Couette flow simulation

Couette Flow Simulation. MD typically simulate systems at thermodynamic equilibrium. For the simulation of systems undergoing flow various methods of nonequilibrium MD have been developed (Ifl iZ.). In all of these methods the viscosity Is calculated directly from the constitutive equation. 

Flow systems. In this subsection we present the results of our Couette flow simulations. Most of these results were first presented In Reference ( ). 

Although only one density profile Is shown In each of Figures 7 and 8 the density profiles of the two systems both at equilibrium and In the presence of flow that have been determined. A conclusion of great importance that is suggested by the Couette flow simulations is that the density profiles of the two systems in the presence of flow coincide with the equilibrium density profiles, even at the extremely high shear rates employed in our simulation. A detailed statistical analysis that Justifies this point was presented In Reference ( ). 

Simulation of the Couette flow of silicon rubber - generalized Newtonian model... 

Figure 5.10 (a,b) Comparison of the simulated die swell in a Couette flow for the power-... 

Figure 3. Finite element simulation of plane Couette flow with thermal dissipation and conductive heat transfer. (f) — fixed temperature condition (c) — convective boundary condition.

The attractive feature of LADM Is that once the fluid structure Is known (e.g., by solution of the YBG equations given In the previous section or by a computer simulation) then theoretical or empirical formulas for the transport coefficients of homogeneous fluids can be used to predict flow and transport In Inhomogeneous fluid. For diffusion and Couette flow In planar pores LADM turns out to be a surprisingly good approximation, as will be shown In a later section. 

The nonequilibrium MD method we employed ( ) is the reservoir method (iff.) which simulates plane Couette flow. The effective viscosity Is calculated from the constitutive relation... 

The shear stress Is uniform throughout the main liquid slab for Couette flow ( ). Therefore, two Independent methods for the calculation of the shear stress are available It can be calculated either from the y component of the force exerted by the particles of the liquid slab upon each reservoir or from the volume average of the shear stress developed Inside the liquid slab from the Irving-Kirkwood formula (JA). For reasons explained In Reference (5) the simpler version of this formula can be used In both our systems although this version does not apply In general to structured systems. The Irvlng-Klrkwood expression for the xy component of the stress tensor used In our simulation Is... 

The simulation value for the effective viscosity Is almost half the viscosity of the bulk fluid. According to the LADM the effective viscosity for plane Couette flow can be Identified as... 

Lundbladh, A. and Johansson, A.V. (1991). Direct simulation of turbulent spots in plane Couette flow. J. Fluid Mech, 229, 499-576. 

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 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. 

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. 

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 ... 

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. 

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. 

T. Yamada and S. Nose, Phys. Rev. A, 42,6282 (1990). Two-Phase Coexistence of String and Liquid Phases Nonequilibrium Molecular-Dynamics Simulations of Couette Flow. 

Wang, L., MARcrasio, D. L., Vigil, R. D. Fox, R. O. 2005a CFD simulation of shear-induced aggregation and breakage in laminar Taylor-Couette flow. Journal of Colloid and Interface Science 282, 380-396. 

Barrat and Bocquet [6] carried out the molecular dynamics simulation of Couette and Poiseuille flows. In Couette flow, the upper wall is moved with a cmistant velocity, and in Poiseuille flow an external force drives the flow. Sample results from molecular dynamics simulation are reproduced in Fig. 7. The application of no-slip boundary condition leads to the expected linear and parabolic profiles, respectively, for Couette and Poiseuille flows. However, the velocity profile obtained from molecular dynamics simulation shows a sudden change of velocity in the near-waU region indicating the slip flow. The velocity profile for Couette flow away from the solid surface is linear with different slope than that of the no-slip case. The velocity for slip flow case is higher than that observed in the no-slip case for Poiseuille flow. For both Couette and Poiseuille flows, the partial slip boundary condition at the wall predict similar bulk flow as that observed by molecular dynamics simulation. Some discrepancy in the velocity profile is observed in the near-wall region. 

The slip length also depends on the shear rate imposed on the fluid particle. Thompson and Troian [8] have reported the molecular dynamics simulation of Couette flow at different shear rates. At lower shear rate, the velocity profile follows the no-slip boundary condition. The slip length increases with increase in shear rate. The critical shear rate for slip is very high for simple liquids, i.e., 10 s for water, indicating that slip flow can be achieved experimentally in very small devices at very high speeds. Experiments performed with the SEA and AFM have also showed shear dependence slip in the hydrodynamic force measurements. 

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]. 

Next, we consider the special case of a plane Couette flow with the velocity given by v(r) = 7J/e, where e is a unit vector in ar-direction and 7 = dvx/dy = const is the shear rate. Furthermore, for simplicity, the motion of the particle is restricted to the xy-plane. Then the equations of motion correspond to the (two-dimensional version) of the SLLOD algorithm used in NEMD simulation studies of the viscous properites of fluids [10] ... 

---