This notes sheet looks at the foundations of fluid dynamics, including the difference between fluids & solids and liquids & gasses, the continuum viewpoint, field descriptions for movement of fluids, as well as streamlines, pathlines and streaklines.

In fluid mechanics, ‘flow’ is what we call a fluid’s tendency to deform continuously under a shear force. This response to tangential forces and not normal forces is what differentiates fluids from solids:

Fundamentals of fluid dynamics, pressure on a solid vs pressure on a fluid/liquid, compression of a solid vs a fluid

Both the solid and the fluid will deform slightly under the uniform normal force, but then resist any further compression.

Fundamentals of fluid dynamics, shear force on a solid vs a fluid, what is flow

When a shear (tangential) force is applied to a solid, it deforms slightly in the direction of the force, but then resists further deformation. When a shear force is applied to a fluid, however, it immediately begins to deform, and will continue to do so until the force is removed (it will then eventually come to rest due to the frictional force between the fluid and the container wall).

When acted on by a normal force, solids and fluids behave similarly. When acted on by a shear force, they behave very differently.

As well as differentiating solids and fluids, we must differentiate between liquids and gasses:

  • A gas will fill the whole space it is given, a liquid will fill the bottom
  • Gasses are far more compressible, as their density is dependent on pressure

The Continuum Viewpoint

Fluids, like solids, are made up of a vast number of molecules. If we really wanted to, we could therefore investigate the flow and deformation of fluids by looking at the motion of each individual molecule. This is called the molecular viewpoint and is the most fundamental viewpoint – it is used in the pure sciences, but in engineering it is far too complex.

Instead, we make the continuum assumption:

A body consists of infinitely many homogeneous elements, each one significantly smaller than the body itself, but significantly larger than the individual molecules.

We model the elements as homogeneous, as this allows us to define important properties, such as density, temperature etc. at the point of the element.

Assuming that the body is not homogenous on the whole, we can find the average density of the body:

    \[  \rho = \frac{m}{v}  \]

If we define an element in the body to have mass \delta m and volume \delta V, where the volume tends to zero (the element is infinitely small), the density of said element is:

    \[ \rho = \lim_{\delta V \to 0} \frac{\delta m}{\delta V}\]

The continuum assumption suggests that as the element gets smaller and smaller, the density does not change.

This assumption is only valid at certain scales.

Foundations of fluid dynamics, the continuum scale

If \delta V gets too close to zero, we intrude on the molecular scale where density fluctuates hugely between molecules, if \delta V is too far from zero, the assumption is vastly inaccurate.

Field Descriptions

The continuum viewpoint allows us to model properties such as density as a function in space. This is because they are not uniform throughout the body, but they are distributed continuously:

Scalar quantity field description, field description in 3D space, density field description, foundations of fluid dynamics

Density is a scalar quantity, so the above graph is a ‘scalar field’. If we want to model vector properties, then each plot is a vector from a point, not just a point: it has magnitude and direction as well as location. This is known as a ‘vector field’:

Vector quantity field description, vector description in 3D space, velocity field description, foundations of fluid dynamics

Scalar and vector fields can be steady or unsteady:

  • Steady functions do not change with respect to time
  • Unsteady functions do change with time

Therefore, we can write functions as:

Steady Scalar Functionsf(x,y,z)
Unsteady Scalar Functionsf(x,y,z,t)
Steady Vector Functions\vec u (x,y,z)
Unsteady Vector Functions\vec u (x,y,z,t)

The most common example in fluid mechanics is the velocity field:

Vector quantity field description, vector description in 3D space, velocity field description, foundations of fluid dynamics

    \[ \vec u (x,y,z,t) = u(x,y,z,t) \hat i + v(x,y,z,t) \hat j + w(x,y,z,t) \hat k \]

This is often written in shorthand as:

    \[ \vec u = u \hat i + v \hat j + w \hat k\]

Generally, velocity fields are three dimensional, however sometimes they can be one or two dimensional:

one and two dimensional velocity fields, 2D velocity field, foundations of fluid dynamics

A common example of a one-dimensional velocity field is for steady laminar pipe flow. We can reduce this three-dimensional system to be one dimensional by plotting it in polar form, assuming that the cross-sectional profile is the same everywhere and also axisymmetric (circular – does not vary with θ):

steady laminar pipe flow, fully developed pipe flow velocity field, foundations of fluid dynamics

Streamlines & Pathlines

To simplify complex velocity fields, we can plot streamlines. These are lines that are always tangential to the instantaneous velocity vectors, and can be thought of as ‘joining the dots’ (ish).

Streamlines, streamlines on a vector field, streamlines in a velocity field, foundations of fluid dynamics

These streamlines are for the same two-dimensional velocity field shown higher in the page.

Streamlines can never cross

This means that the streamlines in a tube are always contained in the tube:

Streamlines in a tube, streamlines in a vessel, streamlines crossing, streamlines cannot cross, foundations of fluid dynamics

Steady Streamlines

For a steady two-dimensional velocity field, u(x, y), the equation for the streamlines can be found by integrating:

\vec u (x,y) = u(x,y) \hat i + v(x,y) \hat j

\frac{dy}{dx} = \frac{v(x,y)}{u(x,y)}

    \[ \int \frac{1}{v(x,y)} dy = \int \frac{1}{u(x,y)} dx \]

This will give a family of streamlines (because of the ‘+c’ constant of integration). To find a specific streamline, you need to know coordinates.

In steady flow, the streamline is constant. This means that the particles always follow it, and so we can investigate it by observing how some marked fluid particles (e.g. dye) travel.

Unsteady Streamlines & Pathlines

In unsteady flow, the fluid particles always follow the streamline, but the streamline is constantly changing.

This means that the particles eventually deviate from the observed instantaneous streamline. The actual trajectory of a particle is therefore not shown by the streamline, but by another line: the pathline.

For two-dimensional fields, the pathline can be found in parametric form, with parameter t (time):

\vec u(x,y,t) = u(x,y,t) \hat i + v(x,y,t) \hat j

u(x,y,t) = \frac{dx}{dt}, therefore: x = \int u(x,y,t) dt

v(x,y,t) = \frac{dy}{dt}, therefore: y = \int v(x,y,t) dt


Streaklines are a third type of line we can plot from unsteady velocity fields: these are lines that join particles that pass through the same point, at different times. For example, smoke particles passing through a chimney.

In steady flow, streamline, pathlines and streaklines are all the same.

  • Flow is a fluid’s tendency to continuously deform under a shear stress
  • In engineering, we apply the continuum viewpoint to simplify fluid models
  • Properties such as density can be modelled as scalar fields, and quantities like velocity can be modelled in vector fields
  • Steady functions do not change with respect to time; unsteady functions do
  • Streamlines are tangential to the instantaneous vectors on the velocity profile
    • For steady flow in the form \vec u(x,y,t) = u(x,y,t) \hat i + v(x,y,t) \hat j, the family of streamlines is given as \int \frac{1}{v(x,y)} dy = \int \frac{1}{u(x,y)} dx
    • For unsteady flow in the form \vec u(x,y,t) = u(x,y,t) \hat i + v(x,y,t) \hat j, the two-dimensional streamlines are given as x = \int u(x,y,t) dt and y = \int v(x,y,t) dt
  • Pathlines are used in unsteady flow to model the actual trajectory of a fluid particle
  • Streaklines join particles that pass through the same point in space at different points in time