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In two-dimensional incompressible fluid flow, the continuity equation is essential for ensuring mass conservation, meaning that any change in fluid entering or exiting a region is balanced by a corresponding change elsewhere. For incompressible flow, where density remains constant, this requirement simplifies to the condition that the divergence of the velocity field must be zero. Mathematically, this is expressed as,
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Physiological models in pharmacokinetics are instrumental in understanding the distribution and elimination of drugs within the body. These models describe the drug concentration within target organs, influenced by factors such as drug uptake, tissue volume, and blood flow. Drug uptake is governed by the partition coefficient, which signifies the drug concentration ratio in tissue to that in the blood. The blood flow rate to a specific tissue is expressed as Qt, and the rate of change in tissue...
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Consider a control volume, such as a pipe with solid boundaries, through which fluid flows and changes direction due to the impulse exerted by the resulting force from the pipe walls. In steady flow, the mass of fluid entering the control volume at a given time, t, with velocity v1, is equal to the mass leaving after infinitesimal time dt, with velocity v2.
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Related Experiment Video

Updated: Jan 30, 2026

Modeling the Size Spectrum for Macroinvertebrates and Fishes in Stream Ecosystems
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Estimating Multilevel Models on Data Streams.

L Ippel1, M C Kaptein2, J K Vermunt2

  • 1Institute of Data Science, Maastricht University, Maastricht, The Netherlands. lianne.ippel@maastrichtuniversity.nl.

Psychometrika
|January 24, 2019
PubMed
Summary

This study introduces a new algorithm for analyzing nested data, called Streaming Expectation Maximization Approximation (SEMA). SEMA efficiently handles large and continuously updating datasets, offering a faster alternative for multilevel model analysis.

Keywords:
Data streamsSEMAexpectation maximization algorithmmachine (online) learningmultilevel modelsnested data

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Area of Science:

  • Social Sciences
  • Statistics
  • Computer Science

Background:

  • Social science data frequently exhibit nested structures (e.g., pupils in schools).
  • Traditional multilevel models struggle with extremely large or continuously augmenting datasets due to high computational demands.
  • Real-time prediction with streaming data exacerbates these challenges for multilevel modeling.

Purpose of the Study:

  • To develop an efficient online algorithm for fitting multilevel models.
  • To address the computational limitations of traditional methods for large-scale and streaming nested data.
  • To introduce the Streaming Expectation Maximization Approximation (SEMA) algorithm.

Main Methods:

  • Development of the Streaming Expectation Maximization Approximation (SEMA) algorithm for online, row-by-row multilevel model fitting.
  • Extensive simulation study comparing SEMA's performance against traditional multilevel modeling methods.
  • Application of SEMA to analyze an empirical data stream.

Main Results:

  • SEMA demonstrates competitive accuracy compared to state-of-the-art methods for multilevel modeling.
  • SEMA achieves computational speeds orders of magnitude faster than traditional algorithms.
  • The algorithm effectively handles large and continuously incoming data streams.

Conclusions:

  • SEMA provides a computationally efficient and accurate solution for analyzing nested data in online settings.
  • The algorithm is particularly beneficial for applications requiring real-time predictions from streaming data.
  • SEMA represents a significant advancement in the scalability and applicability of multilevel modeling.