Intuition The one core ideaThe Navier–Stokes equation is nothing more than ==Newton's F = m a vocabulary — the arrows, the little symbols, the slopes — you need before that one sentence can make sense.
This page assumes you know nothing . We will earn every symbol the parent note (the parent topic ) throws at you, in an order where each item leans only on the ones before it. We build the picture first, then the coordinates , then the quantities , and only then the operators that combine them.
Intuition Why we chop the fluid into cubesA river has far too many molecules to track one by one. So we imagine slicing the fluid into a huge grid of tiny cubes. Each cube is small enough that the fluid's properties are almost constant inside it, but big enough to still contain trillions of molecules (so it behaves like smooth "stuff", not individual bouncing dots). We apply Newton's law to one such cube, then let the cube shrink to a point.
The cube in the figure is our hero. Everything that follows — coordinates, quantities, operators — is built so we can write F = m a this cube . To even measure its size and mass, we first need directions to measure along. That is the next section.
Definition The three axis directionsx y z − z u x x u Picture: three arrows meeting at a corner of the room, at right angles.
Definition Position vectorr = ( x , y , z ) where a point sits in space: how far along x y z Picture: an arrow from the origin (corner of the room) to the point.
Definition The differentialsd x , d y , d z d x tiny step along the x d y y d z z Picture: the three coloured edge-lengths of the cube in the figure above.
Now that we have directions and tiny steps along them, we can finally size and weigh the cube.
Definition Volume elementd V d V = d x d y d z length × width × height of an infinitesimally small box (using the tiny steps just defined).
The topic needs it because forces and mass are measured per cube , and we divide by d V mass needs density , which we meet in Section 5.)
Intuition Two kinds of quantitySome quantities are just a number (a scalar ): temperature, for example. Others need a number AND a direction (a vector ): a push, or a velocity. We draw a little arrow over a letter to say "this one is a vector."
a a with an arrow (u F P ρ ∣ u ∣ u Picture: a scalar is a colour shade filling space; a vector is an arrow drawn at each point.
Reveal-check:
Will density ρ A scalar — just a number at each point, no direction.
Will velocity u A vector — it carries both a speed and a direction (note the arrow on top).
Definition Velocity fieldu ( r , t ) vector (a scalar-with-arrow, from Section 3) at position r t . It is written as a function of both space and time.
Picture: the arrow-forest below — an arrow drawn at every point of the flow. Where arrows are long, fluid is fast; where short, slow.
Intuition Why velocity depends on TWO thingsA weather map of wind arrows can change hour to hour (time dependence), and at a single instant the arrows already differ from place to place (space dependence). A drifting leaf feels both effects at once — this double dependence is exactly what forces us to invent the material derivative later.
ρ ρ kg / m 3 scalar . A cube's mass is then d m = ρ d V Picture: how "heavy per box" the fluid is — mercury is a dense (crowded) fluid, air is a thin one.
P P = area of that surface force pushing on a surface pascals , Pa = N / m 2 scalar : at each point it is a single number, the same squeeze in every direction.
Picture: the fluid pressing on each face of our cube. Multiply P force on that face — this is exactly why a difference in P − ∇ P
μ μ sticky / gooey the fluid is — how strongly one moving layer drags its neighbour. Measured in pascal-seconds , Pa ⋅ s μ ≈ 0.001 Pa ⋅ s Picture: two stacked sheets of fluid sliding past each other; μ Pa ⋅ s Pa 1/ s Viscosity & Newton's law of viscosity .
Intuition What a derivativeis , from zero
A derivative answers "if I nudge the input a little, how much does the output change?" It is the steepness of a graph — rise over run when the run shrinks to almost nothing.
Definition Ordinary derivatived x df f x f Picture: the tangent line hugging a curve; its tilt is the derivative.
Definition Partial derivative∂ x ∂ f f x , y , z , t partial derivative means "wiggle only x ∂ Picture: slicing a hilly surface along one direction and reading that slice's slope.
Why the topic needs derivatives: forces come from differences — a pressure difference across a cube, a velocity difference between layers — and a difference-per-distance is a derivative.
Definition The del symbol∇ ∇ packaging of the three partial derivatives into one arrow-shaped operator:
∇ = ( ∂ x ∂ , ∂ y ∂ , ∂ z ∂ ) instruction waiting for something to act on.
