Kinematics of Fluid Flow: Notes, Methods, Problems and Solutions! This article will help you to get the probable answers for the questions related to Kinematics of Fluid Flow.
Kinematics of fluid flow deals with the motion of fluid particles without considering the agency producing the motion. This deals with the geometry of motion of fluid particles. This also deals with the velocity and acceleration of fluid particles in motion. The motion of a fluid can be analysed on the same principles as those applied in the motion of a solid.
There, however exists a basic difference between the motion of a solid and the motion of a fluid. A solid body is compact and moves as one mass. There is no relative motion between the particles of a solid body. Hence, we study the motion of the entire body and there is no necessity to study the motion of any particle of a solid body.
But in the case of a fluid body, the fluid particles are all separately mobile and have motions independently. A fluid particle may have a motion different from those surrounding it. However, it may be possible to obtain a relationship between the motions of neighbouring fluid particles.
Kinematics of Fluid Flow: Notes, Methods, Problems and Solutions
We know that each particle of a fluid in motion has at any instant a certain definite value of its properties like density, velocity, acceleration etc. As the fluid moves on, the values of these properties will change from one position to other positions, from time to time.
Thus, it may be realized that two methods are possible to describe fluid motion. In the first method called the Lagrangian method, we study the velocity, acceleration etc. of an individual fluid particle at every instant of time as the particle moves to different positions.
This method of studying the properties of a single fluid particle is a very tedious process and therefore this method is not generally adopted. In the second method called the Eulerian method, we describe the flow by studying the velocity, acceleration, pressure, density etc. at a fixed point in space. Due to its easy application, this method is most commonly adopted.
Let x, y and z denote the space coordinates and t the time. Let V be the resultant velocity at any point in space in a fluid body. Let u, v and w be the components of the resultant velocity V at any point in the directions of the x, y and z axes. Fig. 6.1 illustrates the notations.
A stream line is a continuous line in a fluid which shows the direction of the velocity of the fluid at each point along the line. The tangent to the stream line at any point on it is in the direction of the velocity at that point. Fluid particles lying on a stream line at an instant move along the stream line.
When a fluid is in motion there are many stream lines and these stream lines indicate the flow pattern at that particular instant. For example, as a fluid flows round a cylindrical body, the stream line pattern will be as shown in Fig. 6.3. In steady flow the velocity at a point does not change in its magnitude and direction.
Hence, there is no change in the direction of the velocity vector at a point. In other words, the stream line is fixed in position. Conversely, if the stream line pattern remains constant the flow is steady. In the case of an unsteady flow, the direction of the velocity changes with time at every point. This means the position of a stream line is not constant. The position of a stream line changes from instant to instant.
A path line means the path or a line actually described by a single fluid particle as it moves during a period of time. The path line indicates the direction of the velocity of the same fluid particle at successive instants of time.
In the case of a steady flow since there are no fluctuations of the velocity, the path line coincides with the stream line. In the case of an unsteady flow the stream lines change their positions at every instant and thus the path line may fluctuate between different stream lines during an interval of time.
Fig. 6.4 shows the path line of a particular fluid particle. It is the locus of the positions of the same particle as it moves.
Fig. 6.5 shows the path lines described by three particles which had passed through the origin at times t0, t1 and t2. Consider the particle which passed through the origin at time to. Fig. 6.5 shows the positions taken by the particle at times t0 + ∆t, t0 +2∆t, t0+3∆t etc. as it has traced its path line.
The streak line is the locus of the positions of fluid particles which have passed through a given point in succession. Suppose A, B, C, D… are fluid particles which passed through a reference point say the origin one after the other in succession. These particles have described their own path lines. Suppose at a time t, these particles A, B, C, D… are at Pa,
Pb, Pc, Pd …. The line Pa, Pb, Pc, Pd …. is the streak line, at time t.
A stream tube is an imaginary tubular space formed by a number of stream lines. It is an assembly or collection of stream lines forming a tabular space. The surface of a stream tube is made up of stream lines. The velocity of a fluid particle on the surface of the stream tube is along the stream line over the surface of the stream tube. This means there can be no flow across the walls of a stream tube.
At any section of the stream tube the velocity at the centroid of the section represents the average velocity of flow through the stream tube at that section. Fig. 6.7 shows a stream tube. Since there can be no flow across the walls of a stream tube, it should be clear that quantity of fluid entering a stream tube is equal to the quantity of the fluid leaving the stream tube.
On a surface consisting of stream lines, we can imagine lines running orthogonally with the stream lines. Such line are called potential lines, See Fig. 6.8.
A set of stream lines and potential lines constitutes a flow net.
We come across the following types of flow:
(i) Laminar flow and turbulent flow.
(ii) Steady flow and unsteady flow.
(iii) Uniform flow and non-uniform flow.
(iv) Rotational and irrotational flow.
