Fluid Kinematics

Learning Objectives

  • Classify fluid flows based on spatial, temporal, and mixing characteristics.
  • Differentiate between Eulerian and Lagrangian descriptions of fluid motion.
  • Apply the Continuity Equation to steady and unsteady flows.
  • Explain stream functions, velocity potentials, and their relationship to irrotational flow.
  • Describe the properties and construction of a flow net.

Study of fluid motion without considering the forces causing it. This includes classifying flow types, understanding Eulerian and Lagrangian approaches, applying the continuity equation (conservation of mass), and utilizing flow nets to evaluate seepage in civil engineering structures like dams and retaining walls.

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.

V⃗=V⃗(r⃗0,t)\vec{V} = \vec{V}( \vec{r}_0, t )

Variables

SymbolDescriptionUnit
Velocity vector of the fluid particlem/s
Initial position vector of the particle at time t = 0m
ttTime elapseds

Eulerian Velocity Description

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

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

Variables

SymbolDescriptionUnit
Velocity vector at the specific pointm/s
x,y,zx, y, zSpatial coordinatesm
ttTimes

Types of Flow

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

Steady Flow

Properties (velocity, pressure, density) at any point in the fluid do not change with time (∂V/∂t=0\partial V/\partial t = 0).

Unsteady Flow

Properties of the fluid at any point change with time (∂V/∂t≠0\partial V/\partial t \neq 0).

Uniform Flow

The velocity vector is constant along a streamline at any given instant (∂V/∂s=0\partial V/\partial s = 0).

Non-Uniform Flow

The velocity vector changes along a streamline (e.g., occurs during a pipe expansion or contraction).

Laminar Flow

Fluid particles move in smooth, parallel layers with negligible mixing; viscous forces dominate over inertial forces.

Turbulent Flow

Fluid particles move erratically with significant mixing; inertial forces dominate over viscous forces. It is characterized by eddies and vortices.

Streamline

A continuous line that is everywhere tangent to the velocity vector of the fluid at a given instant. No flow can cross a streamline. In steady flow, streamlines remain constant in shape and position.

Pathline

The actual geometric path traveled by a single, individual fluid particle over a period of time. It is analogous to a time-exposure photograph of a single illuminated particle.

Streakline

The locus of all fluid particles that have previously passed through a specific prescribed point (e.g., a continuous trail of smoke from a chimney or dye injected into a pipe).

Steady Flow Lines

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.

∂ρ∂t+∇⋅(ρV⃗)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0

Variables

SymbolDescriptionUnit
Fluid densitykg/m3kg/m^3
ttTimes
Velocity vectorm/s
Del (gradient) operator1/m

Steady Flow Mass Balance

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

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

Variables

SymbolDescriptionUnit
Mass flow rate enteringkg/s
Mass flow rate leavingkg/s

One-Dimensional Steady Flow Continuity

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

ρ1A1V1=ρ2A2V2\rho_1 A_1 V_1 = \rho_2 A_2 V_2

Variables

SymbolDescriptionUnit
Fluid density at sections 1 and 2kg/m3kg/m^3
AACross-sectional area at sections 1 and 2m2m^2
VVAverage velocity at sections 1 and 2m/s

Steady, Incompressible Flow Continuity

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

Q=A1V1=A2V2Q = A_1 V_1 = A_2 V_2

Variables

SymbolDescriptionUnit
QQDischarge (volumetric flow rate)m3/sm^3/s
A1,A2A_1, A_2Cross-sectional areas at sections 1 and 2m2m^2
V1,V2V_1, V_2Average velocities at sections 1 and 2m/s

Stream Function and Velocity Potential

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

Stream Function (ψ\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 ψ\psi are streamlines.
  • The difference in ψ\psi between two streamlines equals the volumetric flow rate per unit depth passing between them (Δq=ψ2−ψ1\Delta q = \psi_2 - \psi_1).

Stream Function Relations

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

u=∂ψ∂y,v=−∂ψ∂xu = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}

Variables

SymbolDescriptionUnit
uuVelocity component in the x-directionm/s
vvVelocity component in the y-directionm/s
Stream functionm2/sm^2/s
x,yx, ySpatial coordinatesm

Velocity Potential (ϕ\phi)

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

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

Velocity Potential Relations

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

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

Variables

SymbolDescriptionUnit
uuVelocity component in the x-directionm/s
vvVelocity component in the y-directionm/s
Velocity potentialm2/sm^2/s
x,yx, ySpatial coordinatesm

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.

Flow Net Construction

  1. Streamlines (ψ\psi): Flow paths.
  2. Equipotential Lines (ϕ\phi): Lines of constant total head.

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 (Δq\Delta q) between any two adjacent streamlines (a flow channel) is constant.

Engineering Applications of Flow Nets

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, QQ) must remain constant throughout a single pipe system.
  • Because Q=A×VQ = A \times V, velocity is inversely proportional to the cross-sectional area. As a pipe narrows, the fluid must speed up.
  • The Stream Function (ψ\psi) exists for all 2D incompressible flows, satisfying continuity. Lines of constant ψ\psi represent actual flow paths.
  • Rotational vs. Irrotational Flow: Determines whether fluid particles spin around their own center of mass.
  • The Velocity Potential (ϕ\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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