Entrance Region Research Articles

Italian investments for soil defence: retrieving and visualizing data by the PublicWorksFinanceIT R Package

Abstract The PublicWorksFinanceIT R package enables users to retrieve and analyze financial data related to public works in Italy. Specifically, it focuses on soil defence investments. The data are sourced from three distinct platforms: the OpenCoesione website, which draws its information from the Cohesion Policy, the OpenBDAP website, the Ministry of Economy and Finance’s open data platform, and the ReNDiS database, provided by ISPRA, which exclusively gathers information about interventions in soil defence. This package offers a user-friendly tool that eliminates the need for direct access to the aforementioned institutional platforms and ensures real-time updates. Additionally, all measurements, metadata, and accompanying analytical tools are provided in English, enhancing accessibility for both international and domestic users. The data records from these three sources are linked using the unique project code (CUP), ensuring that there is no duplication. Moreover, the data is geographically referenced, meaning that each financial investment is associated with a specific municipality within a particular Italian region. This allows to provide information on the region, province, and municipality of each dataset entry. Users can select to geo-reference the data by either the coordinates of the municipality’s centroid or by the polygon representing the municipality’s administrative boundaries. In addition to functions for data retrieval, the package includes functions for visualizing the collected data on maps. After providing a detailed explanation of the purpose and operation of the main commands, the paper presents two case studies illustrating the software’s application. These examples serve as a step-by-step guide to using the PublicWorksFinanceIT package.

Direct detection method for cusp thickness in wavy thin-film flow using an optical waveguide film

The wave crest (cusp) of the disturbance wave in thin liquid film flow is an important factor contributing to heat/mass transfer, e.g., fuel rods in boiling water reactors, stator/rotor blades in steam turbines, and cleaning/drying wafer processes in semiconductor manufacturing. We developed a new technique for directly detecting the thickness of wave cusps using array-based sensing with an optical waveguide film (OWF). This technique, based on geometrical optics assumptions, simultaneously obtains information on liquid films’ thickness and their interfacial shape, i.e., whether or not the local interface is convex upward. We first performed a pseudo-film flow measurement using a metal specimen to confirm the basic principle. According to the results, a meaningful signal indicating the wave-cusp passage, along with a thickness signal, was detected simultaneously. The OWF signal processing for cusp thickness detection was newly established based on this fact. We then applied this technique to a wavy liquid film flow formed on a flat plate in the entry region. A series of experiments were performed over a wide range of air speeds (jG = 20–70 m/s). As a result, the cusp thicknesses of relatively large waves on the wavy interface were successfully extracted from the OWF output signal. Further, the major thickness variables (i.e., base film thickness, median film thickness, and cusp thickness) were compared with those of conventional thickness estimation methods, which showed reasonable agreement. This paper provides a framework for wavy thin-film flow measurements via OWF that is specialized for directly detecting local thickness profiles.Graphical abstract

Heat transfer in the entrance region of annular laminar flow with constant and different heat flux on two walls

AbstractHeat transfer differs in the regions where the flow is developed and developing thermally. These regions can be differentiated by using the thermal entry length. Many researchers have presented correlations to determine the thermal entry length for natural and forced convection. In this study, heat transfer in the entrance region of a concentric annuli is investigated. It is accepted that beginning from the inlet of annuli the flow is developed hydrodynamically and it is developing thermally. Heat transfer is investigated where the internal or external surfaces of the annuli are at constant but different heat fluxes. The fluid velocity is assumed to be constant or radially variable. Due to thermal boundary conditions, one thermal boundary layer appears on the outer cylinder surface, another on the inner cylinder surface. The edge of two boundary layers will be adiabatic and naturally, the temperature of fluid between the two edges will be equal to free stream temperature. Transformation, Separation of Variables method, eigenvalue problem, Sturm‐Liouville system, Bessel differential equation and properties of orthogonal functions are used in solution of the problem. Exact and analytical solutions of the momentum and energy equations are presented. Velocity and temperature distributions, local Nusselt numbers and convection heat transfer coefficients are calculated for the internal and external surfaces of annuli.

