Planar Couette flow

For the steady, planar Couette flow to be examined In a later section, the momentum balance equation yields... 

Simple shear (also known as planar Couette flow) is achieved when fluid is contained between two plane parallel plates in relative in-plane motion. If the velocity direction is taken to be x, one has x = y, all other xa 3 zero and... 

Planar Couette flow is difficult to maintain in a steady state. Cylindrical Couette flow is much easier, in which the fluid is contained in the annulus between two cylinders in relative angular motion about their common axes, as shown in Figure 2.8.2. 

For planar Couette flow, the rate of entropy production of an incompressible Newtonian fluid is... 

Fig. 3 The coordinate system of a nematic liquid crystal undergoing planar Couette flow. In a space fixed coordinate system (e, e, e ) the strain rate Vu is applied. This means...

It is also possible to calculate the shear viscosities and the twist viscosities by applying the SLLOD equations of motion for planar Couette flow, Eq. (3.9). If we have a velocity field in the x-direction that varies linearly in the z-direction the velocity gradient becomes Vu=ye ej, see Fig. 3. Introducing a director based coordinate system (Cj, C2, 63) where the director points in the e3-direction and the angle between the director and the stream lines is equal to 0, gives the following expression for the strain rate in the director based coordinate system. 

K. P. Travis, P. J. Daivis and D. J. Evans, Computer Simulation Algorithms for Molecules Undergoing Planar Couette Flow A Nonequilibrium Molecular Dynamics Study, J. Chem. Phys. 103 (1995)... 

The SLLOD equations of motion presented in Eqs. [123] are for the specific case of planar Couette flow. It is interesting to consider how one could write a version of Eqs. [123] for a general flow. One way to do this is introduce a general strain tensor, denoted by Vu. For the case of planar Couette flow, Vu = j iy in dyadic form, where 1 and j denote the unit vector in the x and y directions, respectively. The matrix representation is... 

The interpretation of Eq. [133] for planar Couette flow is as follows at f < 0, the system evolves under normal NVE dynamics (i.e., Vu = 0). At f = 0, the impulsive force 5(f)q, Vu is applied, after which the system continues to evolve under NVE dynamics for f > 0, but the memory of the flow is contained in the definition of q, and p, as specified in Eq. [129]. When periodic boundary conditions are applied on the simulation cell, they must be treated in a way that preserves the flow. This in general will lead to time-dependent boundary conditions for the case of planar Couette flow to be discussed in detail shortly. 

However, it is clear that for a general tensor Vu, trajectory analysis based on the SLLOD dynamics in Eqs. [129] will yield incorrect results. Equation [132] has an extra term in the force, which is equivalent to saying that the momenta in Eqs. [129] are not peculiar with respect to a general flow (indeed, Eqs. [129] yield peculiar velocities for the case of planar Couette flow), and therefore the flow profile produced will not be q Vu as expected. Equations [129] also lead to problems when one is considering definitions of pressure... 

SLLOD Dynamics for Planar Couette Flow in the Canonical Ensemble... 

We wish to study the effects of planar Couette flow on a system that is in the NPT (fully flexible box) ensemble. In this section, we consider the effects of the external field alone on the dynamics of the cell. The intrinsic cell dynamics arising out of the internal stress is assumed implicitly. The constant NPT ensemble can be employed in simulations of crystalline materials, so as to perform dynamics consistent with the cell geometry. In this section, we assume that the shear field is applied to anisotropic systems such as liquid crystals, or crystalline polytetrafluoroethylene. For an anisotropic solid, we assume that the shear field is oriented in such a way that different weakly interacting planes of atoms in the solid slide past each other. The methodology presented is quite general hence it is straightforward to apply for simulations of shear flow in liquids in a cubic box, as well. 

For planar Couette flow, the gradient of the strain rate tensor is given by... 

Although the box matrix has nine elements, only six of them are independent. Thus, in setting up a box matrix for equilibrium simulations from crystal cell parameters, three of the off-diagonal elements can be arbitrarily chosen to be zero. This choice cannot be arbitrary for planar Couette flow. This is because, if the initial values of 21 aud 23 are chosen to be nonzero, and will evolve boundlessly in time. On the other hand, if 21 and f 23 were to be zero at t = 0, then during NPT dynamics they would oscillate around zero, and thus the average rate of change of and hi would be zero. 

The treatment of applying periodic boundary conditions discussed here is markedly different from that traditionally employed in simulations of planar Couette flow. The PBC method that is commonly used is called the Lees-Edwards boundary condition. In its simplified form applied to cubic boxes, it represents a translation of the image boxes in the y direction, at a rate equal to y. Further details on this method can be found elsewhere. In contrast to the method involving the dynamical evolution of h presented here, the Lees-Edwards method is much harder to develop and implement for noncubic simulation cells. Also, in simulations involving charged particles, the Coulom-bic interaction is handled in both real and recipro l spaces. The reciprocal space vectors k of the simulation cell represented by h can be written " " as follows ... 

