Steady Shapes Research Articles

Optimal wall shapes and flows for steady planar convection

We compute steady planar incompressible flows and wall shapes that maximize the rate of heat transfer ( $Nu$ ) between hot and cold walls, for a given rate of viscous dissipation by the flow ( $Pe^2$ ), with no-slip boundary conditions at the walls. In the case of no flow, we show theoretically that the optimal walls are flat and horizontal, at the minimum separation distance. We use a decoupled approximation to show that flat walls remain optimal up to a critical non-zero flow magnitude. Beyond this value, our computed optimal flows and wall shapes converge to a set of forms that are invariant except for a $Pe^{-1/3}$ scaling of horizontal lengths. The corresponding rate of heat transfer $Nu \sim Pe^{2/3}$ . We show that these scalings result from flows at the interface between the diffusion-dominated and convection-dominated regimes. We also show that the separation distance of the walls remains at its minimum value at large $Pe$ .

Axisymmetric lattice Boltzmann model for three-phase fluids and its application to the Rayleigh-Plateau instability

Based on the multi-component phase field theory, in this paper we propose an axisymmetric lattice Boltzmann model for three-phase fluids. The proposed model takes advantage of two particle distribution functions for capturing phase interface among three different fluids, and another particle distribution function for solving the hydrodynamic equations for flow field. In order to describe the axisymmetric effect arising from the coordinate transformation, we elaborately design the equilibrium distribution function and forcing distribution function in the evolution equation, which ensures that the model can accurately recover the macroscopic governing equation for three-phase fluids. Also, the introduced source terms accounting for the axisymmetric effect contain no additional gradient term, which makes it be simpler than the existing lattice Boltzmann model for axisymmetric three-phase fluids. To validate the proposed model, a series of axisymmetric multiphase benchmark examples are performed, including the static double droplets, the spreading of liquid lens, and the binary-fluid Rayleigh-Plateau instability. It is reported that the present model can accurately capture the phase interface, and the predicted steady shapes of the liquid lens agree well with the analytical profiles. Then, the proposed model is used to study the three-phase Rayleigh-Plateau instability and the effects of the wavenumber and the radius ratio of liquid column on the interfacial dynamic behaviour, the breakup time of liquid threads and the size of daughter droplet are investigated in detail. It can be found that the compound liquid thread at a high wavenumber could break up into one main droplet and one satellite droplet, but the multiple satellite droplets can be produced at a low wavenumber, which leads to that the sizes of main and satellite droplets increase with the wavenumber at first and then decrease with it. Besides, we can observe that the inner fluid undergoes the breakup at earlier time than the middle fluid, and the breakup time for both inner and middle fluids increases with the decrease of the wavenumber. Finally, we can find that increasing the radius ratio of liquid column accelerates the breakup of inner-fluid thread, but prevents the breakup of the middle-fluid thread. In addition, the size of the compound main droplet increases with the radius ratio of liquid column, while the size of the compound satellite droplet doest not change much with it.

Shape transformations of red blood cells in the capillary and their possible connections to oxygen transportation.

In this work, a series of numerical simulations have been performed to obtain the steady shapes of red blood cells under a shear force field in the capillary. Two possible classes of steady shapes, the axisymmetric parachute and the non-axisymmetric parachute, are found. If we assume that oxygen diffusion across the red cell membrane is mediated by membrane curvature, it is found that the non-axisymmetric parachute will be more favorable due to its special shape which enables it to have a larger portion of membrane patch capable of releasing oxygen to tissues.