An Approximate Solution for Heat Transfer in the Entrance Region of Laminar Newtonian Pipe Flow

The study of the simultaneously developing pipe flow requires facing nonlinear systems of partial differential equations. In this framework, the aim of this paper is to demonstrate that the integral method can be an effective procedure to obtain analytic-approximate solutions that are easy to handle while allowing the recovery of a satisfactory accordance with the exact solution. To prove the above statement this paper will present a comparison between the approximate solution and the corresponding numerical solution in the entrance region of Newtonian pipe flow. Third-kind thermal boundary conditions are included, while velocity and temperature profiles at the inlet are assumed uniform. Numerical results demonstrate that the proposed approximate solution is quite accurate and readily implemented, both in terms of developing velocity and temperature profiles. Moreover, the expected functional dependence on the main parameters of the problem at hand is retained. As a consequence, the developing Fanning friction coefficient and Nusselt curves are satisfactory and accurate for different thermal boundary conditions at the wall.

Entrance Lengths for Fully Developed Laminar Flow in Eccentric Annulus

Abstract We study the entrance length in eccentric annular geometries, where the axes of the inner and outer pipes forming the annulus are offset from each other. Such geometries are considered relevant for several industrial applications, such as drilling of wells for hydrocarbon production or geothermal energy recovery, as well as biomedical research involving annular vessels. Previous studies have shown that eccentricity increases the entrance length in annular geometries, and this has been attributed to azimuthal redistribution of fluid between the wide and narrow sides of the annulus. The present computational study aims to increase the understanding of entrance lengths in eccentric annuli for laminar flow with Reynolds numbers spanning the range from the creep limit and up to intermediate laminar values, below the regime of annular gap instabilities. We find that the entrance lengths in eccentric annuli generally increase for narrower annular gaps (higher aspect ratios). Moreover, the entrance lengths in the annuli with higher aspect ratios are also more sensitive to the degree of eccentricity compared to wider annuli (lower aspect ratios). We report empirical correlations for the dimensionless entrance length using the same form as earlier studies (Le=[C0n+(C1Re)n]1/n). We find that eccentricity, which breaks the axial symmetry of the annulus and causes azimuthal flow in the entrance region, impacts the value of the exponent n, which has been assigned the value of 1.6 in previous studies of axisymmetric conduits.

Viscous dissipation impacts on a developing thermal field in a saturated porous medium: An effective thermal management

In this study, we investigate the impacts of two viscous dissipation models—the form drag (FD) model and the clear fluid compatible (CFC) model—on the thermal behavior of fluid flow within a porous medium in a parallel plate channel. We analyze temperature distributions under constant heat flux conditions by employing the local thermal non-equilibrium model and assuming unidirectional flow governed by the Darcy–Brinkman model. Our results indicate that an increase in the Brinkman number, Biot number, and thermal conductivity ratio enhances the temperature distribution. Notably, the CFC model demonstrates a higher local Nusselt number compared to that in the FD model, highlighting its potential for applications in heat exchanger design where efficient heat transfer is crucial. Our analysis aids in predicting thermal performance and efficiency, emphasizing the importance of selecting appropriate viscous dissipation models to achieve optimal thermal outcomes. The findings have significant applications in thermal management technologies. A comparative analysis with experimental and computational studies further strengthens these insights, making them applicable to a wide range of thermal systems (see Sec. V B). This study can assist engineers in accounting for viscous dissipation when designing systems that involve fluid flow through porous media, such as heat exchanger devices, particularly in the entrance region of ducts, where the thermal field begins to develop.

Short- and Medium-Range Predictability of Warm-Season Derechos. Part I: Operational Predictability and Composites

