Exploring the concept of circulation in fluids and electromagnetic materials using Stokes’ theorem through diagrams (2023)

Materials Today: Proceedings

Volume 47, Part 14,


, Pages 4206-4209

Author links open overlay panel, ,


Circulation of vector fields is vital to understand research problems in several areas such as fluid dynamics, electromagnetics, atmospheric sciences, and allied fields. Curl is the mathematical tool to describe circulation. Stokes’ Curl theorem involves the relation between surface integral and line integral. Authors have observed that undergraduate students seek clarity on application aspects of this theorem probably since the curl concept is generally discussed mathematically with less emphasis on physical visualization. An elaborate pictorial explanation is found to enable students to develop an appreciation for this concept. A pictorial method of interpreting Stokes’ theorem through diagrams involving real-time rotation situations is presented. It has been observed that this method of teaching arouses more questions, peer interactions, and contributes to imbibe application skills. The method that has been developed for the present study has never been studied and mentioned in any of the previous researches.

(Video) Divergence and curl: The language of Maxwell's equations, fluid flow, and more


A fluid is a material that reshapes continuously under applied stress or outside force. Fluids are a framework, with fluids, gases, and plasmas being used. They are null-shear modular materials or, in plain language, they cannot withstand any shear strength added to them. Fluids on their motion, experience a common phenomenon, circulation. Circulation is naturally induced while the flow is isolated from a neutral grip, i.e. once the surface layer is removed from a bluffed body with a larger number of Reynolds, which contributes to instability marked by turbulent, tumor- and vortices flowing structures. Circulations are a macroscopic rotation measurement for a finite region of the liquid and are an integral scalar number. The sum of force driving over a closed boundary or route may be known as circulation. Circulation is the absolute push obtained in a direction, like a loop. Circulations are caused due to the formation of the negative pressure zones near the boundary due to the drastic removal of the adjacent surface [1]. Circulation is a special phenomenon that occurs in materials. The concept of circulation is of greater importance in the field of study of the human cardiovascular system, in the study of design and development of aircraft, in understanding the formation of eddy currents in electromagnetic materials. In the human cardiovascular system, the circulation flows existing in the blood flow is helpful in the diagnosis of blood flow-related pathologies [2]. Also, the study on circulation is necessary for the designing and development of aircraft [3]. Cyclones are just a few examples of the circulatory nature of a fluid flow occurring at various longitude scales. These include the flow through a wing, pump, ship, blender, and cyclones (integral to Kolmogorov scale). The rotational behavior of the liquid is important due to the chaotic nature of most flows in real life. Thus, the study of circulation on fluids is an important field of research that can aid in production units of several fluid-related industries.

Additionally, the electromagnetic materials exhibiting the properties of Maxwell’s equations, the concept of Stokes’ theorem is extensively used for the physical interpretation of the circulation concept among the magnetic and electric fields. Also, the analysis of the eddy currents formation in the electromagnetic materials is made using Stokes’ theorem. The available literature about the Stokes’ theorem gives a brief theoretical explanation of the Stokes’ theorem and fails to explain the proof of the theorem in an effective manner. Thus in the present work, we pretend to frame a pictorial method to put forth the geometrical aspects behind the integral conversions occurring in the mathematical form of the Stokes’ theorem.

The relation between differentiation and integration can be meaningfully defined using the “Integral Theorems” of vector calculus [4], which are extensively used in various engineering applications. These theorems are intentionally used in dimension reduction of integration. The theorem of Green analysis the area and the geometric center of planar geometry. The flux across a surface or volume can be explored using the theorem of Gauss and the theorem of Stokes’ anatomizes the circulation within a curved surface.

Stokes’ theorem is widely used in Mathematics, Physics, Engineering, and other areas to represent the circulation of vectors [5], [6], [7]. It involves evaluation of curl operation and line integral. Through the author’s interaction with engineering students over many years, it is observed that students are acquiring the skill of solving problems involving Curl but are unable to inculcate application skills. This is possibly related to the mathematical way of teaching this concept with less emphasis on physical visualization. This prompted the authors to propose a pictorial method to visualize the physical significance of Stokes’ theorem.

In the current work, the pictorial method as to how the rotation of a vector is studied through Curl operation is described. In the second section, the method is described with the help of diagrams. An explanation of the fundamental expression used to evaluate curl is provided. Conclusions are mentioned at the end following the simple results and the discussions made on the pictorial method described. The main aim of the paper is to enable an easy understanding of the concept of Stokes’ theorem among the young minds using an effective pictorial method.

