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Thin, roughly horizontal low-permeability layers are a common form of large-scale heterogeneity in geological porous formations. In this paper, the dynamics of a buoyancy-driven plume in a two-dimensional layered porous medium is studied theoretically, with the aid of high-resolution numerical simulations. The medium is uniform apart from a thin, horizontal layer of a much lower permeability, located a dimensionless distance
below the dense plume source. If the dimensionless thickness
of the low-permeability layer are small, the effect of the layer is found to be well parameterized by its impedance
. Five different regimes of flow are identified and characterized. For
, the layer has no effect on the plume, but as
is increased the plume widens and spreads over the layer as a gravity current. For still larger
, the flow becomes destabilized by convective instabilities both below and above the layer, until, for
, the spread of the plume is dominated by convective mixing and buoyancy is transported across the layer by diffusion alone. Analytical models for the spread of the plume over the layer in the various different regimes are presented.
The elastic analogue of the Landau–Levich dip-coating problem, in which a plate is withdrawn from a bath of fluid on whose surface lies a thin elastic sheet, is analysed for angle of withdrawal
to the horizontal. The flow is controlled by the elasticity number,
, which is a measure of the relative importance of viscous and bending stresses, and
. The leading-order solution for small
is a steady profile in which the thickness of the film on the plate is found to vary as
. This prediction is confirmed in the limit
by comparison with numerical simulation. Finally, the circumstances under which the assumption of a steady solution is no longer valid are discussed, and the time-dependent solution is described.
Three related problems of viscoplastic flow around cylinders are considered. First, translating cylinders with no-slip surfaces appear to generate adjacent rotating plugs in the limit where the translation speed becomes vanishingly small. In this plastic limit, analytical results are available from plasticity theory (slipline theory) which indicate that no such plugs should exist. Using a combination of numerical computations and asymptotic analysis, we show that the plugs of the viscoplastic theory actually disappear in the plastic limit, albeit very slowly. Second, when the boundary condition on the cylinder is replaced by one that permits sliding, the plastic limit corresponds to a partially rough cylinder. In this case, no plasticity solution has been previously established; we provide evidence from numerical computations and slipline theory that a previously proposed upper bound (Martin & Randolph, Geotechnique, vol. 56, 2006, pp. 141–145) is actually the true plastic solution. Third, we consider how a prescribed surface velocity field can propel cylindrical squirmers through a viscoplastic fluid. We determine swimming speeds and contrast the results with those from the corresponding Newtonian problem.
Gilt progeny (GP) are born and weaned lighter than sow progeny (SP) and tend to have higher rates of mortality and morbidity. This study quantified the lifetime growth performance differences between GP and SP and, additionally, evaluated whether segregating GP and SP in the grower–finisher period compared to mixing them within common pens reduced this variation. It was hypothesised that GP would be lighter than SP at every stage and segregation would improve growth performance of both GP and SP. All piglets born to 61 gilts (parity 1) and 47 sows (parities 2 to 7; mean 3.5 ± 0.2) were allocated to four treatments at 10 weeks of age: (i) GP housed together (GG), (ii) GP mixed (M) with SP (GM), (iii) SP housed together (SS) and (iv) SP mixed with GP (SM). The GM and SM pigs were housed together in common pens after movement into the grower–finisher facility. Individual live weight of all progeny was recorded at birth, weaning (WWT), 10 weeks of age (10WT) and sale (SWT). Individual hot carcass weight (HCW), fat depth at the head of the last rib (P2) and dressing percentage were measured at slaughter. Gilt progeny were lighter at birth (P = 0.038), weaning (P < 0.001) and through to sale (P = 0.001) than SP. Nursery and grower–finisher performance differences in GP were highly attributable to their lower WWT compared to SP (P < 0.001 when fitted as a covariate). Segregation of GP and SP increased grower–finisher average daily gain (ADG) in SP but decreased ADG and SWT in GP (P < 0.10). Segregated SP had increased average daily feed intake but only in males (P = 0.007); HCW (P < 0.001) and P2 fat depth (P = 0.055) were higher in mixed female GP, but there was no difference (P > 0.10) in female SP, or in males. In conclusion, GP were lighter at every stage than SP and differences after weaning were highly related to the lighter WWT of GP. Under the conditions of this study, overall segregation of GP and SP showed no consistent advantages in growth performance for both groups and differed significantly between males and females.
