Kinematics of Fluid Motion: Eulerian and Lagrangian Views, Material Derivative, Streamlines, Pathlines and Streaklines

This chapter answers Section XE1.3 of XE1 Fluid Mechanics, one of the nine optional sections of the Engineering Sciences (XE) paper, any two of which a candidate chooses in the exam: “Eulerian and Lagrangian descriptions of fluid motion, Concept of local, Convective and material derivatives, Streamline, Streakline, and Pathline.” It teaches the two ways of describing a flow, the velocity and acceleration fields, the local, convective and material derivatives, the three families of flow lines and when they coincide, the dimensionality of a flow, and incompressible continuity as a purely kinematic condition. The numericals use given velocity fields: particle acceleration in unsteady and in steady non-uniform flow, the rate of change of temperature following a particle, a streamline through a point, a pathline, and the missing velocity component from continuity.

1. Lagrangian and Eulerian descriptions

Kinematics describes motion without asking what causes it. There are two ways to describe a flow. In the Lagrangian description you follow individual fluid particles, each labelled by its starting position, and record its position, velocity and properties as functions of time, exactly as in particle mechanics. In the Eulerian description you fix your attention on points of space and record the velocity V(x, y, z, t) and properties such as p(x, y, z, t) and T(x, y, z, t) there as the fluid goes past. The set of such functions is a field. A fluid has too many particles for the first method to be practical, so fluid mechanics uses the Eulerian description almost everywhere; a pressure probe or a laser measurement at a fixed point is Eulerian, while a drifting float or a dye particle is Lagrangian.

The two views are linked. The Lagrangian velocity of the particle that is at position r at time t is the Eulerian field evaluated there: dx/dt = u(x, y, z, t), dy/dt = v, dz/dt = w. The laws of mechanics (Newton’s second law, conservation of energy) apply to particles, so they are first written in Lagrangian form and then converted to the field form by the material derivative of the next section.

2. Velocity and acceleration fields: local, convective and material derivatives

A particle that is at (x, y, z) at time t moves to (x + u dt, y + v dt, z + w dt) at t + dt. The change in any field property, say T, seen by the particle follows from the chain rule, so the rate of change following the particle is the material (substantial) derivative: DT/Dt = ∂T/∂t + u ∂T/∂x + v ∂T/∂y + w ∂T/∂z = ∂T/∂t + (V·∇)T. The first term is the local derivative, the change at a fixed point because the field is unsteady. The remaining terms are the convective derivative, the change because the particle is carried to a place where the field has a different value. Applying this to the velocity gives the acceleration of a fluid particle: a = DV/Dt = ∂V/∂t + (V·∇)V, with components a_x = ∂u/∂t + u ∂u/∂x + v ∂u/∂y + w ∂u/∂z, and similarly for a_y and a_z.

Worked example 1 (unsteady 2-D field). Let u = xt and v = −yt (x, y in m, t in s, u, v in m/s). Then ∂u/∂t = x, ∂u/∂x = t, ∂u/∂y = 0, ∂v/∂t = −y, ∂v/∂x = 0, ∂v/∂y = −t. So a_x = x + (xt)(t) + 0 = x(1 + t²) and a_y = −y + 0 + (−yt)(−t) = −y(1 − t²). At x = 1 m, y = 2 m, t = 2 s: a_x = 5 m/s² and a_y = 6 m/s², so |a| = √(25 + 36) = 7.81 m/s².

Worked example 2 (scalar field). A steady temperature field T = 20 + 5x (°C, x in m) exists in a duct where the air moves along x at u = 2 m/s. A particle sees ∂T/∂t = 0 locally but DT/Dt = u ∂T/∂x = 2 × 5 = 10 °C/s as it is carried into warmer air. A thermometer fixed in the duct reads a steady value, yet each particle warms up.

3. Acceleration in steady non-uniform and unsteady uniform flow

The two terms of a = ∂V/∂t + (V·∇)V can be switched on separately. Steady flow has ∂V/∂t = 0 but a particle still accelerates if the velocity varies from point to point (non-uniform flow): the flow in a converging nozzle is the standard case. For a steady 1-D flow u(x), a = u du/dx = d(u²/2)/dx. Uniform flow has V the same at all points at an instant, so (V·∇)V = 0 and a = ∂V/∂t: water in a long pipe of constant diameter whose speed is being changed has only local acceleration. Both terms vanish only in flow that is both steady and uniform.

