Fluid Mechanics and Fluid Machinery

Introduction

Fluid mechanics is the study of fluids at rest and in motion. A fluid is a substance that deforms continuously when subjected to any sustained shear stress. Liquids and gases are fluids, although liquids are nearly incompressible while gases are generally compressible. Mechanical engineers apply fluid mechanics in water-supply systems, hydraulic machines, aircraft, turbomachinery, heating and ventilation, lubrication, combustion, and biomedical devices.

Fluid machinery transfers energy between a fluid and a mechanical device. Pumps add energy to liquids, while turbines extract energy from fluids. Fans and compressors raise the pressure of gases. Sound engineering analysis requires the combined use of fluid properties, conservation laws, dimensional reasoning, and experimentally established loss relationships.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Define important fluid properties and select suitable models.
2. Analyze hydrostatic forces and pressure distributions.
3. Apply continuity, momentum, and energy equations to fluid systems.
4. Estimate friction and minor losses in internal pipe flow.
5. Explain the operating principles of pumps, turbines, fans, and compressors.
6. Use dimensional analysis and similarity in fluid experiments and design.
7. Recognize limitations, safety requirements, and efficiency issues in practical systems.

1. Fluid Properties

Density, specific weight, and specific gravity

Density is mass per unit volume:

where ρ is density in kg/m³, m is mass in kg, and V is volume in m³. The specific weight is the weight per unit volume:

where γ is in N/m³ and g is gravitational acceleration, approximately 9.81 m/s². Specific gravity is the ratio of a fluid's density to the density of water at a stated reference temperature:

It is dimensionless. For water near ordinary laboratory conditions, ρ_ref is approximately 1000 kg/m³.

Viscosity

Viscosity measures resistance to relative motion between adjacent fluid layers. For a Newtonian fluid, shear stress is proportional to the velocity gradient:

Here τ is shear stress in Pa, μ is dynamic viscosity in Pa·s, u is velocity parallel to the layers, and y is the coordinate normal to the layers. The relation applies to Newtonian fluids such as water and air over ordinary engineering conditions. Non-Newtonian fluids require a different constitutive model.

Kinematic viscosity is

and has units of m²/s. Viscosity usually decreases with temperature for liquids but increases with temperature for gases.

Compressibility and vapor pressure

The bulk modulus describes resistance to compression:

where K is in Pa. A liquid may often be treated as incompressible when pressure-induced density changes are negligible. Gases must be treated as compressible when pressure changes are significant or when velocities approach a substantial fraction of the speed of sound.

Vapor pressure is the pressure at which a liquid begins to vaporize at a given temperature. If local pressure falls below vapor pressure, vapor bubbles may form. Their subsequent collapse can cause cavitation, noise, vibration, and surface damage.

Surface tension and capillarity

Surface tension is the tensile force per unit length at a liquid surface. It is important in droplets, bubbles, small tubes, and microfluidic devices. Capillary rise or depression in a circular tube is approximately

where h is capillary height, σ is surface tension, θ is contact angle, d is tube diameter, and the other symbols have their usual meanings. The equation neglects meniscus details and is most useful for small, clean tubes.

2. Pressure and Hydrostatics

Pressure is normal force per unit area:

In a fluid at rest, pressure acts equally in all directions at a point. The hydrostatic equation follows from force equilibrium on a small fluid element:

where z is measured upward. For a constant-density liquid:

where h is depth below a reference surface and p₀ is pressure at that surface. Gauge pressure is measured relative to atmospheric pressure; absolute pressure is measured relative to a perfect vacuum.

Hydrostatic force on a plane surface

For a plane area submerged in a liquid, the resultant hydrostatic force is

where h_c is the vertical depth of the area centroid and A is its area. The line of action passes through the center of pressure, which lies below the centroid for a vertical or inclined surface. For an inclined plane,

where y_p and y_c are distances measured along the plane from the free surface, I_G is the second moment of area about a centroidal axis parallel to the free surface, and θ is the plane angle to the horizontal. The formula assumes a static, constant-density liquid and a plane surface.

Worked Example 1: Pressure and force

A vertical rectangular gate is 2 m wide and 3 m high, with its top edge 1 m below the free surface of water. Determine the resultant force and the depth of its center of pressure.

