TY - JOUR
AB - In many basic shear flows, such as pipe, Couette, and channel flow, turbulence does not
arise from an instability of the laminar state, and both dynamical states co-exist. With decreasing flow speed (i.e., decreasing Reynolds number) the fraction of fluid in laminar motion increases while turbulence recedes and eventually the entire flow relaminarizes. The first step towards understanding the nature of this transition is to determine if the phase change is of either first or second order. In the former case, the turbulent fraction would drop discontinuously to zero as the Reynolds number decreases while in the latter the process would be continuous. For Couette flow, the flow between two parallel plates, earlier studies suggest a discontinuous scenario. In the present study we realize a Couette flow between two concentric cylinders which allows studies to be carried out in large aspect ratios and for extensive observation times. The presented measurements show that the transition in this circular Couette geometry is continuous suggesting that former studies were limited by finite size effects. A further characterization of this transition, in particular its relation to the directed percolation universality class, requires even larger system sizes than presently available.
AU - Avila, Kerstin
AU - Hof, Björn
ID - 8999
IS - 1
JF - Entropy
TI - Second-order phase transition in counter-rotating taylor-couette flow experiment
VL - 23
ER -
TY - JOUR
AB - In this paper we experimentally study the transitional range of Reynolds numbers in
plane Couette–Poiseuille flow, focusing our attention on the localized turbulent structures
triggered by a strong impulsive jet and the large-scale flow generated around these
structures. We present a detailed investigation of the large-scale flow and show how
its amplitude depends on Reynolds number and amplitude perturbation. In addition,
we characterize the initial dynamics of the localized turbulent spot, which includes the
coupling between the small and large scales, as well as the dependence of the advection
speed on the large-scale flow generated around the spot. Finally, we provide the first
experimental measurements of the large-scale flow around an oblique turbulent band.
AU - Klotz, Lukasz
AU - Pavlenko, A. M.
AU - Wesfreid, J. E.
ID - 9207
JF - Journal of Fluid Mechanics
SN - 0022-1120
TI - Experimental measurements in plane Couette-Poiseuille flow: Dynamics of the large- and small-scale flow
VL - 912
ER -
TY - JOUR
AB - We report the results of an experimental investigation into the decay of turbulence in plane Couette–Poiseuille flow using ‘quench’ experiments where the flow laminarises after a sudden reduction in Reynolds number Re. Specifically, we study the velocity field in the streamwise–spanwise plane. We show that the spanwise velocity containing rolls decays faster than the streamwise velocity, which displays elongated regions of higher or lower velocity called streaks. At final Reynolds numbers above 425, the decay of streaks displays two stages: first a slow decay when rolls are present and secondly a more rapid decay of streaks alone. The difference in behaviour results from the regeneration of streaks by rolls, called the lift-up effect. We define the turbulent fraction as the portion of the flow containing turbulence and this is estimated by thresholding the spanwise velocity component. It decreases linearly with time in the whole range of final Re. The corresponding decay slope increases linearly with final Re. The extrapolated value at which this decay slope vanishes is Reaz≈656±10, close to Reg≈670 at which turbulence is self-sustained. The decay of the energy computed from the spanwise velocity component is found to be exponential. The corresponding decay rate increases linearly with Re, with an extrapolated vanishing value at ReAz≈688±10. This value is also close to the value at which the turbulence is self-sustained, showing that valuable information on the transition can be obtained over a wide range of Re.
AU - Liu, T.
AU - Semin, B.
AU - Klotz, Lukasz
AU - Godoy-Diana, R.
AU - Wesfreid, J. E.
AU - Mullin, T.