Worked example 3 (nozzle). Along the axis of a converging nozzle the steady speed is u = 2(1 + 2x) m/s, x in m. At x = 0.25 m, u = 2 × 1.5 = 3 m/s and du/dx = 4 s⁻¹, so a = u du/dx = 3 × 4 = 12 m/s² although ∂u/∂t = 0. Mean speed rises with x because the area falls (from continuity, uA = constant).

Along and across a streamline. In natural coordinates (s along the streamline, n towards the centre of curvature, radius R) the acceleration components are a_s = ∂V/∂t + V ∂V/∂s and a_n = V²/R. Fluid moving round a bend at constant speed still accelerates towards the centre of curvature, and this normal acceleration needs a pressure gradient across the streamlines; a straight streamline (R → ∞) has no normal acceleration.

⚠️ Trap
Steady is not zero acceleration, and uniform is not steady. A steady flow in a nozzle has a ≠ 0, and a uniform flow in a pipe with changing discharge has a ≠ 0. Say which of ∂V/∂t and (V·∇)V is zero before concluding. Also ∂/∂t of a field at a point is not DV/Dt unless the particle does not move.

4. Streamlines, pathlines and streaklines

Three flow lines
LineDefinitionEquation
StreamlineA curve tangent to the velocity vector at every point at one instantdx/u = dy/v = dz/w, with t held fixed
PathlineThe trajectory traced by one fluid particle over timedx/dt = u, dy/dt = v, dz/dt = w, integrated with t varying
StreaklineThe locus, at one instant, of all particles that have passed through a fixed pointIntegrate the pathlines of particles released at times τ, then set t fixed and vary τ

In a steady flow the velocity field does not change, so each particle follows the streamline through its start, and all particles passing a point follow the same path: streamlines, pathlines and streaklines coincide. In an unsteady flow they differ in general. Dye injected continuously from a point gives a streakline (smoke from a chimney in a changing wind is one); a long-exposure photograph of a single marked particle gives a pathline. Streamlines cannot be photographed directly but are inferred from many short-exposure tracer streaks. Because the velocity at a point has one direction, two streamlines do not cross except at a stagnation point or a singularity, and in steady flow no fluid crosses a streamline.

Worked example 4 (streamline, steady). For u = x, v = −y (s⁻¹, m): dy/dx = v/u = −y/x, so dy/y = −dx/x and ln y = −ln x + c, i.e. xy = constant, a rectangular hyperbola. The streamline through (2, 3) is xy = 6, so at x = 4 m it is at y = 1.5 m. The flow is steady, so this is also the pathline and streakline.

Worked example 5 (unsteady: all three differ). Let u = U = 2 m/s and v = bt with b = 1 m/s² (t in s). Streamline at instant t: dy/dx = bt/U, straight lines of slope t/2 that tilt as time passes. Pathline of the particle at the origin at t = 0: x = Ut = 2t and y = bt²/2 = t²/2, so y = x²/8, a parabola. When x = 4 m, t = 2 s and y = 2 m. Streakline at time t through the origin: a particle released at τ is at x = U(t − τ), y = b(t² − τ²)/2, so y = (bx/2U)(2t − x/U), another parabola that changes with t. At t = 2 s the three curves through the origin have different shapes.

5. Dimensionality of a flow and what uniform and steady mean

A flow is one-, two- or three-dimensional according to the number of space coordinates on which the velocity field depends. (Convention used here: count the coordinates on which the velocity components depend; some textbooks also call a pipe flow 'one-dimensional' when they use only the section-mean speed.) A fully developed laminar flow in a long circular pipe has only an axial velocity u(r), a function of one coordinate: it is one-dimensional. Flow between very wide parallel plates, u(y), is also one-dimensional. Flow past a long cylinder far from its ends, V(x, y), is two-dimensional (plane flow). A flow round a finite wing or a sphere (u, v, w all depend on x, y, z) is three-dimensional. Axisymmetric flow depends on r and z only and is two-dimensional in this sense.

Dimensionality, steadiness and uniformity are three independent labels. Steady means ∂/∂t = 0 at a point; uniform means no variation with position at an instant; one-dimensional means velocity depends on one space coordinate. A flow can be steady and one-dimensional but not uniform (a nozzle, if the area change is described along the axis), or unsteady and uniform (a piston-driven pipe flow). Choosing the right dimension is the main simplification in analysing a flow.

