Conservation Laws and Governing Equations

Introduction

Fluid mechanics is governed by a small number of conservation principles. Mass cannot be created or destroyed; the rate of change of linear momentum equals the resultant force; angular momentum changes in response to torque; and energy is transferred or transformed but is not destroyed. These principles become engineering equations when applied to a carefully selected control volume.

This chapter develops the continuity, momentum, angular-momentum, energy, Euler, Bernoulli, and Navier–Stokes equations. Emphasis is placed on assumptions, boundary conditions, correction factors, and practical application to pipe flow, nozzles, pumps, turbines, jets, and hydraulic systems.

Learning Objectives

By the end of this chapter, the student should be able to:

• distinguish a system from a control volume;
• formulate mass, momentum, angular-momentum, and energy balances;
• derive and apply the continuity equation;
• use the linear-momentum equation to determine forces on bends, nozzles, and vanes;
• explain the conditions leading to Euler’s and Bernoulli’s equations;
• interpret the Navier–Stokes equations and their boundary conditions;
• apply kinetic-energy and momentum correction factors;
• recognize the limitations of ideal-flow equations in engineering design.

1. Systems, Control Volumes, and Flow Descriptions

A system is a fixed quantity of matter. Its boundary moves with the material, so no mass crosses it. A control volume is a selected region in space through which fluid may flow. The control surface is its boundary. Engineering analyses generally use control volumes because pumps, turbines, pipes, valves, and nozzles are stationary devices through which mass flows.

The system approach follows individual fluid particles. The control-volume approach observes what crosses a selected boundary. The two descriptions are connected by the Reynolds transport theorem:

Here, B is an extensive property, b = B/m is the corresponding property per unit mass, ρ is density, V is velocity, n is the outward unit normal, CV denotes control volume, and CS denotes control surface. The theorem assumes a fixed control volume in this form. For a moving or deforming control volume, the relative velocity across the surface must be used.

The scalar product V · n is positive for outflow and negative for inflow. This sign convention should be established before writing any control-volume balance.

2. Conservation of Mass: The Continuity Equation

For mass, B = m and b = 1. Applying the transport theorem gives:

This is the integral continuity equation. The first term is the rate of accumulation of mass inside the control volume; the second is the net mass flow out.

For a one-dimensional inlet–outlet control volume with uniform average properties:

where m_CV is the mass within the control volume and ṁ is mass-flow rate. At steady state, accumulation is zero, so:

For a stream crossing an area A with average normal velocity V̄:

The units are kg/s when ρ is in kg/m³, A in m², and V̄ in m/s. For incompressible flow with constant density, the volumetric-flow equation is:

and, for steady flow through a single stream:

This result does not imply that local velocity is uniform. V̄ is the area-averaged velocity, defined by:

where V_n is the local velocity component normal to the area.

Worked Example 1: Flow through a contraction

Water flows steadily through a horizontal pipe that contracts from 100 mm diameter to 50 mm diameter. If the average velocity in the larger pipe is 2 m/s, determine the smaller-pipe velocity.

For incompressible flow:

Thus,

The smaller-pipe velocity is 8 m/s. This calculation neglects leakage and assumes steady incompressible flow.

3. Conservation of Linear Momentum

Newton’s second law applied to a control volume is:

The left side includes all external forces, including pressure forces, body forces such as weight, and forces exerted by supports or walls. The right side is the rate of momentum accumulation plus net momentum flux.

For steady one-dimensional flow:

This is a vector equation and must normally be resolved into x-, y-, and z-components. Pressure forces should be written using absolute or gauge pressure consistently. Gauge pressure is often convenient when the surroundings are at atmospheric pressure.

For a control volume with one inlet and one outlet:

where R⃗ is the resultant external force. The force exerted by the fluid on the pipe or fitting is opposite to the force exerted by the pipe on the fluid.

Application to a pipe bend

A bend changes the direction of velocity and therefore produces a momentum-force requirement even if speed is unchanged. In addition to momentum flux, the calculation must include pressure forces at inlet and outlet and the weight of fluid inside the bend when significant. Neglecting these terms without justification is a common design error.

For a horizontal 90° bend with equal inlet and outlet areas, negligible losses, and equal speed V, the momentum change has components proportional to ṁV in the two coordinate directions. The support force is obtained only after pressure forces are included in the free-body diagram.

