Intuition The one core ideaA fluid flow is nothing but an arrow pinned to every point of space telling you which way the fluid moves there and how fast. Streamlines, pathlines, and streaklines are just three different ways of drawing pictures out of that field of arrows — so before we can tell them apart, we must be fluent in reading a single arrow.
This page assumes nothing . If the parent note Streamlines, pathlines, streaklines threw a symbol at you and you blinked, it lives here. We list every piece of notation, say what it means in plain words , show the picture it stands for, and explain why the topic can't live without it .
( x , y ) one location on a flat sheet. x y origin ( 0 , 0 )
Picture: a dot on graph paper. x y Why the topic needs it: the fluid fills space , so every question — "where is this leaf?", "where do I inject dye?" — is answered by a pair ( x , y )
We will only ever work in 2D (a flat sheet) to keep every picture drawable. Real flows add a third number z
τ second time-label used only for streaklines: the moment a particular puff of smoke left the chimney.
Picture: each puff of smoke wears a name-tag saying "I left at τ = 1.3 s Why: a streakline mixes puffs born at many different τ now at time t t τ
t τ t present moment (when you take the photo). τ birth moment . In a streakline you freeze t sweep τ τ ≤ t
Intuition A vector = an arrowA vector is an arrow: it has a length (how big) and a direction (which way). Two numbers describe it — how far along x y
v = ( u , v )
u x v y Picture: an arrow with a horizontal leg u v Why: velocity has both speed and heading, so one plain number can't hold it. An arrow can.
Common mistake Two things both called "v v whole velocity vector . Plain v vertical part . The parent note uses both. When in doubt, look for the little arrow.
Now we glue arrows to every point.
Definition Velocity fieldv ( r , t ) = ( u ( x , y , t ) , v ( x , y , t ) ) r = ( x , y ) t
r = ( x , y ) u ( x , y , t ) depends on where you are and when." Same for v Picture: a whole sheet peppered with little arrows, one per point — a "field" of arrows. Drag the time-slider and every arrow may swing.
Because there's a value (an arrow) at every point of the field of view , like temperature at every spot in a room. This is the Eulerian picture — watch fixed points, not fixed particles (see Lagrangian vs Eulerian description ).
Before any equation, here is what each of the three drawings is — one sentence each, no symbols beyond the arrow-field you now understand.
Definition Streamline (the snapshot)A ==streamline is a curve that is everywhere tangent to the velocity field at one fixed instant t
Picture: freeze time, then thread a smooth line through the arrows so the line always points the way the arrow beneath it points.Why: it captures "which way is everything pointing RIGHT NOW" — a picture of the whole field at one moment.
Definition Pathline (the trail)A pathline is the actual trajectory that one tagged fluid particle sweeps out as time runs forward. One particle, followed across all times.
Picture: drop a single dyed leaf and film its whole journey; the wet trail it leaves is the pathline.Why: it answers "where did THIS particle actually go", integrating the arrows the particle meets moment by moment.
Definition Streakline (the plume)A ==streakline is, at one fixed instant t
Picture: blow smoke continuously from one chimney and photograph the hanging plume — every puff, born at a different τ now .Why: it is exactly what a dye or smoke experiment shows, so we must know how it differs from the other two.
The rest of this page builds the symbols those three definitions secretly rely on.
Definition The differentialsd x , d y d x tiny step to the right. d y tiny step up. "Tiny" means so small the curve looks straight over that step.