6. Continuity for incompressible flow as a kinematic condition

A flow is incompressible when the density of each particle does not change, Dρ/Dt = 0. Mass conservation, ∂ρ/∂t + ∇·(ρV) = 0, then reduces to ∇·V = ∂u/∂x + ∂v/∂y + ∂w/∂z = 0 (the continuity equation). This is a constraint on the velocity field alone: a field that does not satisfy it cannot be the velocity field of an incompressible flow, whatever the forces. Physically ∇·V is the rate of volumetric dilatation of a particle, so zero divergence means a fluid element can change shape but not volume. For plane flow ∂u/∂x + ∂v/∂y = 0, and given u one integrates to find v, with the constant of integration fixed by a boundary condition. It does not need the density to be uniform in space, since Dρ/Dt = 0 allows a stratified fluid.

Worked example 6. A plane incompressible flow has u = 2xy and v = 0 on the line y = 0. Continuity gives ∂v/∂y = −∂u/∂x = −2y, so v = −y² + f(x); v = 0 at y = 0 gives f = 0 and v = −y². At (x = 1 m, y = 3 m) the magnitude is |v| = 9 m/s. The field u = x², v = y² fails the test, since ∂u/∂x + ∂v/∂y = 2x + 2y is not zero everywhere, while u = 3x, v = −3y passes. In a 3-D case with u = 4x and v = −y, ∂w/∂z = −(4 − 1) = −3, so w = −3z.

🎯 Why this is kinematic
The condition ∇·V = 0 follows from volume conservation of a fluid particle and says nothing about forces. Streamlines of a steady incompressible 2-D flow crowd together where the speed is high, because the same flux must pass through a narrower strip between neighbouring streamlines. This is the reason a nozzle accelerates the fluid.

Key takeaways

  • The Lagrangian description follows particles; the Eulerian description records fields V(x, y, z, t) at fixed points and is the one used in fluid mechanics.
  • The material derivative is D/Dt = ∂/∂t + V·∇ (local plus convective); the particle acceleration is a = ∂V/∂t + (V·∇)V, with components such as a_x = ∂u/∂t + u ∂u/∂x + v ∂u/∂y + w ∂u/∂z.
  • Steady flow can still have acceleration (nozzle: a = u du/dx) and uniform flow can still have acceleration (a = ∂V/∂t); both terms vanish only for flow that is steady and uniform.
  • Streamlines solve dx/u = dy/v = dz/w at fixed t, pathlines solve dx/dt = u etc., streaklines are the locus of particles through a fixed point; all three coincide in steady flow.
  • Dimensionality is the number of space coordinates on which the velocity depends; fully developed pipe flow u(r) is one-dimensional and flow past a long cylinder is two-dimensional.
  • Incompressible continuity is the kinematic condition ∇·V = 0, which any admissible velocity field must satisfy and which lets one find a missing velocity component by integration.

Practice questions (17)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. Which statement describes the Eulerian description of a flow?

    1. Each fluid particle is labelled and its position is tracked as a function of time
    2. The velocity is recorded only along the streamlines, which are assumed to be closed curves
    3. Flow properties such as velocity and pressure are given as functions of fixed position and time
    4. The description applies only to steady flows, because time does not appear in the fields it uses and so no change with time can be described
    Show answer

    Answer: C — Flow properties such as velocity and pressure are given as functions of fixed position and time

    In the Eulerian view the fields are functions of (x, y, z, t) at fixed points; time appears, so unsteady flows are included. Labelling and following particles is the Lagrangian description, which is the tempting reversal.

  2. In a steady flow, which part of the particle acceleration a = ∂V/∂t + (V·∇)V is zero everywhere?

    1. The local acceleration ∂V/∂t
    2. The convective acceleration (V·∇)V
    3. Both parts, because a steady flow has zero acceleration
    4. Neither part, since steady describes pressure and not velocity
    Show answer

    Answer: A — The local acceleration ∂V/∂t

    Steady means no change with time at a fixed point, so ∂V/∂t = 0. The convective part is generally non-zero (a particle in a nozzle speeds up). Concluding that steady flow has zero acceleration forgets this convective term.

  3. A steady temperature field T = 20 + 5x (T in °C, x in m) exists in a duct. Air moves in the x direction at a uniform u = 2 m/s. What is the rate of change of temperature of a fluid particle, DT/Dt?

    1. 0 °C/s, because the field is steady
    2. 10 °C/s
    3. 5 °C/s, the gradient of the temperature field
    4. 20 °C/s, the temperature at the origin multiplied by u
    Show answer

    Answer: B — 10 °C/s

    DT/Dt = ∂T/∂t + u ∂T/∂x = 0 + 2 × 5 = 10 °C/s. The local derivative is zero, but the particle is carried into warmer air. Answering zero uses only the local derivative; the gradient 5 °C/m has the wrong units for a rate.

