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Related Concept Videos

Steady, Laminar Flow Between Parallel Plates01:17

Steady, Laminar Flow Between Parallel Plates

Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.
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Couette flow represents the flow of fluid between two parallel plates, with one plate fixed and the other moving with a constant velocity. This configuration allows for a simplified analysis using the Navier-Stokes equations, which govern fluid motion under conditions of viscosity and incompressibility. For Couette flow, the assumptions include a steady, laminar, incompressible flow with a zero-pressure gradient in the flow direction. This flow type is beneficial for understanding shear-driven...
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The application of the linear momentum equation can be used to analyze the forces needed to hold a 180-degree pipe bend in place with flowing water. In this case, water flows through the bend with a constant cross-sectional area of 0.01 square meters and a flow velocity of 15 meters per second. The pressure at the entrance is 0.2 Megapascals and the pressure at the exit is 0.16 Megapascals.
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Stream Function01:20

Stream Function

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,
Steady, Laminar Flow in Circular Tubes01:23

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Hagen-Poiseuille flow describes a viscous fluid's steady, incompressible flow through a cylindrical tube with a constant radius R. This flow profile is often applied to understand fluid transport in narrow channels, such as capillaries. It serves as a foundational example of laminar flow. In this model, cylindrical coordinates (r,θ,z) are used to describe the radial (r), angular (θ), and axial (z) dimensions within the tube. For Hagen-Poiseuille flow, the velocity profile is purely axial,...
Xylem and Transpiration-driven Transport of Resources02:03

Xylem and Transpiration-driven Transport of Resources

The xylem of vascular plants distributes water and dissolved minerals that are taken up by the roots to the rest of the plant. The cells that transport xylem sap are dead upon maturity, and the movement of xylem sap is a passive process.

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Related Experiment Video

Updated: May 12, 2026

Isolation and Time-Lapse Imaging of Primary Mouse Embryonic Palatal Mesenchyme Cells to Analyze Collective Movement Attributes
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A closed-form solution for steady-state coupled phloem/xylem flow using the Lambert-W function.

A J Hall1, P E H Minchin

  • 1The New Zealand Institute for Plant & Food Research Limited (PFR), Palmerston North, New Zealand.

Plant, Cell & Environment
|April 27, 2013
PubMed
Summary

This study presents a new analytical model for plant water transport, integrating phloem and xylem flow. The model offers a more realistic understanding of how water moves within plants, crucial for plant physiology research.

Keywords:
Münch flowphloem-xylem interactionssolute transportwater potential equilibrium

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Area of Science:

  • Plant Physiology
  • Biophysics
  • Mathematical Biology

Background:

  • Plant water transport relies on coupled phloem and xylem flow.
  • Existing models often simplify complex interactions and variations.
  • The Münch flow and cohesion models are foundational but have limitations.

Purpose of the Study:

  • To develop a closed-form analytical solution for steady-state coupled phloem/xylem flow.
  • To incorporate realistic variations in xylem water potential and lateral water flow.
  • To provide a more general and accurate model of plant water transport.

Main Methods:

  • Developed a closed-form solution using the Lambert-W function.
  • Incorporated variable phloem resistance and osmotic potential deviations.
  • Validated model predictions against a numerical time-dependent model.

Main Results:

  • The analytical solution accurately predicts steady-state coupled phloem/xylem flow.
  • The model successfully accounts for variations in xylem water potential and lateral flow.
  • The solution handles complex plant architectures with multiple sources/sinks and sink competition.

Conclusions:

  • The new model offers significant advancements over previous analytical approaches.
  • It provides novel insights into Münch flow by integrating xylem interactions.
  • This solution is valuable for understanding whole-plant water transport and can be integrated into larger numerical models.