Kinematics of Fluid Flow Notes

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 $a_{n} = \frac{V ^{2}}{r}$ .

Downloaded from civilgyantests.com · Not for resale. Verify every code clause and date against the official source.

Kinematics of Fluid Flow