  4. Which statement about a streamline is correct?

    1. It is the path traced by one fluid particle as time passes in any flow
    2. It is the locus of all particles that have passed through a given fixed point
    3. It is always a closed curve around the body in a steady flow
    4. At each point on it, the velocity vector at that instant is tangent to the curve
    Show answer

    Answer: D — At each point on it, the velocity vector at that instant is tangent to the curve

    A streamline is defined by tangency to the instantaneous velocity. The path of one particle is a pathline and the locus through a fixed point is a streakline; only in steady flow do these coincide with streamlines. Streamlines need not be closed.

  5. Smoke is released continuously from a fixed point in an unsteady wind, and a photograph is taken at one instant. What does the visible smoke trail show?

    1. A pathline, the complete history of one particle that has left the source
    2. A streakline, the present positions of the particles that passed through the source
    3. A streamline, because streamlines are always the lines that tracers and dyes make visible in any flow
    4. A timeline, the positions at one later instant of particles all marked together at one earlier time
    Show answer

    Answer: B — A streakline, the present positions of the particles that passed through the source

    Particles released at different times from one fixed point and photographed together form a streakline. A pathline follows one particle over time, a streamline is tangent to the instantaneous velocity, and a timeline needs particles marked all at the same time, not released continuously.

  6. A flow has velocity components u = U (constant) and v = bt, with b a constant and t the time. Which statement is correct?

    1. The streamlines and pathlines coincide, because u is constant and a constant velocity component makes the whole flow steady, so every line is the same
    2. The streamlines at every instant are parabolas, because the vertical velocity grows linearly with time and the particle paths are curved
    3. The streakline and pathline are the same, but both differ from the streamline, which holds in a steady flow of this kind
    4. The streamlines at an instant are straight lines of slope bt/U, but a particle’s pathline is a parabola, so the two differ
    Show answer

    Answer: D — The streamlines at an instant are straight lines of slope bt/U, but a particle’s pathline is a parabola, so the two differ

    At fixed t, dy/dx = v/u = bt/U is constant, so each streamline is straight. A particle has x = Ut, y = bt²/2 (from the origin at t = 0), a parabola. Since v depends on t the flow is unsteady, so the lines differ; constant u does not make the flow steady.

  7. Which of the following two-dimensional velocity fields (u, v) can represent an incompressible flow?

    1. u = 2x, v = 2y
    2. u = x², v = y²
    3. u = 3x, v = −3y
    4. u = xy, v = xy
    Show answer

    Answer: C — u = 3x, v = −3y

    The test is ∂u/∂x + ∂v/∂y = 0. For u = 3x, v = −3y it is 3 − 3 = 0. The field 2x, 2y gives 4, the field x², y² gives 2x + 2y and the field xy, xy gives y + x, none of which is zero everywhere.

  8. Count the space coordinates on which the velocity components depend. Steady, fully developed laminar flow in a long straight circular pipe has only an axial velocity u(r), with r the radial distance from the axis. How is the flow classified?

    1. One-dimensional, because the velocity depends on a single coordinate r
    2. Two-dimensional, because a circular cross-section has two coordinates and the flow fills the whole section
    3. Three-dimensional, because the pipe is a body that occupies three-dimensional space
    4. Unsteady, because the velocity varies from the wall to the centre
    Show answer

    Answer: A — One-dimensional, because the velocity depends on a single coordinate r

    By the stated count, u depends only on r, so the flow is one-dimensional. The pipe being a 3-D object, or its section having two coordinates, does not matter; and variation with position is non-uniformity, not unsteadiness, since nothing changes with time.

  9. Water in a long straight pipe of constant diameter has a mean speed V(t) = 2 + 0.5t (m/s, t in s), the same at all points along the pipe. What is the acceleration of a fluid particle at t = 4 s?

    1. 0.5 m/s²
    2. 0 m/s², because the flow is uniform along the pipe
    3. 2.0 m/s², the product of the speed at t = 4 s and 0.5
    4. 4.0 m/s², the speed at t = 4 s
    Show answer

    Answer: A — 0.5 m/s²

    The flow is uniform, so the convective term V ∂V/∂x = 0, and a = ∂V/∂t = 0.5 m/s². Zero forgets the local acceleration; V × 0.5 multiplies by speed with no basis, and the speed itself (4 m/s) is not an acceleration.

  10. Statements about flow lines are given.
    P: In a steady flow, streamlines, pathlines and streaklines coincide.
    Q: In an unsteady flow, a streamline drawn at one instant is always the path of a particle.
    R: A streakline at time t is the locus of positions at t of all particles that have passed through a fixed point.
    S: The equation of a streamline is found by integrating dx/dt = u and dy/dt = v with t varying.
    Select all correct statements.