ID - 9297
JF - Journal of Fluid Mechanics
SN - 0022-1120
TI - Decay of streaks and rolls in plane Couette-Poiseuille flow
VL - 915
ER -
TY - JOUR
AB - Turbulence in the flow of fluid through a pipe can be suppressed by buoyancy forces. As the suppression of turbulence leads to severe heat transfer deterioration, this is an important and undesirable phenomenon in both heating and cooling applications. Vertical flow is often considered, as the axial buoyancy force can help drive the flow. With heating measured by the buoyancy parameter 𝐶, our direct numerical simulations show that shear-driven turbulence may either be completely laminarised or it transitions to a relatively quiescent convection-driven state. Buoyancy forces cause a flattening of the base flow profile, which in isothermal pipe flow has recently been linked to complete suppression of turbulence (Kühnen et al., Nat. Phys., vol. 14, 2018, pp. 386–390), and the flattened laminar base profile has enhanced nonlinear stability (Marensi et al., J. Fluid Mech., vol. 863, 2019, pp. 50–875). In agreement with these findings, the nonlinear lower-branch travelling-wave solution analysed here, which is believed to mediate transition to turbulence in isothermal pipe flow, is shown to be suppressed by buoyancy. A linear instability of the laminar base flow is responsible for the appearance of the relatively quiescent convection driven state for 𝐶≳4 across the range of Reynolds numbers considered. In the suppression of turbulence, however, i.e. in the transition from turbulence, we find clearer association with the analysis of He et al. (J. Fluid Mech., vol. 809, 2016, pp. 31–71) than with the above dynamical systems approach, which describes better the transition to turbulence. The laminarisation criterion He et al. propose, based on an apparent Reynolds number of the flow as measured by its driving pressure gradient, is found to capture the critical 𝐶=𝐶𝑐𝑟(𝑅𝑒) above which the flow will be laminarised or switch to the convection-driven type. Our analysis suggests that it is the weakened rolls, rather than the streaks, which appear to be critical for laminarisation.
AU - Marensi, Elena
AU - He, Shuisheng
AU - Willis, Ashley P.
ID - 9467
JF - Journal of Fluid Mechanics
SN - 00221120
TI - Suppression of turbulence and travelling waves in a vertical heated pipe
VL - 919
ER -
TY - JOUR
AB - High impact epidemics constitute one of the largest threats humanity is facing in the 21st century. In the absence of pharmaceutical interventions, physical distancing together with testing, contact tracing and quarantining are crucial in slowing down epidemic dynamics. Yet, here we show that if testing capacities are limited, containment may fail dramatically because such combined countermeasures drastically change the rules of the epidemic transition: Instead of continuous, the response to countermeasures becomes discontinuous. Rather than following the conventional exponential growth, the outbreak that is initially strongly suppressed eventually accelerates and scales faster than exponential during an explosive growth period. As a consequence, containment measures either suffice to stop the outbreak at low total case numbers or fail catastrophically if marginally too weak, thus implying large uncertainties in reliably estimating overall epidemic dynamics, both during initial phases and during second wave scenarios.
AU - Scarselli, Davide
AU - Budanur, Nazmi B
AU - Timme, Marc
AU - Hof, Björn
ID - 9407
IS - 1
JF - Nature Communications
TI - Discontinuous epidemic transition due to limited testing
VL - 12
ER -
TY - JOUR
AB - We show that turbulent dynamics that arise in simulations of the three-dimensional Navier--Stokes equations in a triply-periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the original system with more than 10^5 degrees of freedom to a 17-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system's observables as weighted sums over the periodic orbits.
AU - Yalniz, Gökhan
AU - Hof, Björn
AU - Budanur, Nazmi B
ID - 9558
IS - 24
JF - Physical Review Letters
SN - 0031-9007
TI - Coarse graining the state space of a turbulent flow using periodic orbits
VL - 126
ER -
TY - JOUR
AB - With decreasing Reynolds number, Re, turbulence in channel flow becomes spatio-temporally intermittent and self-organises into solitary stripes oblique to the mean flow direction. We report here the existence of localised nonlinear travelling wave solutions of the Navier–Stokes equations possessing this obliqueness property. Such solutions are identified numerically using edge tracking coupled with arclength continuation. All solutions emerge in saddle-node bifurcations at values of Re lower than the non-localised solutions. Relative periodic orbit solutions bifurcating from branches of travelling waves have also been computed. A complete parametric study is performed, including their stability, the investigation of their large-scale flow, and the robustness to changes of the numerical domain.
