Fluid Kinematics

Fluid Kinematics

Study of fluid motion without considering forces, including flow types, continuity equation, and flow nets.

Concept Overview

Fluid kinematics deals with the geometry of motion: velocity, acceleration, and flow patterns, without considering the forces causing the motion. It focuses on how fluid moves.

Lagrangian vs. Eulerian

Most fluid mechanics problems use the Eulerian approach.

Lagrangian Approach: Follows individual fluid particles as they move through space and time. Like tracking a specific car on a highway.
Eulerian Approach: Observes the flow field at fixed points in space as fluid passes through. Like a traffic camera at a specific location.

Lagrangian Velocity Description

Expresses the velocity of an individual fluid particle as a function of its initial position and time.

\vec{V} = \vec{V}( \vec{r}_0, t )

Variables

Symbol	Description	Unit
	Velocity vector of the fluid particle	m/s
	Initial position vector of the particle at time t = 0	m
$t$	Time elapsed	s

Eulerian Velocity Description

Expresses the fluid velocity at a fixed point in space as a function of coordinates and time.

\vec{V} = \vec{V}( x, y, z, t )

Variables

Symbol	Description	Unit
	Velocity vector at the specific point	m/s
$x, y, z$	Spatial coordinates	m
$t$	Time	s

Types of Flow

Fluid flow is classified by whether its properties change with time, position, and internal mixing pattern.

Steady vs. Unsteady Flow

Steady Flow: Properties (velocity, pressure, density) at any point do not change with time ( $\partial V/\partial t = 0$ ).
Unsteady Flow: Properties change with time ( $\partial V/\partial t \neq 0$ ).

Uniform vs. Non-Uniform Flow

Uniform Flow: Velocity vector is constant along a streamline at any instant ( $\partial V/\partial s = 0$ ).
Non-Uniform Flow: Velocity changes along a streamline (e.g., pipe expansion).

Laminar vs. Turbulent Flow

Laminar Flow: Fluid particles move in smooth, parallel layers; viscous forces dominate.
Turbulent Flow: Fluid particles move erratically; inertial forces dominate. Characterized by eddies and mixing.

Flow Visualization

Streamline: A line everywhere tangent to the velocity vector at a given instant. No flow crosses a streamline. In steady flow, streamlines remain constant in shape and position.
Pathline: The actual path traveled by a single fluid particle over time. Think of it as a time-exposure photograph of a single illuminated particle.
Streakline: The locus of particles that have earlier passed through a prescribed point (e.g., smoke from a chimney or dye injected continuously into a pipe).

Note

In a steady flow, streamlines, pathlines, and streaklines are all identical. They only diverge in unsteady flow.

Rotational vs. Irrotational Flow

Rotational and irrotational classifications describe whether fluid particles spin about their own centers as they translate through the flow field.

Rotational Flow

Fluid particles rotate about their own mass centers as they move along a streamline.
Occurs when viscous forces or uneven boundary shear forces are present.

Irrotational Flow

Fluid particles do not rotate about their own mass centers as they flow. They translate and deform, but their orientation remains parallel to their original position.
Often assumed for ideal (inviscid) fluids outside of the boundary layer.

Continuity Equation Concept

The continuity equation is the mathematical statement of the Conservation of Mass. It applies to any control volume, ensuring that mass cannot be created or destroyed.

General Continuity Equation

The differential form of the continuity equation expressing mass conservation in a fluid flow.

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0

Variables

Symbol	Description	Unit
	Fluid density	$k g / m^{3}$
$t$	Time	s
	Velocity vector	m/s
	Del (gradient) operator	1/m

Steady Flow Mass Balance

For steady flow, the total mass flow rate entering a control volume equals the total mass flow rate leaving.

\dot{m}_{in} = \dot{m}_{out}

Variables

Symbol	Description	Unit
	Mass flow rate entering	kg/s
	Mass flow rate leaving	kg/s

One-Dimensional Steady Flow Continuity

Relates density, area, and velocity at two sections in a steady flow.

\rho_1 A_1 V_1 = \rho_2 A_2 V_2

Variables

Symbol	Description	Unit
	Fluid density at sections 1 and 2	$k g / m^{3}$
$A$	Cross-sectional area at sections 1 and 2	$m^{2}$
$V$	Average velocity at sections 1 and 2	m/s

Steady, Incompressible Flow Continuity

Relates the volumetric flow rate (discharge) of a fluid in steady, incompressible flow where density is constant.