    1. P
    2. Q
    3. R
    4. S
    Show answer

    Answer: A — P; C — R

    P and R are correct. Q is wrong because in unsteady flow the particle path differs from the instantaneous streamline. S describes a pathline; a streamline integrates dx/u = dy/v with t held fixed.

  11. Statements about derivatives following a particle are given.
    P: DT/Dt = ∂T/∂t + V·∇T gives the rate of change of T following a fluid particle.
    Q: The local derivative is zero in a steady flow.
    R: The convective acceleration is zero in a flow where the velocity has the same value at every point at an instant.
    S: The material derivative of the velocity is zero in every steady flow.
    Select all correct statements.

    1. P
    2. Q
    3. R
    4. S
    Show answer

    Answer: A — P; B — Q; C — R

    P, Q and R follow from the definitions: in a uniform flow all spatial gradients of V vanish. S is wrong: a steady nozzle flow has DV/Dt = (V·∇)V ≠ 0.

  12. Statements about continuity are given.
    P: For incompressible flow ∂u/∂x + ∂v/∂y + ∂w/∂z = 0.
    Q: The two-dimensional field u = y, v = x satisfies incompressible continuity.
    R: Incompressible flow requires the density to be the same at every point of the fluid.
    S: The incompressible continuity equation expresses conservation of linear momentum.
    Select all correct statements.

    1. P
    2. Q
    3. R
    4. S
    Show answer

    Answer: A — P; B — Q

    P is the continuity equation, and for Q ∂u/∂x + ∂v/∂y = 0 + 0 = 0. R is wrong because incompressibility is Dρ/Dt = 0 and allows a stratified fluid. S is wrong: continuity is mass conservation.

  13. A two-dimensional velocity field is u = xt and v = −yt, with x and y in metres, t in seconds and u, v in m/s. The magnitude of the acceleration of a fluid particle at x = 1 m, y = 2 m and t = 2 s is ____ m/s² (to two decimal places).

    Numerical answer — type the value.

    Show answer

    Answer: 7.81

    a_x = ∂u/∂t + u ∂u/∂x + v ∂u/∂y = x + (xt)(t) = 1 + 1 × 2 × 2 = 5. a_y = ∂v/∂t + u ∂v/∂x + v ∂v/∂y = −y + (−yt)(−t) = −2 + 2 × 2 × 2 = 6. |a| = √(25 + 36) = √61 = 7.81 m/s².

  14. Along the axis of a converging nozzle, the steady velocity is u = 2(1 + 2x) m/s, where x is in metres. The acceleration of a fluid particle at x = 0.25 m is ____ m/s².

    Numerical answer — type the value.

    Show answer

    Answer: 12

    Steady, so a = u du/dx. u(0.25) = 2 × 1.5 = 3 m/s and du/dx = 4 s⁻¹, so a = 3 × 4 = 12 m/s².

  15. A steady two-dimensional flow has u = x and v = −y, in consistent SI units. The streamline passing through the point (x, y) = (2 m, 3 m) crosses the line x = 4 m at y = ____ m (to one decimal place).

    Numerical answer — type the value.

    Show answer

    Answer: 1.5

    dy/dx = v/u = −y/x gives xy = constant. At (2, 3) the constant is 6, so at x = 4, y = 6/4 = 1.5 m.

  16. The velocity field is u = 2 m/s and v = t m/s (t in seconds). A fluid particle is at the origin at t = 0. At the instant when its x coordinate is 4 m, its y coordinate is ____ m (to one decimal place).

    Numerical answer — type the value.

    Show answer

    Answer: 2.0

    Pathline: dx/dt = 2 gives x = 2t, and dy/dt = t gives y = t²/2. At x = 4, t = 2 s, so y = 4/2 = 2.0 m. Equivalently y = x²/8 = 16/8 = 2. Using the streamline slope at t = 2 instead of integrating over time is the usual slip.

  17. An incompressible plane flow has u = 2xy, and v = 0 along the line y = 0. The magnitude of v at the point (x, y) = (1 m, 3 m) is ____ m/s.

    Numerical answer — type the value.

    Show answer

    Answer: 9

    Continuity: ∂v/∂y = −∂u/∂x = −2y. Integrating, v = −y² + f(x); v = 0 at y = 0 gives f(x) = 0. At y = 3, v = −9 m/s, so the magnitude is 9 m/s.