# Newtonian fluids

Newtonian fluids are the simplest fluids of everyday life, such as water or light oils. In contrast to complex fluids, Newtonian fluids do not have elastic properties, do not have long memories, and have a constant viscosity (when using reasonable deformation rates). Despite their simplicity, such fluids can still display unexpected phenomena. In my work I focus on two cases: phase-separating fluids in shear flow and microdevices and fluids with actively swimming organisms.

# phase separation in microdevices

When two initially mixed fluids repel each other they tend to separate into macroscopic phases. A well-known example is a mixture of water and oil. Phase separated domains grow in time until there are only two domains, with the oil floating on top of the water.

Shear flow causes an additional pattern formation in phase separating mixtures which is very interesting and not completely understood. Pattern formation plays a central role in the formation of various structures in complex fluids such as polymers, colloids, liquid crystals, and self-assembling membranes and micelles. The (shear) rheology of phase separating complex liquids is thus very important from both a physical and an engineering point of view.

Simulations may help in gaining insight in the dynamics of phase separating systems. During the last decade simulations of phase separating systems undergoing shear have been carried out. In these studies it was observed that the interplay between surface tension and deformation due to shear gives rise to various growth patterns like stretched domains and string phases in the direction of shear. Very recently it has been shown that a nonequilibrium (dynamical) steady state is reached, in which the domains attain a finite length instead of growing indefinitely. The finite length is a result of interference of the shear flow with the transition from an interfacial/viscous to an interfacial/inertial regime, and is found to decrease with increasing shear rate.

In our work we have focused on the influence of a different mechanism, namely one where (thermal) diffusive growth of domains is counterbalanced by convection with the flow. Suppression of domain growth at high shear rates by this mechanism is relevant to phase separating colloidal systems in, e.g., microdevices. Moreover, whereas most Lattice Boltzmann simulations have focused on the bulk behaviour of phase separating systems under planar shear, we have taken the opposite stand and focused on the influence of confining walls, and the curvature of these walls. This is interesting because, even without shear, walls can structure and orient the phase separating domains.

We used molecular dynamics simulations to study phase separation of a fluid mixture in a
confined and curved (Taylor-Couette) geometry, consisting of two concentric cylinders (see picture to the left). The inner cylinder could be rotated to achieve a shear flow.
In non-sheared systems we observed that, for all cases under consideration, the final equilibrium state has a stacked structure.
Depending on the lowest free energy in the geometry the stack may be either flat, with its normal in the *z* (vorticity) direction,
or curved, with its normal in the *r* (gradient) or theta (flow) direction (see figure below).

In sheared systems we made several observations. First, when starting from a pre-arranged stacked structure, we found that sheared gradient and vorticity stacks retain their character for the durations of the simulation, even when another configuration is preferred (as found when starting from a randomly mixed configuration). This slow transition to another configuration is attributed to a large free energy barrier between the two states. In case of stacks with a normal in the gradient direction, we find interesting interfacial waves moving with a prescribed angular velocity in the flow direction (picture to the right). Because such a wave is not observed in simulations with a flat geometry at similar shear rates, the curvature of the wall is an essential ingredient of this phenomenon. Secondly, when starting from a randomly mixed configuration, stacks are also observed, with an orientation that depends on the applied shear rate. Such transitions to other orientations are reminiscent of observations in microphase separated diblock copolymer melts. At higher shear rates complex patterns emerge, accompanied by deviations from a homogeneous flow profile. The transition from steady stacks to complex patterns takes place around a shear rate 1/tau, where tau is the cross-over time from diffusive to viscous driven growth of phase-separated domains, as measured in equilibrium simulations.

- A.K. Thakre, J.T. Padding, W.K. den Otter and W.J. Briels,
*Molecular Dynamics simulation of phase separating binary liquids in cylindrical Couette flow*, J. Chem. Phys.**128**, 154707 (2008).

# asymmetric mixtures

It is interesting to study what happens when the relaxation times of the two liquids are different. We have studied the spinodal decomposition of quenched polymer/solvent and liquid-crystal/solvent mixtures in the above miniature Taylor-Couette cell. Three stacking motifs, each reflecting the geometry and symmetry of the cell, are most abundant among the fully phase-separated stationary states. At zero or low angular velocity of the inner cylindrical drum, the two segregated domains have a clear preference for the stacking with the lowest free energy and hence the smallest total interfacial tension. For high shear rates the steady state appears to be determined by a minimum dissipation mechanism, i.e. the mixtures are likely to evolve into the stacking demanding the least mechanical power by the rotating wall. The partial slip at the polymer-solvent interfaces then gives rise to a new pattern: a stack of three concentric cylindrical shells with the viscous polymer layer sandwiched between two solvent layers. Neither of these mechanisms can explain all simulation results, as the separating mixture easily becomes kinetically trapped in a long-lived sub-optimal configuration. The phase separation process is observed to proceed faster under shear than in a quiescent mixture.