The centroid is at

h_c = 1 + 3/2 = 2.5 m.

The area is A = 2 × 3 = 6 m². Thus

F_R = 1000 × 9.81 × 2.5 × 6 = 147,150 N ≈ 147 kN.

For the rectangle, I_G = bh³/12 = 2(3³)/12 = 4.5 m⁴. Therefore

y_p = 2.5 + [4.5/(2.5 × 6)] = 2.8 m below the free surface.

The force acts 0.30 m below the centroid.

3. Description of Fluid Motion

A control volume is a chosen region through which fluid may flow. The control-volume approach is particularly useful for valves, nozzles, pumps, turbines, and pipe junctions. Flow may be steady or unsteady, uniform or nonuniform, and one-, two-, or three-dimensional.

The volume flow rate is

For one-dimensional uniform velocity, Q = AV, where A is cross-sectional area and V is mean velocity. Mass flow rate is

For incompressible steady flow through a pipe, Q is constant. Consequently, a reduction in area produces an increase in mean velocity.

Continuity equation

The general conservation of mass equation for a control volume is

For steady one-dimensional flow with one inlet and one outlet,

For incompressible flow, A₁V₁ = A₂V₂. This assumes no accumulation and uses mean velocities when the velocity profile is nonuniform.

Worked Example 2: Continuity

Water flows through a pipe that contracts from 100 mm diameter to 50 mm diameter. If the upstream mean velocity is 2 m/s, find the downstream velocity.

Because area is proportional to diameter squared,

V₂ = V₁(D₁/D₂)² = 2(0.10/0.05)² = 8 m/s.

The result assumes steady incompressible flow and negligible leakage.

4. Momentum and Energy Equations

Linear momentum

The control-volume momentum equation is

For steady one-dimensional flow, external force equals the vector difference between outlet and inlet momentum fluxes. Pressure forces, weight, support reactions, and shear forces must be included. Momentum analysis is essential for pipe bends, jets, nozzles, and vanes.

The momentum correction factor may be required when velocity is not uniform. For many turbulent pipe flows it is close to unity; for laminar flow it is larger and should not be ignored in precise calculations.

Bernoulli equation

For steady, incompressible flow along a streamline with negligible viscosity and no shaft work,

The three terms are pressure head, velocity head, and elevation head. The equation is an ideal mechanical-energy relation. It does not automatically include friction, pumps, turbines, or heat transfer.

The extended energy equation between sections 1 and 2 is

Here h_p is pump head added, h_t is turbine head extracted, h_L is total head loss, and α is the kinetic-energy correction factor. For a uniform turbulent profile, α is commonly approximated as 1. The equation assumes a single-phase flow and consistent average conditions at the selected sections.

Mechanical efficiency

Pump hydraulic power delivered to the fluid is

where H is pump head. If the pump efficiency is η_p, input shaft power is P_shaft = P_h/η_p. Turbine output power is approximately η_tρgQH available, where η_t is turbine efficiency.

5. Internal Flow in Pipes

Reynolds number

The Reynolds number compares inertial and viscous effects:

For a circular pipe, laminar flow generally occurs below Re ≈ 2300, transitional flow occurs approximately from 2300 to 4000, and turbulent flow generally occurs above about 4000. These ranges are indicative rather than universal because disturbances, entrance conditions, and geometry affect transition.

Laminar pipe flow

For fully developed laminar flow in a circular pipe, the Hagen–Poiseuille relation is

where ∆p is pressure drop and L is pipe length. The velocity profile is parabolic, and the mean velocity is half the centerline velocity. This relation assumes steady, incompressible, Newtonian flow, a straight circular pipe, no-slip walls, and fully developed conditions.

Darcy–Weisbach equation

Head loss due to wall friction is

where f is the Darcy friction factor. Do not confuse this with the Fanning friction factor, which is one quarter of the Darcy factor.

For laminar flow, f = 64/Re. For turbulent flow, f depends on Reynolds number and relative roughness ε/D. The Colebrook equation is

It is implicit and is commonly solved iteratively or with an accepted explicit approximation. The Moody chart provides the same relationship graphically.