Section snippets

Description of the method

Stokes’ theorem is stated as for a given vector field, net circulation of a vector over a surface is equal to the circulation of the vector along the boundary enclosing the area. It is expressed as [8], [9], [10],s(×A)·ds=LA·dl

L.H.S of this expression is a measure of net rotation [10]. It involves a curl of vector A integrated over a surface S. The R.H.S is the line integral of the vector A over the boundary enclosing the surface S. In the R.H.S, the notation L is used to represent the

Results and discussion

Stokes’ theorem is necessary to study variation in field patterns which are linked to fluid dynamics parameters, current distribution [10]. Authors in their two-decade experience in undergraduate classes have noticed that with a normal mode of teaching approach, students acquire mathematical skills to solve textbook problems. However, they are unable to visualize application aspects of this theorem for real fluid dynamics or electromagnetics problems. When they are taught through the


A geometrical method to visualize Stokes’ theorem is presented. This method has multiple applications as it provides a pictorial interpretation of curl and relates surface and line integral. This method when experimented with in undergraduate classes yielded positive results through greater peer discussion, arousing questions, and appreciation of the subtle nature of circulating vectors. Students realize that to evaluate net circulation for a surface, effectively, the curl along the boundary is

CRediT authorship contribution statement

Raveesha K.H.: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing. Shankar Narayan S.: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing. Sunrose Shrestha: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Validation, Visualization, Writing - original draft,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (13)

  • Z. Li et al.



  • M. Roy et al.

    Proc. Comput. Sci.


  • K. Kanistras et al.

    Foundations of Circulation Control Based Small-Scaled Unmanned aircraft


  • M. Antoni

    Calculus with Curviliear Coordinates


  • D. Cheng

    Field and wave Electromagnetics


  • John D. Kraus



There are more references available in the full text version of this article.

(Video) The Incorrect Assumption Made by Fluid Dynamics, and Why It Still Works - Fluids by Parth G

Cited by (0)

Recommended articles (6)

  • Research article

    Impact of expansion level on flow field of a circular duct at high Mach numbers

    Materials Today: Proceedings, Volume 47, Part 14, 2021, pp. 4396-4408

    (Video) Stokes Theorem and Definition of Curl

    The study discusses the impact of different nozzle pressure ratios and the duct sizes on the duct’s flow field. The duct diameter was 16 mm, and the inertia levels at which the experiments were done are M = 1.87, 2.2, and 2.58. Dynamic regulators as an orifice stationed in the wake region at 6.5 mm from the central jet axis. At M = 1.87, when the jets are operated at NPR = 3, the duct size 1D and 2D, the wall pressure assumes high, and control is not helpful. There is no impact on the flow control management at Mach 2.2 for duct length 1D and 2D at NPR = 3, 5, and 7. But for this Mach, this trend is limited at NPR = 3 and 5. When the Mach value is 2.58, the wall pressure assumes ambient condition, and control has minimal impact on the duct’s flow pattern. When the duct length is 10D, the stream field turns oscillatory for Mach investigated. The control device does not impact the inside flow negatively.

  • Research article

    Experimental study on the effect of biogas diesel engines driven by silkworm larval litter, cyperus rotundus and cattle urine

    Materials Today: Proceedings, Volume 47, Part 14, 2021, pp. 4617-4623

    The auto ignition problems in I.C Engine are due to the poor thermal properties of biogas, such as flash point, spark ignition temperature, heating value. This leads to increases in fuel consumption and hydrocarbon emissions. In diesel engine mode, it is necessary to boost engine performance and reduce emissions. In order to reduce the emissions and diesel consumption in CI engines the biogas has been implemented. The biogas has been generated from different biomass such as Cyperus rotundus and silkworm larval litter were used to produce biogas. Various blends of biogas and diesel were used to power a four-stroke single-cylinder diesel engine at a compression ratio of 16 and a speed of 1500rpm, the engine was tested at different load conditions (20%, 40%, 60%, 80%, and 100%) respectively. Variables such as Brake Thermal Performance (BTHE), Brake Specific Fuel Consumption (BSFC), and Heat Release at maximum pressure were calculated under different conditions of load. Exhaust gases were studied as part of the emission testing. According to the findings, at an 80 percent load condition, brake thermal performance fell from 22 percent (100 percent diesel) to 17 percent (70 percent diesel+30 percent biogas). A reduction in NOx and CO values was observed when the engine was run in dual-fuel mode.