AU - Paranjape, Chaitanya S
AU - Duguet, Yohann
AU - Hof, Björn
ID - 8043
JF - Journal of Fluid Mechanics
SN - 00221120
TI - Oblique stripe solutions of channel flow
VL - 897
ER -
TY - THES
AB - Cytoplasm is a gel-like crowded environment composed of tens of thousands of macromolecules, organelles, cytoskeletal networks and cytosol. The structure of the cytoplasm is thought to be highly organized and heterogeneous due to the crowding of its constituents and their effective compartmentalization. In such an environment, the diffusive dynamics of the molecules is very restricted, an effect that is further amplified by clustering and anchoring of molecules. Despite the jammed nature of the cytoplasm at the microscopic scale, large-scale reorganization of cytoplasm is essential for important cellular functions, such as nuclear positioning and cell division. How such mesoscale reorganization of the cytoplasm is achieved, especially for very large cells such as oocytes or syncytial tissues that can span hundreds of micrometers in size, has only begun to be understood.
In this thesis, I focus on the recent advances in elucidating the molecular, cellular and biophysical principles underlying cytoplasmic organization across different scales, structures and species. First, I outline which of these principles have been identified by reductionist approaches, such as in vitro reconstitution assays, where boundary conditions and components can be modulated at ease. I then describe how the theoretical and experimental framework established in these reduced systems have been applied to their more complex in vivo counterparts, in particular oocytes and embryonic syncytial structures, and discuss how such complex biological systems can initiate symmetry breaking and establish patterning.
Specifically, I examine an example of large-scale reorganizations taking place in zebrafish embryos, where extensive cytoplasmic streaming leads to the segregation of cytoplasm from yolk granules along the animal-vegetal axis of the embryo. Using biophysical experimentation and theory, I investigate the forces underlying this process, to show that this process does not rely on cortical actin reorganization, as previously thought, but instead on a cell-cycle-dependent bulk actin polymerization wave traveling from the animal to the vegetal pole of the embryo. This wave functions in segregation by both pulling cytoplasm animally and pushing yolk granules vegetally. Cytoplasm pulling is mediated by bulk actin network flows exerting friction forces on the cytoplasm, while yolk granule pushing is achieved by a mechanism closely resembling actin comet formation on yolk granules. This study defines a novel role of bulk actin polymerization waves in embryo polarization via cytoplasmic segregation. Lastly, I describe the cytoplasmic reorganizations taking place during zebrafish oocyte maturation, where the initial segregation of the cytoplasm and yolk granules occurs. Here, I demonstrate a previously uncharacterized wave of microtubule aster formation, traveling the oocyte along the animal-vegetal axis. Further research is required to determine the role of such microtubule structures in cytoplasmic reorganizations therein.
Collectively, these studies provide further evidence for the coupling between cell cytoskeleton and cell cycle machinery, which can underlie a core self-organizing mechanism for orchestrating large-scale reorganizations in a cell-cycle-tunable manner, where the modulations of the force-generating machinery and cytoplasmic mechanics can be harbored to fulfill cellular functions.
AU - Shamipour, Shayan
ID - 8350
TI - Bulk actin dynamics drive phase segregation in zebrafish oocytes
ER -
TY - JOUR
AB - In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating that turbulent flows in both experiment and numerics transiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). We show that UPOs we computed are also statistically significant, with turbulent flows spending a sizable fraction of the total time near these solutions. As a result, the average rates of energy input and dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.
AU - Suri, Balachandra
AU - Kageorge, Logan
AU - Grigoriev, Roman O.
AU - Schatz, Michael F.
ID - 8634
IS - 6
JF - Physical Review Letters
KW - General Physics and Astronomy
SN - 0031-9007
TI - Capturing turbulent dynamics and statistics in experiments with unstable periodic orbits
VL - 125
ER -
TY - JOUR
AB - We present nsCouette, a highly scalable software tool to solve the Navier–Stokes equations for incompressible fluid flow between differentially heated and independently rotating, concentric cylinders. It is based on a pseudospectral spatial discretization and dynamic time-stepping. It is implemented in modern Fortran with a hybrid MPI-OpenMP parallelization scheme and thus designed to compute turbulent flows at high Reynolds and Rayleigh numbers. An additional GPU implementation (C-CUDA) for intermediate problem sizes and a version for pipe flow (nsPipe) are also provided.