Following Eqs. [202], the equation of motion for the particle coordinate in a planar Couette flow can be written as follows ... 

A difference between the perturbation considered here and that in the section on LRT considered earlier is the term involving qo, the position at which the drift velocity (i.e., the velocity contribution from the external field) of the fluid is zero. This term was chosen to be zero in the treatment for bulk fluids for simplicity it must be used here because confinement has broken the translational invariance of the system. The perturbation generates a planar Couette flow in the fluid between two surfaces ... 

As we have already demonstrated, the SLLOD equations have been highly successful for studying moderate shear rate systems. To review, the equations of motion for planar Couette flow, with Nose-Hoover thermostats, - " are ... 

Alternatively, one could use SLLOD equations to do direct simulations, such as shear a system under planar Couette flow and measure the shear stress. As we have already discussed, this approach has been used successfully to calculate a host of transport properties. It is important to remember, however, that direct simulation is often unable to simulate realistic materials at experimentally accessible shear rates. At low shear rates, the nonequilibrium response becomes small compared to the magnitude of the equilibrium fluctuations that naturally arise. The extremely small signal-to-noise ratio would demand prohibitively long simulations before any meaningful answers could be obtained. 

D. Evans and G. Morriss, Rhys. Rev. A., 30, 1528 (1984). Nonlinear-Response Theory for Steady Planar Couette Flow. 

R. Bhupathiraju, P. T. Cummings, and H. D. Cochran, Mol. Phys., 88, 1665 (1996). An Efficient Parallel Algorithm for Non-Equilibrium Simulations of Very Large Systems in Planar Couette Flow. 

Problem 3-9. Oscillating Planar Couette Flow. We consider an initially motionless incompressible Newtonian fluid between two infinite solid boundaries, one at y = 0 and the other at y = d. Beginning at t = 0, the lower boundary oscillates back and forth in its own plane with a velocity ux = U sin cot (t > 0). 

Problem 4-4. Asymptotic Analysis of Oscillating Planar Couette Flow. When we considered Problem 3-9, we obtained an exact analytic solution (which can then be evaluated to determine the form of the solution for large and small frequencies). Here we consider the... 

Problem 4-9. Steady Planar Couette Flow With Porous Boundaries. Newtonian fluid of viscosity p and density p flows between two rigid boundaries y = 0 and y = h, the lower boundary moving in the x direction with constant speed U, the upper boundary being at rest. The boundaries are porous and there is a vertical velocity v = —voey with [<>o a constant at both the upper and lower boundaries, so that there is an imposed downflow across the channel. 

A planar Couette flow of gaseous mixtures is considered in [8]. 

The most detailed simulation study of the orientational ordering of simple dipolar fluids undergoing planar Couette flow at a constant shear rate y has been presented in a series of papers by McWhirter and Patey [205-208]. hi their work the translational motion of the particles is obtained from the so-called SLLOD equations given by [209]... 

Whether the string phase corresponds to a real situation or is an artifact of the simulations due to the use of an inaccurate expression for the secondary flow in regimes where the hydrodynamic stability of planar Couette flow is lost is difficult to ascertain at the present time [205]. Experiments on colloidal suspensions have not provided a clear answer, though at moderate strain rates structural behavior similar to that found in the simulations is observed [222]. 

Finally, McWhirter and Patey [206] investigated the change in orientational order entailed by planar Couette flow in a DSS system T = 1.35, p = 0.80 and fi = 3.0) which is ferroelectricaUy ordered in the imsheared state. Only low shear rates, where Ws(r, t) could be taken as equal to zero, were considered. Contrary to the state with the lower dipole moment where orientational order builds up at low shear rates as a response to spatial structure, here the system looses the orientational order present at y = 0 when sheared. The net dipole moment M(f) = A i/I IZi shown to rotate continuously, but in a nonuniform way, about the z axis with an average angular velocity roughly equal to the vorticity - yez and no steady state is obtained. The orientations of M t), at which the order parameter Pi drops rapidly, are encoimtered more frequently at large y which explains the decrease of (Pi) with increasing shear rate. 

Box 6 Periodic boundary conditions used for nonequiiibrium MD simuiations of planar Couette flow... 

If we design the coupling of the external field to the system in such a way that the dissipative flux is equal to one of the Navier-Stokes fluxes (such as the shear stress in planar Couette flow or the heat flux in thermal conductivity), it can be shown - provided the system satisfies a number of fairly simple conditions (Evans Morriss 1990) - that the response is proportional to the Green-Kubo time integral for the corresponding Navier-Stokes transport coefficient. This means that the linear response of the system to the fictitious external field is exactly related to linear response of a real system to a real Navier-Stokes force, thereby enabling the calculation of the relevant transport coefficient. 

---