## What is meant by Fluid Kinematics?

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid statics) or in motion (posing some velocity). In fluid kinematics, studies focus on the associated motion of fluid particles or groups of particles, and its associated parameters like displacements, velocity, and acceleration without concerning about the forces that caused the motion. All the laws of classical mechanics are applied in the study of fluid kinematics.

From the elementary level of science, a solid is a substance that can resist deformations, that is, a solid can resist shear forces. However, a fluid cannot resist any shear forces, it keeps on deforming continuously under the action of the external shear force. The fluid flow is characterized by the amount of shear force it receives.

The properties of fluids can be recalled as; a fluid that has no definite shape, but it conforms to the shape of the container. A fluid in any substance is capable of flowing. There are various classifications of fluids such as liquid, gas, vapor, ideal fluids, and real fluids.

## The Lagrangian description of fluid flow

There are two distinct ways to describe the flow of fluid, the primary way of description is the Lagrangian description of fluid flow. In the Lagrangian method of fluid flow, an individual fluid particle is chosen and its properties such as the path traced by the particle, its velocity, and acceleration are studied.

In the language of calculus, from a fluid flow field, an individual fluid element is considered and tracked. Various properties such as the velocity field, temperature field, and acceleration field are determined.

According to the Lagrangian motion, a single fluid particle in Cartesian space $\left(x,y,z\right)$ at any time instant, $t$, from its position $\left(a,b,c\right)$ is given by,

$x=f\left(a,b,c,t\right)$

$y=g\left(a,b,c,t\right)$

$z=h\left(a,b,c,t\right)$

## The Eulerian description of fluid flow

In the Eulerian approach, certain space in the fluid system is considered. All the changes in velocity, acceleration and other fluid properties and equations are concluded with respect to that space.

The method of the Eulerian approach is extensively used in fluid kinematics problems. In fluid kinematics, the study of individual motion of fluid particles requires tedious mathematical calculations, hence for simplicity, the Eulerian approach is usually preferred.

The velocities at any point $\left(x,y,z\right)$ are written as,

$u=f\left(x,y,z,t\right)$

$v=g\left(x,y,z,t\right)$

$w=h\left(x,y,z,t\right)$

## Basic concepts in fluid kinematics

Some of the basic concepts in fluid kinematics are summarized below,

### Control volume

A control volume is a particular volume in space, where special interests lie in analyzing problems. Mass and properties such as energy can change within the control volume.

### Boundary condition

The boundary conditions are a set of constraints that are necessary to solve boundary value problems. A differential equation or a set of differential equations is known as a boundary value problem. These equations are to be solved in a domain whose set of constraints or conditions are known.

### Streamline

It is an imaginary line within the fluid flow field such that, a tangent to any point on the fluid flow, indicates the velocity of the fluid at that point. A streamline cannot intersect itself, also two streamlines cannot cross each other. Any fluid mass cannot move across the streamline, it always moves along the streamline. The spacing of a stream line varies inversely with the velocity of the fluid.

### Rotational and irrotational flow

When the fluid particles moving along the direction of the flow, rotate about their mass centers, the fluid flow is known as a rotational flow. And, if the fluid particles do not rotate about their mass centers, the type of flow is termed an irrotational flow.

#### Uniform flow and non-uniform flow

It is a type of flow in which the velocity at any given time does not change with respect to time. If the velocity of a fluid particle at any given time changes with respect to time, then the type of flow is termed as non-uniform flow.

#### One, two, and three-dimensional flows

When the flow parameter, like the velocity, is a function of time, the type of flow is called one-dimensional flow. When the flow parameter, like the velocity, is a function of time and any two of the spatial coordinates, the kind of flow is termed as two-dimensional flows. And, if the velocity parameter is a function of time and three spatial coordinates, then the type of flow is called, three-dimensional flow.

#### Path line

It is the path followed by a fluid particle that is in motion. Through the path line, the direction of a fluid particle can be traced as it moves. It is always a three-dimensional curve in space.

#### Continuity equation

It is also termed a transport equation. This equation generally describes the transport of a quantity. The continuity equations are a form of conservation equations. When a fluid is in motion, its mass always remains conserved.

#### Stream tube

When a fluid mass is bounded by several streamlines, it forms a stream tube. The stream tube contents are termed stream filaments.

## Velocity potential function

Velocity potential function is a scalar function of space and time, where, the negative of the partial derivative with respect to any of the directions gives the fluid velocity magnitude along that direction. It is given by $\varphi$

If $\varphi =\left(x,y,z\right)$

Then, $u=-\frac{\partial \varphi }{\partial x}$

$v=-\frac{\partial \varphi }{\partial y}$

$w=-\frac{\partial \varphi }{\partial z}$

Where, $u,v,w$ are the velocity components along the $x,y,z$ directions.

## Stream function

The stream function is defined as the function of time and space, such that the negative of the partial derivative with respect to any of the directions, gives the velocity component complementary to that direction. It is given by $\psi$.

If, $\psi =f\left(x,y\right)$

Then, $u=-\frac{\partial \psi }{\partial y}$

and, $v=-\frac{\partial \psi }{\partial x}$

## Flow nets

Flow nets are grids obtained by drawing a series of streamlines and equipotential lines. The idea of flow nets is to provide a simple graphical technique for studying two-dimensional irrotational flows. During the case of unavailability of mathematical relations for stream functions and velocity functions, the concept of flow nets becomes useful.

## Context and Application

Knowledge in fluid kinematics is extensively required for the design of hydraulic dams used for the production of electricity. The topic finds its wide application in the oil production and exploration sector.

• Masters in Science (Physics)
• Bachelors in Science (Physics)

## Practice Problems

1. Which of the following notations is used to signify a velocity potential function?
1. $\varphi$
2. $\psi$
3. $\phi$
4. $\omega$

Correct option- a

Explanation: The velocity potential function is usually represented by the notation $\varphi$.

2. Which of the following is true for the Lagrangian method of fluid flow description?

1. A certain point in space is considered and all the relevant properties of the fluid are studied with respect to that point.
2. An individual fluid particle is considered and all the properties related to that particle are studied.
3. A certain path in the flow field is considered and the related properties of the fluid are studied.
4. All of these

Correct option- b

Explanation: The idea behind the Lagrangian method of fluid flow analysis is to consider an individual fluid particle, and study the related properties of the fluid in relation to that individual fluid particle.

3. Which of the following option is true for a streamline?

1. Flow can occur across the streamline
2. No flow can occur across a streamline
3. A tangent to any point on the streamline gives the acceleration of the particle along that direction.
4. None of these

Correct option- b

Explanation: Streamlines are the combination of the fluid flow field, and no flow can occur across a streamline.

4. Which of the following is true for the continuity equation?

1. It is an equation of conservation of momentum
2. It is an equation of conservation of mass
3. It is an equation of conservation of energy
4. It is an equation of mass-momentum conservation

Correct option- b

Explanation: Continuity equations are also known as transport equations. The equations relate to some quantity to be conserved. The equation is generally used in accordance with the conservation of mass.

5. Which of the following is true for an irrotational flow?

1. The fluid particles move along with the flow field
2. The fluid particles along with their motion rotate about their mass centers
3. The fluid particles do not rotate about their mass centers
4. The fluid particles rotate about a specific point in space

Correct option- c

Explanations: Rotational flow is that flow while undergoing motion along the flow field, the particles do not rotate about their mass centers.

• Equipotential lines

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