1 of 93

Fluid Mechanics and Machinery

Md. Mohiuddin

Lecturer

Department of Mechanical Engineering

ME 3219

Fluid Dynamics

3 of 93

Viscosity

Rate of deformation or rate of shear strain

Thus, the rate of deformation of a fluid element is equivalent to the velocity gradient
From Newton’s law of viscosity shear stress on a fluid element layer is directly proportional to the rate of shear strain

4 of 93

CLASSIFICATION OF FLUID FLOWS

Viscous versus Inviscid
Internal versus External Flow
Compressible versus Incompressible Flow
Laminar versus Turbulent Flow
Steady versus Unsteady Flow
Uniform versus nonuniform flow
One-, Two-, and Three-Dimensional Flows
Rotational versus Irrotational Flows

5 of 93

Viscous versus Inviscid Regions of Flow

When two fluid layers move relative to each other, a friction force develops between them and the slower layer tries to slow down the faster layer.
This internal resistance to flow is quantified by the fluid property viscosity, which is a measure of internal stickiness of the fluid.
There is no fluid with zero viscosity, and thus all fluid flows involve viscous effects to some degree.
Flows in which the frictional effects are significant are called viscous flows.
However, in many flows of practical interest, there are regions (typically regions not close to solid surfaces) where viscous forces are negligibly small compared to inertial or pressure forces.
In these regions, we can ignore the effects of viscosity, which makes the analysis much simpler without sacrificing much accuracy.
The region of the flow field where the effect of viscosity is neglected, the flow is called inviscid flow.

6 of 93

Internal versus External Flow

External flow: occurs when a fluid moves over an unbounded surface like a plate, wire, or pipe. For example, when wind blows over a ball or a pipe, it's external flow.
Internal flow: happens when the fluid is completely surrounded by solid surfaces, like water flowing inside a pipe or duct.
Open channel flow: The flow of liquids in a duct is called open-channel flow if the duct is only partially filled with the liquid and there is a free surface.
Internal flows are mainly influenced by viscosity throughout the flow area.
In external flows, viscous effects are only significant close to solid surfaces and in regions behind objects in the flow, called wake regions.

7 of 93

Compressible versus Incompressible Flow

The flow is said to be incompressible if the density remains nearly constant throughout.
The fluid does not experience significant changes in density with variations in pressure or temperature.
Incompressible flow is typically applicable to liquids, as they generally have negligible compressibility.
A pressure of 210 atm, for example, causes the density of liquid water at 1 atm to change by just 1 percent.

Incompressible Flow

8 of 93

Compressible versus Incompressible Flow

Compressible Flow

9 of 93

Laminar versus Turbulent Flow

A flow is said to be laminar when the various fluid particles move in layers (or laminae) with one layer of fluid sliding smoothly over an adjacent layer.
The flow of high-viscosity fluids such as oils at low velocities is typically laminar.

Laminar Flow

A fluid motion is said to be turbulent when the fluid particles move in an entirely haphazard or disorderly manner, that results in a rapid and continuous mixing of the fluid leading to momentum transfer as flow occurs.
The flow of low-viscosity fluids such as air at high velocities is typically turbulent.

Turbulent Flow

10 of 93

Laminar versus Turbulent Flow

11 of 93

Steady versus Unsteady Flow

Fluid flow is said to be steady if at any point in the flowing fluid various characteristics such as velocity, pressure, density, temperature, etc., which describe the behavior of the fluid in motion, do not change with time.
During steady flow, the fluid properties can change from point to point within a device, but at any fixed point they remain constant.

Steady Flow

Unsteady Flow

Fluid flow is said to be unsteady if at any point in the flowing fluid any one or all the characteristics which describe the behavior of the fluid in motion change with time.

12 of 93

Steady versus Unsteady Flow

13 of 93

Steady versus Unsteady Flow

(a) is an instantaneous image, while photo (b) is a long-exposure (time-averaged) image

One of the most important jobs of an engineer is to determine whether it is sufficient to study only the time-averaged “steady” flow features of a problem, or whether a more detailed study of the unsteady features is required.

14 of 93

Uniform versus non-uniform flow

When the velocity of flow of fluid does not change, both in magnitude and direction, from point to point in the flowing fluid, for any given instant of time, the flow is said to be uniform.
The term uniform implies no change with location over a specified region.