On the shape of air–liquid interfaces with surface tension that bound rigidly rotating liquids in partially filled containers

Abstract The interface shape of a fluid in rigid body rotation about its axis and partially filling the container is often the subject of a homework problem in the first graduate fluids class. In that problem, surface tension is neglected, the interface shape is parabolic and the contact angle boundary condition is not satisfied in general. When surface tension is accounted for, the shapes exhibit much richer dependencies as a function of rotation velocity. We analyze steady interface shapes in rotating right-circular cylindrical containers under rigid body rotation in zero gravity. We pay special attention to shapes near criticality, in which the interface, or part thereof, becomes straight and parallel to the axis of rotation at certain specific rotational speeds. We examine geometries where the container is axially infinite and derive properties of their solutions. We then examine in detail two special cases of menisci in a cylindrical container: a meniscus spanning the cross-section and a meniscus forming a bubble. In each case, we develop exact solutions for the respective axial lengths as infinite series in powers of appropriate rotation parameters, and we find the respective asymptotic behaviors as the shapes approach their critical configuration. Finally, we apply the method of asymptotic approximants to yield analytical expressions for the axial lengths of the menisci over the whole range of rotation speeds. In this application, the analytical solution is employed to examine errors introduced by the assumption that the interface is a right circular cylinder; this assumption is key to the spinning bubble method used to measure surface tension.

Internal circulation and mixing within tight-squeezing deformable droplets.

The internal flow and mixing properties inside deformable droplets, after reaching the steady state within two types of passive droplet traps, are visualized and analyzed as dynamical systems. The first droplet trap (constriction) is formed by three spheres arranged in an equilateral triangle, while the second consists of two parallel spherocylinders (capsules). The systems are assumed to be embedded in a uniform far-field flow at low Reynolds number, and the steady shapes and interfacial velocities on the drops are generated using the boundary-integral method. The internal velocity field is recovered by solving the internal Dirichlet problem, also via a desingularized boundary-integral method. Calculation of 2D streamlines within planes of symmetry reveals the internal equilibria of the flow. The type of each equilibrium is classified in 3D and their interactions probed using passive tracers and their Poincaré maps. For the two-capsule droplet, saddle points located on orthogonal symmetry planes influence the regular flow within the drop. For the three-sphere droplet, large regions of chaos are observed, embedded with simple periodic orbits. Flow is visualized via passive dyes, using material lines and surfaces. In 2D, solely the interface between two passive interior fluids is advected using an adaptive number of linked tracer particles. The reduction in dimension decreases the number of required tracer points, and also resolves arbitrarily thin filaments, in contrast to backward cell-mapping methods. In 3D, the advection of a material surface, bounded by the droplet interface, is enabled using an adaptive mesh scheme. Off-lattice 3D contour advection allows for highly resolved visualizations of the internal flow and quantification of the associated degree of mixing. Analysis of the time-dependent growth of material surfaces and 3D mixing numbers suggests the three-sphere droplet exhibits superior mixing properties compared to the two-capsule droplet.

Eliminating injection and memory effects in bubble rise experiments within yield stress fluids

Bubbles rising steadily in Carbopol often have an inverted teardrop shape with pointed tail, in contrast to steady bubble rise computations using simple yield stress fluid models. The cause of the pointed tail is suspected to be viscoelasticity, but another possibility is that the tail arises during the bubble formation and the shape is retained via some form of fluid memory effect (elastic, thixotropic or yield stress). Here we present results of a study in which we eliminate all possibility of memory effects using a multi-layered experimental design, but still find the pointed tail in steady bubble propagation. Thus, the tail shape can only result from a combination of the rheology and steady fluid deformation around the rising bubble. Further experiments show that the same steady shapes are found for quite different initial bubble shapes. • Removing the effect of the injection method on the steady-state propagation of bubbles in Carbopol. • Effect of the initial shape of bubbles on their steady propagation in Carbopol. • Controlling the size of bubbles by changing the yield stress of the host fluid. • Bubble rise through a three-layered yield stress–Newtonian-yield stress fluid system.

Nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading in the subcritical buckling regime

Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton’s principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated.

