Intuition The big pictureImagine wind blowing leaves across a pond. There are three different ways to "draw the flow," and they only look the same when the flow is steady (not changing in time).
A streamline is a snapshot — "which way is everything pointing RIGHT NOW?"
A pathline is the trail of one particle over time — "where did THIS leaf travel?"
A streakline is the smoke plume — "all particles that ever passed through ONE point."
WHY care? Because experiments give you streaklines (dye/smoke), theory gives you streamlines (the velocity field), and tracking gives you pathlines. Confusing them gives wrong answers in unsteady flow.
Definition Velocity fieldA fluid flow is described by a velocity vector at every point in space and time:
v ⃗ ( r ⃗ , t ) = ( u ( x , y , t ) , v ( x , y , t ) ) \vec{v}(\vec{r},t) = \big(u(x,y,t),\, v(x,y,t)\big) v ( r , t ) = ( u ( x , y , t ) , v ( x , y , t ) ) WHAT it means: stick a tiny probe at ( x , y ) (x,y) ( x , y ) t t t v ⃗ \vec v v
A flow is steady ∂ v ⃗ ∂ t = 0 \dfrac{\partial \vec v}{\partial t}=0 ∂ t ∂ v = 0 even though individual particles still move .
Intuition WHAT is a streamline?A curve drawn so that at one instant , its tangent is everywhere parallel to the local velocity. No fluid crosses a streamline (by construction the velocity points along it).
Worked example Worked: rotating fieldv ⃗ = ( − y , x ) \vec v = (-y,\,x) v = ( − y , x ) Step 1 Write equation: d x − y = d y x \dfrac{dx}{-y}=\dfrac{dy}{x} − y d x = x d y Why? Plug u = − y , v = x u=-y,\ v=x u = − y , v = x d x u = d y v \frac{dx}{u}=\frac{dy}{v} u d x = v d y Step 2 Cross-multiply: x d x = − y d y x\,dx = -y\,dy x d x = − y d y Why? Just rearranging the ratios to separate variables.
Step 3 Integrate: x 2 2 = − y 2 2 + C ⇒ x 2 + y 2 = R 2 \dfrac{x^2}{2} = -\dfrac{y^2}{2}+C \Rightarrow x^2+y^2 = R^2 2 x 2 = − 2 y 2 + C ⇒ x 2 + y 2 = R 2 Why? Both sides are exact differentials. Result: streamlines are concentric circles — makes sense, the field swirls.
The actual trajectory traced by one fluid particle as time advances. WHAT: tag a single particle and film it; the film's trail is the pathline.
Worked example Worked: unsteady fieldu = 1 , v = t u=1,\ v=t u = 1 , v = t Step 1 d x d t = 1 ⇒ x = x 0 + ( t − t 0 ) \dfrac{dx}{dt}=1 \Rightarrow x = x_0 + (t-t_0) d t d x = 1 ⇒ x = x 0 + ( t − t 0 ) Why? Integrate constant horizontal speed.
Step 2 d y d t = t ⇒ y = y 0 + 1 2 ( t 2 − t 0 2 ) \dfrac{dy}{dt}=t \Rightarrow y = y_0 + \tfrac{1}{2}(t^2-t_0^2) d t d y = t ⇒ y = y 0 + 2 1 ( t 2 − t 0 2 ) Why? Integrate t t t t t t Step 3 Eliminate t t t t = t 0 + ( x − x 0 ) t = t_0 + (x-x_0) t = t 0 + ( x − x 0 ) y y y Result: a parabola for the released particle.
The locus, at a fixed time t t t all particles that have passed through a fixed point ( x p , y p ) (x_p,y_p) ( x p , y p ) τ ≤ t \tau \le t τ ≤ t WHAT: inject continuous dye at one point; the visible streak is the streakline.
Worked example Same fieldu = 1 , v = t u=1,\ v=t u = 1 , v = t Step 1 Particle released at τ \tau τ x = ( t − τ ) x=(t-\tau) x = ( t − τ ) y = 1 2 ( t 2 − τ 2 ) y=\tfrac12(t^2-\tau^2) y = 2 1 ( t 2 − τ 2 ) τ \tau τ Why? Integrate from τ \tau τ x 0 = y 0 = 0 x_0=y_0=0 x 0 = y 0 = 0 Step 2 Fix observation time, say t = 2 t=2 t = 2 τ ≤ 2 \tau\le 2 τ ≤ 2 ( x , y ) (x,y) ( x , y ) Result: a different curve than the pathline — proving the three differ in unsteady flow.
Intuition The golden ruleSteady flow ⇒ streamlines = pathlines = streaklines \boxed{\textbf{Steady flow} \Rightarrow \text{streamlines}=\text{pathlines}=\text{streaklines}} Steady flow ⇒ streamlines = pathlines = streaklines WHY: if v ⃗ \vec v v
Common mistake Steel-man the common errorsMistake 1: "A streamline shows where a particle goes."