4. Conservation of Angular Momentum

Taking moments about a fixed point or axis gives the angular-momentum equation:

Here, T_O is torque about point O and r is the position vector from O. For steady flow through rotating machinery:

where r is radius and V_θ is the tangential component of absolute velocity. The power transferred to or from a rotor is:

where ω is angular velocity in rad/s. This relation is the Euler turbomachinery equation in torque form. Sign conventions depend on whether power is supplied to the fluid, as in a pump, or extracted from the fluid, as in a turbine.

For an ideal radial-flow pump with negligible inlet whirl, V_θ1 ≈ 0, and the theoretical specific work is:

where U₂ = ωr₂ is blade speed at the outlet. Real machines depart from this ideal result because of slip, hydraulic losses, mechanical losses, leakage, and nonuniform velocity distributions.

5. Conservation of Energy

For a control volume, the first law of thermodynamics may be written:

Here, E_CV is total stored energy, Q̇ is heat transfer into the control volume, Ẇ is shaft work out of the control volume, e is internal energy per unit mass, p is pressure, ρ is density, V is speed, g is gravitational acceleration, and z is elevation. The combination e + p/ρ is enthalpy per unit mass, h.

For steady one-inlet, one-outlet flow, using shaft work positive out of the control volume:

For hydraulic devices, it is convenient to express energy per unit weight. Including head losses h_L and machine head H_M:

H_P is head added by a pump, H_T is head extracted by a turbine, and α is the kinetic-energy correction factor. All terms have units of metres of fluid. For incompressible flow, this equation is the extended Bernoulli equation.

Kinetic-energy correction factor

Because velocity profiles are generally nonuniform, the true kinetic-energy flux is not exactly based on V̄³A. The correction factor is:

For fully developed laminar flow in a circular pipe, α = 2. For a uniform profile, α = 1. In turbulent pipe flow, α is usually close to 1, commonly about 1.03–1.10 depending on the profile. The value should not be assumed to be exactly 1 when precision is important.

6. Euler’s Equation and Bernoulli’s Equation

Euler’s equation for inviscid flow is obtained by applying Newton’s second law to a differential fluid element while neglecting viscous stresses. In vector form for an incompressible fluid subject only to conservative body forces:

Here, D/Dt is the material derivative, ∇p is the pressure gradient, and g⃗ is gravitational acceleration. If the flow is steady, incompressible, and integrated along a streamline, Euler’s equation gives:

Dividing by g gives Bernoulli’s equation:

The terms are pressure head, velocity head, and elevation head. Bernoulli’s equation is applicable along a streamline for steady, incompressible, inviscid flow with no pump, turbine, or significant heat-transfer work between the two points. For irrotational flow, the constant is the same throughout a connected flow field; otherwise it may differ from streamline to streamline.

Real flows require losses and machine terms. Applying ideal Bernoulli’s equation across a long pipe, valve, or sudden expansion without including losses produces physically incorrect pressure predictions.

7. Navier–Stokes Equations

The Navier–Stokes equations are the differential momentum equations for viscous fluid motion. For a Newtonian, incompressible fluid of constant viscosity:

The symbols are ρ, density; μ, dynamic viscosity in Pa·s; p, pressure; V, velocity vector; and g⃗, body-force acceleration. The terms represent, respectively, inertia, pressure force, viscous diffusion, and body force.

The corresponding incompressible continuity equation is:

For compressible flow, density varies and continuity is instead:

The Navier–Stokes equations are derived from local momentum conservation plus a constitutive relation for Newtonian fluids. They are not universally solvable in closed form. Engineering solutions commonly use simplifications such as steady flow, symmetry, fully developed flow, negligible inertia, or computational fluid dynamics.

Boundary conditions

A viscous flow requires appropriate boundary conditions. At a stationary solid wall, the no-slip condition is:

At an impermeable moving wall, the normal fluid velocity equals the wall’s normal velocity, and the tangential fluid velocity equals the wall’s tangential velocity for ordinary viscous flow. At an inlet, velocity or mass flow may be prescribed; at an outlet, pressure or a suitable fully developed condition may be specified. Symmetry planes require zero normal velocity and zero normal gradient of tangential quantities where appropriate.

Incorrect boundary conditions can produce mathematically converged but physically meaningless results.

8. Momentum and Energy Correction Factors

The momentum correction factor β accounts for nonuniform momentum flux:

For a uniform profile, β = 1. For fully developed laminar flow in a circular pipe, β = 4/3. The momentum equation for a steady one-dimensional stream may therefore use βṁV rather than ṁV when velocity is nonuniform.

The energy correction factor α and momentum correction factor β are different because energy flux depends on V³ whereas momentum flux depends on V². Using α in a momentum balance, or β in an energy balance, is incorrect.