Uniform Flow

Non-Uniform Flow

If the velocity of the flow of fluid changes from point to point in the flowing fluid at any instant, the flow is said to be non-uniform.

15 of 93

One-, Two-, and Three-Dimensional Flows

The flow is two-dimensional in the entrance region and becomes one-dimensional downstream when the velocity profile fully develops and remains unchanged in the flow direction

16 of 93

SYSTEM AND CONTROL VOLUME

A system is a collection of matter of fixed identity, which may move, flow, and interact with its surroundings.
The real or imaginary surface that separates the system from its surroundings is called the boundary
it always contains the same mass, no mass can cross its boundary.
It may continually change size and shape.

System

The figure is showing piston–cylinder device which is a system that consists of a fixed mass of gas.
Notice that energy may cross the boundary, and part of the boundary may move.

17 of 93

SYSTEM AND CONTROL VOLUME

Control volume is a selected region in space.
Any arbitrary region in space can be selected as a control volume.
Both mass and energy can cross the boundary (the control surface) of a control volume.
The matter within a control volume may change with time as the fluid flows through it. Similarly, the amount of mass within the volume may change with time.
A control volume is separated from the surrounding by the control surface.
In control volume approach the fluid flow is analyzed within, through, or around that volume.
Heat, work, and mass can cross the CS.
CS may be real or imaginary.
CS may be at rest or in motion

Control Volume

Control surface (CS)

18 of 93

SYSTEM AND CONTROL VOLUME

Why do we use a control volume approach more often instead of a system approach in fluid mechanics?

One of the important concepts used in the study of statics and dynamics is that of the free body diagram. That is, we identify an object, isolate it from its surroundings, replace its surroundings by the equivalent actions that they put on the object, and apply Newton’s laws of motion. The body in such cases is our system—an identified portion of matter.
In fluid mechanics, it is often quite difficult to identify and keep track of a specific quantity of matter. Imagine you have a small amount of liquid, like water or oil. Inside this tiny amount of liquid, there are a huge number of tiny particles moving around. These particles move very freely, unlike the particles in a solid that usually stay together and are easier to recognize.
We care more about the forces on a fan, airplane, or car caused by the airflow than following individual air particles as they move.
For these situations we often use the control volume approach and limit our focus on a volume associated with the fan, airplane, or automobile, for example.

19 of 93

Lagrangian Description

Lagrangian analysis is analogous to the (closed) system analysis where we follow a mass of fixed identity.
The Lagrangian description requires us to track the position and velocity of each individual fluid particle
In the Lagrangian method, we choose a single fluid particle and follow it as it moves through space.
We pay close attention to how this particle behaves during its journey.
This helps us to understand the overall motion of the fluid and how it changes over time.
It's like tracking the path of one specific water droplet in a flowing river to understand how the entire river flows.

20 of 93

Eulerian Description

Instead of following individual particles, we look at a specific region in space, called a "control volume," where the fluid flows in and out.
From this method we obtain information about the flow in terms of what happens at fixed points in space as the fluid flows through those points.
It's like looking at a particular area of a flowing river and noting how the water's speed and other characteristics vary at different points and moments within that particular area.

21 of 93

Eulerian Description

For example, the pressure field

The velocity field

The acceleration field

Instead focusing on a particular fluid particle/ particles, the focus is given on a specific volume through which fluid flows.

22 of 93

Eulerian Vs Lagrangian

In the Eulerian method, one may attach a temperature-measuring device to the top of the chimney (point 0) and record the temperature at that point as a function of time. At different times there are different fluid particles passing by the stationary device. Thus, one would obtain the temperature, T (x,y,z,t), for that location and as a function of time.

In the Lagrangian method, one would attach the temperature-measuring device to a particular fluid particle (particle A) and record that particle’s temperature as it moves. Thus, one would obtain that particle’s temperature as a function of time.

23 of 93

Eulerian Vs Lagrangian

Which approach, system or control volume, is more associated with the Lagrangian description, and which one is linked with the Eulerian description?

In the system or Lagrangian description, we follow the fluid and observe its behavior as it moves about.
In the control volume or Eulerian description we remain stationary and observe the fluid’s behavior at a fixed location.

