1. Fluid kinematics refers to features of a fluid in motion.
Consideration of velocity, acceleration, flow rate, nature of flow and flow visualisation are taken up in fluid kinematics.

2. There are two ways to analyse fluid motion:
Lagrangian method, and 2) Eulerian method
In the Lagrangian method, a particle or an element of fluid is identified and followed during the course of motion with time, as shown in Fig. 1.
The identified lump of fluid may change in shape, size and state as it moves.
The laws of Mechanics must be applied to it at all times.
Difficulty in tracing the lump of fluid rules out the possibility of applying the Lagrangian approach
In the Eulerian method, the fluid is observed by setting up fixed stations or observatories in the flow field. Motion of the fluid is specified by velocity components expressed as functions of space and time,

3. In the Eulerian approach, the fluid motion at all points in the flow field is determined by applying the laws of mechanics at all fixed stations.
This is considerably easier than the Lagrangian approach and is followed in the study of Fluid Mechanics.

4. Dimensions of Flow
A fluid flow is said to be one, two, or three dimensional depending upon the number of independent space coordinates, i.e. one, two, or three respectively, required to describe the flow.
One dimensional Flow: When the dependent variables are functions of only one space coordinate, say x, it is one dimensional flow.
Examples: Flow through pipes, channels and variable area ducts if the velocity distribution is considered constant at each cross section.
Dependent variables such as velocity, pressure, density, temperature vary only with x, the only independent variable in one dimensional flow as shown in Fig. 2.
The axis of the passage does not have to be a straight line for a flow to be one dimensional. For example, the flow shown in Fig. 2(b) is one dimensional if s, the streamwise direction is chosen as the independent coordinate.

5. The axis of the passage does not have to be a straight line for a flow to be one dimensional. For example, the flow shown in Fig. 2(b) is one dimensional if s, the streamwise direction is chosen as the independent coordinate.

7. Two dimensional Flow:
When the dependent variables in a fluid flow vary with only two space coordinates, the flow is said to be two dimensional. The flow does not vary along the third coordinate direction
Example: The flow around a circular cylinder of infinite length (as shown in Fig. 2c) is two dimensional in the x-y plane.
Axisymmetric Flow: A flow is said to be axisymmetric if the velocity profile is symmetrical about the axis of symmetry. In other words, the velocity profile is the same at different diametral planes drawn through the passage.
Example: Velocity profiles at two locations for an axisymmetric flow through a conical passage are shown in Fig. 2(d).

8. Steady Flow: If the dependent fluid variables at any point in the flow do not change with time, the flow is said to be steady flow. Thus
It follows that
Unsteady Flow: If the dependent fluid variables change with the passage of time at a position in the flow, the flow is called unsteady flow. Thus

9. Steadiness of flow means that the flow-pattern does not change with time whereas unsteadiness refers to changing flow-pattern with the passage of time at the same point in space.
Uniform Flow: If the velocity, in magnitude, direction and sense is identical throughout the flow field, the flow is said to be uniform flow. This requires the velocity components to be the same at different positions in the flow.
In other words, the space rate of change of velocity components at that time must vanish.
It follows that
for uniform flow.

10. It is conventional to define the term “uniform flow” only in terms of the velocity components rather than in terms of other dependent fluid variables.
Further, a flow may be considered uniform over the cross sections although it may not be uniform longitudinally.
Non-uniform Flow: If the velocity components at different locations are different at the same instant of time, the flow is said to be non-uniform.
Note: Steadiness refers to ‘no change with time’ and uniformity refers to ‘no change in space’. Therefore, a flow can be steady or unsteady quite independent of its being uniform or non-uniform. All the four combinations are possible.

11. Acceleration in Fluid Flow: The velocity components in a fluid flow are, in general, functions of space and time
From differential calculus, an infinitesimal change in u is given by
And the acceleration components are given by

13. Total acceleration = Convective acceleration + Local acceleration
Total acceleration is also called as substantial, material or particle derivative.
For steady flow: Only Local acceleration components are zero.
For uniform flow: Only Convective acceleration components vanish.
For steady uniform flow: Total acceleration becomes zero and the flow is non-accelerating in space and time.
Expressions for the components of acceleration in cylindrical coordinates for an axisymmetric flow are given below:

16. dy/dx = tan θ = v/u or u dy – v dx = 0
Streamlines, Pathlines and Streaklines
A streamline in a fluid flow is a line tangent to which at any point is in the direction of velocity at that point at that instant.
Streamlines are, therefore, equivalent to an instantaneous snap-shot indicating the directions of velocity in the entire flow field as shown in Fig. 4(a) and (b).
There can be no flow across a streamline since the component of velocity normal to a streamline is zero.
A streamline cannot intersect itself nor can any streamline intersect another streamline.
In steady flow, the same streamline-pattern should hold at all times, since the velocity vector cannot change with time.
Consider, first a streamline in a plane flow in the x-y plane as shown in Fig. 3. By definition, the velocity vector U at a point P must be tangential to the streamline at that point. It follows that
dy/dx = tan θ = v/u or u dy – v dx = 0
where u and v are the velocity components along x and y directions respectively.