Minor losses

Fittings and changes in geometry cause additional losses:

where K is a dimensionless loss coefficient based on the selected reference velocity. Total loss is the sum of distributed and local losses. Sudden expansions may also be analyzed using the Borda–Carnot relation:

for an abrupt expansion with V₁ upstream and V₂ downstream.

Worked Example 3: Pipe friction

Water flows through a 50 m long, 0.10 m diameter pipe at 2 m/s. Take f = 0.020. Estimate friction head loss.

Thus h_f = 2.04 m of water. The corresponding pressure loss is ρgh_f ≈ 20.0 kPa. This calculation excludes fittings, entrance losses, and elevation changes.

6. Dimensional Analysis and Similarity

Dimensional analysis reduces the number of independent variables and assists the design of experiments. The Buckingham π theorem states that if a problem contains n dimensional variables involving r independent fundamental dimensions, it can be represented using n − r independent dimensionless groups.

Important dimensionless numbers include:

Dynamic similarity requires corresponding dimensionless groups to be equal between model and prototype. Exact matching of all groups is often impossible, so engineers identify the dominant phenomena and match the most important parameter. For open-channel flows, Froude similarity is usually important; for viscous pipe flow, Reynolds similarity is central.

7. Fluid Machinery

Pumps

A pump converts mechanical energy into hydraulic energy. Centrifugal pumps use a rotating impeller to increase fluid angular momentum and pressure. Positive-displacement pumps trap and displace a nearly fixed volume per cycle and are suitable for high pressure, accurate metering, and viscous fluids.

Pump performance is represented by head, discharge, efficiency, and power. The operating point is the intersection of the pump characteristic curve and the system resistance curve. Increasing system resistance generally reduces flow rate and changes the operating point.

The affinity laws for geometrically similar centrifugal pumps, at constant fluid density, are approximately

where N is rotational speed and D is impeller diameter. These relations are most reliable for moderate changes and dynamically similar operation.

Net positive suction head and cavitation

The available net positive suction head is the absolute total head at the pump suction above vapor-pressure head. The pump manufacturer specifies required NPSH. Safe operation requires available NPSH to exceed required NPSH by an appropriate margin. Cavitation prevention may involve lowering suction losses, reducing liquid temperature, increasing suction pressure, or reducing pump speed.

Turbines

A turbine extracts fluid energy and produces shaft power. Hydraulic turbines are commonly classified as impulse or reaction machines. In an impulse turbine, most pressure conversion occurs before the runner; in a reaction turbine, pressure changes occur through the runner as well.

The Euler turbomachinery equation expresses ideal specific work:

where U is blade speed and V_w is the tangential component of absolute velocity. Sign conventions must be defined consistently. The equation follows from angular-momentum conservation and is applied using velocity triangles.

Fans and compressors

Fans produce relatively small pressure rises, while compressors produce larger pressure increases and often require compressible-flow analysis. Fan selection depends on flow rate, pressure rise, efficiency, noise, speed, and operating environment. In ventilation systems, excessive resistance increases power consumption and may reduce delivered airflow.

8. External Flow and Drag

When a fluid flows around a body, boundary layers develop along the surface. Viscous effects are concentrated in the boundary layer, while pressure variations outside it may often be approximated using inviscid theory. Flow separation occurs when the boundary layer loses momentum against an adverse pressure gradient. Separation increases pressure drag and may produce unsteady vortex shedding.

Drag and lift are expressed using dimensionless coefficients:

where F_D and F_L are drag and lift, V is reference velocity, and A is reference area. Coefficients depend on Reynolds number, surface roughness, geometry, and angle of attack. Streamlining reduces pressure drag, while smooth surfaces reduce skin-friction drag only within the limits of boundary-layer behavior.

9. Illustrative Case: Selecting a Pump for a Water-Supply System

Background

An illustrative municipal workshop must deliver water from a storage tank to an elevated tank. The required flow is 0.020 m³/s. The pipe is 100 m long with an internal diameter of 0.10 m, and the elevation difference is 18 m. The system contains valves and bends represented by a combined minor-loss coefficient K = 4.0. Assume water density 1000 kg/m³, g = 9.81 m/s², and Darcy friction factor f = 0.025.