24 of 93

Velocity Field

At any particular moment, we can describe various properties of a fluid, such as density, pressure, velocity, and acceleration, by using its location. This way of representing fluid properties based on spatial coordinates (like x, y, z) is called a "field representation" of the flow.
However, it's important to note that these representations can change over time. So, to fully describe a fluid flow, we need to determine how these parameters vary not only with the spatial coordinates (like x, y, z) but also with time (t).
One of the most important fluid variables is the velocity field

25 of 93

Acceleration Field

26 of 93

Acceleration Field

Since this is valid for any particle, acceleration field can be written as

This is a vector result whose scalar components can be written as

27 of 93

Acceleration Field

28 of 93

Flow Visualization

Streamlines

A streamline is a curve that is everywhere tangent to the local velocity vector at an instant of time.

Since streamlines are everywhere parallel to the local velocity, fluid cannot cross a streamline by definition.
If the flow is steady, streamlines are fixed lines in space.
But for unsteady flows, streamlines may change shape with time.

Equation of Streamline

Consider an infinitesimal arc length dr along a streamline
dr must be parallel to the local velocity vector V by definition of the streamline.

From similar triangle

If the velocity field is known as a function of x and y, this equation can be integrated to give the equation of the streamlines

29 of 93

Flow Visualization

Streakline

A streakline consists of all particles in a flow that have previously passed through a common point.

Pathline

A pathline is the actual path traveled by an individual fluid particle over some time period.
A pathline is a Lagrangian concept in that we simply follow the path of an individual fluid particle as it moves around in the flow field

A streakline is formed by the continuous introduction of dye or smoke from a point in the flow

A pathline is formed by following the actual path of a fluid particle.

30 of 93

Flow Visualization

Timeline

Timeline is the line that a number of adjacent fluid particles form in a flow filed at a particular instant

Timelines are formed by marking a line of fluid particles, and then watching that line move (and deform) through the flow field; timelines are shown at t=0, t1, t2, and t3.

31 of 93

Flow Visualization

Relation between streamline, streakline, and pathline

If the flow is steady, each successively injected particle follows precisely behind the previous one, forming a steady streakline that is exactly the same as the streamline through the injection point.
If the flow is steady, the path taken by a marked particle (a pathline) will be the same as the line formed by all other particles that previously passed through the point of injection (a streakline).
Hence, pathlines, streamlines, and streaklines are the same for steady flows
However, in unsteady flow, streaklines, pathlines, and streamlines do not coincide.

32 of 93

Basic laws of fluid motion can be formulated in terms of infinitesimal and finite systems/control volumes
Finite system or control volume formulation gives integral equations.
Integral approach works with a finite region as fluid flows through it and determines the gross effect of the flow on a device, i.e., force or torque on a body, pressure drop or increase etc.

For example, the overall lift a wing produces

Infinitesimal formulations give differential equations of fluid motion.
Differential approach seeks detailed information at every point (x,y,z) in the flow field i.e., details of velocities and pressures in the domain.

For example, pressure distribution on a wing surface

Integral Versus Differential Approach

33 of 93

Mass and Volume Flow Rates

34 of 93

Extensive and Intensive Properties

Let B represent any fluid parameters and b represent the amount of that parameter per unit mass.

m is the mass of the portion of fluid of interest

The parameter B is termed an extensive property
The parameter b is termed an intensive property
B is directly proportional to the amount of the mass being considered
The value of b is independent of the amount of mass
Most of the laws governing fluid motion involve the time rate of change of an extensive property of a fluid system
For example, the rate at which the momentum of a system changes with time, the rate at which the mass of a system changes with time

35 of 93

Reynolds transport theorem (RTT)

It is needed to describe the laws governing fluid motion using both system concepts and control volume concepts.
To do this we need an analytical tool to shift from one representation to the other.
The Reynolds transport theorem provides an analytical tool to link between the system approach and the control volume approach which describes the governing laws of fluid motion using both concepts.

