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The purpose of this article is to derive the Navier–Stokes equations specialized to incompressible, steady, fully developed laminar pipe flow and to obtain the classic Poiseuille velocity profile and the Hagen–Poiseuille pressure drop relation in a form that can be directly used for engineering analysis.
1. Why this derivation matters for laminar pipe flow design
Incompressible laminar pipe flow is one of the few internal flow problems where the Navier–Stokes equations admit an exact closed-form solution under clear, physically meaningful assumptions.
The result is the Poiseuille flow solution, which provides a direct bridge from fundamental fluid mechanics to practical calculations of flow rate, pressure drop, wall shear stress, and laminar friction factor.
Because many search queries focus on “Navier–Stokes derivation,” “Hagen–Poiseuille equation,” and “laminar pipe flow pressure drop,” this derivation is presented from first principles with the exact steps that reduce the full three-dimensional equations to a solvable ordinary differential equation.
2. Physical model and assumptions for incompressible laminar pipe flow
Consider a straight circular pipe of radius R and diameter D = 2R aligned with the axial z-direction.
A Newtonian fluid flows through the pipe driven by a constant pressure gradient along the pipe length.
The goal is to start from the general incompressible Navier–Stokes equations and derive the specialized form for fully developed laminar pipe flow, then solve for the axial velocity profile u(r) and related performance equations.
2.1 Assumptions used to reduce the Navier–Stokes equations
| Assumption | Mathematical statement | Implication for the equations |
|---|---|---|
| Steady flow | ∂( )/∂t = 0. | All unsteady terms vanish. |
| Incompressible fluid | ρ = constant and ∇·V = 0. | Continuity becomes divergence-free constraint and density can be taken outside derivatives. |
| Newtonian viscosity | τ = μ(∇V + (∇V)T). | Viscous term in momentum becomes μ∇²V for constant μ. |
| Axisymmetric internal flow | ∂( )/∂θ = 0. | No circumferential variation. |
| Fully developed pipe flow | ∂u/∂z = 0 and v = 0. | Axial velocity depends only on r and radial convection terms simplify strongly. |
| No slip at the wall | u(R) = 0. | Sets the integration constant for the velocity profile. |
| Finite velocity at centerline | u is finite at r = 0. | Eliminates singular solutions and sets derivative behavior at r = 0. |
Note : Fully developed laminar pipe flow means the velocity profile does not change with axial position, which is distinct from entrance-region flow where ∂u/∂z is not zero and the solution is not the simple Poiseuille parabola.
3. Governing equations before specialization
Let the velocity field be V = (u, v, w) in Cartesian coordinates or V = (vr, vθ, vz) in cylindrical coordinates.
For pipe flow it is natural to use cylindrical coordinates (r, θ, z) with the pipe centerline as the z-axis.
3.1 Continuity equation for incompressible flow
The incompressible continuity equation is the divergence-free condition.
∇·V = 0.In cylindrical coordinates, the incompressible continuity equation is.
(1/r) ∂(r v_r)/∂r + (1/r) ∂v_θ/∂θ + ∂v_z/∂z = 0.3.2 Momentum equation for incompressible Newtonian flow
The incompressible Navier–Stokes momentum equation in vector form for constant viscosity μ is.
ρ ( ∂V/∂t + (V·∇)V ) = -∇p + μ ∇²V + ρ g.For many horizontal pipe-flow derivations, gravity is either neglected in the axial momentum balance or treated via a hydraulic grade line, and the driving mechanism is represented by the axial pressure gradient.
Note : If the pipe is inclined, gravity can be retained by combining p and ρgz into a modified pressure, and the same laminar Poiseuille velocity profile in r is obtained when the net axial driving potential is constant along z.
4. Specializing the equations to incompressible laminar pipe flow
4.1 Symmetry and velocity structure for Poiseuille flow
For a straight circular pipe with axisymmetric, fully developed laminar pipe flow, the only nonzero velocity component is the axial component vz.
Define u(r) as the axial velocity, using u as a standard pipe-flow notation.
The specialized velocity field is.
v_r = 0, v_θ = 0, v_z = u(r).Axisymmetry implies ∂( )/∂θ = 0, and fully developed flow implies ∂u/∂z = 0.
4.2 Continuity check under the fully developed assumption
Substitute vr = 0, vθ = 0, and vz = u(r) into incompressible continuity.
(1/r) ∂(r · 0)/∂r + (1/r) ∂(0)/∂θ + ∂u/∂z = 0 0 + 0 + 0 = 0.The continuity equation is satisfied identically, which confirms that the assumed velocity structure is kinematically admissible.
4.3 Axial momentum equation in cylindrical coordinates
Under the adopted assumptions, the z-momentum equation is the only equation needed to determine u(r).
The key steps are to evaluate the convection term (V·∇)V and the viscous diffusion term ∇²V for the axial component.
4.4 Evaluating the convection term for fully developed laminar pipe flow
The convective acceleration for the axial component in cylindrical coordinates includes terms proportional to vr∂u/∂r, vθ(1/r)∂u/∂θ, and vz∂u/∂z.
