What the Navier–Stokes equation is balancing
For an incompressible Newtonian fluid with constant density and viscosity, the momentum equation says: mass density × fluid acceleration equals the sum of pressure, viscous, and body-force effects.
ρ [ ∂u/∂t + (u · ∇)u ] = −∇p + μ∇²u + ρgTransient inertia
Acceleration because velocity at a fixed location changes with time. It vanishes in steady flow.
Convective inertia
Acceleration because a fluid parcel moves into a region where velocity is different.
Pressure force
Fluid accelerates from higher pressure toward lower pressure. In 1-D flow, this is represented by −dp/dx.
Viscous diffusion
Momentum is diffused by viscosity. This term becomes important when velocity changes sharply over short distances.
Body force
Force acting throughout the fluid volume, most commonly gravity.
Continuity
For incompressible flow, the velocity field must also satisfy conservation of mass: the divergence of velocity is zero.
How a difficult PDE becomes a useful engineering equation
Define the geometry
Pipe, parallel plates, moving wall, external flow, and so on. Geometry determines the coordinate system and boundary conditions.
State the assumptions
Examples: steady, incompressible, fully developed, laminar, one-dimensional velocity, constant viscosity.
Delete the terms that become zero
For fully developed pipe flow, for example, the axial velocity does not vary with x, so axial convective acceleration disappears.
Apply boundary conditions
No slip sets the fluid velocity equal to the wall velocity. Symmetry or finite-velocity conditions remove nonphysical integration constants.
A useful sign convention
All calculators use +x as the nominal downstream direction. A pressure decrease in +x therefore has dp/dx < 0.
If dp/dx = −500 Pa/m, then −dp/dx = +500 N/m³ drives the fluid in +x.Keeping this sign convention visible prevents one of the most common errors in pressure-driven-flow calculations.
Navier–Stokes calculators
All inputs use SI units. Values may be entered in ordinary decimal or scientific notation, such as 1.0e-3. Each calculator now includes real-world presets so the numbers begin with a physical story rather than an arbitrary scalar value.
Real-world scale: what do the inputs feel like?
| Density ρ | Air near sea level ≈ 1.2 kg/m³; air near 40,000 ft ≈ 0.31 kg/m³; water ≈ 998 kg/m³; many oils ≈ 800–900 kg/m³. |
| Dynamic viscosity μ | Air ≈ 1.8×10⁻⁵ Pa·s; water ≈ 1.0×10⁻³ Pa·s; lubricating oils are commonly tens to hundreds of times more viscous than water. |
| Velocity | Room/duct air is often single-digit to tens of m/s; water in small laboratory channels can be cm/s to m/s; the atmospheric presets range from about 102 m/s at Mach 0.3 sea level to about 2.95 km/s at Mach 10 and 40,000 ft. |
| Velocity derivative | A value such as ∂u/∂x = 10 s⁻¹ means the x-velocity changes by about 10 m/s over 1 m, or about 1 m/s over 0.1 m. |
Important model boundary: transonic, supersonic & hypersonic flight
Calculator A now includes an atmospheric preset series from Mach 0.3 through Mach 10. These presets map altitude and Mach number to approximate Standard Atmosphere density, viscosity, local speed of sound, and flight speed so students can compare physically meaningful scales. They are input-context examples, not compressible-flow solutions. Once compressibility becomes important, the incompressible constant-property equation on this page is no longer a physically complete flight model. Supersonic and hypersonic analysis requires compressible Navier–Stokes plus the energy equation; shocks, temperature-dependent properties, and at sufficiently high temperatures real-gas chemistry may also matter.Try this comparison: load Mach 3, 6, 8, and 10 at 40,000 ft to isolate the effect of Mach number. Then compare Mach 3 at 40,000 and 60,000 ft, or the higher-altitude Mach 6 and Mach 8 cases, to see how rapidly atmospheric density falls with altitude.
Calculator A — Local 3-D x-momentum balance
Use a local 3-D velocity field and its derivatives to find the pressure gradient required to satisfy the x-component of Navier–Stokes. This is the most general calculator on the page.
dp/dx = −ρ(∂u/∂t + u∂u/∂x + v∂u/∂y + w∂u/∂z) + μ∇²u + ρgₓ| Contribution | Acceleration form | Meaning |
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Calculator B — Fully developed laminar flow in a circular pipe
This is the Hagen–Poiseuille solution obtained directly from Navier–Stokes for steady, incompressible, axisymmetric, fully developed flow of a Newtonian fluid in a straight circular pipe.
u(r) = −(dp/dx)(R² − r²)/(4μ)Calculator C — Pressure-driven flow between fixed parallel plates
Plane Poiseuille flow is the planar cousin of pipe flow. The plates are stationary, the velocity varies only across the gap, and a pressure gradient drives a parabolic profile.
u(y) = (dp/dx)(y² − Hy)/(2μ)Calculator D — Combined Couette–Poiseuille flow
The lower plate is stationary, the upper plate moves at speed U, and a pressure gradient may assist or oppose that motion. The result is a linear Couette component plus a parabolic Poiseuille component.
u(y) = U(y/H) + (dp/dx)(y² − Hy)/(2μ)Reduced equations and boundary conditions
Circular pipe
For axial velocity u(r), steady and fully developed flow removes the time derivative and axial convection. The axial equation becomes
0 = −dp/dx + μ(1/r) d/dr [ r du/dr ]Finite velocity at r = 0 removes the logarithmic integration term. No slip, u(R)=0, supplies the second boundary condition. The result is a parabola in r².
Two parallel plates
For axial velocity u(y), the x-momentum equation reduces to
0 = −dp/dx + μ d²u/dy²With two fixed walls, u(0)=u(H)=0 gives plane Poiseuille flow. Replacing the upper condition by u(H)=U produces the combined Couette–Poiseuille solution.
Pressure-driven scaling
The characteristic velocity scales like |dp/dx|L²/μ. Doubling the geometric length scale can therefore have a much larger effect than doubling the pressure gradient.
Viscosity
In these laminar solutions, velocity and flow rate are inversely proportional to μ. More viscous fluids resist deformation and move more slowly for the same forcing.
No-slip condition
At a solid wall the fluid velocity equals the wall velocity. That is why fixed-wall profiles reach zero at the boundary and Couette flow reaches U at the moving plate.
What students should notice
Why is pipe velocity maximum at the centerline?
The center is farthest from the no-slip wall, and symmetry requires du/dr = 0 at r = 0. Viscous shear is zero at the center and grows in magnitude toward the wall.
Why does a negative dp/dx drive positive flow?
The pressure term in Navier–Stokes is −∇p. If pressure decreases downstream, dp/dx is negative, making −dp/dx positive.
What does Reynolds number tell us here?
Re compares inertial and viscous effects. A large Re means inertia is relatively stronger. For pipe flow, the classic parabolic solution is physically appropriate for fully developed laminar conditions; transition and turbulence require different modeling.
Can Navier–Stokes always be solved analytically?
No. These exact solutions exist because the geometry and assumptions remove many terms. General 2-D and 3-D flows usually require numerical methods such as finite volume, finite difference, or finite element discretization.
Where this fits in a fluid-mechanics course
The equations and analytic solutions used here are standard results for incompressible Newtonian fluid mechanics. Useful textbook references include Frank M. White, Fluid Mechanics; Fox, McDonald & Pritchard, Introduction to Fluid Mechanics; and Panton, Incompressible Flow.