  • Research article

    Physico-mechanical characterizations of epoxy composites reinforced with lathe waste materials

    Materials Today: Proceedings, Volume 47, Part 14, 2021, pp. 4326-4329

    Nowadays, the use of industrial, municipal and agricultural wastes in reinforced polymers has become a common practice in composite industry. This paper discusses the feasibility of adding lathe waste as reinforcement of polymer composite. The effects of the incorporation of two types of lathe waste (mild steel and aluminium alloy 6063; 10–40wt%) on the physico-mechanical behavior of epoxy composites were experimentally studied. The finding indicates that higher content of waste steel and aluminium alloy content in the fabricated composite resulted in increased density, void content and hardness. The composites containing 30 and 20wt% aluminium and steel waste content showed higher mechanical properties.

  • Research article

    Spectral investigations of dysprosium (Dy3+) ions doped ZnBiNaPSr oxyfluoride glasses for intense white light emitting diodes (w-LEDs)

    Materials Today: Proceedings, Volume 47, Part 14, 2021, pp. 4364-4372

    (Video) A math/physics view of ocean circulation

    Oxyfluoride (P2O5 + Bi2O3 + Na2CO3 + SrCO3 + ZnF2) glasses doped with different dysprosium (Dy3+) ion concentrations (0.5, 1.0, 1.5 & 2 mol %) have been synthesized using conventional melt quenching technique. The Physical and optical parameters such as density (ρ), refractive index (n), polaron radius (rp), interionic distance (ri), field strength (F), electron polarizability (αe), molar refractivity (RM), reflection loss (R), etc., were evaluated. The X-ray diffractograms were recorded in the 10–900 region, showed amorphous nature. Surface morphological studies were performed with SEM and elemental analysis has also been performed with EDS. The absorption spectra consist of a few well-defined high intense bands in the UV–Vis-NIR region. The optical band gaps were estimated from the tauc’s plots drawn from the absorption spectra and obtained values exhibited nonlinear behaviour with dopant ion concentration. The emission spectra were recorded by monitoring the excitation wavelength at 453 nm which consists of three emission bands at 487, 557 and 667 nm, due to f-f characteristic transitions of Dy3+ ion. The Judd-Ofelt and radiative parameters have been calculated from absorption and luminescence spectra. It has been noticed that the concentration quenching effect was observed for higher concentrations (1.5 and 2.0 mol %) in the present glass series. The emission intensity ratio between yellow and blue color bands and CIE color coordinates were evaluated. The correlated color temperature (CCT) and color purity (CP) has been evaluated using CIE 1931 chromaticity diagram. The ZnBiNaPSr: Dy1.0 glass exhibits superior luminescent properties than other concentrations and it was found to be the most prominent device for white light applications.

  • Research article

    Low velocity impact analysis of carbon/glass/epoxy hybrid composite pipes

    Materials Today: Proceedings, Volume 47, Part 14, 2021, pp. 4181-4188

    Composite pipes are widely used in automotive and transportation of liquid in various industries. Impact analysis of composites gives the knowledge to understand failure and damage resistance of the material during low velocity impact like tool drops and designing new component. In this work carbon and glass fibers are used to produce hybrid composite pipes. The effect of impact on hybrid composite pipes was carried out by finite element analysis with varying orientations and for varying thicknesses, fixing the internal diameter. The experimental validation is done for one of the orientation the results are compared with the finite element analysis results. The experimental and analysis results closely match each other. It was found that the impact resistance decreases with increase in the helix angle of orientation such as 15°, 30°, 45°, 60°, 75°, and 90° respectively. The failure of the specimen determined by experimental method for (CG90/CG60)3 the orientation, it takes a peak force of 4358N and 6935N for 4mm and 5.3mm thickness respectively. The thicknesses 4mm and 5.3mm of the specimen increases the stiffness of the specimen, which further increases the impact resistance and reduces the deformation of the specimen.

  • Research article

    Hydrothermal synthesis of Zn2SnO4/ZnO composite for the degradation of organic pollutant Methylene Blue under UV irradiation

    Materials Today: Proceedings, Volume 47, Part 14, 2021, pp. 4566-4570

    In this work, Zn2SnO4/ZnO composite was synthesized by hydrothermal method with the support of Na2CO3 mineralizer. Structure, morphology and optical properties of the synthesized composite was analysed using characterization techniques such as powder x-ray diffraction (PXRD), Raman spectroscopy, UV–visible diffuse reflectance spectroscopy (UV–Vis. DRS) and Scanning electron microscopy (SEM). SEM micrographs shown distinct morphology of Zn2SnO4 in polygonal shape and ZnO particles in tetragonal morphology. The applicability of this prepared composite as a photocatalyst was tested for photodegradation of organic pollutant Methylene Blue (MB) dye. The study shown 91.5% degradation of MB dye in 120min under UV irradiation, this indicates that Zn2SnO4/ZnO composite as an effective photocatalyst.