With vr = 0, vθ = 0, and ∂u/∂z = 0, all axial convection terms vanish.
(V·∇)u = v_r ∂u/∂r + (v_θ/r) ∂u/∂θ + v_z ∂u/∂z = 0.4.5 Evaluating the viscous term for the axial velocity
For an axisymmetric field where u = u(r) only, the Laplacian of u in cylindrical coordinates reduces to the radial diffusion operator.
∇²u = (1/r) ∂/∂r ( r ∂u/∂r ) + (1/r²) ∂²u/∂θ² + ∂²u/∂z² = (1/r) ∂/∂r ( r ∂u/∂r ).4.6 Resulting differential equation from the Navier–Stokes equations
Using steady flow, zero convection, and the reduced Laplacian, the axial momentum equation becomes a balance between pressure gradient and viscous diffusion.
0 = -∂p/∂z + μ (1/r) ∂/∂r ( r ∂u/∂r ).Rearrange into a standard form.
(1/r) ∂/∂r ( r ∂u/∂r ) = (1/μ) ∂p/∂z.For fully developed laminar pipe flow, the axial pressure gradient ∂p/∂z is constant with respect to r and, for a straight uniform pipe under steady conditions, is typically constant along z as well.
Define a constant G as the negative of the pressure gradient to emphasize that flow is from high pressure to low pressure.
G = -∂p/∂z.Then the governing equation becomes.
(1/r) d/dr ( r du/dr ) = -(G/μ).Note : The derivation up to this point is the central reduction of the incompressible Navier–Stokes equations to the Poiseuille flow ordinary differential equation, and it is the step most often searched as “Navier–Stokes for laminar pipe flow.”
5. Solving for the Poiseuille velocity profile
5.1 Integrate the differential equation
Multiply both sides by r and integrate with respect to r.
d/dr ( r du/dr ) = -(G/μ) r.Integrate once.
r du/dr = -(G/μ) (r²/2) + C1.Divide by r for r > 0.
du/dr = -(G/μ) (r/2) + C1/r.Integrate again.
u(r) = -(G/μ) (r²/4) + C1 ln(r) + C2.5.2 Apply the centerline regularity condition
The term C1 ln(r) diverges as r approaches zero unless C1 = 0.
Finite velocity at the centerline therefore requires.
C1 = 0.The solution simplifies to.
u(r) = -(G/μ) (r²/4) + C2.5.3 Apply the no-slip boundary condition at the wall
No slip at r = R requires u(R) = 0.
0 = -(G/μ) (R²/4) + C2 C2 = (G/μ) (R²/4).Substitute C2 back into u(r).
u(r) = (G/4μ) (R² - r²).Using G = -∂p/∂z yields the standard Poiseuille flow velocity profile.
u(r) = -(1/4μ) (∂p/∂z) (R² - r²).The result is a parabolic velocity profile with maximum velocity at the centerline and zero velocity at the wall.
6. Flow rate, mean velocity, and the Hagen–Poiseuille equation
6.1 Volumetric flow rate from the velocity profile
The volumetric flow rate Q is the area integral of u(r) over the circular cross-section.
Q = ∬_A u dA = ∫_0^{2π} ∫_0^R u(r) r dr dθ.Insert u(r) = (G/4μ)(R² - r²) and integrate.
Q = ∫_0^{2π} ∫_0^R (G/4μ)(R² - r²) r dr dθ = 2π (G/4μ) ∫_0^R (R² r - r³) dr = 2π (G/4μ) [ (R² r²/2) - (r^4/4) ]_0^R = 2π (G/4μ) ( R^4/2 - R^4/4 ) = 2π (G/4μ) ( R^4/4 ) = (π G R^4) / (8 μ).Using G = -∂p/∂z and Δp = pin - pout over length L so that -∂p/∂z = Δp/L gives the Hagen–Poiseuille equation.
Q = (π R^4 / (8 μ)) (Δp / L).In diameter form with R = D/2.
Q = (π D^4 / (128 μ)) (Δp / L).6.2 Mean velocity and maximum velocity relation
The mean axial velocity U is Q divided by cross-sectional area A = πR².
U = Q/(πR²) = [ (π G R^4)/(8 μ) ] / (πR²) = (G R²)/(8 μ).The maximum velocity occurs at r = 0.
u_max = u(0) = (G/4μ) R².Therefore, the classic laminar pipe flow relation is.
u_max = 2 U.Note : The relationship u_max = 2U is a distinctive signature of fully developed laminar pipe flow and is frequently used to validate experimental or numerical results for Poiseuille flow.
7. Wall shear stress, pressure drop, and laminar friction factor
7.1 Wall shear stress from the velocity gradient
For a Newtonian fluid, the shear stress τrz is μ du/dr.
From u(r) = (G/4μ)(R² - r²), the derivative is.
du/dr = (G/4μ) (-2r) = -(G r)/(2μ).Thus the shear stress distribution is.
τ_rz(r) = μ du/dr = -(G r)/2.At the wall r = R, the wall shear stress magnitude is.