Analysis

The pipe velocity is

V = Q/A = 0.020/[π(0.10)²/4] = 2.55 m/s.

The friction head is

h_f = 0.025(100/0.10)[2.55²/(2×9.81)] = 8.28 m.

Minor losses are

h_m = 4.0[2.55²/(2×9.81)] = 1.32 m.

Neglecting reservoir surface velocity heads, the required pump head is

H_p = 18 + 8.28 + 1.32 = 27.6 m.

Hydraulic power is

P_h = ρgQH_p = 1000×9.81×0.020×27.6 = 5.42 kW.

If the pump efficiency is 72%, shaft input power is

P_shaft = 5.42/0.72 = 7.53 kW.

Approach and outcome

A suitable pump should deliver approximately 0.020 m³/s at 27.6 m head, with a motor rating exceeding the calculated shaft requirement after considering starting conditions, service factor, and manufacturer recommendations. The engineer must also check NPSH, water temperature, pipe pressure rating, surge, maintenance access, and control-valve operation.

Learning points

The static elevation is only part of the required head. Friction and fittings can be substantial, and the selected pump must operate near its best-efficiency region rather than merely satisfy one point on a catalog curve. Oversizing can cause throttling, energy waste, and unstable operation.

10. Practical Design, Safety, and Sustainability

Fluid systems should be designed with adequate pressure ratings, isolation valves, relief devices, guarding, and protection against leakage. Hydraulic oil injection injuries, rotating-shaft hazards, hot compressed gases, and stored pressure require formal risk assessment and lockout procedures.

Energy efficiency is improved by selecting appropriate pipe diameters, minimizing unnecessary fittings, controlling pump speed, maintaining clean filters, repairing leaks, and operating equipment near its best-efficiency point. Engineers should account for lifecycle energy and maintenance rather than minimizing initial purchase cost alone.

Computational fluid dynamics can predict velocity, pressure, separation, and heat transfer, but results depend on mesh quality, turbulence models, boundary conditions, and validation. Artificial intelligence can assist in anomaly detection and predictive maintenance using pressure, vibration, and flow data; it should supplement, not replace, conservation laws, physical inspection, and engineering judgment.

Limitations of Basic Models

The equations in this chapter commonly use steady, one-dimensional, incompressible, or fully developed assumptions. Real systems may involve transient water hammer, multiphase flow, non-Newtonian rheology, compressibility, thermal effects, roughness changes, and complex three-dimensional turbulence. When assumptions are violated, engineers should use transient analysis, compressible-flow relations, experimentally validated correlations, or numerical simulation.

Chapter Summary

Fluid mechanics describes fluids at rest and in motion, while fluid machinery transfers energy between fluids and mechanical devices. Density, viscosity, pressure, vapor pressure, and surface tension determine fluid behavior. Hydrostatics provides pressure distributions and forces on submerged surfaces. Continuity expresses conservation of mass; the momentum equation predicts forces and reactions; and the extended Bernoulli equation accounts for elevation, pressure, velocity, pumps, turbines, and losses.

Internal pipe-flow analysis uses Reynolds number, friction factors, the Darcy–Weisbach equation, and minor-loss coefficients. Dimensional analysis provides dimensionless groups and supports model testing. Pumps add head, turbines extract energy, and fans or compressors raise gas pressure. Practical design requires attention to operating points, efficiency, cavitation, safety, sustainability, and the limitations of simplified models.

Key Terms

Density: Mass per unit volume of a fluid.

Specific weight: Weight per unit volume.

Viscosity: Resistance of a fluid to relative shearing motion.

Newtonian fluid: Fluid whose shear stress is proportional to velocity gradient.

Pressure: Normal force per unit area.

Gauge pressure: Pressure measured relative to atmospheric pressure.

Hydrostatics: Study of fluids at rest.

Control volume: Defined region used for conservation-law analysis.

Continuity equation: Conservation of mass applied to fluid flow.

Reynolds number: Ratio indicating relative importance of inertia and viscosity.

Laminar flow: Smooth, highly ordered flow dominated by viscous effects.

Turbulent flow: Fluctuating flow with strong mixing and inertia effects.

Darcy friction factor: Dimensionless factor used to calculate pipe-friction head loss.