The effect of permeability heterogeneities and viscosity variations on miscible displacement processes in porous media is examined using high-resolution numerical simulations and reduced theoretical modelling. The planar injection of one fluid into a fluid-saturated, two-dimensional porous medium with a permeability that varies perpendicular to the flow direction is studied. Three cases are considered, in which the injected fluid is equally viscous, more viscous or less viscous than the ambient fluid. In general it is found that the flow in each case evolves through three regimes. At early times, the flow exhibits the concentration evolves diffusively, independent of both the permeability structure and the viscosity ratio. At intermediate times, the flow exhibits different dynamics including channelling and fingering, depending on whether the injected fluid is more or less viscous than the ambient fluid, and depending on the relative magnitude of the viscosity and permeability variations. Finally, at late times, the flow becomes independent of the viscosity ratio and dominated by shear-enhanced (Taylor) dispersion. For each of the regimes identified above, we develop reduced-order models for the evolution of the transversely averaged concentration and compare them to the full numerical simulations.
Chloris spp. are warm-season grasses that outcompete crops for scarce resources throughout Australia. In Queensland, mild winters and increased adoption of conservation tillage practices have led to an increase of this warm-season grass family in winter crops. The objective of this study is to understand whether droplet size (nozzle type) effects herbicide efficacy of summer perennial grasses, as previous research found no effect of droplet size (nozzle type) on herbicide efficacy of winter annual grasses. A study to compare droplet-size (nozzle type) effects on control of windmillgrass and its domesticated relative, rhodesgrass, was conducted at the University of Queensland in Gatton, QLD, Australia. Results showed little difference in dry weight reductions for windmillgrass or rhodesgrass across droplet size (nozzle type). Paraquat applications with the TTI nozzle resulted in significantly lower dry weight reductions compared with other droplet-size sprays (nozzle types) for rhodesgrass. Glyphosate, imazamox plus imazapyr, and clodinafop resulted in commercially acceptable control for both species, regardless of the droplet size (nozzle type) selected, indicating droplet size (nozzle type) has relatively little impact on the efficacy of these herbicides. Proper nozzle selection can result in control of Chloris spp., a hard to control weed species, while reducing the occurrence of spray drift to nearby sensitive areas.
The theory of slow viscous flow around a slender body is generalized to the situation where the ambient fluid has a yield stress. The local flow around a cylinder that is moving along or perpendicular to its axis, and rotating, provides a first step in this theory. Unlike for a Newtonian fluid, the nonlinearity associated with the viscoplastic constitutive law precludes one from linearly superposing solutions corresponding to each independent component of motion, and instead demands a full numerical approach to the problem. This is accomplished for the case of a Bingham fluid, along with a consideration of some asymptotic limits in which analytical progress is possible. Since the yield stress of the fluid strongly localizes the flow around the body, the leading-order slender-body approximation is rendered significantly more accurate than the equivalent Newtonian problem. The theory is applied to the sedimentation of inclined cylinders, bent rods and helices, and compared with some experimental data. Finally, the theory is applied to the locomotion of a cylindrical filament driven by helical waves through a viscoplastic fluid.
Recent advances in biotechnology have resulted in crops that are tolerant to the synthetic auxin 2,4-D, expanding the weed management versatility of this herbicide. With potential expansions of use, concerns have been raised about the increased risk of herbicide drift, leading to damage to nontarget crops. A field-scale study was conducted with the objective to measure drift deposition and the potential for drift reduction conferred by a proprietary pre-mixture formulation of 2,4-D choline salt plus glyphosate dimethylammonium salt compared to an in-tank mixture of 2,4-D dimethylamine salt plus glyphosate potassium salt. Treatments were made with field-scale spray equipment under typical application conditions in McCook, NE, using three widely used nozzle tips. Deposition was captured in triplicate downwind collector lines and assayed for tracer dye and 2,4-D. In comparison to the in-tank mixture, the pre-mixture formulation exhibited lower downwind depositions when applied through a flat-fan (TeeJet Extended Range; XR) and air induction (TeeJet Air Induction Extended Range; AIXR) nozzles, but not with a pre-orifice (TeeJet TurboTeeJet Induction; TTI) nozzle. Based upon median deposition at 30 m downwind, the pre-mixture formulation reduced drift by 62% and 91%, for the XR and AIXR nozzles, respectively. From a drift reduction perspective, the pre-mixture formulation performance with the AIXR nozzle was equivalent to a much coarser TTI nozzle while still offering sufficient foliar coverage for acceptable weed control.
