Fluid Mechanics

E. Energy, impulse, and momentum equations

Continuity, Bernoulli, and Momentum

The three control-volume workhorses — conservation of mass, the Bernoulli energy equation in head form, and the impulse-momentum principle for forces on bends, nozzles, and jets.

Three conservation laws — mass, energy, and momentum — carry most of the Fluid Mechanics points on the FE-OTH, and the handbook's "Principles of One-Dimensional Fluid Flow" section is where they live. Continuity tells you how velocity changes when a pipe narrows; Bernoulli trades pressure, velocity, and elevation head along a streamline; and the impulse-momentum principle gives the force a moving fluid exerts on a bend, nozzle, or vane. The recurring exam errors are mechanical, not conceptual: dropping a head term, using gauge where absolute is needed, or — most often — forgetting that momentum is a ·vector· so pressure forces and momentum flux must be resolved by component. Build the habit of writing the full equation, then crossing out the terms the problem zeroes.

Conservation of mass — continuity

Mass cannot accumulate in a steady flow, so the mass flow rate is the same at every cross section:

\dot m = \rho A v

is constant. When the fluid is incompressible (

\rho

constant) this reduces to constant volumetric flow

Q = Av

, which means velocity rises where the pipe narrows. This single relation links the two velocities in almost every Bernoulli problem.

\dot m = \rho_1 A_1 v_1 = \rho_2 A_2 v_2, \qquad Q = A_1 v_1 = A_2 v_2

The Bernoulli equation

For steady, incompressible, frictionless flow along a streamline with no pump or turbine, the sum of pressure energy, kinetic energy, and potential energy per unit weight is constant between two points. Each term is a ·head· — a height of fluid — so the whole equation has units of length. It is the energy budget of a streamline: speed up and the pressure must fall.

\frac{P_1}{\gamma} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + z_2

The energy equation with losses and machines

Real systems lose head to friction and may add or remove it with pumps and turbines. The extended energy equation inserts a pump head

h_p

(energy in), a turbine head

h_t

(energy out), and a friction head loss

h_f

Energy grade line versus hydraulic grade line

The energy grade line (EGL) plots the total head — pressure + velocity + elevation head — along the system; the hydraulic grade line (HGL) plots only pressure + elevation head. The two differ by exactly the velocity head

v^2/2g

, so the HGL sits below the EGL by that amount and the gap widens where the flow speeds up. In frictionless flow the EGL is horizontal; friction makes it slope downward in the direction of flow.

EGL - HGL = \frac{v^2}{2g}

Torricelli's result and stagnation

Two famous special cases fall straight out of Bernoulli. A tank draining through a small hole a depth

h

below the surface gives the efflux speed

v = \sqrt{2gh}

(atmospheric on both ends, negligible surface velocity). And bringing a stream to rest at a stagnation point converts all velocity head to pressure: the stagnation pressure exceeds the static pressure by the dynamic pressure

\tfrac12\rho v^2

The impulse-momentum principle

Newton's second law for a control volume says the net external force equals the rate of change of momentum flux through it. For steady one-inlet, one-outlet flow this is

\sum \vec F = \dot m(\vec v_2 - \vec v_1) = \rho Q(\vec v_2 - \vec v_1)

Forces on bends, nozzles, and jets

To find the anchoring force on a bend or nozzle, draw the control volume, mark inlet/outlet pressures and velocities, and apply momentum component-by-component. In the

x

-direction,

P_1 A_1 - P_2 A_2\cos\alpha - F_x = \rho Q(v_2\cos\alpha - v_1)

Exam strategy

Pick your two points where you know the most (free surfaces and atmospheric exits zero out pressure and velocity heads). Write the full energy equation, then strike the zero terms. Use continuity to relate the two velocities ·before· substituting. For force problems, switch tools: continuity + Bernoulli give the velocities and pressures, then momentum (resolved into

x

and

y

, with

PA

terms included) gives the force — and remember the force on the structure is the negative of the force on the fluid. Keep everything in absolute or gauge consistently, and carry units of head (m or ft) throughout.

Key equations

Continuity (mass)

\dot m = \rho A v = \text{const}

Steady-flow mass conservation; $\dot m$ in kg/s. Same at every cross section.

Worked examples

Continuity through a contraction

Problem. Water flows in a horizontal pipe that contracts from $D_1 = 100\,\text{mm}$ to $D_2 = 50\,\text{mm}$

Common pitfalls

•Applying Bernoulli where it does not hold: it requires steady, incompressible, frictionless flow with no pump/turbine. Across a pump, fitting, or shock, use the energy or momentum equation instead.
•Dropping a head term — forgetting the elevation head on a non-horizontal run, or the velocity head when areas differ — unbalances the equation.
•Mixing gauge and absolute pressure between the two points. Pick one convention and keep it.
•Treating momentum as a scalar. It is a vector: resolve into $x$ and $y$ and include the $PA$ pressure forces on each face.
•Sign error on the reaction force: the force on the structure is equal and opposite to the force on the fluid.

References

NCEES FE Reference Handbook — Fluid Mechanics (One-Dimensional Fluid Flow)
NCEES FE Reference Handbook — Fluid Mechanics (Impulse-Momentum Principle)

A. Fluid properties

Fluid Properties and Flow Regimes

B. Dimensionless numbers

Dimensionless Numbers and Similitude

D. Fluid statics

Fluid Statics

E. Energy, impulse, and momentum equations

Continuity, Bernoulli, and Momentum

Conservation of mass — continuity

The Bernoulli equation

The energy equation with losses and machines

Energy grade line versus hydraulic grade line

Torricelli's result and stagnation

The impulse-momentum principle

Forces on bends, nozzles, and jets

Exam strategy

Key equations

Worked examples

Common pitfalls

References

F. Pipe and duct flow and friction losses

Pipe Flow and Friction Losses

G. Open-channel flow

Open-Channel Flow

H. Fluid transport systems

Fluid Transport Systems

I. Flow measurement

Flow Measurement

J. Turbomachinery

Pumps, Turbines, and Affinity Laws

K. Ideal gas law

Ideal and Real Gas Behavior