AU - Lopez Alonso, Jose M
AU - Feldmann, Daniel
AU - Rampp, Markus
AU - Vela-Martín, Alberto
AU - Shi, Liang
AU - Avila, Marc
ID - 7364
JF - SoftwareX
TI - nsCouette – A high-performance code for direct numerical simulations of turbulent Taylor–Couette flow
VL - 11
ER -
TY - JOUR
AB - In the past two decades, our understanding of the transition to turbulence in shear flows with linearly stable laminar solutions has greatly improved. Regarding the susceptibility of the laminar flow, two concepts have been particularly useful: the edge states and the minimal seeds. In this nonlinear picture of the transition, the basin boundary of turbulence is set by the edge state's stable manifold and this manifold comes closest in energy to the laminar equilibrium at the minimal seed. We begin this paper by presenting numerical experiments in which three-dimensional perturbations are too energetic to trigger turbulence in pipe flow but they do lead to turbulence when their amplitude is reduced. We show that this seemingly counterintuitive observation is in fact consistent with the fully nonlinear description of the transition mediated by the edge state. In order to understand the physical mechanisms behind this process, we measure the turbulent kinetic energy production and dissipation rates as a function of the radial coordinate. Our main observation is that the transition to turbulence relies on the energy amplification away from the wall, as opposed to the turbulence itself, whose energy is predominantly produced near the wall. This observation is further supported by the similar analyses on the minimal seeds and the edge states. Furthermore, we show that the time evolution of production-over-dissipation curves provides a clear distinction between the different initial amplification stages of the transition to turbulence from the minimal seed.
AU - Budanur, Nazmi B
AU - Marensi, Elena
AU - Willis, Ashley P.
AU - Hof, Björn
ID - 7534
IS - 2
JF - Physical Review Fluids
SN - 2469-990X
TI - Upper edge of chaos and the energetics of transition in pipe flow
VL - 5
ER -
TY - JOUR
AB - We introduce “state space persistence analysis” for deducing the symbolic dynamics of time series data obtained from high-dimensional chaotic attractors. To this end, we adapt a topological data analysis technique known as persistent homology for the characterization of state space projections of chaotic trajectories and periodic orbits. By comparing the shapes along a chaotic trajectory to those of the periodic orbits, state space persistence analysis quantifies the shape similarity of chaotic trajectory segments and periodic orbits. We demonstrate the method by applying it to the three-dimensional Rössler system and a 30-dimensional discretization of the Kuramoto–Sivashinsky partial differential equation in (1+1) dimensions.
One way of studying chaotic attractors systematically is through their symbolic dynamics, in which one partitions the state space into qualitatively different regions and assigns a symbol to each such region.1–3 This yields a “coarse-grained” state space of the system, which can then be reduced to a Markov chain encoding all possible transitions between the states of the system. While it is possible to obtain the symbolic dynamics of low-dimensional chaotic systems with standard tools such as Poincaré maps, when applied to high-dimensional systems such as turbulent flows, these tools alone are not sufficient to determine symbolic dynamics.4,5 In this paper, we develop “state space persistence analysis” and demonstrate that it can be utilized to infer the symbolic dynamics in very high-dimensional settings.
AU - Yalniz, Gökhan
AU - Budanur, Nazmi B
ID - 7563
IS - 3
JF - Chaos
SN - 1054-1500
TI - Inferring symbolic dynamics of chaotic flows from persistence
VL - 30
ER -
TY - JOUR
AB - Pulsating flows through tubular geometries are laminar provided that velocities are moderate. This in particular is also believed to apply to cardiovascular flows where inertial forces are typically too low to sustain turbulence. On the other hand, flow instabilities and fluctuating shear stresses are held responsible for a variety of cardiovascular diseases. Here we report a nonlinear instability mechanism for pulsating pipe flow that gives rise to bursts of turbulence at low flow rates. Geometrical distortions of small, yet finite, amplitude are found to excite a state consisting of helical vortices during flow deceleration. The resulting flow pattern grows rapidly in magnitude, breaks down into turbulence, and eventually returns to laminar when the flow accelerates. This scenario causes shear stress fluctuations and flow reversal during each pulsation cycle. Such unsteady conditions can adversely affect blood vessels and have been shown to promote inflammation and dysfunction of the shear stress-sensitive endothelial cell layer.
