The veins distributing oxygenated blood from the placenta to the fetal body have been given much attention in clinical Doppler velocimetry studies, in particular the ductus venosus. The ductus venosus is embedded in the left liver lobe and connects the intra-abdominal portion of the umbilical vein (IUV) directly to the inferior vena cava, such that oxygenated blood can bypass the liver and flow directly to the fetal heart. In the current work, we have developed a mathematical model to assist the clinical assessment of volumetric flow rate at the inlet of the ductus venosus. With a robust estimate of the velocity profile shape coefficient (VC), the volumetric flow rate may be estimated as the product of the time-averaged cross-sectional area, the time-averaged cross-sectional maximum velocity and the VC. The time average quantities may be obtained from Doppler ultrasound measurements, whereas the VC may be estimated from numerical simulations. The mathematical model employs a 3D fluid structure interaction model of the bifurcation formed by the IUV, the ductus venosus and the left portal vein. Furthermore, the amniotic portion of the umbilical vein, the right liver lobe and the inferior vena cava were incorporated as lumped model boundary conditions for the fluid structure interaction model. A hyperelastic material is used to model the structural response of the vessel walls, based on recently available experimental data for the human IUV and ductus venous. A parametric study was constructed to investigate the VC at the ductus venosus inlet, based on a reference case for a human fetus at 36 weeks of gestation. The VC was found to be [Formula: see text] (Mean [Formula: see text] SD of parametric case study), which confirms previous studies in the literature on the VC at the ductus venosus inlet. Additionally, CFD simulations with rigid walls were performed on a subsection of the parametric case study, and only minor changes in the predicted VCs were observed compared to the FSI cases. In conclusion, the presented mathematical model is a promising tool for the assessment of ductus venosus Doppler velocimetry.

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The fluid dynamics in the human fetal ductus venosus in the early stage of pregnancy is not well explored. Consequently, there is an uncertainty in the interpretation of the temporal and spatial velocity variation in the ductus venosus. A robust estimation procedure for non-invasive measurement of the blood flow, based on conventional Doppler ultrasound measurements, is therefore missing. The aim of the present study was to describe the spatial and temporal velocity distribution at the ductus venosus bifurcation for boundary condition typical for fetuses at 11-13 weeks of gestation by means of a mathematical model. In particular we wanted to investigate velocity profiles at the ductus venosus inlet region in early pregnancy under normal conditions, to assess whether robust estimates of velocity profile shape coefficients may be given in order to provide noninvasive volumetric flow rate assessment in the ductus venosus. Such information will be useful in a clinical assessment of the fetus. Our model predicted a close to parabolic velocity profile in the inlet section of the ductus venosus during the cardiac cycle, with a shape factor of 0.53. Our simulations also showed that during atrial contraction (the A-wave), transient simultaneous positive and negative velocities may be observed in the same cross-section, in Womersley-like velocity profiles. Thus, as previous clinical investigators have reported these velocities as either positive or negative, our findings challenge clinical interpretation.

Long slugs arriving in separators/slug catchers is a major flow assurance concern in the offshore oil production industry, potentially causing flooding and/or severe separation problems. The sizing of the receiving facilities is determined by the longest slugs, so the economic implications of slug length predictions can be substantial. Slugs may also over time cause serious fatigue issues in free-span pipe sections, as large load variations can drastically reduce the lifetime of the flange connections. In most laboratory experiments reported in the literature, slugs rarely become longer than around 30-40 pipe diameters, while in many oil production fields, slugs can be considerably longer. Consequently, there is a clear need to better understand how and why such long slugs appear in production systems, and in this paper we present results that shed some light on this matter. We present a unique set of two- and three-phase slug flow experiments conducted in a 766 meter long 8" pipe at 45 bara pressure. The first half of the pipe was horizontal, while the second half was inclined by 0.5 degrees. A total of ten narrow-beam gamma densitometers were mounted on the pipe to study flow evolution, and in particular slug length development. In addition, the average phase fractions were measured using two traversing gamma densitometers, and one 160 meter long section with shut-in valves. The pressure drop was also measured along the loop using a total of twelve pressure transmitters. The results show that the mean slug length initially increases with the distance from the inlet, but this increase slows down and the mean slug length typically reaches a value between 20 and 50 diameters at the outlet. At low flow rates, the slug length distributions tend to be extremely wide, sometimes with standard deviations approaching 100%. The longest slugs that we observed were over 250 pipe diameters (50 meters). At higher flow rates, the slug length distributions are generally narrower. The effect of the water cut on the slug length distribution is significant, but complex, and it is difficult to establish any general trends regarding this relationship. Finally, it was observed that slug flow often requires a very long distance to develop. Specifically, in most of the slug flow experiments, the flow regime 50 meters downstream of the inlet was not slug flow. The reported experiments are the first three-phase slug flow experiments ever conducted in a large-scale setup. By using a long, heavily instrumented pipe, we were able to study the evolution of slug length distributions over a long distance. We believe that these experiments can be of considerable value for developing tools for predicting slug lengths in multiphase transport systems, which is a critical matter for oil field operators.

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