IV. Kinematics of Fluid Motion

Contents 1. Specification of Fluid Motion 2. Material Derivatives 3. Geometric Representation of Flow 4. Terminology 5. Motion and Deformation of Fluid Element 6. Rotational and Potential Flows 7. Continuity Equation

1. Specification of Fluid Motion

Lagrangian View Study fluid motion by tracing the motion of fluid particles

Identify a representative fluid particle Determine its position instantaneously Determine the velocity and acceleration Determine other physical quantities

Eulerian View Study fluid motion by investigating the temporal and spatial variation of the flow field

2. Material Derivatives

Definition The rate of change one observed when following the motion of a fluid particle

Local / Temporal Advective / Spatial Material Derivative

Acceleration of Fluid particles

3. Geometric Representation of Flow

Pathline

A pathline is the trajectory of a fluid particle

Mathematical representation

Streamline

A streamline is a line whose tangent always represents the direction of velocity

Mathematical representation

Example Find the pathline and streamline of the following flow field:

Pathline

Streamline

Streamline is identical to pathline if the velocity is invariable with time In general, streamlines will not intercross and will not end at a solid wall, etc.

4. Terminology

Discharge and Mass flux

Streamtube, Stream filament, Total flow

Fluid system and Control volume

Steady flow and Unsteady flow

Uniform flow and Non-uniform flow

The streamlines of a uniform flow is necessarily straight lines and parallel to each other

Gradually-varying flow and Rapidly-varying flow Curvature of all streamlines are small Curvature of all streamlines are small Streamlines are nearly parallel Streamlines are nearly parallel

5. Motion and Deformation of Fluid Elements

Motion of a fluid element can be decomposed into Translation Translation Rotation Rotation Deformation Deformation

The translation is described by

The rotation is described by

The angular velocity

The deformation is described by

Rate of strain

Helmholtz’s theorem of velocity decomposition

Translation Rotation Deformation

6. Rotational and Potential Flows

Physical Interpretation

Example

Velocity Potential Irrotational flowPotential flow

7. Continuity Equation

Conservation of Mass: Mass in a closed system is invariant

Net outflow of mass through the surface of the control volume

Decrease of mass within the control volume

Mass Conservation

For incompressible fluid Bulk expansion

Continuity Equation for Steady Total Flows SoSo SeSe

Continuity Equation for Potential Flows