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Stokes' Law

Stokes' Law

Viscous forces, like friction, are intermolecular forces that resist the relative motion of molecules over each other. When a solid body moves through a liquid, viscous forces drag it in the opposite direction. The force's magnitude depends on the solid's shape and size, as well as its speed and the liquid's coefficient of viscosity, density and temperature.
The expression for the force on a solid spherical object in a fluid is called Stokes' law. Stokes' law is valid only...

Divergence and Stokes' Theorems

Divergence and Stokes' Theorems

The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...

Steady, Laminar Flow in Circular Tubes

Steady, Laminar Flow in Circular Tubes

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,...

Steady, Laminar Flow Between Parallel Plates

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.

Poiseuille's Law and Reynolds Number

Poiseuille's Law and Reynolds Number

Any fluid in a horizontal tube can flow due to pressure differences—fluid flows from high to low pressure. The flow rate (Q) is the ratio of pressure difference and resistance through a horizontal tube. The greater the pressure difference, the higher the flow rate. The flow resistance is expressed as:

Reynolds Transport Theorem

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...

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Related Experiment Video

Updated: Dec 19, 2025

Non-equilibrium Microwave Plasma for Efficient High Temperature Chemistry

Non-equilibrium Microwave Plasma for Efficient High Temperature Chemistry

Published on: August 1, 2017

Use of Stokes' theorem for plasma confinement.

¹Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|June 9, 2020

Summary

This summary is machine-generated.

Stokes

Keywords:

Stokes’ theorem magnetic fields plasma confinement

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Area of Science:

Plasma physics
Magnetohydrodynamics
Differential geometry

Background:

Stokes' theorem is vital for designing magnetic fields to confine plasma.
Understanding plasma behavior requires accurate magnetic field design.

Purpose of the Study:

To illustrate the application of Stokes' theorem in designing quasi-symmetric magnetic fields.
To investigate if quasi-symmetric fields can be created with minimal toroidal current.

Main Methods:

Applying Stokes' theorem and its generalization (Cartan's generalization).
Analyzing the conditions for integrable guiding-centre motion in magnetic fields.

Main Results:

Demonstrated the utility of Stokes' theorem in magnetic field design.
Provided insights into the feasibility of low-current quasi-symmetric fields.

Conclusions:

Stokes' theorem is a powerful tool for plasma confinement magnetic field design.
The study contributes to the understanding of quasi-symmetric magnetic field generation.