17. It may also be noticed that the velocity vector is expressed as
U = U (s, t)
which shows that the velocity may vary along a streamline direction s as well as with the passage of time.
Consider next an elementary displaced element δs along a general streamline where the velocity is U such that
U = u i+ v j + w k
and δs = δx i+ δy j + δz k

18. whence δx/u = δy/v = δz/w
By the definition of a streamline, U must be directed along δs. For collinear vectors, the cross product must vanish.
Hence, U x δs = 0
Or, (vδz – wδy) i– (uδz – wδx) j + (uδy – vδx) k = 0
whence δx/u = δy/v = δz/w
which is the equation of a streamline in a three dimensional flow, steady or unsteady, uniform or non-uniform, viscous 0r inviscid, compressible or incompressible.

19. A strteam surface is generated by a large number of closely spaced streamlines which pass through an arbitrary curve AB as in Fig. 4(c).
A stream surface can be plane, axisymmetric or spatial.
If the arbitrary line constitutes a closed curve, e.g., ABCD as in Fig. 4(d), the closely spaced streamlines passing through the curve constitute a tubular stream surface called a stream tube.
No flow can penetrate a stream surface or a stream tube.

21. A pathline in a fluid flow is the trajectory of a fluid particle, say P1 as it advances with the passage of time, say from ti to final time tf as in Fig. 4(e).
Pathlines are, therefore, history lines of individual fluid particles over a period of time.
Tangent at a point on a pathline must be in the direction of velocity at that point at the time when the particle passes that point.
A pathline, therefore, intersect itself at different times and in general pathlines may constitute a pattern of intersecting zig-zag lines.
In steady flow, the streamlines and pathlines coincide.
A streakline is the locus of locations, Fig. 4(f), at an instant of time, of all the fluid particles that have passed through a fixed point in a flow field.
A little consideration will show that the streamlines, pathlines and streaklines are coincident in a steady flow.

22. Law of Conservation of mass: Conservation of mass or Continuity Equation
Consider a control volume within the boundary layer as shown in Fig. 7(b) and assume that steady state conditions prevail.
There are no gradients in the z direction and the fluid is incompressible.
Rate of mass flow into the control volume in the x-direction
Rate of mass flow out of the control volume in the x-direction
Rate of mass flow into the C.V. in the y-direction
Rate of mass flow out of the C.V. in the y-direc.ion 7 7

23. From Fig. 7, the net mass flow into the element in the x direction is,
Similarly, the net mass flow into the control volume in the y direction is,
Since the net mass flow rate out of the control volume must be zero, we obtain
From which it follows that in two-dimensional steady flow, conservation of mass requires that
(13)

24. Example:
Does a velocity field given by
U = 5 x3 i – 15 x2 y j +t k
Represent a possible fluid motion of an incompressible fluid ?
Two components of velocity in an incompressible fluid flow are given by
u = x3 – y3 and v = z3 – y3
Determine the third component, assuming that the origin is a stagnation point.
Do the following velocity components represent a possible flow ? Discuss the possibilities.
u = 2 x + 3 y and v = 3 y – 4 x.

25. Kinematics of a Fluid element
A fluid element may move in a flow and undergo
Pure or irrotational translation
Pure rotation or rotational translation
Pure distortion or deformation; angular or linear.
These are illustrated in Fig. 9 (a), (b) and (c) respectively by taking an upright cubical fluid element to start with in each case.

26. Irrotational Flow
For irrotational flow, each component of rotation or of vorticity (rotation = ½ vorticity) must vanish, i.e., ωx = 0 = ωy = ωz.
In particular, for plane flow in x-y plane,
δu/δy - δv/δx = 0
is the condition of irrotationality.

27. EULER’S EQUATION OF MOTION ALONG A STREAMLINE
To derive a relation between velocity, pressure, elevation and density along a streamline.
Fig. shows a short section of a streamtube surrounding the streamline and having a small cross sectional area so that velocity be considered constant over the cross section.
AB and CD are two cross sections separated by a short distance δs.
At AB, the area is A, velocity v, pressure p and elevation z, while at CD, the corresponding values are, A+ δA, v+ δv, p+ δp and z+ δz.
The surrounding fluid will exert a pressure pside on the sides of the element.
The fluid is assumed to be inviscid. Thus, there will be no shear stresses on the sides of the element and pside will act normally.
The weight of the element mg will act vertically downward at an angle θ to the streamline.
Mass per unit time flowing = ρAv
Rate of increase of momentum from AB to CD = ρAv[(v+ δv) – v)] = ρAv δv

