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Dimensionless Groups in Fluid Mechanics01:15

Dimensionless Groups in Fluid Mechanics

Dimensionless groups in fluid mechanics provide simplified ratios that help analyze fluid behavior without relying on specific units. The Reynolds number (Re), which represents the ratio of inertial to viscous forces, distinguishes between laminar and turbulent flows, making it essential in the design of pipelines and aerodynamic surfaces. The Froude number (Fr), the ratio of inertial to gravitational forces, is particularly useful in predicting wave formation and hydraulic jumps in...
Control Volume and System Representations01:16

Control Volume and System Representations

Two key frameworks are employed to analyze mass, energy, and momentum transfer: the control volume approach and the system approach. These frameworks offer different perspectives, depending on whether the focus is on a specific region in space (control volume approach) or a defined mass of fluid (system approach).
The control volume approach considers a stationary region in space through which fluid flows. This region is bounded by a control surface.  For instance, in the case of water flowing...
Reynolds Transport Theorem01:24

Reynolds Transport Theorem

The Reynolds transport theorem provides a framework to relate the time rate of change of an extensive property within a system to that in a control volume, which is crucial for analyzing fluid dynamics. Extensive properties, such as mass, velocity, acceleration, temperature, and momentum, can be expressed in terms of the mass of a fluid portion. These properties are called extensive because they depend on the system's size, while intensive properties are their corresponding values per unit mass.
The Buckingham Pi Theorem01:09

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Typical Model Studies01:30

Typical Model Studies

Fluid mechanics model studies often utilize scaled-down systems to predict fluid behavior in full-scale environments, such as river flows, dam spillways, and structures interacting with open surfaces. Maintaining Froude number similarity in river models is crucial, as it replicates surface flow features like wave patterns and velocities.
Newtonian Fluid: Problem Solving01:18

Newtonian Fluid: Problem Solving

Newtonian fluids exhibit a constant viscosity, meaning their shear stress and shear strain rate are directly proportional. This property ensures a predictable and stable response to applied forces, maintaining a linear relationship between force and flow. Examples include water, air, and light oils, consistently demonstrating this proportional behavior regardless of external conditions.
A velocity gradient forms within the fluid when a Newtonian fluid is placed between two parallel plates, with...

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Related Experiment Video

Updated: May 21, 2026

Controlled Synthesis and Fluorescence Tracking of Highly Uniform Poly(N-isopropylacrylamide) Microgels
11:34

Controlled Synthesis and Fluorescence Tracking of Highly Uniform Poly(N-isopropylacrylamide) Microgels

Published on: September 8, 2016

Generalized approach to global renormalization-group theory for fluids.

A Sai Venkata Ramana1, S V G Menon

  • 1Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India. asaivenk@gmail.com

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 12, 2012
PubMed
Summary

A new global renormalization-group theory (GRGT) for fluids accurately models density fluctuations. This advanced GRGT approach avoids Maxwell

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Area of Science:

  • Statistical Mechanics
  • Physical Chemistry
  • Computational Fluid Dynamics

Background:

  • Traditional fluid theories often rely on approximations for density fluctuations.
  • Existing global renormalization-group theory (GRGT) models have limitations, such as assuming cosine variation of density fluctuations.
  • Accurate modeling of fluid behavior across all length scales is crucial for understanding phase transitions.

Purpose of the Study:

  • To derive a more general global renormalization-group theory (GRGT) for fluids.
  • To relax assumptions made in previous GRGT formulations regarding density fluctuations.
  • To develop a theory that inherently accounts for fluctuations and avoids empirical constructions like Maxwell's.

Main Methods:

  • Derivation of GRGT using the square-gradient approximation for the Helmholtz free energy functional.
  • Utilization of Wilson's functions to represent density fluctuations, removing the cosine variation constraint.
  • Application of the derived GRGT to model square-well and Lennard-Jones fluids.

Main Results:

  • The new GRGT formulation successfully produces a flat isotherm in the two-phase region, indicating accurate fluctuation modeling.
  • The theory naturally incorporates fluctuations across all length scales, eliminating the need for Maxwell's construction.
  • Analysis of phase diagrams and critical constants for various potentials shows good agreement with simulation and experimental data.

Conclusions:

  • The generalized GRGT provides a robust framework for fluid behavior, accurately capturing critical phenomena.
  • By incorporating fluctuations and avoiding empirical constructions, this theory offers improved predictive power for fluid properties.
  • The developed GRGT, when combined with appropriate mean-field approximations and coarse-graining lengths, aligns well with empirical and simulated fluid data.