Q = A_1 V_1 = A_2 V_2

Variables

Symbol	Description	Unit
$Q$	Discharge (volumetric flow rate)	$m^{3} / s$
$A_{1}, A_{2}$	Cross-sectional areas at sections 1 and 2	$m^{2}$
$V_{1}, V_{2}$	Average velocities at sections 1 and 2	m/s

Stream Function and Velocity Potential

In 2D incompressible flow, two scalar functions are used to describe the flow field mathematically.

Stream Function ( $\psi$ )

A mathematical function defined such that the velocity components are given by its partial derivatives. It satisfies the continuity equation identically.

Lines of constant $\psi$ are streamlines.
The difference in $\psi$ between two streamlines equals the volumetric flow rate per unit depth passing between them ( $\Delta q = \psi_2 - \psi_1$ ).

Stream Function Relations

Relates the 2D velocity components to the partial derivatives of the stream function.

u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}

Variables

Symbol	Description	Unit
$u$	Velocity component in the x-direction	m/s
$v$	Velocity component in the y-direction	m/s
	Stream function	$m^{2} / s$
$x, y$	Spatial coordinates	m

Velocity Potential ( $\phi$ )

A mathematical function defined for irrotational flow, such that velocity components are given by its gradient.

Lines of constant $\phi$ are equipotential lines.
Flow is irrotational ( $\nabla \times \vec{V} = 0$ ) if and only if a velocity potential exists. This condition leads to Laplace's equation for $\phi$ ( $\nabla^2 \phi = 0$ ).

Velocity Potential Relations

Relates the 2D velocity components to the partial derivatives of the velocity potential.

u = \frac{\partial \phi}{\partial x}, \quad v = \frac{\partial \phi}{\partial y}

Variables

Symbol	Description	Unit
$u$	Velocity component in the x-direction	m/s
$v$	Velocity component in the y-direction	m/s
	Velocity potential	$m^{2} / s$
$x, y$	Spatial coordinates	m

Flow Nets

A flow net is a graphical technique for 2D irrotational flow problems such as seepage under a dam. It consists of a grid of streamlines and equipotential lines.

Procedure

STEP-BY-STEP

Streamlines ( $\psi$ ): Flow paths.
Equipotential Lines ( $\phi$ ): Lines of constant total head.

Note

Properties of Flow Nets:

Streamlines and equipotential lines intersect at 90 degrees (they are orthogonal).
Ideally, the grid elements form "curvilinear squares" where the ratio of lengths of opposite sides approaches 1 as the grid becomes finer.
The flow rate ( $\Delta q$ ) between any two adjacent streamlines (a flow channel) is constant.

Key Takeaways

The Eulerian approach (focusing on a fixed spatial window) is predominantly used in fluid mechanics, rather than the Lagrangian approach (tracking individual particles).
Flow can be categorized by how it behaves over time (steady vs. unsteady), over space (uniform vs. non-uniform), and by its internal mixing (laminar vs. turbulent).
Streamlines, pathlines, and streaklines are key visualization tools that coincide perfectly only when the flow is steady.
The Continuity Equation is the fundamental mathematical expression of the principle of conservation of mass.
For steady, incompressible flow, the volumetric flow rate (discharge, $Q$ ) must remain constant throughout a single pipe system.
Because $Q = A \times V$ , velocity is inversely proportional to the cross-sectional area. As a pipe narrows, the fluid must speed up.
The Stream Function ( $\psi$ ) exists for all 2D incompressible flows, satisfying continuity. Lines of constant $\psi$ represent actual flow paths.
Rotational vs. Irrotational Flow: Determines whether fluid particles spin around their own center of mass.
The Velocity Potential ( $\phi$ ) only exists for irrotational flow (zero vorticity).
A Flow Net is a powerful graphical grid formed by orthogonal streamlines and equipotential lines, used to visualize flow patterns and estimate seepage or pressure distribution in complex geometries.

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Concept Overview

Lagrangian vs. Eulerian

Lagrangian Velocity Description

Eulerian Velocity Description

Types of Flow

Steady vs. Unsteady Flow

Uniform vs. Non-Uniform Flow

Laminar vs. Turbulent Flow

Flow Visualization

Note

Rotational vs. Irrotational Flow

Rotational Flow

Irrotational Flow

Continuity Equation Concept

General Continuity Equation

Steady Flow Mass Balance

One-Dimensional Steady Flow Continuity

Steady, Incompressible Flow Continuity

Stream Function and Velocity Potential

Stream Function (ψ\psiψ)

Stream Function Relations

Velocity Potential (ϕ\phiϕ)

Velocity Potential Relations

Flow Nets

Procedure

Note

Stream Function ( $\psi$ )

Velocity Potential ( $\phi$ )