τ_w = |τ_rz(R)| = (G R)/2 = (R/2)(Δp/L).Equivalently, the pressure drop relates to wall shear stress by.
Δp = (2 τ_w L)/R = (4 τ_w L)/D.7.2 Darcy friction factor for laminar pipe flow
Define Reynolds number based on mean velocity U and diameter D as Re = ρUD/μ.
For fully developed incompressible laminar pipe flow in a circular pipe, combining the Hagen–Poiseuille equation with the Darcy–Weisbach form yields the well-known laminar friction factor relation.
f = 64 / Re.This expression is a direct consequence of the Navier–Stokes solution for Poiseuille flow and is valid in the laminar regime where the assumptions remain accurate.
8. Practical engineering form and common pitfalls
8.1 Key equations summary for incompressible laminar pipe flow
| Quantity | Result for Poiseuille flow | Typical usage |
|---|---|---|
| Velocity profile | u(r) = (Δp/(4μL))(R² - r²). | Predict local speed, residence time, shear profile. |
| Flow rate | Q = (πR⁴/(8μ))(Δp/L). | Size pump and pipe for a required Q under laminar conditions. |
| Mean velocity | U = (R²/(8μ))(Δp/L). | Compute Re and check laminar validity. |
| Centerline speed | umax = 2U. | Sanity check for measured or computed profiles. |
| Wall shear stress | τw = (R/2)(Δp/L). | Estimate shear-sensitive process constraints. |
| Darcy friction factor | f = 64/Re. | Alternative pressure drop calculations and comparisons. |
8.2 Dimensional and physical checks
The velocity profile scales as u ~ (Δp/L)R²/μ, which matches the idea that stronger pressure gradient increases velocity, higher viscosity decreases velocity, and larger radius strongly increases velocity.
The R⁴ dependence of Q is a defining feature of the Hagen–Poiseuille equation and explains why small diameter changes drastically affect pressure drop in laminar pipe flow.
Note : The Poiseuille solution is exact for the stated assumptions, but real piping systems can depart from those assumptions due to entrance effects, fittings, non-Newtonian rheology, wall slip, roughness-induced transition, or pulsation.
8.3 Entrance length awareness for fully developed laminar pipe flow
Fully developed laminar pipe flow requires sufficient axial distance for the boundary layers to merge and for ∂u/∂z to become negligible.
When the flow is not fully developed, the Navier–Stokes reduction used above is not valid because axial derivatives and radial velocity components can no longer be neglected.
For design work, checking whether a region is fully developed is essential when applying the Poiseuille velocity profile and the Hagen–Poiseuille pressure drop relation.
9. Worked symbolic example using the derived equations
Assume a circular pipe of radius R, length L, viscosity μ, and pressure drop Δp.
The Poiseuille velocity profile gives the local velocity at radius r as.
u(r) = (Δp/(4μL))(R² - r²).The mean velocity is.
U = (R²/(8μ))(Δp/L).The flow rate is.
Q = πR²U = (πR⁴/(8μ))(Δp/L).The wall shear stress is.
τ_w = (R/2)(Δp/L).These expressions are mutually consistent and all stem directly from the incompressible Navier–Stokes equations specialized to steady, fully developed laminar pipe flow.
FAQ
How the Navier–Stokes equations reduce to a single equation for laminar pipe flow.
The reduction follows from assuming steady conditions, incompressibility, axisymmetry, and fully developed flow where vr and vθ are zero and u depends only on r.
These assumptions eliminate unsteady terms, eliminate all convection terms in the axial momentum equation, and reduce the viscous operator to (1/r)d/dr(r du/dr).
The result is a balance between the axial pressure gradient and radial viscous diffusion that can be integrated twice to obtain the Poiseuille profile.
What boundary conditions are required to obtain the Poiseuille velocity profile.
No slip at the wall imposes u(R) = 0, and finite velocity at the centerline eliminates logarithmic singularities and forces the ln(r) integration constant to be zero.
These two conditions uniquely determine the parabolic velocity profile for incompressible laminar pipe flow.
When the Hagen–Poiseuille equation is valid for pressure drop calculations.
The Hagen–Poiseuille equation is valid for Newtonian fluids in steady, incompressible, fully developed laminar pipe flow in a straight circular pipe of constant radius.
Departures from these conditions can require corrections or alternative models, including entrance-region analysis, non-Newtonian constitutive relations, or more general computational solutions of the Navier–Stokes equations.
How to connect the Navier–Stokes solution to the laminar friction factor expression.
By combining the exact pressure drop relation from the Poiseuille solution with the Darcy–Weisbach form, one obtains the laminar friction factor scaling inversely with Reynolds number.
The resulting expression f = 64/Re is a direct consequence of the incompressible Navier–Stokes equations under laminar pipe flow assumptions.
Why the velocity profile is parabolic in laminar pipe flow.
The axial pressure gradient is uniform in r, while viscous diffusion of momentum acts through the radial Laplacian operator.
The equation (1/r)d/dr(r du/dr) = constant integrates to a quadratic function of r once singular terms are excluded by centerline regularity, and the no-slip wall condition fixes the remaining constant.