This is a type of flow in which the fluid particles move in layers, gilding smoothly over adjacent layers. There is no transportation of fluid particles from one layer to another. The fluid particles in any layer move along well defined stream lines.
The paths of the individual particles do not cross each other. This type of flow is also called stream line flow or viscous flow. This is a smooth flow of one layer of fluid over another. This type of flow occurs in viscous fluids where viscosity influences the flow.
This is the most common type of flow that occurs in nature. This flow is characterised by random, erratic, unpredictable motion of fluid particles which result in eddy currents. There is a general mixing up of fluid particles, in motion. The velocity changes in direction and magnitude from point to point.
There is a continuous collision between particles resulting in transference of momentum between them. The eddy currents cause a considerable loss of energy compared to the loss of energy in laminar flow. This greater loss of energy is due to the fact that turbulent shear stresses are very much greater than laminar shear stress given by Newton’s law of viscosity.
In a turbulent flow the distinguished characteristic of turbulence is its irregularity, indefinite frequency and no definite observable pattern. This type of flow cannot be truly mathematically analysed and any analysis is possible by statistical evaluation. Flow of water in rivers is generally turbulent. Flow of water in pipes at high velocity is turbulent. Flow of thick oil in narrow tubes, flow of ground water, flow of blood in blood vessels are laminar.
As the velocity of water in a pipe is gradually increased the flow will change from laminar to turbulent flow. The velocity at which the flow changes from laminar to turbulent flow in a pipe is called the critical velocity. The type of flow that exists in any case depends upon the value of a non-dimensional number dν/γ called the Reynolds’s number, where d is the diameter of the pipe, ν is the mean velocity of flow in the pipe and γ is the kinematic viscosity of the fluid.
When the Reynolds number is less than 2000, the flow is generally laminar. When the Reynolds number is greater than 2800, the flow is generally turbulent. If the Reynolds number lies between the above limits the flow may be either laminar or turbulent. Thus the critical velocity has no fixed or definite value.
The velocity corresponding to Reynolds number equal to 2000, is called the lower critical velocity and the velocity corresponding to Reynolds number equal to 2800 is called the upper critical velocity.
If the flow characteristics like, velocity, density, pressure etc. at a given point in a flowing mass of fluid does not change with the passage of time, the flow is said to be steady. On the contrary, if these flow characteristics at a given point change with respect to time, the flow is said to be unsteady flow.
Since velocity is a commonly adopted characteristic of flow it is quite sufficient to regard the flow to be steady if the velocity at a given point does not change with respect to time.
Suppose V is the velocity at a point (x1, y1, z1). At this point if V remains constant at all times the flow is a steady flow. But, if at this point, V changes with time the flow is unsteady flow i.e.,
(iii) Uniform and Non-Uniform Flow:
If the flow characteristics like velocity, density, pressure etc. at a given instant remain the same at all points, the flow is uniform. If V is chosen as a flow characteristic, then, at a given instant V has the same value at all points and is independent of the space position. If the flow characteristics have different values at different points at a given instant of time, the flow is non-uniform flow.
We use the terms uniform and non-uniform flow often in connection with open channels. In a channel where the section of the channel is uniform and the depth of flow is uniform the flow will be uniform as the velocity will be the same at all sections. But if the sectional dimensions of the channel are different at different sections, the depths of flow will be different at different sections.
Obviously, the velocity will be different at different sections and the flow will be non-uniform, whether the flow is uniform or non-uniform if the rate of flow is constant the flow is steady and if the rate of flow changes with time, the flow is unsteady. Thus, we may come across steady or unsteady or uniform or non-uniform flow. Any type of flow can exist independently of the other.
A combination of two types of flow is also possible.
Some combinations are:
a. Steady uniform flow.
b. Steady non-uniform flow.
c. Unsteady uniform flow.
d. Unsteady non-uniform flow.
As a fluid moves the fluid particles may be subjected to translatory or rotatory displacements. Suppose a particle which is moving along a stream line rotates about its own axis also then the particle is said to have a rotational motion. Whereas if the particle as it moves along the stream line does not rotate about its own axis the particle is said to have irrotational motion.
Fig. 6.11 (a) shows a rotational motion. Consider the fluid particle AB. As this particle moves along the stream line it rotates about its own axis also. Fig 6.11 (b) shows an irrotational motion. The fluid particle AB in this case, as it moves along the stream line, does not rotate about its own axis.
A fluid element may undergo four types of movements, namely,
(i) A pure translation,
(ii) A linear deformation,
(iii) A pure rotation,
(iv) An angular (shearing) deformation
(i) Pure Translation:
Fig. 6.12 shows a movement of a fluid element by pure translation.
The solid line figure represents a very small fluid element at a certain instant. After a pure translation the position of the fluid element is shown is dotted lines. During this displacement the dimensions of the fluid element do not change and they continue to retain their original orientations.
(ii) Linear Deformation:
Fig. 6.13 shows a movement of a fluid element by linear deformation.