Large-amplitude membrane flutter in inviscid flow

We study the large-amplitude flutter of membranes (of zero bending rigidity) with vortex sheet wakes in two-dimensional inviscid fluid flows. We apply small initial deflections and track their exponential decay or growth and subsequent large-amplitude dynamics in the space of three dimensionless parameters: membrane pretension, mass density and stretching modulus. With both ends fixed, all the membranes converge to steady deflected shapes with single humps that are nearly fore-aft symmetric, except when the deformations are unrealistically large. With leading edges fixed and trailing edges free to move in the transverse direction, the membranes flutter periodically at intermediate values of mass density. As mass density increases, the motions are increasingly aperiodic, and the amplitudes increase and spatial and temporal frequencies decrease. As mass density decreases from the periodic regime, the amplitudes decrease and spatial and temporal frequencies increase until the motions become difficult to resolve numerically. With both edges free to move in the transverse direction, the membranes flutter similarly to the fixed–free case, but also translate vertically with steady, periodic or aperiodic trajectories, and with non-zero slopes that lead to small angles of attack with respect to the oncoming flow.

Spatio-temporal image-based parametric water surface reconstruction: a novel methodology based on refraction

This paper presents a low-cost and easy-to-implement image-based reconstruction technique for laboratory experiments, which results in a temporal description of the water surface topography. The distortion due to refraction of a known pattern, located below the water surface, is used to fit a low parameter surface model that describes the time-dependent and three-dimensional surface variation. Instead of finding the optimal water depth for characteristic points on the surface, the deformation of the entire pattern is compared to its original shape. This avoids the need for feature tracking adopted in similar techniques, which improves the robustness to suboptimal optical conditions and small-scale, high-frequency surface perturbations. Experimental validation, by comparison with water depth measurements using a level gauge and pressure sensor, proves sub-millimetre accuracy for smooth and steady surface shapes. Although such accuracy cannot be achieved in case of highly dynamic surface phenomena, the low-frequency and large-scale free surface oscillations can still be measured with a temporal and spatial resolution mostly limited by the available optical set-up. The technique is initially intended for periodic surface phenomena, but the results presented in this paper indicate that also irregular surface shapes can robustly be reconstructed. Therefore, the presented technique is a promising tool for other research applications that require non-intrusive, low-cost surface measurements while maintaining visual access to the water below the surface. The latter ensures that the suggested surface reconstruction is compatible with simultaneous image-based velocity measurements, enabling a detailed study of the flow.

Mechanisms Governing Protein Clustering and Shape Changes in Biological Membranes

A characteristic feature of eukaryotic cells is the variety of membrane bound organelles, distinguished by their unique morphology and chemical composition. Despite the differences in membrane composition across organelles, ramified, tubular or sheet-like shapes are generic large scale morphologies observed in internal membranes, which suggest involvement of common underlying principles. While there is detailed knowledge of the molecular processes involved in membrane remodelling at short scales, our understanding of the underlying physical principles governing large scale morphogenesis is still rudimentary. One common aspect is that membranes are subject to the action of curvature sensing and curvature generating proteins which modulate local membrane shape and lipid composition. Even at relatively small values of surface coverage these proteins can cluster through to a variety or processes and trigger morphological changes which are important for singling. Another important feature, specific to organelle membranes, especially those in the trafficking pathways, is that they are subject to and driven by a continuous flux of membrane bound materials, on time scales comparable to membrane relaxation times. Here we present Monte Carlo Simulation models that tests the effect of the above processes on the morphology of the membranes. We find that the steady state shapes obtained as a result of such active processes, bear a striking resemblance to the ramified morphologies of organelles in vivo, pointing to the relevance of nonequilibrium fission-fusion in organelle morphogenesis. We show that, due to membrane curvature and composition interactions, the curvature-inducing membrane-nematogens can aggregate spontaneously, even at low concentrations, leading to a variety of membrane morphologies such as tubular and the sheet conformations. Strong lipid-protein interaction can result in fast protein clustering indicating a route to a lipid mediated signal amplification.

Sustainable Slopes: Satisfying Rainfall-Erosion Equilibrium and Mechanical Stability

Abstract. While manmade slopes are traditionally constructed with planar cross-sections, natural stable slopes are usually curvilinear with significant concavity of the profile. This concavity occurs as a result of evolutionary processes in which rain-driven erosion and sediment transport are balanced through slope shape adjustments. At the point of equilibrium, a relatively steady concavity with a uniform erosion rate is observed over time. Nevertheless, a true equilibrium state is possible only if mechanical stability is satisfied. In this study, concave profiles in rainfall erosion equilibrium were found based on the principles of the well-known RUSLE2 model. Results showed the existence of a family of steady shapes satisfying the condition of uniform normalized erosion rate. Those steady concave shapes that also satisfied long-term mechanical stability were then investigated. The overall results suggest that concave slopes can be constructed to achieve both minimal steady-state erosion equilibrium and mechanical stability, leading to more natural and sustainable landforms with minimal sediment delivery during initial slope adjustments. Keywords: Concave slopes, Erosion, Sustainable design, Slope stability.