Abstract This study analyzes the operational predictability of warm-season (May–August) progressive derechos. A subset of 47 derechos occurring between 2010 and 2022 were selected based on NCEI Storm Data and radar imagery. The Storm Prediction Center Convective Outlooks issued from 5 days before each derecho to the day of the event were used to determine whether the severe weather and derecho predictability was low, moderate, or high, for each case. The cases were also categorized based on the synoptic-scale midlevel flow pattern. Composite maps of synoptic patterns associated with the derecho cases and distributions of severe weather parameters in the derecho inflow region were generated. About 72% of the 47 derechos selected for this study were associated with low predictability, and the remainder were categorized with moderate predictability. None of the cases had high predictability. Most derechos occurring under northwest- and zonal-flow regimes (80% and 85%, respectively) had low predictability. Composites of low- and moderate-predictability derechos indicate that derechos formed in the equatorward entrance region of the upper-level jet, where low-level warm advection exists in conjunction with an equivalent potential temperature maximum. A midlevel trough is located upstream of the derecho initiation point in many of the moderate-predictability composites but is absent in the low-predictability composites, which indicates weaker synoptic-scale forcing for ascent in low-predictability cases. The distributions of severe weather parameters for low- and moderate-predictability derechos are similar, suggesting that these parameters alone are not particularly useful as a discriminator of derecho predictability. Significance Statement Derechos are characterized by swaths of damaging wind gusts caused by an organized group of long-lived convective storms. These events are more frequent in the summer and are often difficult to predict far in advance. This study analyzes a subset of derecho cases to better understand the differences between cases with lower and higher predictability. About 72% of the 47 derechos examined in this study had low predictability. In these cases, the development and maintenance of storms are typically associated with larger uncertainty, but if persistent storms do form, there is potential to produce a derecho. These findings are part of ongoing work that examines ways to improve derecho forecasts with greater lead time.

Investigation of the developmental characteristics of isothermal supercritical CO2 flow in the hydrodynamic entrance region of a circular straight tube

Investigation of the developmental characteristics of isothermal supercritical CO2 flow in the hydrodynamic entrance region of a circular straight tube

EFFECT OF FLOW DISTORTIONS ON PIN FINS IN THE ENTRY REGION OF CIRCULAR CHANNELS

Abstract Utilizing heat transfer augmentation features at the inlet of aircraft engines as an additional heat sink requires further understanding of entry region heat transfer and flow field interactions. The effect of asymmetric flow fields or distortions on entry region heat transfer has not been studied. Because the addition of augmentation features will create a near-wall distortion that can impact downstream aircraft components, comparing entry region features to standard distortion screens can provide insight on the performance impact to aircraft engines. This study investigates the effect of inlet distortion screens on pin fin entry region heat transfer, in order to understand if flow field disturbances disrupt the benefits of augmented entry region heat transfer. The superposition of the distortion caused by pin fin features onto a screen distorted smooth entry region is also tested. Three distortion screens are investigated with pin fin arrays, each based on SAE standard designs. Reynolds numbers ranging from 1.5*105 to 3.50*105 are investigated. From this study, inlet heat transfer augmentation due to pin fins is not significantly impacted by inlet distortion flowfields. This is due to the near wall boundary layer impact of augmentation features overshadowing the effect of distortion. Due to the insensitivity of the near wall boundary layer for pin fin arrays with distortions, superimposing the smooth entry region distorted flow with the non-distorted pin fin array flow provides an adequate estimate of the flow field of the pin fin array with a distorted inlet.

Structural insights into the recognition of tetrapyrrole substrates by ancestral class II chelatase CfbA.

Nickel-chelatase CfbA, unlike descendant chelatases, is an ancestral class II chelatase with a symmetric active site architecture. CfbA utilizes sirohydrochlorin (SHC) as a physiological substrate in the biosynthesis of coenzyme F430. CbiXS, a structural analog of CfbA, can use uroporphyrin III (UPIII) and uroporphyrin I (UPI) as non-physiological substrates. Owing to the broad tetrapyrrole specificity of the unique active site of ancestral class II chelatases, the substrate recognition mechanism of CfbA has garnered interest. Herein, we conducted an X-ray crystallographic analysis of CfbA in complex with UPIII and UPI. Interestingly, the binding sites for UPIII and UPI were distinct. UPI was bound at the entrance of the active site, whereas UPIII was bound deep inside the active site cavity in a manner similar to SHC. Despite the difference in the binding positions of UPIII and UPI, Ser11 at the active site provided critical polar interactions for recognizing UPIII and UPI. Several CfbA variants with a Ser11 mutation were studied to confirm the significance of Ser11's position in the context of tetrapyrrole recognition. The CfbA S11T variant showed Ni2+-chelatase activity against coproporphyrin I (CPI), which is a more hydrophobic tetrapyrrole than UPIII and UPI. Using a CPI-docked model of the S11T variant, we proposed that balancing the hydrophobic/polar interactions at residue 11 could alter substrate selectivity. The structural and mutational analyses reported here highlight the importance of polar and hydrophobic interactions at the entry region of the active site for substrate tetrapyrrole recognition by ancestral and descendant class II chelatases.