    (Video) Fluid Dynamics in Liquid Metals (Magnetohydrodynamics) - Rainer Hollerbach
View full text

© 2021 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Futuristic Research in Engineering Smart Materials.


What is Stokes theorem in electromagnetic theory? ›

The Stoke's theorem states that “the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface.” Where, C = A closed curve.

What is the circulation theorem in fluid mechanics? ›

In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869) states: In a barotropic, ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time.

What is the application of Stokes theorem in electromagnetism? ›

Through Stokes' theorem, line integrals can be evaluated using the simplest surface with boundary C. Faraday's law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokes' theorem can be used to derive Faraday's law.

What is the formula for calculating circulation? ›

That circulation is a measure of rotation is demonstrated readily by considering a circular ring of fluid of radius R in solid-body rotation at angular velocity Ω about the z axis angular velocity Ω about the z axis. In this case, U = Ω × R, where R is the distance from the axis of rotation to the ring of fluid.

What is Stokes law used for quizlet? ›

Stokes law can be used to determine the viscosity of a fluid by dropping a spherical object in it and measuring the terminal velocity of the object in that fluid. This can be done by plotting the distance traveled against time and observing when the curve becomes linear.

What is the definition of circulation of fluid? ›

The flow or motion of a fluid in or through a given area or volume. A precise measure of the average flow of fluid along a given closed curve. Mathematically, circulation is the line integral. about the closed curve, where v is the fluid velocity and dr is a vector element of the curve.

What are the three types of flow in fluid mechanics? ›

Key Takeaways. There are three fluid flow regimes: laminar, turbulent, and a transition region. The conditions that lead to each type of flow behavior are system-specific.

What is Stokes theorem requirements? ›

Stokes' Theorem is about tiny spirals of circulation that occurs within a vector field (F). The vector field is on a surface (S) that is piecewise-smooth. Additionally, the surface is bounded by a curve (C). The curve must be simple, closed, and also piecewise-smooth.

What is the circulation of a magnetic field? ›

Magnetic field lines always are closed loops. The fourth of Maxwell's equations is called Ampere's law. It tells us that the circulation of the magnetic field B, namely the integral of the tangential component of B along a closed curve Γ, is proportional to the current flowing through the area enclosed by the curve.

How do you find the circulation flow rate of the working fluid? ›

The flow rate formula is the velocity of the fluid multiplied by the area of the cross-section: Q=v×A Q = v × A .

What is the velocity of circulation? ›

Definition: Velocity of circulation is the amount of units of money circulated in the economy during a given period of time. Description: Velocity of circulation is measured by dividing GDP by the country's total money supply. A high velocity of circulation in a country indicates a high degree of inflation.

What is the physics of circulation? ›

In physics, circulation is the line integral of a vector field around a closed curve. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field.

What is the circulation of the electric field? ›

The circulation of an electric field is proportional to the rate of change of the magnetic field. This is a statement of the Maxwell-Faraday Equation (Section 8.8), which includes as a special case Kirchoff's Voltage Law for electrostatics (Section 5.11).

What is Stokes law simplified? ›

What is Stokes' law? Stokes' law is a law in physics that states that the force that resists a sphere's fall in a viscous fluid is directly proportional to the velocity of the sphere, the radius of the sphere, and the viscosity of the fluid.

What is Stokes law examples? ›

When any object rises or falls through a fluid it will experience a viscous drag (frictional force) due to the fluid. This object can be skydiver falling through air, a stone falling through water or a bubble rising through water.

What is Stokes law easy? ›

1. : a law in physics: the frequency of luminescence excited by radiation does not exceed that of the exciting radiation. : a law in physics: the force required to move a sphere through a given viscous fluid at a low uniform velocity is directly proportional to the velocity and radius of the sphere.

What are the 3 types of circulation and how are they different? ›

Pulmonary veins move oxygen-rich blood from the lungs to the heart's left atrium. Systemic arteries take oxygen-rich blood from the left ventricle to the body's tissues. Systemic veins move blood with low levels of oxygen from the body's tissues to the heart's right atrium.