We develop a model of the rapid propagation of water at the contact between elastic glacial ice and a poroelastic subglacial till, motivated by observations of the rapid drainage of supraglacial lakes in Greenland. By treating the ice as an elastic bending beam, the fluid dynamics of contact with the subglacial hydrological network, which is modelled as a saturated poroelastic till, can be examined in detail. The model describes the formation and dynamics of an axisymmetric subglacial cavity, and the spread of pore pressure, in response to injection of fluid. A combination of numerical simulation and asymptotic analysis is used to describe these dynamics for both a rigid and a deformable porous till, and for both laminar and turbulent fluid flow. For constant injection rates and laminar flow, the cavity is isostatic and its spread is controlled by bending of the ice and suction of pore water in the vicinity of the ice–till contact. For a deformable till, this control can be modified: generically, a flexural wave that is initially trapped in advance of the contact point relaxes over time by diffusion of pore pressure ahead of the cavity. While the dynamics are found to be relatively insensitive to the properties of the subglacial till during injection with a constant flux, significant dependence on the till properties is manifest during the subsequent spread of a constant volume. A simple hybrid turbulent–laminar model is presented to account for fast injection rates of water: in this case, self-similar turbulent propagation can initially control the spread of the cavity, but there is a transition to laminar control in the vicinity of the ice–till contact point as the flow slows. Finally, the model results are compared with recent geophysical observations of the rapid drainage of supraglacial lakes in Greenland; the comparison provides qualitative agreement and raises suggestions for future quantitative comparison.
We examine the full ‘life cycle’ of miscible viscous fingering from onset to shutdown with the aid of high-resolution numerical simulations. We study the injection of one fluid into a planar two-dimensional porous medium containing another, more viscous fluid. We find that the dynamics are distinguished by three regimes: an early-time linearly unstable regime, an intermediate-time nonlinear regime and a late-time single-finger exchange-flow regime. In the first regime, the flow can be linearly unstable to perturbations that grow exponentially. We identify, using linear stability theory and numerical simulations, a critical Péclet number below which the flow remains stable for all times. In the second regime, the flow is dominated by the nonlinear coalescence of fingers which form a mixing zone in which we observe that the convective mixing rate, characterized by a convective Nusselt number, exhibits power-law growth. In this second regime we derive a model for the transversely averaged concentration which shows good agreement with our numerical experiments and extends previous empirical models. Finally, we identify a new final exchange-flow regime in which a pair of counter-propagating diffusive fingers slow exponentially. We derive an analytic solution for this single-finger state which agrees well with numerical simulations. We demonstrate that the flow always evolves to this regime, irrespective of the viscosity ratio and Péclet number, in contrast to previous suggestions.
Drift reduction technologies aim to eliminate the smaller droplets that occur with some sprays because these small droplets can move off-target in the wind. Commonly used drift reduction technologies such as air-induction nozzles and spray additives impact on reducing off-target movement is well documented, however, the impact on herbicide penetration into an established crop canopy is not well known. This experiment evaluated the canopy penetration and efficacy of glyphosate treatments applied using four nozzle types (XR11005, AIXR11005, AITTJ11005, and TTI11005), two carrier volume rates (94 and 187 L ha-1), and glyphosate applications with and without a commercial drift reducing adjuvant. Applications were made to corn and soybean fields using glyphosate applied at 1.26 kg ae ha-1 with liquid ammonium sulfate at 5% v/v. A rhodamine dye was added (0.025% v/v) to the spray tank of each mixture as a tracer. MylarTM cards were placed in the field above the canopy, in the middle canopy, and on the ground for corn and above and below canopy for soybean. Five cards were at each position in the canopy arranged across the crop row. The addition of a drift reducing adjuvant did not impact canopy penetration. Doubling the carrier volume increased the amount of penetration proportionally and as such the percent reduction was not different. The TTI11005 nozzle had the greatest amount of spray penetration (28%) in the soybean canopies and the XR nozzle had the greatest amount (50%) in the corn canopies. Deposition across the row, beginning in-between the row crop and ending in the row of the crop was 44, 18, and 8% for soybean and 59, 50, and 36% for corn. For both crops, more than half of the herbicide application was captured in the crop canopy. Proper nozzle selection for canopy type can increase herbicide penetration and increasing the carrier volume will increase penetration proportionally.