- A.K. Thakre, W.K. den Otter, J.T. Padding and W.J. Briels,
*Spinodal decomposition of asymmetric binary fluids in a micro-Couette geometry simulated with molecular dynamics*, J. Chem. Phys.**129**, 074505 (2008).

# capillary waves in confinement

We have studied the relaxation dynamics of capillary waves in the interface between two confined liquid layers by means of molecular dynamics simulations. We measured the autocorrelations of the interfacial fourier modes and found that the finite thickness of the liquid layers leads to a marked increase of the relaxation times as compared to the case of fluid layers of infinite depth. The simulation results are in good agreement with a theoretical first-order perturbation derivation which starts from the overdamped Stokes' equation. The theory also takes into account an interfacial friction, but the difference with no-slip interfacial conditions is small. When the walls are sheared, it is found that the relaxation times of modes perpendicular to the flow are unaffected. Modes along the flow direction are relatively unaffacted as long as the equilibrium relaxation time is sufficiently short compared to the rate of deformation. These results have consequences for experiments on thin layers, experiments on ultra-low surface tension fluids, as well as most computer simulations on the dynamics of fluid interfaces.

- A.K. Thakre, J.T. Padding, W.K. den Otter and W.J. Briels,
*Finite system size effects in the interfacial dynamics of binary liquid films*, J. Chem. Phys.**129**, 044701 (2008).

# actively swimming organisms

The swimming of organisms such as fish is a topic of broad interest, not only to biologists but also to engineers. In this context, undulatory swimming is important because it seems to be an efficient mode of locomotion, whether the organisms swim alone or in groups. However, it is still unknown what influences the efficiency of an undulating organism. It has been difficult, both empirically and theoretically, to accurately determine this. The most promising approach to this problem appears to be the use of simulation models.

We have employed the Stochastic Rotation Dynamics (SRD) method to study swimming organisms because it is computationally fast (especially when implemented for calculations on a graphics card), simple to implement, and has a continuous representation of space. As regards the study of hydrodynamics of moving organisms, the SRD method has until now only been applied at low Reynolds numbers (below 120) for soft, permeable bodies, and static fish-like shapes. We used it to study the hydrodynamics of undulating fish at Reynolds numbers 1100-1500, after confirming its performance for a moving insect wing at Reynolds number 75. The figure above shows the vorticity field in the fluid around fish swimming in a diamond pattern (one periodic box is shown). Below a movie of a single fish swimming at Re=1300.

We measured 1) flow and drag, thrust and lift forces, 2) swimming efficiency and spatial structure of the wake and 3) distribution of forces along the fish body. We confirmed the resemblance between the simulated undulating fish and empirical data. In contrast to common simplifying assumptions of theoretical models, our model showed that for steadily undulating fish, thrust is produced along the rear 2/3ds of the body, that fish which are constrained from accelerating sideways resemble unconstrained fish that have a higher tailbeat frequency, and that the slip ratio U/V (between forwards swimming speed and backwards velocity of the body wave) correlates negatively with the actual Froude efficiency of swimming. Besides, we found the unexpected result that the slip ratio increases with tailbeat frequency.

- D.A.P. Reid, J.T. Padding, H. Hildenbrandt and C.K. Hemelrijk,
*Study of flow around fish-like shapes using Multi-Particle Collision Dynamics*, Phys. Rev. E**79**, 046313 (2009). - D. Reid, J.T. Padding, H. Hildenbrandt and C.K. Hemelrijk,
*Modelling hydrodynamics and swimming efficiency of fish*, Comparative Biochemistry and Physiology - Part A: Molecular & Integrative Physiology**153**, S125 (2009). - D.A.P. Reid, H. Hildenbrandt, J.T. Padding and C.K. Hemelrijk,
*Fluid dynamics of moving fish in a two-dimensional multiparticle collision dynamics model in 2D*, Phys. Rev. E**85**, 021901 (2012). - C.K. Hemelrijk, D.A.P. Reid, H. Hildenbrandt, and J.T. Padding,
*The increased efficiency of fish swimming in a school*, Fish and Fisheries**16**, 511 (2015). This paper was added as a research highlight in Nature (13 February 2014).

# collaboration

The work on phase separating fluids has been performed mainly by our former PhD student Amol Thakre at the University of Twente. He received his PhD in September 2008. For the work on actively swimming organisms, I am collaborating with Prof. Charlotte Hemelrijk and her former PhD student (now Dr) Daan Reid and postdoc Dr Hanno Hildenbrandt at the Behavioural Ecology and Self-organization group of the University of Groningen in The Netherlands.