Films, layers, and droplets: The effect of near-wall fluid structure on spreading dynamics.

We present a study of the spreading of liquid droplets on a solid substrate at very small scales. We focus on the regime where effective wetting energy (binding potential) and surface tension effects significantly influence steady and spreading droplets. In particular, we focus on strong packing and layering effects in the liquid near the substrate due to underlying density oscillations in the fluid caused by attractive substrate-liquid interactions. We show that such phenomena can be described by a thin-film (or long-wave or lubrication) model including an oscillatory Derjaguin (or disjoining or conjoining) pressure and explore the effects it has on steady droplet shapes and the spreading dynamics of droplets on both an adsorption (or precursor) layer and completely dry substrates. At the molecular scale, commonly used two-term binding potentials with a single preferred minimum controlling the adsorption layer height are inadequate to capture the rich behavior caused by the near-wall layered molecular packing. The adsorption layer is often submonolayer in thickness, i.e., the dynamics along the layer consists of single-particle hopping, leading to a diffusive dynamics, rather than the collective hydrodynamic motion implicit in standard thin-film models. We therefore modify the model in such a way that for thicker films the standard hydrodynamic theory is realized, but for very thin layers a diffusion equation is recovered.

A slender drop in a nonlinear extensional flow

The deformation of a slender drop in a nonlinear axisymmetric extensional and creeping flow has been theoretically studied. This problem, which was first suggested by Sherwood (J. Fluid Mech., vol. 144, 1984, pp. 281–295), is being revisited, and new results are presented. The problem is governed by three dimensionless parameters: the capillary number ($\mathit{Ca}\gg 1$), the viscosity ratio ($\unicode[STIX]{x1D706}\ll 1$), and the nonlinear intensity of the flow ($E\ll 1$). Contrary to linear extensional flow ($E=0$), where the local radius of the drop decreases monotonically (in the positive $z$ direction), in a nonlinear extensional flow ($E\neq 0$), two possible steady shapes exist: steady shapes (stable or unstable) with the local radius decreasing monotonically, and steady shapes (unstable) where the local radius of the drop has a local maximum, besides the one at the centre of the drop. Similar to linear extensional flow, the addition of nonlinear extensional effects does not change the end shape of the steady drop, which has pointed ends. A stability analysis has been done to distinguish between stable and unstable steady shapes and to determine the breakup point. Time-dependent studies reveal three types of breakup mechanism: a centre pinching mode, indefinite elongation, and a mechanism that remind us of tip-streaming, where a cusp is developed at the end of the drop.

Electrohydrodynamics of a viscous drop with inertia.

Most of the existing numerical and theoretical investigations on the electrohydrodynamics of a viscous drop have focused on the creeping Stokes flow regime, where nonlinear inertia effects are neglected. In this work we study the inertia effects on the electrodeformation of a viscous drop under a DC electric field using a novel second-order immersed interface method. The inertia effects are quantified by the Ohnesorge number Oh, and the electric field is characterized by an electric capillary number Ca_{E}. Below the critical Ca_{E}, small to moderate electric field strength gives rise to steady equilibrium drop shapes. We found that, at a fixed Ca_{E}, inertia effects induce larger deformation for an oblate drop than a prolate drop, consistent with previous results in the literature. Moreover, our simulations results indicate that inertia effects on the equilibrium drop deformation are dictated by the direction of normal electric stress on the drop interface: Larger drop deformation is found when the normal electric stress points outward, and smaller drop deformation is found otherwise. To our knowledge, such inertia effects on the equilibrium drop deformation has not been reported in the literature. Above the critical Ca_{E}, no steady equilibrium drop deformation can be found, and often the drop breaks up into a number of daughter droplets. In particular, our Navier-Stokes simulations show that, for the parameters we use, (1) daughter droplets are larger in the presence of inertia, (2) the drop deformation evolves more rapidly compared to creeping flow, and (3) complex distribution of electric stresses for drops with inertia effects. Our results suggest that normal electric pressure may be a useful tool in predicting drop pinch-off in oblate deformations.