The stability of steady convective exchange flow with a rectangular planform in an unbounded three-dimensional porous medium is explored. The base flow comprises a balance between vertical advection with amplitude
in interleaving rectangular columns with aspect ratio
and horizontal diffusion between the columns. Columnar flow with a square planform (
) is found to be weakly unstable to a large-scale perturbation of the background temperature gradient, irrespective of
, but to have no stronger instability on the scale of the columns. This result provides a stark contrast to two-dimensional columnar flow (Hewitt et al., J. Fluid Mech., vol. 737, 2013, pp. 205–231), which, as
is increased, is increasingly unstable to a perturbation on the scale of the columnar wavelength. For rectangular planforms with
, a critical aspect ratio is identified, below which a perturbation on the scale of the columns is the fastest growing mode, as in two dimensions. Scalings for the growth rate and the structure of this mode are identified, and are explained by means of an asymptotic expansion in the limit
. The difference between the stabilities of two-dimensional and three-dimensional exchange flow provides a potential explanation for the apparent difference in dominant horizontal scale observed in direct numerical simulations of two-dimensional and three-dimensional statistically steady ‘Rayleigh–Darcy’ convection at high Rayleigh numbers.
A yield stress is added to Taylor’s (Proc. R. Soc. Lond. A, vol. 209, 1951, pp. 447–461) model of a two-dimensional flexible sheet swimming through a viscous fluid. Both transverse waves along the sheet, as in Taylor’s original model, and longitudinal waves are considered as means of locomotion. In each case, numerical solutions are provided over a range of the two key parameters of the problem: the wave amplitude relative to the wavelength and a Bingham number which describes the strength of the yield stress. The numerical solutions are supplemented with discussions of various limits of the problem in which analytical progress is possible. When the yield stress is large, the swimming speed for low wave amplitude is exactly double that for a Newtonian fluid, for either type of wave.
This theoretical and numerical study presents three-dimensional boundary-layer solutions for laminar incompressible flow adjacent to a semi-infinite flat plate, subject to a uniform free-stream speed and injection through the plate surface. The novelty in this case arises from a fully three-dimensional formulation, which also allows for slot injection over a spanwise length scale comparable to the boundary-layer thickness. This approach retains viscous effects in both the spanwise and transverse directions, and effectively results in a parabolised Navier–Stokes system (sometimes referred to as the ‘boundary-region equations’). Any injection profile can be described in this approach, but we restrict attention to three-dimensional states driven by a finite-width slot aligned with the flow direction and self-similar in their downstream development. The classical two-dimensional states are known to only exist up to a critical (‘blow off’) injection amplitude, but the three-dimensional solutions here appear possible for any injection velocity. These new states take the form of low-speed streamwise-aligned streaks whose geometry depends on the amplitude of injection and the spanwise width of the injection slot; intriguingly, although very low wall shear is typically obtained, streamwise flow reversal is not observed, however hard the blowing. Asymptotic descriptions are provided in the limit of increasing slot width and fixed injection velocity, which allow for classification of the solutions according to two bounding injection rates.
In the limit of a large yield stress, or equivalently at the initiation of motion, viscoplastic flows can develop narrow boundary layers that provide either surfaces of failure between rigid plugs, the lubrication between a plugged flow and a wall or buffers for regions of predominantly plastic deformation. Oldroyd (Proc. Camb. Phil. Soc., vol. 43, 1947, pp. 383–395) presented the first theoretical discussion of these viscoplastic boundary layers, offering an asymptotic reduction of the governing equations and a discussion of some model flow problems. However, the complicated nonlinear form of Oldroyd’s boundary-layer equations has evidently precluded further discussion of them. In the current paper, we revisit Oldroyd’s viscoplastic boundary-layer analysis and his canonical examples of a jet-like intrusion and flow past a thin plate. We also consider flow down channels with either sudden expansions or wavy walls. In all these examples, we verify that viscoplastic boundary layers form as envisioned by Oldroyd. For each example, we extract the dependence of the boundary-layer thickness and flow profiles on the dimensionless yield-stress parameter (Bingham number). We find that, while Oldroyd’s boundary-layer theory applies to free viscoplastic shear layers, it does not apply when the boundary layer is adjacent to a wall, as has been observed previously for two-dimensional flow around circular obstructions. Instead, the boundary-layer thickness scales in a different fashion with the Bingham number, as suggested by classical solutions for plane-parallel flows, lubrication theory and, for flow around a plate, by Piau (J. Non-Newtonian Fluid Mech., vol. 102, 2002, pp. 193–218); we rationalize this second scaling and provide an alternative boundary-layer theory.