What are the two circulating fluids? ›

Blood and lymph are the two most important body fluids in the human body. Blood comprises plasma, white blood cells, red blood cells, and platelets. Lymph is a colourless fluid that circulates inside the lymphatic vessels.

What is circulation and why is it important? ›

The circulatory system delivers oxygen and nutrients to cells and takes away wastes. The heart pumps oxygenated and deoxygenated blood on different sides. The types of blood vessels include arteries, capillaries and veins.

What is an example of a fluid flow? ›

A river flowing down a mountain; air passing over a bird's wing; blood moving through a circulatory system; fuel moving through an engine. These are all examples of fluid flow.

What are the types of fluid flows and explain them? ›

The Different Types of Flow
Physiological occurrence
Pulsatile laminar flowIn large arterial vessels due to fluctuations caused by the heartbeat
Oscillatory laminar flowAccepted as a means of turbulence simulation using flow chambers
Turbulent flowRare, during pathophysiological processes
2 more rows

What are the characteristics of fluid flow? ›

The fluid flow characteristics of all reservoirs are governed by a complex relationship between: (1) pore size range and distribution, (2) matrix and fracture permeability, (3) gravity segregation, (4) wettability, (5) pressure and temperature, and (6) gravity.

What are the four conditions of Stokes law? ›

Stokes' law is valid under the following conditions. Particles must be solid, smooth, and spherical. Particles must be of uniform density. Particles must be sufficiently large (>0.001mm) as compared to molecules of fluid so that the thermal (Brownian) motion of the fluid molecules does not affect the particles.

How do you find the circulation around a rectangle? ›

The circulation per unit area is the integral divided by the area of the rectangle, which is ΔxΔy ∫CF⋅dsΔxΔy=F2(a+Δx,b)Δy−F2(a,b)Δy−(F1(a,b+Δy)Δx−F1(a,ΔxΔy. Half of the numerator is multiplied by Δy and half is multiplied byΔx.

What is the formula for velocity in fluid flow? ›

Flow rate and velocity are related by Q = Av where A is the cross-sectional area of the flow and v is its average velocity.

What is the velocity of circulation equal to quizlet? ›

The velocity of circulation is the average speed with which money is loaned to businesses and households.

What does the velocity of circulation refers to? ›

The velocity of circulation is the frequency at which one unit of currency (i.e., a one-Euro coin) is exchanged in an economy within a given time period. If it is rising, then more transactions are taking place between individuals in an economy. It is also known as the velocity of money.

What is a circulation figure? ›

The number of copies of a publication distributed by the publishing company. Circulation figures are available for all major publications. A newspaper's circulation is the amount of copies it distributes on an average day. Circulation is one of the key elements used to determine advertising rates.

How do you find the circulation of a vector field around a circle? ›

Γ=∫C→F⋅ˆTds. Given the vector field →F(→r)=−yˆi+xˆj find circulation over a circle. ˆT=→T|→T|=−Rsin(t)ˆi+Rcos(t)ˆjR=−sin(t)ˆi+cos(t)ˆj.

What is the circulation of a curve? ›

In physics, circulation is the line integral of a vector field around a closed curve. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field.


1. | Part 4 | Mathematical tools for Electrodynamics | Vaisakhan Thampi
(Standard Model)
2. Electromagnetic Stirring Simulation with THERCAST® | G. Puaux, Transvalor
3. Fluid 08 || Stokes Theorem and Terminal Velocity IIT JEE MAINS / NEET ||
(Physics Wallah - Alakh Pandey)
4. The Most Important Reason To Learn Anything
(The Math Sorcerer)
5. Using Vector Calculus to Solve Problems in Electricity and Magnetism, Steven L. Richardson, Lec. 12
(Center for Integrated Quantum Materials)
6. Astrophysical fluid dynamics - James Stone
(Institute for Advanced Study)


Top Articles
Latest Posts
Article information

Author: Kelle Weber

Last Updated: 23/08/2023

Views: 6265

Rating: 4.2 / 5 (73 voted)

Reviews: 88% of readers found this page helpful

Author information

Name: Kelle Weber

Birthday: 2000-08-05

Address: 6796 Juan Square, Markfort, MN 58988

Phone: +8215934114615

Job: Hospitality Director

Hobby: tabletop games, Foreign language learning, Leather crafting, Horseback riding, Swimming, Knapping, Handball

Introduction: My name is Kelle Weber, I am a magnificent, enchanting, fair, joyous, light, determined, joyous person who loves writing and wants to share my knowledge and understanding with you.