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.

Approaches to Describe Flow

Lagrangian vs. Eulerian

  • Lagrangian Approach: Follows individual fluid particles as they move through space and time. Like tracking a specific car on a highway.

    Lagrangian vs. Eulerian

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

    V⃗=V⃗(r⃗0,t)\vec{V} = \vec{V}( \vec{r}_0, t )
  • Eulerian Approach: Observes the flow field at fixed points in space as fluid passes through. Like a traffic camera at a specific location.
    V⃗=V⃗(x,y,z,t)\vec{V} = \vec{V}( x, y, z, t )
Most fluid mechanics problems use the Eulerian approach.

Types of Flow

Steady vs. Unsteady Flow

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

Uniform vs. Non-Uniform Flow

  • Uniform Flow: Velocity vector is constant along a streamline at any instant (∂V/∂s=0\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

Classifying flow based on the rotation of fluid particles.

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

The continuity equation is the mathematical statement of the Conservation of Mass.

Continuity Equation

For a control volume:

Continuity Equation

Classifying flow based on the rotation of fluid particles.

∂ρ∂t+∇⋅(ρV⃗)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0
For Steady Flow: The mass flow rate entering equals the mass flow rate leaving.
m˙in=m˙out\dot{m}_{in} = \dot{m}_{out}
ρ1A1V1=ρ2A2V2\rho_1 A_1 V_1 = \rho_2 A_2 V_2
For Steady, Incompressible Flow (ρ=constant\rho = \text{constant}): Volume flow rate (QQ) is constant.
Q=A1V1=A2V2Q = A_1 V_1 = A_2 V_2

Variables

SymbolDescriptionUnit
QQDischarge$m^3/s$
AACross-sectional area$m^2$
VVAverage velocity$m/s$

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.

Stream Function ($\psi$)

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

Velocity Potential (ϕ\phi)

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

Velocity Potential ($\phi$)

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

Flow Nets

A graphical technique for 2D irrotational flow problems (like seepage under a dam). It consists of a grid of:

Procedure

  1. Streamlines (ψ\psi): Flow paths.
  2. Equipotential Lines (ϕ\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 (Δq\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, 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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