Why it feels right: the particle does move along the local velocity. The flaw: by the time the particle moves to the next point, in unsteady flow the field there has changed, so its path peels off the original streamline. Fix: streamline = instantaneous snapshot; pathline = time-integrated trajectory.
Mistake 2: "Streaklines and pathlines are the same thing."
Why it feels right: dye injection seems to "trace a particle." The flaw: a streakline connects different particles released at different times , all viewed now . A pathline is one particle viewed over all times . Fix: remember "streak = many particles, one source, one instant."
Mistake 3: keeping t t t
Fix: freeze t t t d x u = d y v \frac{dx}{u}=\frac{dy}{v} u d x = v d y
Recall Feynman: explain to a 12-year-oldPicture a river with leaves and a smoke machine.
Streamline: Take a photo. Draw little arrows showing which way the water pushes everywhere. Connect the arrows — that's a streamline. It's "the now."Pathline: Drop ONE leaf and watch where it floats all day. The wet trail it leaves is its pathline. It's "one leaf's whole story."Streakline: Keep blowing smoke from the SAME chimney. The smoke you see hanging in the air is the streakline — all the smoke puffs from one spot, seen together.
If the river never changes its mood (steady), all three drawings end up identical!
Mnemonic Remember the three"SNAP, TRAIL, SMOKE"
S treamline = SNAP shot (now, whole field)P athline = TRAIL of one particle (one particle, all time)StreaK line = SMOKE from one chimney (many particles, one spot, now)
What is a streamline? A curve whose tangent is everywhere parallel to the instantaneous velocity field (a snapshot); no flow crosses it.
What is a pathline? The actual trajectory of a single fluid particle traced over time.
What is a streakline? At a fixed instant, the locus of all particles that have passed through one fixed point.
Streamline ODE in 2D? d x u = d y v \frac{dx}{u}=\frac{dy}{v} u d x = v d y with
t t t held constant.
Pathline equations? d x d t = u ( x , y , t ) , d y d t = v ( x , y , t ) \frac{dx}{dt}=u(x,y,t),\ \frac{dy}{dt}=v(x,y,t) d t d x = u ( x , y , t ) , d t d y = v ( x , y , t ) , integrate in time from start point.
When do all three coincide? In steady flow (
∂ v ⃗ / ∂ t = 0 \partial \vec v/\partial t=0 ∂ v / ∂ t = 0 ).
In the streamline equation, is time a variable or a constant? A constant (frozen) parameter.
For v ⃗ = ( − y , x ) \vec v=(-y,x) v = ( − y , x ) Concentric circles
x 2 + y 2 = x^2+y^2= x 2 + y 2 = const.
Which curve does an experiment with continuous dye injection show? A streakline.
Key difference streakline vs pathline? Streakline = many particles (one source) at one instant; pathline = one particle over all time.
Velocity field and material derivative Steady vs unsteady flow Continuity equation Stream function ψ Lagrangian vs Eulerian description Flow visualization techniques (dye, smoke, PIV)
all particles through one point
Intuition Hinglish mein samjhoDekho, fluid flow ko samajhne ke liye teen tareeke hote hain, aur exam mein log inhe confuse kar dete hain. Streamline ek "photo" ki tarah hai — abhi is waqt har point par velocity kis direction mein point kar rahi hai, uske tangent ko jodo, bas wahi streamline hai. Iska equation d x u = d y v \frac{dx}{u}=\frac{dy}{v} u d x = v d y t t t constant freeze karke integrate karte ho.
Pathline matlab ek single particle ki poori journey. Ek leaf ko paani mein chhodo aur din bhar dekho — uska poora raasta pathline hai. Iska equation d x d t = u , d y d t = v \frac{dx}{dt}=u,\ \frac{dy}{dt}=v d t d x = u , d t d y = v Streakline matlab ek hi jagah se lagatar smoke ya dye chhodte raho — jo plume dikhti hai woh streakline hai. Yeh alag-alag time par release hue particles ko ek hi instant par jodti hai.
Sabse important baat: agar flow steady hai (matlab field time ke saath change nahi hota, ∂ v ⃗ / ∂ t = 0 \partial \vec v/\partial t=0 ∂ v / ∂ t = 0 same ho jaate hain. Isiliye textbook examples mein aksar teeno overlap karte dikhte hain. Lekin jaise hi flow unsteady ho jata hai (jaise v = t v=t v = t streakline dikhti hai, isliye theory ke streamline se compare karte waqt savdhaan raho. Mnemonic yaad rakho: SNAP, TRAIL, SMOKE .