Characterization and comparative appraisal of two novel annular channel designs with variable flow areas for supercritical heat transfer

Deterioration of heat transfer is a common concern for any supercritical heat transport system, regardless of the geometric orientation. Present study proposes two novel designs of annular channel with variable cross-sectional area and numerically assesses their respective performances, with the primary objective being the enhancement of overall heat transport characteristics in comparison with an equivalent traditional plain annular channel. Both the configurations exhibit substantial gain in terms of overall heat transfer coefficient and prevailing temperature level, while also eliminating deterioration over the entire parametric ranges explored here, with the diverging one demonstrating relatively superior characteristics. The converging channel generally maintains a flatter temperature profile and comparatively lower maximum temperature, and hence can be employed at larger power-to-mass-flux ratios. Taper angle is earmarked as the most influencing design variable, illustrating enhanced performance with greater tapering, albeit at the cost of nominal increase in pressure drop. Both the designs are found to be insensitive to flow acceleration, which is a primary reason of not encountering deterioration. Strong buoyancy effect can be present within the entrance region of the converging design, affecting its overall performance. Local thermalhydraulics have been noted to be contingent to the effective level of turbulence and distribution of specific heat in the radial direction, which also contribute to the favorable response from the diverging design.

Mass transfer in developing flow of a reactive mixture through a curved cylindrical tube.

Mass transfer to the reactive wall of a curved cylindrical tube is analyzed in the combined entry region where both momentum and concentration boundary layers are developing. The velocity and concentration distributions developing from uniform inlet profiles are obtained using perturbation analysis, accounting for acceleration of the inviscid flow outside the boundary layer arising from channel curvature and flow displacement. The enhancement in the average flux of the reactive species at the tube wall downstream of the inlet is determined and compared to predictions of numerical simulations. A regression model is presented for the cross-sectional average of the Sherwood number, valid for distances up to ten tube diameters downstream of the tube inlet over a wide range of Reynolds and Schmidt numbers.

Thermal Transport in the Entry Region of a Microchannel in the Presence of Electro-osmotic Flow

Abstract The motion of a polar fluid in microducts induced by an external electric field, known as electro-osmotic flow, enables fluid circulation without the need for mechanical devices. This feature makes it particularly appealing for the thermal management of electronic components, as microchannels with almost any cross-sectional shape can be easily integrated on the chips. Therefore, it is essential to evaluate how the channel’s geometry influences heat transfer performance. This study investigates the thermal entry region and fully developed electro-osmotic flow in a microchannel with a rectangular cross-section and smoothed corners, with one adiabatic wall and uniform temperature elsewhere. The paper proposes correlations for the Poiseuille and Nusselt numbers, taking into account the aspect ratio and non-dimensional smoothing radius under fully developed thermal and hydrodynamic conditions, which can be valuable for practical design purposes. The study also emphasizes how Joule heating may lead to the reversal of heat flow and how the thermal entry length is linearly dependent on the logarithm of the non-dimensional Joule heating in thermally developing flow. Additionally, it demonstrates that smoothing the corners may increase the local Nusselt number over sharp corners, but it may also shorten the thermal entry length.

Nonlinear evolution of vortical disturbances entrained in the entrance region of a circular pipe

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the low-frequency disturbances that induce streamwise elongated structures. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances. Their amplitude is assumed to be intense enough for nonlinear interactions to occur inside the pipe. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations in the cylindrical coordinate system, derived and solved herein for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions and the initial–boundary value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions. A parametric study is carried out to evince the effect of the Reynolds number, the streamwise and azimuthal wavelengths, and the radial length scale of the inlet disturbance on the nonlinear flow evolution. Elongated pipe-entrance nonlinear structures (EPENS) occupying the whole pipe cross-section are discovered. EPENS with $h$ -fold rotational symmetry comprise $h$ high-speed streaks positioned near the wall, and $h$ low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. Good agreement of our mean-flow and root mean square data with experimental measurements is obtained.