Minor loss: Local energy loss caused by fittings or changes in geometry.

Cavitation: Formation and collapse of vapor bubbles due to low pressure.

Pump: Machine that adds mechanical energy to a fluid.

Turbine: Machine that extracts mechanical energy from a fluid.

NPSH: Net positive suction head available above vapor-pressure head.

Dimensional analysis: Method of organizing variables into dimensionless groups.

Boundary layer: Thin region near a solid surface where viscous effects are important.

Review Questions

Short Answer Questions

1. Define dynamic viscosity and state its SI unit.
2. Distinguish between absolute and gauge pressure.
3. State the assumptions of the Bernoulli equation in its ideal form.
4. What physical effects are compared by the Reynolds number?
5. Why can cavitation damage a pump?

Long Answer Questions

1. Derive the hydrostatic pressure relation for a constant-density liquid and explain its engineering use.
2. Explain the continuity, momentum, and energy equations and identify one application of each.
3. Describe laminar and turbulent pipe flow, including the role of friction factor.
4. Compare centrifugal and positive-displacement pumps in terms of operating characteristics and applications.
5. Explain dimensional analysis, dynamic similarity, and the significance of Reynolds and Froude numbers.

Analytical / Application Questions

1. Water flows at 0.015 m³/s through a 75 mm pipe. Calculate the mean velocity and Reynolds number using μ = 1.0 × 10⁻³ Pa·s and ρ = 1000 kg/m³. Classify the flow.
2. A pump supplies 0.025 m³/s against a total head of 32 m at 75% efficiency. Determine the hydraulic power and shaft input power.
3. Design an outline method for diagnosing unexpectedly low flow in a piping system. Identify measurements, equations, likely faults, and safety precautions.

Multiple Choice Questions with Answers

Question 1

For a Newtonian fluid, shear stress is proportional to which quantity?

A. Pressure itself

B. Velocity gradient

C. Fluid density only

D. Pipe diameter only

Correct Answer: B. Velocity gradient

Question 2

A 100 mm diameter pipe carries incompressible water at 2 m/s and contracts to 50 mm diameter. What is the downstream mean velocity?

A. 0.5 m/s

B. 1 m/s

C. 4 m/s

D. 8 m/s

Correct Answer: D. 8 m/s

Question 3

Which equation directly represents conservation of mass in steady incompressible one-dimensional pipe flow?

A. p + ρgz = constant

B. A₁V₁ = A₂V₂

C. F = ma

D. τ = μdu/dy

Correct Answer: B. A₁V₁ = A₂V₂

Question 4

The Darcy–Weisbach equation calculates which quantity?

A. Hydrostatic force

B. Capillary rise

C. Pipe-friction head loss

D. Pump efficiency

Correct Answer: C. Pipe-friction head loss

Question 5

For fully developed laminar flow in a circular pipe, the Darcy friction factor is:

A. 16/Re

B. 32/Re

C. 64/Re

D. Re/64

Correct Answer: C. 64/Re

Question 6

A pump has flow rate 0.020 m³/s and head 25 m. Using ρ = 1000 kg/m³ and g = 9.81 m/s², its hydraulic power is approximately:

A. 0.49 kW

B. 4.91 kW

C. 49.1 kW

D. 490 kW

Correct Answer: B. 4.91 kW

Question 7

Which condition most directly promotes cavitation at a pump inlet?

A. Higher absolute suction pressure

B. Lower liquid temperature

C. Local pressure below vapor pressure

D. Reduced flow velocity in the suction line

Correct Answer: C. Local pressure below vapor pressure

Question 8

The Reynolds number primarily compares:

A. Pressure and elevation effects

B. Inertial and viscous effects

C. Gravity and surface tension only

D. Heat and mass transfer

Correct Answer: B. Inertial and viscous effects

Question 9

A turbine primarily performs which function?

A. Adds mechanical energy to a fluid

B. Converts fluid energy into shaft power

C. Prevents all pipe losses

D. Measures fluid density

Correct Answer: B. Converts fluid energy into shaft power

Question 10

Which dimensionless number is especially important for free-surface flows governed by gravity?

A. Froude number

B. Mach number

C. Prandtl number

D. Euler number

Correct Answer: A. Froude number