Materials Today: Proceedings

Volume 47, Part 14,

2021

, Pages 4206-4209

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## Abstract

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.

## Introduction

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],$\underset{s}{\iint}(\mathrm{\nabla}\times A)\xb7ds={\int}_{L}A\xb7dl$

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

## Conclusions

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.

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

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## FAQs

### 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? ›

Physiological occurrence | |
---|---|

Pulsatile laminar flow | In large arterial vessels due to fluctuations caused by the heartbeat |

Oscillatory laminar flow | Accepted as a means of turbulence simulation using flow chambers |

Turbulent flow | Rare, during pathophysiological processes |

**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.