← Hydraulics & Water Resources Engineering · TNPSC AE Civil

Chapter 7 of 15

Momentum equation applications; kinematics of flow

In the TNPSC AE Civil syllabus under Hydraulics & Water Resources Engineering · 2 parts

Impulse-Momentum Equation & Its Applications · Kinematics of Fluid Flow

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📑 Contents (17 sections)

Part 1 of 2

Impulse-Momentum Equation & Its Applications

Last reviewed 16 Sept 2026 · 6 min read

Impulse-momentum principle

Newton's second law applied to a flowing fluid: the net external force on the fluid in a control volume equals the rate of change of momentum of the fluid passing through it.

FormulaMomentum equation (steady flow, one inlet and one outlet)

\sum F_{x} = ρQ (V_{2 x} - V_{1 x}), \sum F_{y} = ρQ (V_{2 y} - V_{1 y})

Impulse form: $F d t = d (mV)$ .

$\sum F$ includes pressure forces on the inlet/outlet sections, gravity (weight of fluid) and the force from the solid boundary. The force exerted by the fluid on the boundary is equal and opposite.

The momentum equation is a vector equation — resolve into components.
Unlike Bernoulli's equation, it needs no information about internal losses; it is therefore used where losses are unknown (hydraulic jump, sudden enlargement, jets).
For non-uniform velocity, multiply momentum flux by the momentum correction factor $β$ (4/3 laminar pipe flow; ≈1.0–1.04 turbulent).

Force on pipe bends and reducers

For a bend turning the flow through angle $θ$ (horizontal plane), with inlet pressure $p_{1}$ , area $A_{1}$ , velocity $V_{1}$ and outlet $p_{2}$ , $A_{2}$ , $V_{2}$ , the force exerted by the fluid on the bend:

FormulaPipe bend

F_{x} = ρQ (V_{1} - V_{2} cos θ) + p_{1} A_{1} - p_{2} A_{2} cos θ

F_{y} = - (ρQ V_{2} sin θ + p_{2} A_{2} sin θ)

F = F_{x}^{2} + F_{y}^{2}

For a reducer (straight, $θ$ = 0): $F_{x} = ρQ (V_{1} - V_{2}) + p_{1} A_{1} - p_{2} A_{2}$ .
These forces are resisted by anchor (thrust) blocks at bends, tees, dead ends and reducers in water mains.

Force exerted by a jet

Jet of area $a$ , velocity $V$ , density $ρ$ .

Stationary plates

Plate	Force in the direction of the jet
Flat plate normal to the jet	$F = ρ a V^{2}$
Flat plate inclined at $θ$ to the jet	Normal to plate: $F_{n} = ρ a V^{2} sin θ$ ; along jet: $F_{n} sin θ$
Curved plate, jet at the centre (symmetric), deflected through angle $(180° - θ)$	$F = ρ a V^{2} (1 + cos θ)$
Hemispherical cup (jet turned back through 180°)	$F = 2 ρ a V^{2}$
Curved plate, jet striking at one tip (unsymmetrical, tangential entry)	$F_{x} = ρ a V^{2} (1 + cos θ)$

Moving plates

If the plate moves in the jet direction with velocity $u$ , the relative velocity is $(V - u)$ :

Case	Force	Work done per second
Single flat plate moving away	$ρ a (V - u)^{2}$	$ρ a (V - u)^{2} u$
Series of flat radial plates (wheel) — the whole jet is used	$ρ aV (V - u)$	$ρ aV (V - u) u$
Series of hemispherical (curved) vanes	$2 ρ aV (V - u)$	$2 ρ aV (V - u) u$

FormulaEfficiency of a series of vanes

Flat radial vanes: $η = \frac{2 u ( V - u )}{V ^{2}}$ → maximum 50% at $u = V /2$ .
Hemispherical vanes (Pelton-type): $η = \frac{4 u ( V - u )}{V ^{2}}$ → maximum 100% (theoretical) at $u = V /2$ .
Single moving flat plate: maximum work at $u = V /3$ ; maximum efficiency $8/27$ ≈ 29.6% (based on kinetic energy of the jet).

Jets on curved vanes with inlet and outlet angles

For turbines, velocity triangles at inlet and outlet give the work done per second per unit weight:

\frac{1}{g} (V_{w 1} u_{1} \pm V_{w 2} u_{2})

( $V_{w}$ = whirl component; + when the outlet whirl is opposite to vane motion). Detailed in Hydraulic Turbines.

Part 2 of 2

Kinematics of Fluid Flow

Last reviewed 16 Sept 2026 · 6 min read

Describing fluid motion

Kinematics describes velocity and acceleration of fluid without considering forces.

Lagrangian method — follows an individual fluid particle as it moves.
Eulerian method — observes velocity at fixed points in space as a function of position and time: $V = V (x, y, z, t)$ . Used almost universally in fluid mechanics.

Types of flow

Classification	Definition
Steady	Flow properties at a point do not change with time: $\partial V / \partial t = 0$
Unsteady	Properties at a point change with time
Uniform	Velocity does not change along the flow direction at a given instant: $\partial V / \partial s = 0$
Non-uniform	Velocity changes from point to point along the flow
Laminar	Particles move in smooth layers; viscous forces dominate (low Reynolds number)
Turbulent	Irregular, eddying motion with mixing (high Reynolds number)
Compressible / incompressible	Density varies / is constant
Rotational / irrotational	Fluid particles rotate / do not rotate about their own axes
One-, two-, three-dimensional	Velocity varies with one, two or three space coordinates

Combinations: steady uniform (flow at constant rate in a long straight pipe of constant diameter), steady non-uniform (constant flow in a tapering pipe), unsteady uniform (accelerating flow in a constant-diameter pipe), unsteady non-uniform (accelerating flow in a tapering pipe).

Flow patterns

Streamline — an imaginary line whose tangent at every point gives the direction of velocity at that instant. No flow crosses a streamline; streamlines cannot intersect. Equation: $\frac{d x}{u} = \frac{d y}{v} = \frac{d z}{w}$ .
Pathline — the actual path traced by one particle over time.
Streakline — the locus of all particles that have passed through a fixed point (e.g. dye injected continuously).
Streamtube — a tube formed by streamlines through a closed curve; fluid cannot cross its walls.

In steady flow, streamlines, pathlines and streaklines coincide.

Continuity equation

Conservation of mass:

FormulaContinuity

Along a streamtube (steady flow): $ρ_{1} A_{1} V_{1} = ρ_{2} A_{2} V_{2}$

Incompressible: $A_{1} V_{1} = A_{2} V_{2} = Q$ (discharge, m³/s)

Differential form (3-D, incompressible):

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0

Compressible, general: $\frac{\partial ρ}{\partial t} + \frac{\partial ( ρ u )}{\partial x} + \frac{\partial ( ρ v )}{\partial y} + \frac{\partial ( ρw )}{\partial z} = 0$

Acceleration

The acceleration of a fluid particle has two parts:

a_{x} = local (temporal) \frac{\partial u}{\partial t} + convective u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}

(similarly $a_{y}$ , $a_{z}$ ).

Local acceleration — due to unsteadiness; zero in steady flow.
Convective acceleration — due to change of velocity with position; zero in uniform flow.
Along a streamline: tangential $a_{s} = \frac{\partial V}{\partial t} + V \frac{\partial V}{\partial s}$ ; normal $a_{n} = \frac{V ^{2}}{r}$ .

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