In this case the fluid element has changed its shape. But the directions of the principal axes of the fluid element have not changed.
(iii) Pure Rotation:
Fig. 6.14 (a) and (b) show pure rotation of a fluid element.
The fluid element has rotated by an angle θ without any deformation.
(iv) Angular Deformation or Shearing Deformation:
This is another way of describing fluid motion. The velocity of a fluid element in the most general case is dependent upon its position. If any point in space be defined in terms of the space coordinates (x, y, z), then at any given instant the velocity at the point is given by V = ƒ (x, y, z). The flow in such a case is called a three-dimensional flow.
Sometimes, the flow condition may be such that the velocity at any point depends only on two space coordinates say (x, y) at a given instant, i.e., in this case, at the given instant V = ƒ (x, y). In this case the flow conditions are potential in planes normal to the Z-axis. This type of flow is called a two-dimensional flow.
In a two-dimensional flow, the flow is identical in parallel planes. Fig. 6.16 shows a two-dimensional flow. In this figure is shown a channel whose walls are perpendicular to the plane of the diagram. Note the velocity vectors at sections 1-1 and 2-2. At section 1-1 the velocity varies across the channel. Similarly at section 2-2 the velocity varies across the channel. But the flow is identical in all planes parallel to the plane of the figure.
There is no component of the velocity perpendicular to the plane of the figure. In quite a number of cases it is usual to consider the motion as one-dimensional. This is no doubt a simplification over the two- dimensional and three-dimensional fluid motions. In this type of flow the velocity V at a given instant is a function of one space coordinate say x only i.e., at a given instant, V = ƒ (x)
Fig. 6.17 shows a one-dimensional flow. At section 1-1 the velocity is constant over the entire section. Similarly at section 2-2 the velocity is constant over the entire section.
A one-dimensional or two-dimensional or three-dimensional flow may be a steady flow or an unsteady flow.
These types of flow may be expressed as follows:
Rate of Flow or Discharge:
The quantity of the fluid flowing per unit of time across any section of a pipe or conduit is called the rate of flow or discharge. The rate of flow may be expressed as the weight of the fluid flowing per second across a section or the mass of the fluid flowing across the section per second or the volume of the fluid flowing across the section per second. Accordingly, the rate of flow may be expressed in such units like (i) N per second, (ii) kg per second, (iii) cubic metre per second (iv) litres per second.
In this case of flow of incompressible fluids the rate of flow is usually expressed as the volume of the fluid flowing across a section per second. In the case of compressible fluids the rate of flow is usually expressed as the weight of the fluid flowing across the section per second.
This is a certain well defined extent of space. For the purpose of understanding the changes that take place in the fluid characteristics we may introduce a control volume so that we may compare the flow characteristics of a fluid just before it enters the control volume and just after it leaves the control volume.
This is an equation based on the principle of conservation of mass. Suppose we consider a stream tube. Since the stream tube is always full of the fluid, the quantity of the fluid entering the stream tube at one end per unit of time should be equal to the quantity of the fluid leaving the stream tube at the other end per unit of time.
Let V be the average velocity at any section and A the area of the section. If w be the specific weight of the fluid, the quantity of the fluid flowing per second across the section
Consider an infinitesimal parallelopiped of space in a fluid body. Let the sides of parallelopiped have length dx, dy and dz respectively. See Fig. 6.20. Let u, v and w be the inlet velocity components in the directions of the X, Y and Z axes. Mass of the fluid entering the left face = ρu dy dz. Mass of the fluid leaving the right face
In the most general case of fluid motion, the resultant velocity V at any point is a function of not only the displacement s along a stream line but also the time t, that is-
Velocity potential function:
This is a function which is devised to expedite the analytical study of velocity fields. If ɸ is some function of the coordinates x and y in a two-dimensional flow, such that
Let us consider a two-dimensional steady incompressible flow. Fig. 6.26 shows two stream lines PP’ and QQ’ close to each other.
We know a stream line is a line of constant stream function. Let the coordinates of P and Q be (x, y) and [(x + dx), (y + dy)].
Consider the flow across any line from P to Q say a line PRQ. If the flow is steady, the flow across the line PRQ or any such line should be the same. Consider the flow towards the positive direction of X as positive and the flow towards the positive direction of Y as positive, we have, the flow across the line PRQ = – udy + vdx
Hence dΨ numerically represents the flow across the line PRQ, i.e., dΨ represents the flow between the stream lines PP’ and QQ’. Thus, we conclude that the flow across a line joining two points is equal to the change in the stream function between these two points.
Consider a closed line or contour in a two-dimensional flow. Let V represent the resultant velocity at any point on the contour. Let θ be the angle between the velocity V and the elemental contour element ds. The integral of the product (contour length ds x the component of the velocity in the direction of ds) is called the line integral. The line integral of the velocity around a closed contour is called circulation (usually denoted by γ).