Picture: zoom into any curve until it's a straight micro-segment; d x d y d r = ( d x , d y ) along the curve.Why: a streamline is a curve , and "the curve points the same way as the velocity" is a statement about its tiny direction arrow d r
Definition The scaling factorλ λ λ ∈ R stretches or shrinks one arrow into another pointing the same way. Saying d r = λ v d r v λ
Picture: the tiny step arrow d r v λ
Common mistake Division by zero — whenu = 0 v = 0 u d x = v d y both components are nonzero . If u = 0 not divide by u d x = λ u = 0 vertically (x v = 0 horizontal streamline (y both u = 0 v = 0 stagnation point — where no single direction exists and the streamline can branch. Always handle these pure-horizontal, pure-vertical, and zero cases from d x = λ u , d y = λ v
Definition Position vector of a moving particler ( t ) = ( x ( t ) , y ( t )) functions of time : x ( t ) , y ( t ) r ( t ) = ( x ( t ) , y ( t ))
Picture: an arrow rooted at the origin whose tip rides along with the particle, sweeping out the pathline as t Why: the pathline is the set of tips of r ( t )
d t d x x d t x d x rightward speed .
Picture: the slope of the x t Why the topic needs it: the particle's velocity is the time-derivative of its position vector, d t d r = ( d t d x , d t d y ) d t d x = u , d t d y = v
Common mistake Streamline ratio vs pathline derivative — the key confusionu d x = v d y t d t d x = u t
Definition Integrating (adding up tiny steps)The reverse of the derivative: given the rate , rebuild the whole journey by summing all the tiny steps. The symbol ∫
Picture: stacking many tiny d x
x 0 t 0 C t 0 initial time , the instant you start tracking (the clock reading when the particle sits at its start point). x 0 initial position at that instant, so x ( t 0 ) = x 0
The leftover ==constant C starting information the rate alone can't recover; fixing it to match x ( t 0 ) = x 0 which curve of the whole family you are on.
Why: solving u d x = v d y d t d x = u means integrating. Different ( x 0 , t 0 ) C
Worked example Tiny warm-upTake d t d x = 1 x 0 t 0 x ( t 0 ) = x 0 t 0 t x = x 0 + ∫ t 0 t 1 d t ′ = x 0 + ( t − t 0 ) . C x 0 x ( t 0 ) = x 0
∂ t ∂ v ∂ position fixed , wiggle only time , and ask if the arrow at that fixed spot changes.
Picture: stand at one dot, stare at its arrow, and watch the clock. If the arrow never moves, this quantity is zero .Why: the parent's golden rule hinges on this. If ∂ t ∂ v = 0 steady and all three curves coincide; otherwise they split apart (see Steady vs unsteady flow ).
Intuition Steady still means motion!Steady does not mean "nothing moves." Particles still race along. It means the pattern of arrows is frozen — each fixed point always shows the same arrow, so newcomers all trace the same route.
Tiny step d r along a curve
Streamline ratio dx/u = dy/v
Integration and constant C
Partial dv/dt steady test
Streamlines pathlines streaklines
Test yourself — cover the right side.
I can read ( x , y ) Yes: x y
I know the difference between t τ t τ t τ ≤ t
I can tell v v The little arrow
means the full vector
( u , v ) ; bare
v is only the
y -component.
I can state in words what a streamline, pathline, and streakline are Streamline = curve tangent to the field at one instant; pathline = one particle's trajectory over time; streakline = all particles through one point, seen at one instant.
I can picture a velocity field as an arrow at every point A sheet peppered with arrows, one per location, possibly swinging as time passes.
I know d r = ( d x , d y ) along a curve Zoom in until the curve is a straight micro-segment; its legs are d x , d y
I know λ λ ∈ R d x = λ u , d y = λ v u d x = v d y
I know what to do when u = 0 v = 0 Do not divide; read from d x = λ u , d y = λ v u = 0 v = 0
I know r ( t ) = ( x ( t ) , y ( t )) An origin-rooted arrow whose tip rides the particle and sweeps the pathline.
I can read d t d x The slope of x t x
I know why a constant C x 0 , t 0 Integration can't recover the start; setting C x ( t 0 ) = x 0
I know ∂ t ∂ v = 0 The arrow at each fixed point never changes; steady =
Velocity field and material derivative Steady vs unsteady flow Lagrangian vs Eulerian description Stream function ψ Continuity equation Flow visualization techniques (dye, smoke, PIV) 