On the Complex Flow Dynamics of Shear Thickening Fluids Entry Flows.

Due to their nature, using shear thickening fluids (STFs) in engineering applications has sparked an interest in developing energy-dissipating systems, such as damping devices or shock absorbers. The Rheinforce technology allows the design of customized energy dissipative composites by embedding microfluidic channels filled with STFs in a scaffold material. One of the reasons for using microfluidic channels is that their shape can be numerically optimized to control pressure drop (also known as rectifiers); thus, by controlling the pressure drop, it is possible to control the energy dissipated by the viscous effect. Upon impact, the fluid is forced to flow through the microchannel, experiencing the typical entry flow until it reaches the fully developed flow. It is well-known for Newtonian fluid that the entrance flow is responsible for a non-negligible percentage of the total pressure drop in the fluid; therefore, an analysis of the fluid flow at the entry region for STFs is of paramount importance for an accurate design of the Rheinforce composites. This analysis has been numerically performed before for shear-thickening fluids modeled by a power-law model; however, as this constitutive model represents a continuously growing viscosity between end-viscosity plateau values, it is not representative of the characteristic viscosity curve of shear-thickening fluids, which typically exhibit a three-region shape (thinning-thickening-thinning). For the first time, the influence of these three regions on the entry flow on an axisymmetric pipe is analyzed. Two-dimensional numerical simulations have been performed for four STFs consisting of four dispersions of fumed silica nanoparticles in polypropylene glycol varying concentrations (7.5-20 wt%).

Carbon ion beam dosimetry in magnetic fields using Farmer-type ionization chambers of different radii: measurements and simulations

Objective.To investigate magnetic field effects on the dose distribution and ionization chambers response in carbon ion reference fields and determine magnetic field correction factors for chambers of different volumes.Approach.The response of six Farmer-type chambers with varying radii (1-6 mm, termed as R1-R6) was measured in magnetic fields up to 1 T in 0.1 T increments using an experimental electromagnet and compared with Monte Carlo simulations. Chamber readings were measured in the entrance region of a monoenergetic carbon ion beam of 390.75 MeV u-1. A lower energy of 200.28 MeV u-1was applied to chamber R3 for comparison. Polarity and recombination corrections were investigated for the R3 chamber. The local dose change induced by the magnetic field was calculated by Monte Carlo, which together with change of the chamber's response, was used to calculate the final magnetic field correction factors.Main results.The dependence of the chamber response on the magnetic field was non-linear and volume-dependent. Maximum changes ranged from 0.30% (R4) to 0.62% (R5) at 0.2 T. For R3, the response for the lower energy was systematically decreased by 0.2% in the range of 0.2 T to 0.7 T. No significant effect of the magnetic field on polarity and ion recombination correction was found. The maximum variation of the local dose was found to be (0.03 ± 0.08) % at 0.2 T for beam energy of 390.75 MeV u-1. Magnetic field correction factors for the different chambers ranged from 0.28% (R4) to 0.60% (R5).Significance.This study provides the first detailed analysis of chambers' response to magnetic flux densities of up to 1 T using chambers of different radii and comparison with simulations. By combining the chamber response alterations with local dose changes magnetic field correction factors were calculated for all six chambers, including the commercial Farmer-type chamber.

Response and adaptation of Chlorella pyrenoidosa to 6PPD: Physiological and genetic mechanisms