Non-thermal radio emission has been detected from dMe stars, RS CVn binaries and W T Tauri stars. Polarisation and intensity measurements of the quiescent (i.e. non-flaring) emission indicate that the emission is gyrosynchrotron emission from mildly relativistic electrons spiralling in a magnetic field. A three-dimensional dipole magnetic field model for the stellar field is presented and the quiescent gyrosynchrotron emission from such a model is calculated and compared with observations. The model can account for many phenomenological features of quiescent emission. Quantitative comparisons of model results with observations indicate that the electron distribution in the emission region may be a magnetic mirroring distribution.
Ground-based observations of Jupiter’s decametric radio emission (DAM) have been reviewed by Ellis (1965), Warwick (1967, 1970) and Carr and Gulkis (1969). A startling feature of DAM is the modulating effect of Io, and interpretation of the Io effect has dominated theoretical discussions of DAM until quite recently, specifically until the fly-by s of Voyagers 1 and 2. The Voyager data showed that the DAM appears as nested arcs in the frequency-Jovian longitude plane (Warwick et al. 1979, Boischot et al. 1981). The interpretation of this arc structure has been of primary theoretical interest over the past two years. The most widely adopted explanation is that the emission from each point is confined to the surface of a hollow cone (Goldstein and Thieman 1981). This idea is not new: emission on the surface of a cone was discussed by Ellis and McCulloch (1963); Dulk (1967) derived detailed parameters for the cone (half angle 79° width 1°) from the occurrence pattern of DAM; and Goldreich and Lynden-Bell (1969) presented a theoretical interpretation of it. More recently Goldstein et al. (1979) used observational data on the Jovian magnetic field in deriving properties of the required emission cone. It seems that one requires the properties of the emission cone to vary with position in the Jovian magnetosphere to account for the nested arc pattern (Goldstein and Thieman 1981; Gurnett and Goertz 1981).
A theoretical study is presented of the flow of viscoplastic fluid through a Hele-Shaw cell that contains various kinds of obstructions. Circular and elliptical blockages of the cell are considered together with stepwise contractions or expansions in slot width, all within the simplifying approximation of a narrow gap. Specific attention is paid to the flow patterns that develop around the obstacles, particularly any stagnant plugged regions, and the asymptotic limits of relatively small or large yield stress. Periodic arrays of circular contractions or expansions are studied to explore the interference between obstructions. Finally, viscoplastic flow through a cell with randomly roughened walls is examined, and it is shown that constructive interference of local contractions and expansions leads to a pronounced channelization of the flow. An optimization algorithm based on minimization of the pressure drop is derived to construct the path of the channels in the limit of relatively large yield stress or, equivalently, relatively slow flow.
POST weed control in soybean in the United States is difficult because weed resistance to herbicides has become more prominent. Herbicide applicators have grown accustomed to low carrier volume rates that are typical with glyphosate applications. These low carrier volumes are efficient for glyphosate applications and allow applicators to treat a large number of hectares in a timely manner. Alternative modes of action can require greater carrier volumes to effectively control weeds. Glyphosate, glufosinate, lactofen, fluazifop-P, and 2,4-D were evaluated in field and greenhouse studies using 47, 70, 94, 140, 187, and 281 L ha−1 carrier volumes. Spray droplet size spectra for each herbicide and carrier volume combination were also measured and used to determine their impact on herbicide efficacy. Glyphosate efficacy was maximized using 70 to 94 L ha−1 carrier volumes using droplets classified as medium. Glufosinate efficacy was maximized at 140 L ha−1 and decreased as droplet diameter decreased. For 2,4-D applications, efficacy increased when using carrier volumes equal to or greater than 94 L ha−1. Lactofen was most responsive to changes in carrier volume and performed best when applied in carrier volumes of at least 187 L ha−1. Carrier volume had little impact on fluazifop-P efficacy in this study and efficacy decreased when used on taller plants. Based on these data, applicators should use greater carrier volumes when using contact herbicides in order to maximize herbicide efficacy.