AU - Xu, Duo
AU - Varshney, Atul
AU - Ma, Xingyu
AU - Song, Baofang
AU - Riedl, Michael
AU - Avila, Marc
AU - Hof, Björn
ID - 7932
IS - 21
JF - Proceedings of the National Academy of Sciences of the United States of America
SN - 00278424
TI - Nonlinear hydrodynamic instability and turbulence in pulsatile flow
VL - 117
ER -
TY - THES
AB - Many flows encountered in nature and applications are characterized by a chaotic motion known as turbulence. Turbulent flows generate intense friction with pipe walls and are responsible for considerable amounts of energy losses at world scale. The nature of turbulent friction and techniques aimed at reducing it have been subject of extensive research over the last century, but no definite answer has been found yet. In this thesis we show that in pipes at moderate turbulent Reynolds numbers friction is better described by the power law first introduced by Blasius and not by the Prandtl–von Kármán formula. At higher Reynolds numbers, large scale motions gradually become more important in the flow and can be related to the change in scaling of friction. Next, we present a series of new techniques that can relaminarize turbulence by suppressing a key mechanism that regenerates it at walls, the lift–up effect. In addition, we investigate the process of turbulence decay in several experiments and discuss the drag reduction potential. Finally, we examine the behavior of friction under pulsating conditions inspired by the human heart cycle and we show that under such circumstances turbulent friction can be reduced to produce energy savings.
AU - Scarselli, Davide
ID - 7258
SN - 2663-337X
TI - New approaches to reduce friction in turbulent pipe flow
ER -
TY - JOUR
AB - During bacterial cell division, the tubulin-homolog FtsZ forms a ring-like structure at the center of the cell. This Z-ring not only organizes the division machinery, but treadmilling of FtsZ filaments was also found to play a key role in distributing proteins at the division site. What regulates the architecture, dynamics and stability of the Z-ring is currently unknown, but FtsZ-associated proteins are known to play an important role. Here, using an in vitro reconstitution approach, we studied how the well-conserved protein ZapA affects FtsZ treadmilling and filament organization into large-scale patterns. Using high-resolution fluorescence microscopy and quantitative image analysis, we found that ZapA cooperatively increases the spatial order of the filament network, but binds only transiently to FtsZ filaments and has no effect on filament length and treadmilling velocity. Together, our data provides a model for how FtsZ-associated proteins can increase the precision and stability of the bacterial cell division machinery in a switch-like manner.
AU - Dos Santos Caldas, Paulo R
AU - Lopez Pelegrin, Maria D
AU - Pearce, Daniel J. G.
AU - Budanur, Nazmi B
AU - Brugués, Jan
AU - Loose, Martin
ID - 7197
JF - Nature Communications
SN - 2041-1723
TI - Cooperative ordering of treadmilling filaments in cytoskeletal networks of FtsZ and its crosslinker ZapA
VL - 10
ER -
TY - JOUR
AB - Polymer additives can substantially reduce the drag of turbulent flows and the upperlimit, the so called “maximum drag reduction” (MDR) asymptote is universal, i.e. inde-pendent of the type of polymer and solvent used. Until recently, the consensus was that,in this limit, flows are in a marginal state where only a minimal level of turbulence activ-ity persists. Observations in direct numerical simulations using minimal sized channelsappeared to support this view and reported long “hibernation” periods where turbu-lence is marginalized. In simulations of pipe flow we find that, indeed, with increasingWeissenberg number (Wi), turbulence expresses long periods of hibernation if the domainsize is small. However, with increasing pipe length, the temporal hibernation continuouslyalters to spatio-temporal intermittency and here the flow consists of turbulent puffs sur-rounded by laminar flow. Moreover, upon an increase in Wi, the flow fully relaminarises,in agreement with recent experiments. At even larger Wi, a different instability is en-countered causing a drag increase towards MDR. Our findings hence link earlier minimalflow unit simulations with recent experiments and confirm that the addition of polymersinitially suppresses Newtonian turbulence and leads to a reverse transition. The MDRstate on the other hand results from a separate instability and the underlying dynamicscorresponds to the recently proposed state of elasto-inertial-turbulence (EIT).