28. The forces acting to produce this increase of momentum in the direction of motion are:
Force due to p in direction of motion = pA
Force due to p+ δp opposing motion = (p+ δp)(A+ δA)
Force due to pside producing a component
In the direction of motion = pside δA
Force due to mg producing a component
Opposing motion = mgcosθ
Resultant force in the direction of motion pA – (P+ δp)(A+ δA)+pside δA-mgcosθ
The value of pside will vary from p at AB to p+ δp at CD and can be taken up as p+k δp, where k is a fraction.
Weight of element, mg = ρg*Volume = ρg(A+½ δA) δs,
cosθ = δz/ δs
Resultant force in the direction of motion = -p δA – A δ p - δp δA + p δA + k δp. δA – ρg(A+½ δA) δs.(δz/ δs)

29. Neglecting products of small quantities,
Resultant force in the direction of motion = -A δp - ρgA δz
Applying Newton’s second law of motion, we get,
ρAv δv = -A δp - ρgA δz
Dividing by ρAδs, we get
Or in the limit as δs 0,
(A)
This is known as Euler’s Equation, giving, in differential form, the relationship between pressure p, velocity v, density ρ and elevation z along a streamline for steady flow.

30. It cannot be integrated until the relationship between density ρ and pressure p is known.
For an incompressible flow, for which ρ is constant, integration of the above equation (A) along the streamline, w.r.t. to s, gives
In which the terms represent the energy per unit weight. The above equation is known as Bernoulli’s Equation and states the relationship between pressure, velocity and elevation for steady flow of a frictionless fluid of constant density

31. An alternative form is:
In which the terms represent the energy per unit volume.
These equations apply to a single streamline. The sum of the three terms is constant along any streamline, but the value of the constant may be different for different streamlines in a given stream.
If the earlier integration of Equation (A) is carried along the streamline between any two points indicated by suffixes 1 and 2, then
For a compressible fluid, the integration of Equation (A) can only be partially completed, to give

32. Euler’s Equation of Motion
Let us obtain the Euler’s momentum equation for two dimensional inviscid steady flow in a vertical plane with reference to an elementary fluid control volume in Figure below. The external forces acting on an element are classified as surface forces and body forces.
We use the term surface force = A force expressible
as a stress multiplied by the area of a surface
e.g. pressure force.
We use the term body force = A force proportional
to the volume of the fluid element e.g. gravity force.
With reference to the figure, the net surface force
In the x-direction = =
There is no body force in the x-direction. δx A B D
p + (∂p/∂z)δz p
p + (∂p/∂x)δx δz C CG a x

33. Similarly the net surface force plus the body force in the z-direction =
Recalling the acceleration components for steady flow in the x-direction =
And the acceleration component for steady flow in the z-direction =
The Newton’s second law in the differential form can be written as
Therefore,
along the x-axis (1)
along the z-axis (2)

34. Finally, for the two-dimensional steady flow of an inviscid fluid in a vertical plane, the Euler’s momentum equations are:
(3)
And
(4)
Case (a): Along a streamline in an inviscid and steady flow
Multiply each term of equation (3) by and each term of equation (4) by and adding them: =
The right hand side term of the above equation can be simplified as follows:
Using the equation of a streamline in the x-z plane:

35. The L.H.S. of equation (5) becomes:
Thus, equation (5) becomes:
Case (b) Between any two points in a Potential/Irrotational and steady flow
Using the equation of irrotationality in the x-z plane,

36. The left hand side of equation (5) becomes:
Which is precisely the same as for the case (a).
Now, for either case (a) or case (b), it can be written as
The assumption of two-dimensionality, as above, in case (a) or case (b) is not necessary. It was made merely for the sake of reducing the number of terms for convenience.
Integration of the Euler’s equation results in the familiar form of the BERNOULLI equation.

37. Or, on rearranging the terms and then integrating, we get:
Constant
This is the general form of the Bernoulli equation, which may now be simplified for different cases.
For an incompressible fluid flow, ρ = constant,
(ii) For a compressible fluid flow, undergoing an adiabatic process,
And

38. The Bernoulli equation appears as:
The assumptions made in the derivation of the Bernoulli equation may now be summarized as follows:
Inviscid flow
Steady flow
Either the flow is along a streamline or the flow is irrotational
Two-dimensional flow, in the presence of gravitational forces.
For incompressible flow, the form is constant
For the compressible flow, the form is
The Bernoulli equation may, therefore, be applied where these assumptions can be justified.

39. Energy Analysis of Steady Flows
Where,hpump,u is the useful head delivered to the fluid by the pump.
Where is Shaft Power input through the pump’s shaft.

40. Similarly hturbine,e is the extracted head removed from the fluid by the turbine. Also,
Where, is Shaft Power output through the turbine’s shaft. hL is the irreversible head loss between 1 and 2 due to all components of the piping system other than pump or turbine.

41. Problem 1: The pump of Figure 1 is to increase the pressure of water from 200 kPa to 600 kPa. If the pump efficiency is 85%, how much electrical power will the pump require? The exit area is 20 cm above the inlet area. Assume inlet and exit areas are equal.
Problem 2: The venturimeter shown reduces the pipe diameter from 10 cm to a minimum of 5 cm (Fig. 2). Calculate the flow rate and the mass flux assuming ideal conditions.