The extensive contamination of the tire antidegradant N-(1,3-dimethylbutyl)-N′-phenyl-p-phenylenediamine (6PPD) in aquatic environments have raised concerns about its potential threats to aquatic organisms. Here, the responses of green algae Chlorella pyrenoidosa (C. pyrenoidosa) to 6PPD exposure were investigated for the first time. The growth of C. pyrenoidosa experienced three sequential phases, including inhibition, recovery and stimulation. Physiological and transcriptome analysis suggested that the growth inhibition was associated with the suppressed nitrogen assimilation and amino acid biosynthesis pathways, among which nitrate transporter (NRT) 2.1 was a key target of 6PPD. Molecular docking revealed the steadily binding of 6PPD to the substrate entry region of NRT 2.1 via hydrogen bonds and π − cation interaction, blocking the acquisition of extracellular inorganic nitrogen. Along with the removal of 6PPD through abiotic processes and biodegradation, an adaptive metabolic shift in cells not only facilitated growth recovery but also triggered a compensatory stimulation phase. With regard to microalgal adaptation, upregulated DNA replication and repair pathways served to maintain the integrity of the genetic information, enhanced photosynthesis cascades and central carbon metabolism improved carbon flux and energy conversion to microalgal biomass, recovered amino acid biosynthesis produced essential proteins for multiple metabolisms. The results provide new insights into microalgal molecular responses to 6PPD exposure, facilitating a better understanding of ecological consequences of 6PPD in the environment.

Multi-stage automatic and rapid ablation and needle trajectory planning method for CT-guided percutaneous liver tumor ablation.

Computer-assisted planning methods have increasingly contributed to preoperative ablation planning; however, these methods cannot automatically obtain the final optimal solution within a short time and are rarely validated in practice, greatly limiting their clinical applicability. We aimed to propose a full-automatic multi-stage ablation and needle trajectory planning method for CT-guided percutaneous liver ablation to attain the final optimal plans under multiple clinical constraints rapidly. Our proposed method integrates the ablation zone planning fulfilling complete tumor coverage and critical structure avoidance while reaching a trade-off between ablation number and healthy tissue damage, and needle trajectory planning under multiple clinical constraints. Our needle trajectory planning determines feasible skin entry regions based on hard constraints, where the multi-objective optimization (MOO) considering soft constraints is performed using the Pareto Optimality and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) methods for the final optimal solution. The performance of our proposed method was evaluated on 30 tumors of various characteristics from 23 patients and clinically validated in five clinical cases. Our proposed method achieved 99.8% treatment zone coverage and 40.5% ablation efficiency without involving critical structures, and completely satisfied multiple clinical constraints in all needle trajectory planning results. The average planning time was 23.6 s for tumors of different sizes. All the plans were considered clinically acceptable by the doctors' evaluation. Our method achieved complete tumor coverage without complications in clinical case validation. Our proposed planning method can generate a final optimal plan satisfying multiple clinical constraints within a short time, potentially facilitating preoperative planning for hepatic tumor ablation.

Quantifying a common inconsistency in RANS-VoF modeling of water and oil core annular flow

Current trends in modeling CAF of oil and water combine the interface capturing methodology of VoF with the computational savings of a RANS approach. As demonstrated mathematically, this results in an inconsistency in the overall RANS-VoF treatment and incurs a number of omissions in the solution of oil fraction advection and momentum. To quantify these inconsistencies, five CAF cases with increasing Rew are considered and solved via DNS. Symptoms of the inconsistencies include qualitative errors in the prediction of the flow behavior in the entrance and fully-developed regions, as well as in the transition between these two regions. From a momentum perspective, the more troubling issue is the absence of unclosed terms resulting from fluctuating viscous and surface tension forces in the RANS equations. For momentum advection, terms associated with fluctuating density can be safely ignored due to the similar magnitude between oil and water density.

Power-law fluid flow in the entrance region between two parallel plates

Flow of non-Newtonian fluids between parallel plates has many uses including heat transfer applications. Many non-Newtonian fluids can be categorized as pseudoplastic fluids. A broader category, power-law fluids, covers in addition dilatant and Newtonian fluids. Models for developing power-law fluid flow between parallel plates include computational and integral boundary layer methods. Most published studies include partial results and apply to one of the three categories. The present work addresses the three cases and exclusively shows the transition from developing to fully developed flow. In the inlet region, viscous effects are restricted to the boundary layer. The present approach is based on the inlet-filled region concept, and includes an additional zone, called filled region in which viscous effects are present in the whole region, with the flow asymptotically reaching the fully developed solution. The integral boundary layer method is used in both regions to solve for the flow. The results are found to be in good agreement with published Newtonian, pseudoplastic and dilatant fluid results for boundary layer thickness, centerline velocity, local friction factor, pressure drop profiles, and entrance region size. The present model is fundamentally analytical with minor computational work required.