AU - Lopez Alonso, Jose M
AU - Choueiri, George H
AU - Hof, Björn
ID - 7397
JF - Journal of Fluid Mechanics
SN - 0022-1120
TI - Dynamics of viscoelastic pipe flow at low Reynolds numbers in the maximum drag reduction limit
VL - 874
ER -
TY - JOUR
AB - We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the formation of a global attractor. In this final regime, the droplet’s angular momentum spontaneously changes its sign as observed in the experiments of Perrard et al.
AU - Budanur, Nazmi B
AU - Fleury, Marc
ID - 5878
IS - 1
JF - Chaos: An Interdisciplinary Journal of Nonlinear Science
SN - 1054-1500
TI - State space geometry of the chaotic pilot-wave hydrodynamics
VL - 29
ER -
TY - JOUR
AB - The hairpin instability of a jet in a crossflow (JICF) for a low jet-to-crossflow velocity ratio is investigated experimentally for a velocity ratio range of R ∈ (0.14, 0.75) and crossflow Reynolds numbers ReD ∈ (260, 640). From spectral analysis we characterize the Strouhal number and amplitude of the hairpin instability as a function of R and ReD. We demonstrate that the dynamics of the hairpins is well described by the Landau model, and, hence, that the instability occurs through Hopf bifurcation, similarly to other hydrodynamical oscillators such as wake behind different bluff bodies. Using the Landau model, we determine the precise threshold values of hairpin shedding. We also study the spatial dependence of this hydrodynamical instability, which shows a global behaviour.
AU - Klotz, Lukasz
AU - Gumowski, Konrad
AU - Wesfreid, José Eduardo
ID - 5943
JF - Journal of Fluid Mechanics
TI - Experiments on a jet in a crossflow in the low-velocity-ratio regime
VL - 863
ER -
TY - JOUR
AB - Speed of sound waves in gases and liquids are governed by the compressibility of the medium. There exists another type of non-dispersive wave where the wave speed depends on stress instead of elasticity of the medium. A well-known example is the Alfven wave, which propagates through plasma permeated by a magnetic field with the speed determined by magnetic tension. An elastic analogue of Alfven waves has been predicted in a flow of dilute polymer solution where the elastic stress of the stretching polymers determines the elastic wave speed. Here we present quantitative evidence of elastic Alfven waves in elastic turbulence of a viscoelastic creeping flow between two obstacles in channel flow. The key finding in the experimental proof is a nonlinear dependence of the elastic wave speed cel on the Weissenberg number Wi, which deviates from predictions based on a model of linear polymer elasticity.
AU - Varshney, Atul
AU - Steinberg, Victor
ID - 6014
JF - Nature Communications
SN - 2041-1723
TI - Elastic alfven waves in elastic turbulence
VL - 10
ER -
TY - JOUR
AB - Electron transport in two-dimensional conducting materials such as graphene, with dominant electron–electron interaction, exhibits unusual vortex flow that leads to a nonlocal current-field relation (negative resistance), distinct from the classical Ohm’s law. The transport behavior of these materials is best described by low Reynolds number hydrodynamics, where the constitutive pressure–speed relation is Stoke’s law. Here we report evidence of such vortices observed in a viscous flow of Newtonian fluid in a microfluidic device consisting of a rectangular cavity—analogous to the electronic system. We extend our experimental observations to elliptic cavities of different eccentricities, and validate them by numerically solving bi-harmonic equation obtained for the viscous flow with no-slip boundary conditions. We verify the existence of a predicted threshold at which vortices appear. Strikingly, we find that a two-dimensional theoretical model captures the essential features of three-dimensional Stokes flow in experiments.
AU - Mayzel, Jonathan
AU - Steinberg, Victor
AU - Varshney, Atul
ID - 6069
JF - Nature Communications
SN - 2041-1723
TI - Stokes flow analogous to viscous electron current in graphene
VL - 10
ER -