36 of 93

Reynolds transport theorem (RTT)

………………………….....(i)

…......(ii)

37 of 93

Reynolds transport theorem (RTT)

………………………….....(i)

…......(ii)

...(iii)

Here,

…......(a)

…........(b)

….........(c)

….........(d)

Where A1 and A2 are the cross-sectional areas at locations 1 and 2

38 of 93

Reynolds transport theorem (RTT)

Equation (ii) becomes,

…......(iv)

39 of 93

Reynolds transport theorem (RTT)

From vector classes we have,

Then the net rate of outflow through the entire control surface is determined by integration to be

40 of 93

Reynolds transport theorem (RTT)

The total amount of property B within the control volume must be determined by integration

∴

In the case of properties being uniform at sections and velocity being normal to the control surfaces (inlet/exit). The Reynolds transport theorem simplifies to

41 of 93

Reynolds transport theorem (RTT)

Theorem states that “The time rate of change of the property B of the system is equal to the time rate of change of B within the control volume plus the net flux of B out through the control surface by mass.”

So now it would be easier for one to understand the RTT statement

42 of 93

Conservation of Mass

According to the conservation of mass principle for a system, the mass of the system remains constant during a process
When a system undergoes any change mass of the system remains the same.
Mathematically,
From RTT,

∴

43 of 93

Conservation of Mass

The equation can be rearranged as

The equation implies the rate of increase of mass in the control volume is due to the net inflow of mass through the CS.

44 of 93

Conservation of Mass

For a fixed control volume, where the size of the control volume does not change

If the flow within the control volume is steady or the fluid is incompressible

Do not confuse between steady and incompressible. They two have completely different meanings. Though here we can see both of them have the same effect, in most cases, they could have different impacts.

Further simplification can be made when there are a definite number of inlets and outlets and the flow is normal to the inlet and outlet.

The total rate of mass entering a control volume is equal to the total rate of mass leaving it.

45 of 93

Linear Momentum Equation- Newton’s 2^nd law

Newton’s second law of motion for a system is

46 of 93

Linear Momentum Equation- Newton’s 2^nd law

When a control volume is coincident with a system at an instant of time, the forces acting on the system and the forces acting on the contents of the coincident control volume are instantaneously identical. From this argument we can write

47 of 93

Linear Momentum Equation- Newton’s 2^nd law

The total force acting on the control volume

The momentum equation states that “the total force acting on the control volume leads to a rate of change of momentum within the control volume and/or a net rate of flow of linear momentum through the control surface.”

48 of 93

Linear Momentum Equation- Newton’s 2^nd law

The x, y and z momentum equations can be obtained by setting the respective velocity components as

49 of 93

Linear Momentum Equation- Newton’s 2^nd law

50 of 93

Linear Momentum Equation- Newton’s 2^nd law

51 of 93

Linear Momentum Equation- Newton’s 2^nd law

A water jet of velocity V_j impinges normal to a flat plate which moves to the right at velocity V_c, as shown in the figure. Find the force required to keep the plate moving at constant velocity if the jet density is 1000 kg/m³, the jet area is 3 cm^2,and V_j and V_c are 20 and 15 m/s, respectively. Neglect the weight of the jet and plate, and assume steady flow with respect to the moving plate with the jet splitting into an equal upward and downward half-jet.

52 of 93

Bernoulli Equation

The Bernoulli equation is an approximate relation between pressure, velocity, and elevation, and is valid in regions of steady, incompressible flow where net frictional forces are negligible.
Despite its simplicity, it has proven to be a very powerful tool in fluid mechanics.
Approximations of Bernoulli equation

53 of 93

Bernoulli Equation

Consider the velocity V(s,t) of a fluid particle to be a function of s and t.

For steady flow,

54 of 93

Bernoulli Equation

Applying Newton’s second law in the s-direction

55 of 93

Bernoulli Equation

Divide both side by dA

For steady flow:

For steady, incompressible flow:

This is the famous Bernoulli equation

56 of 93

Bernoulli Equation

So the equation states that the sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow when compressibility and frictional effects are negligible.

Other two forms of Bernoulli equation

57 of 93

Bernoulli Equation

Different forms of Bernoulli equations represent different units of energy

58 of 93

Bernoulli Equation

The sum of the static, dynamic, and hydrostatic pressures is called the total pressure.
The sum of the static and dynamic pressures is called the stagnation pressure.

59 of 93