical slip, the coupling between orientation and flow leads to flow profiles with an apparent slip. This has consequences for the macroscopically measurable effective velocity. In analytical investigations, we consider the simplified case of an isotropic fluid in the Newtonian and stationary flow regime. For special geometries such as plane and cylindrical Couette flow, plane Poiseuille flow, and a flow down an inclined plane, we demonstrate explicitly how the boundary conditions lead to an apparent slip. Furthermore, we discuss the dependence of the effective viscosity and of the effective slip length on the model parameters.

Figure 1

Non-Newtonian viscosity for the plane Couette flow in the isotropic phase for $Q=8$. Here $G=\gamma A^{-1}{\tau }_a$ is the shear rate and $H$ is the viscosity in units of the first Newtonian viscosity $\eta$. The values for $\kappa$ are, from left to right, 0, 0.4, 1.0, and 1.2 (dashed curve).

Figure 2

The velocity vs the distance in units of $h$ is plotted for plane Couette flow, cf. (43). The model parameters are $Q=8$, $h∕\ell =9$, and $c_a=0$ (thick dashed line), $c_a=1$ (thin line), $c_a=10$ (thick line).

Figure 3

The effective slip length in units of $l$ is plotted as a function of the parameter $c_a$ for the same conditions as in Fig. 2. The parameters are $Q=8$, $\ell =2$, and $h=3$ (thick dashed line), $h=30$ (thin line), $h=300$ (thick line).

Figure 4

The effective slip length (47) in units of $h$ vs $h$ is displayed for plane Couette flow. The model parameter are $Q=8$ and $c_a=0$ (dashed line), $c_a=1$ (thin line), $c_a=10$ (thick line).

Figure 5

The reduced effective viscosity ${\eta }_{\mathrm{eff}}∕\eta$ as a function of $h∕l$ for plane Couette flow is displayed. The parameters are $Q=2$ and $c_a=0$ (thick dashed line), $c_a=1$ (thin line), $c_a=10$ (thick line).

Figure 6

The Poiseuille flow profile (59) is displayed for several values of the parameter $c_a$: $c_a=0$ (thick line), $c_a=1$ (thick dashed line), $c_a=10$ (thin line), and $c_a=\infty$ (thin dashed line). The remaining parameters are $Q=8$, $h∕\ell =9$.

Figure 7

The effective slip length (63) vs $h$ is plotted. The same conditions as in Fig. 6 are considered with parameters $Q=8$ and $c_a=0$ (thick line), $c_a=1$ (thick dashed line), $c_a=10$ (thin line).

Figure 8

The reduced effective viscosity ${\eta }^{\mathrm{eff}}∕\eta$ (65) as a function of $h∕\ell$ is plotted. The parameters are $Q=8$ and $c_a=0$ (thick line), $c_a=1$ (thick dashed line), $c_a=10$ (thin line).

Figure 9

Sketch of the inclined plane.

Figure 10

The flow profile of the flow down an inclined plane is displayed for several values of the parameter $h∕\ell =8$, 12, 20 (from the right to the left). The remaining parameters are $c_0=c_a=0$, $c_h=\infty$, $Q=8$. The dashed line shows one-half of the Poiseuille flow profile for the same parameter.

Figure 11

The $xy$ component of the alignment tensor is plotted for plane Couette flow for several values of the parameter $c_a$. The parameters are chosen as $Q=8$, $A=1$, $\xi =0.3$, $\ell =0.2$, ${\tau }_{ap}=0.5$, $v^w=1$, $h∕\ell =1$, $a\left(h\right)=0$ and $c_a=0$ (thick line), $c_a=10$ (middle thick line), $c_a=20$ (thin line), $c_a=\infty$ (dashed line).

Figure 12

The $xy$ component of the alignment tensor is plotted for plane Poiseuille flow for several values of the parameter $c_a$. The parameters are $Q=8$, $A=1$, $\xi =0.1$, $l=0.2$, ${\tau }_{ap}=0.5$, $v^w=1$, $h=1$, $a\left(h\right)=0$ and $c_a=0$ (thick line), $c_a=1$ (middle thick line), $c_a=\infty$ (dashed line).

Figure 13

The velocity vs the distance in units of $h$ is plotted for Couette flow in cylindrical geometry. The chosen model parameters are $Q=8$, $h∕\ell =9$, $r_m∕\ell =10$, and $c_a=0$ (thick line), $c_a=3$ (thick dashed line), $c_a=100$ (thin line).

Figure 14

Same as Fig. 12 but for parameters $r_m∕\ell =11$, $h∕\ell =9$, $c_a=0$, and $Q=10$ (thick line), $Q=1$ (thick dashed line), $Q=0$ (thin line).