Definition Gradient of a scalar:∇ P ∇ P = ( ∂ x ∂ P , ∂ y ∂ P , ∂ z ∂ P ) vector pointing in the direction pressure rises fastest, with length = how steeply it rises.
Picture: on a hill, the gradient arrow points straight uphill. Fluid gets pushed down the pressure hill, which is why the pressure force per volume is − ∇ P P
∇ ⋅ u dot between ∇ scalar : ∇ ⋅ u = ∂ x ∂ u x + ∂ y ∂ u y + ∂ z ∂ u z spreading out from a point (a source) or piling in (a sink).
Picture: arrows all pointing outward from a spot → positive divergence. For an incompressible fluid nothing piles up, so ∇ ⋅ u = 0 Continuity equation .
Definition The advection operator( u ⋅ ∇ ) u ⋅ ∇ = u x ∂ x ∂ + u y ∂ y ∂ + u z ∂ z ∂ as you move along with the flow ." Apply it to u ( u ⋅ ∇ ) u Material derivative .
Picture: riding downstream and feeling the arrows around you change.
Definition The Laplacian:∇ 2 ∇ 2 = ∇ ⋅ ∇ = ∂ x 2 ∂ 2 + ∂ y 2 ∂ 2 + ∂ z 2 ∂ 2 second derivative in every direction. It compares a point's value to the average of its neighbours.
Picture: a bump that is higher than its surroundings has negative ∇ 2 ∇ 2 u difference of a difference (the change in the sliding-rate across the cube).
∇ 2 vector u ∇ 2 scalar . On a vector it acts component by component :
∇ 2 u = ( ∇ 2 u x , ∇ 2 u y , ∇ 2 u z ) separately on each of the three parts u x , u y , u z μ ∇ 2 u
∇ P ∇ ⋅ u Why it feels right: both use ∇ The flaw: ∇ on a scalar makes a vector (gradient); ∇ ⋅ on a vector makes a scalar (divergence). The little dot changes everything.
Fix: read the neighbours: no dot + scalar → gradient (vector out); dot + vector → divergence (scalar out).
Definition Gravitational accelerationg g 9.81 m/s 2 g = ( 0 , 0 , − g ) = ρ g d V Picture: an arrow pointing straight down, the same everywhere.
F F = m a F m a
Read this as a short chain, not a spider-web: picture → coordinates → quantities → operators → the equation .
operators grad div laplacian
Each foundation becomes one word in the final sentence: acceleration (D u / D t − ∇ P μ ∇ 2 u ρ g
Test yourself — you are ready for the derivation only if you can answer each without peeking.
What are the differentials d x , d y , d z d V Tiny steps along the x y z d V = d x d y d z
What is the mass of the cube d V d m = ρ d V ρ
What is the difference between a scalar and a vector, with one fluid example of each? A scalar is just a number (pressure
P ); a vector has magnitude and direction (velocity
u , marked with an arrow).
Why does the velocity field u ( r , t ) r t The flow can differ from place to place (space) and can change over time (time); a moving particle feels both.
What is pressure P Force per unit area, P = force / area Pa = N / m 2
What does μ Viscosity — the stickiness/drag between fluid layers, measured in pascal-seconds Pa ⋅ s
What is the difference between d x df ∂ x ∂ f The ordinary derivative varies the single input; the partial derivative wiggles one variable while holding the others fixed.
What kind of object does ∇ P A vector, pointing in the direction pressure increases fastest (fluid is pushed the opposite way, hence − ∇ P
What does ∇ ⋅ u = 0 The fluid is incompressible — no net spreading-out or piling-up at any point (continuity).
How does ∇ 2 u Component by component:
∇ 2 u = ( ∇ 2 u x , ∇ 2 u y , ∇ 2 u z ) .
Why does viscosity use the second derivative and not the first? Net viscous force is a difference of stresses across the cube; stress is already one derivative, so the net force needs one more.
What are the three forces on a fluid cube, and what equation ties them to acceleration? Pressure (
− ∇ P ), viscosity (
μ ∇ 2 u ), gravity (
ρ g ); together they equal
ρ D u / D t (Newton's
F = m a ).