Open water bells

A smooth circular moving liquid sheet is formed by the head-on impingement of two equal laminar water jets. We subject such a liquid sheet to uniform laminar air flow from one side such that the direction of air velocity is perpendicular to the liquid sheet. The pressure of the moving air deforms the liquid sheet giving rise to an open water bell. The water bell is symmetric suggesting that the gas flow around the bell is also symmetric and that the gravitational force is negligible. We have captured the shape of the water bells for varying air flow rates and for varying Weber numbers, and compared the measurements with theoretical predictions obtained from a force balance involving liquid inertia, surface tension, and pressure difference across the sheet. The pressure exerted by the gas phase on the front and the rear surface of the deformed liquid sheet is obtained from known results of flow past flat circular discs. The predicted steady state shapes match well with the measurements at low Weber numbers but differences are observed at high Weber numbers, where the sheet flaps and is no longer smooth. Interestingly, the shape predicted by assuming a constant pressure difference equal to the stagnation pressure over the whole of the front face of the sheet and free stream value over the whole of the rear face yields nearly identical results suggesting that an open water bell is similar to a closed water bell in that, to a good approximation, the pressure on either sides of the water bell is homogeneous.

Numerical study of surfactant effects on the buoyancy-driven motion of a drop in a tube

We present results of our numerical study of the effect of surfactants on buoyancy-driven motion of drops in a tube at intermediate Reynolds numbers. The drop and bulk phases are treated as incompressible Newtonian fluids and simulated using a front-tracking method. The steady shapes and velocity–volume curves for drops ranging in drop size from 0.2 to 1.3 of tube radius are determined numerically. At small Bond numbers, the velocity–volume curve shows a maximum before the velocity plateaus for large drops. As the Bond number increases, the maximum in the velocity–volume curve disappears with elongated, more streamlined drop shapes consistent with previous experimental studies. For increasing Weber numbers, the rear of the drop shows a flattening followed by the development of a negative curvature. Surfactants are modeled using a Langmuir equation of state in the adsorption–desorption limit and the effect of surfactant mass transfer, fractional coverage of surfactants, and the interfacial Peclet number on the velocity–volume curve is determined. Marangoni stress generated due to the non-uniform distribution of surfactants at the interface reduces drop mobility. Reduced drop mobility is more prominent for drop sizes that are comparable to the tube diameter and is maximum when mass transfer to and from the interface is inhibited. As the fractional coverage of soluble surfactants, or the interfacial Peclet number increases, larger Marangoni stresses are generated along the interface that lead to greater retardation of the drop motion.

Lattice Boltzmann modeling of contact angle and its hysteresis in two-phase flow with large viscosity difference.