9. Worked Example 2: Pump head and power

Water flows from a reservoir through a pump to a higher reservoir. The elevation difference is 18 m, pipe and fitting losses total 4 m, and the required flow rate is 0.050 m³/s. Determine the ideal hydraulic power delivered by the pump. Take ρ = 1000 kg/m³ and g = 9.81 m/s².

Both reservoirs have negligible surface velocity and are open to the atmosphere. The required pump head is:

Hydraulic power is:

Therefore,

This is hydraulic power. If pump efficiency is η = 0.75, the shaft power required is:

The calculation assumes steady incompressible water flow and correctly includes head loss.

10. Applied Case: Force on a Hydraulic Nozzle

Illustrative case. A fire-protection system discharges water through a reducing nozzle. The nozzle is anchored to a support, but the support has begun to loosen. Engineers must estimate the axial reaction force.

Background and problem

The inlet diameter is 100 mm, the outlet diameter is 50 mm, and the flow rate is 0.020 m³/s. The inlet gauge pressure is 300 kPa, and the outlet discharges to atmosphere. Weight and friction within the short horizontal nozzle are neglected.

Analysis

The inlet and outlet average velocities are:

The mass flow rate is ṁ = ρQ = 20 kg/s. Taking the positive x-direction downstream, the momentum equation is:

Because outlet gauge pressure is zero, p₂A₂ = 0. The pressure force at inlet is:

The momentum-flux change is:

Thus the force exerted by the nozzle wall and support on the fluid is:

The fluid therefore exerts approximately 2.20 kN downstream on the nozzle and its support. The anchorage must resist this force in the upstream direction.

Outcome and learning points

The pressure force is much larger than the net momentum-flux term in this example. Both terms are essential. A design based only on velocity change would substantially underestimate the anchorage load. In an actual installation, nozzle weight, friction, transient water hammer, nonuniform profiles, and safety factors would also be examined.

11. Practical Limitations and Best Practices

The governing equations are exact conservation statements, but engineering models are approximate because properties, profiles, losses, and boundary conditions must be represented. Important limitations include:

• Bernoulli’s equation cannot replace a loss model in viscous pipe flow.
• The incompressible assumption is inappropriate when pressure changes cause significant density variation or when Mach number is appreciable.
• One-dimensional equations require area-averaged quantities and may fail near separation, strong recirculation, or rapidly varying geometry.
• Steady equations cannot describe water hammer, pump startup, valve closure, or other transients.
• Pressure and velocity measurements must be taken at locations where the stated model applies.
• Numerical solutions of Navier–Stokes equations require mesh, time-step, turbulence-model, and boundary-condition verification.

Good engineering practice is to draw the control volume, label all inlets and outlets, define axes and signs, list assumptions, check dimensions, and compare the result with an order-of-magnitude estimate. Conservation equations should be supplemented by appropriate constitutive laws, property data, empirical loss coefficients, and experimental validation.

Modern computational fluid dynamics can solve complex geometries and provide detailed fields, but it does not remove the need for conservation-law reasoning. AI-assisted design and automated optimization are useful only when models are physically constrained, validated against measurements, and reviewed by qualified engineers. In safety-critical hydraulic systems, traceability and conservative design remain essential.

Chapter Summary

Control volumes provide the practical framework for applying conservation laws to flowing fluids. The continuity equation follows from mass conservation and relates density, area, and average velocity. The linear-momentum equation determines forces associated with pressure, weight, and changes in momentum. Angular momentum provides the basis for torque and power relations in pumps and turbines.

The energy equation accounts for pressure, velocity, elevation, heat transfer, shaft work, and losses. Bernoulli’s equation is a restricted ideal-flow form and must be extended with pump head, turbine head, and head loss for real hydraulic systems. Euler’s equation neglects viscous stresses, while the Navier–Stokes equations include viscous effects and require appropriate initial and boundary conditions. Correction factors α and β account for nonuniform velocity profiles in energy and momentum balances, respectively.

Reliable analysis depends less on memorizing equations than on selecting a suitable control volume, stating assumptions, retaining relevant forces and losses, and checking units and physical plausibility.

Key Terms

Control volume: A region in space selected for analysis.

Control surface: The boundary of a control volume.

System: A fixed quantity of matter followed through time.

Mass-flow rate: Mass crossing a section per unit time, ṁ.

Continuity equation: Conservation equation for mass.

Momentum flux: Transport of linear momentum across a control surface.

Angular momentum: Moment of linear momentum about a point or axis.

Torque: Moment of force or rate of angular-momentum transfer.