Contact angle hysteresis is an important physical phenomenon omnipresent in nature and various industrial processes, but its effects are not considered in many existing multiphase flow simulations due to modeling complexity. In this work, a multiphase lattice Boltzmann method (LBM) is developed to simulate the contact-line dynamics with consideration of the contact angle hysteresis for a broad range of kinematic viscosity ratios. In this method, the immiscible two-phase flow is described by a color-fluid model, in which the multiple-relaxation-time collision operator is adopted to increase numerical stability and suppress unphysical spurious currents at the contact line. The contact angle hysteresis is introduced using the strategy proposed by Ding and Spelt [Ding and Spelt, J. Fluid Mech. 599, 341 (2008)JFLSA70022-112010.1017/S0022112008000190], and the geometrical wetting boundary condition is enforced to obtain the desired contact angle. This method is first validated by simulations of static contact angle and dynamic capillary intrusion process on ideal (smooth) surfaces. It is then used to simulate the dynamic behavior of a droplet on a nonideal (inhomogeneous) surface subject to a simple shear flow. When the droplet remains pinned on the surface due to hysteresis, the steady interface shapes of the droplet quantitatively agree well with the previous numerical results. Four typical motion modes of contact points, as observed in a recent study, are qualitatively reproduced with varying advancing and receding contact angles. The viscosity ratio is found to have a notable impact on the droplet deformation, breakup, and hysteresis behavior. Finally, this method is extended to simulate the droplet breakup in a microfluidic T junction, with one half of the wall surface ideal and the other half nonideal. Due to the contact angle hysteresis, the droplet asymmetrically breaks up into two daughter droplets with the smaller one in the nonideal branch channel, and the behavior of daughter droplets is significantly different in both branch channels. Also, it is found that the contact angle hysteresis is strengthened with decreasing the viscosity ratio, leading to an earlier droplet breakup and a decrease in the maximum length that the droplet can reach before the breakup. These simulation results manifest that the present multiphase LBM can be a useful substitute to Ba etal. [Phys. Rev. E 88, 043306 (2013)PLEEE81539-375510.1103/PhysRevE.88.043306] for modeling the contact angle hysteresis, and it can be easily implemented with higher computational efficiency.

Phenomenology based multiscale models as tools to understand cell membrane and organelle morphologies.

An intriguing question in cell biology is "how do cells regulate their shape?" It is commonly believed that the observed cellular morphologies are a result of the complex interaction among the lipid molecules (constituting the cell membrane), and with a number of other macromolecules, such as proteins. It is also believed that the common biophysical processes essential for the functioning of a cell also play an important role in cellular morphogenesis. At the cellular scale-where typical dimensions are in the order of micrometers-the effects arising from the molecular scale can either be modeled as equilibrium or non-equilibrium processes. In this chapter, we discuss the dynamically triangulated Monte Carlo technique to model and simulate membrane morphologies at the cellular scale, which in turn can be used to investigate several questions related to shape regulation in cells. In particular, we focus on two specific problems within the framework of isotropic and anisotropic elasticity theories: namely, (i) the origin of complex, physiologically relevant, membrane shapes due to the interaction of the membrane with curvature remodeling proteins, and (ii) the genesis of steady state cellular shapes due to the action of non-equilibrium forces that are generated by the fission and fusion of transport vesicles and by the binding and unbinding of proteins from the parent membrane.

Premixed-flame shapes and polynomials

The nonlinear nonlocal Michelson–Sivashinsky equation for isolated crests of unstable flames is studied, using pole-decompositions as starting point. Polynomials encoding the numerically computed 2N flame-slope poles, and auxiliary ones, are found to closely follow a Meixner–Pollaczek recurrence; accurate steady crest shapes ensue for N≥3. Squeezed crests ruled by a discretized Burgers equation involve the same polynomials. Such explicit approximate shapes still lack for finite-N pole-decomposed periodic flames, despite another empirical recurrence.

Shape variation in a benthic stream fish across flow regimes

Evolution of fish body shapes in flowing and non-flowing waters have been examined for several species. Flowing water can select for fish body shapes that increase steady swimming efficiency, whereas non-flowing water can favor shapes that increase unsteady swimming efficiency. Benthic stream fishes often use areas near the substrate that exhibit reduced or turbulent flow, thus it is unclear which swimming forms would be favored in such environments, and how shape might change across flow regimes. To test the relationship between fish body shape and flow regime in a benthic stream fish, we used geometric morphometric techniques to characterize lateral body shape in mountain sucker (Catostomus platyrhynchus) across flow rates, using stream gradient as an indicator of stream flow. Mountain suckers from low-flow environments were more streamlined, consistent with steady swimming body shapes, whereas mountain suckers from high flows had deeper bodies, consistent with unsteady swimming body shapes. In addition, smaller individuals tended to have more robust body shapes. These patterns are opposite to those predicted for stream fishes in the mid-water column. The benthic stream environment represents a distinct selective environment for fish shape that does not appear to conform to the simple dichotomy of flowing versus non-flowing water.