Bernoulli equation: Mechanical-energy relation for restricted ideal flow.

Euler equation: Inviscid differential momentum equation.

Navier–Stokes equations: Differential momentum equations including viscosity.

Head loss: Mechanical energy dissipated per unit weight of fluid.

Pump head: Energy per unit weight added to fluid by a pump.

Turbine head: Energy per unit weight extracted by a turbine.

Kinetic-energy correction factor: Factor α correcting nonuniform kinetic-energy flux.

Momentum correction factor: Factor β correcting nonuniform momentum flux.

No-slip condition: Equality of fluid and wall tangential velocity at a viscous wall.

Incompressible flow: Flow in which density is treated as constant.

Steady flow: Flow properties at fixed locations do not change with time.

Review Questions

5 Short Answer Questions

1. Distinguish between a system and a control volume.
2. State the assumptions required for the simple Bernoulli equation.
3. Why are pressure forces important in a bend or nozzle momentum balance?
4. Define the kinetic-energy correction factor α.
5. State the no-slip boundary condition at a stationary solid wall.

5 Long Answer Questions

1. Derive the steady one-dimensional continuity equation from the integral mass-conservation equation.
2. Explain the linear-momentum equation for a control volume and apply it conceptually to a 90° pipe bend.
3. Derive Bernoulli’s equation from Euler’s equation and discuss its limitations.
4. Explain every term in the incompressible Navier–Stokes equation and identify the physical process represented by each term.
5. Compare the energy equation, Bernoulli equation, and extended Bernoulli equation.

3 Analytical / Application Questions

1. Water flows at 0.030 m³/s through a 75 mm pipe that contracts to 40 mm. Determine both average velocities and explain whether the continuity equation requires equal velocities.
2. A pump delivers 0.080 m³/s against a total head of 35 m. If its efficiency is 78%, calculate the required shaft power using ρ = 1000 kg/m³ and g = 9.81 m/s².
3. Develop a control-volume force analysis for a horizontal reducing nozzle, identifying pressure forces, momentum fluxes, support reaction, and assumptions needed for a defensible design calculation.

Multiple Choice Questions with Answers

Question 1

For steady incompressible flow through a pipe, which relation is correct?

A. A₁V₁ = A₂V₂

B. A₁/V₁ = A₂/V₂

C. ρ₁A₁ = ρ₂A₂

D. V₁ + V₂ = constant

Correct Answer: A. A₁V₁ = A₂V₂

Question 2

The mass-flow rate through an area is given by:

A. ṁ = ρA/V̄

B. ṁ = ρAV̄

C. ṁ = AV̄/ρ

D. ṁ = ρV̄/A

Correct Answer: B. ṁ = ρAV̄

Question 3

Which condition is required for the simple Bernoulli equation along a streamline?

A. Highly viscous flow

B. Steady incompressible inviscid flow with no machine work

C. Unsteady compressible flow

D. Flow with large head losses

Correct Answer: B. Steady incompressible inviscid flow with no machine work

Question 4

For fully developed laminar flow in a circular pipe, the kinetic-energy correction factor α is:

A. 0.5

B. 1

C. 4/3

D. 2

Correct Answer: D. 2

Question 5

The momentum correction factor β accounts for nonuniform:

A. Pressure distribution only

B. Momentum flux

C. Fluid density only

D. Elevation head

Correct Answer: B. Momentum flux

Question 6

Which term represents viscous diffusion in the incompressible Navier–Stokes equation?

A. −∇p

B. ρg⃗

C. μ∇²V

D. ρV

Correct Answer: C. μ∇²V

Question 7

A pump adds head to a hydraulic system primarily to:

A. Reduce mass conservation

B. Increase the fluid’s mechanical energy

C. Eliminate density

D. Remove gravity

Correct Answer: B. Increase the fluid’s mechanical energy

Question 8

The no-slip condition at a stationary solid wall states that the fluid:

A. Has zero pressure

B. Has zero density

C. Has the wall velocity at the wall

D. Has infinite velocity gradient everywhere

Correct Answer: C. Has the wall velocity at the wall

Question 9

If Q = 0.050 m³/s and pump head is 22 m, the hydraulic power for water is approximately:

A. 1.08 kW

B. 10.8 kW

C. 108 kW

D. 0.108 kW

Correct Answer: B. 10.8 kW

Question 10

The angular-momentum equation is especially important for determining:

A. Torque in turbomachinery

B. Fluid density only

C. Pipe roughness only

D. Temperature at a wall only

Correct Answer: A. Torque in turbomachinery