Navier–Stokes Equations and Exact Viscous Flows

Introduction and context

The Navier–Stokes equations are the central differential equations of classical fluid mechanics. They express conservation of mass, linear momentum, and—when coupled with an energy equation—thermodynamic energy. Their importance lies not only in their broad applicability, but also in the fact that they connect measurable material properties, such as density and viscosity, with velocity, pressure, and force fields.

For postgraduate study, the equations should be understood at three levels: as conservation laws, as constitutive models for Newtonian fluids, and as mathematical problems requiring appropriate initial and boundary conditions. Exact viscous-flow solutions are especially valuable because they reveal the physical meaning of individual terms and provide benchmark solutions for numerical methods and experiments.

Learning objectives

After studying this chapter, the reader should be able to:

1. Derive the differential continuity and momentum equations from conservation principles.
2. Interpret the local, convective, pressure, viscous, and body-force terms in the Navier–Stokes equations.
3. State the assumptions underlying the Newtonian-fluid model.
4. Reduce the general equations to classical exact solutions.
5. Solve and interpret plane Couette flow, plane Poiseuille flow, Hagen–Poiseuille pipe flow, and start-up flow between parallel plates.
6. Distinguish fully developed, steady, creeping, and unsteady viscous flows.
7. Assess assumptions, scaling parameters, and limitations of exact solutions.
8. Use exact solutions as verification cases for computational fluid dynamics.

1. Kinematics and conservation principles

Let the velocity field be u = (u, v, w), with Cartesian coordinates (x, y, z) and time t. The material derivative of a scalar φ is

The first term is the local rate of change, while the remaining terms represent convection. For a vector field, the corresponding acceleration is

where ∇ is the gradient operator. In Cartesian coordinates, the x-component is

1.1 Conservation of mass

For a control volume, conservation of mass gives

where ρ is density in kg m⁻³. Equivalently,

For an incompressible fluid of constant density,

This condition is a kinematic constraint; it does not necessarily imply that pressure is constant or that the fluid is liquid. Gases may also be treated as incompressible when their Mach number is sufficiently small and density variations are negligible.

1.2 Conservation of linear momentum

Newton’s second law applied to a differential fluid element states that mass times acceleration equals the sum of surface and body forces. In tensor form,

Here 𝛔 is the Cauchy stress tensor in Pa and 𝐛 is body force per unit mass in m s⁻². The stress tensor is decomposed as

where p is thermodynamic pressure, 𝐈 is the identity tensor, and 𝛕 is the deviatoric or viscous stress tensor.

2. Newtonian constitutive behaviour

For a Newtonian fluid, viscous stress is linearly related to the rate of deformation. The general three-dimensional relation is

where μ is dynamic viscosity in Pa·s, λ is the second viscosity coefficient, and

is the rate-of-deformation tensor. The frequently used Stokes hypothesis is λ = −2μ/3. This hypothesis is convenient but is not a universal law; it is an approximation whose accuracy depends on the fluid and flow regime.

For a constant-viscosity incompressible fluid, substitution into the momentum equation gives the Navier–Stokes equation:

The terms have units of force per unit volume, N m⁻³. The equation applies to a continuum Newtonian fluid with constant μ; for variable viscosity, the viscous term must be retained in divergence form as ∇ · 𝛕 rather than automatically written as μ∇²𝐮.

For an incompressible fluid with constant ρ and μ, the governing system is therefore

The pressure is determined as part of the solution so that continuity is satisfied. Taking the divergence of the momentum equation yields a pressure-Poisson equation. Since ∇ · 𝐮 = 0,

for constant density and viscosity. This equation illustrates why pressure is often described as a constraint variable in incompressible flow.

3. Boundary and initial conditions

The differential equations alone do not define a unique flow. A complete problem requires:

• initial velocity, and sometimes pressure or temperature, in unsteady problems;
• velocity conditions on solid surfaces;
• inlet, outlet, periodic, symmetry, or far-field conditions where appropriate.

For a stationary impermeable solid wall, the classical no-slip and no-penetration condition is

At a stationary wall, 𝐮 = 0. This condition is strongly supported for ordinary continuum flows, although apparent slip may occur in rarefied gases, microflows, hydrophobic surfaces, and certain non-Newtonian or nanofluidic systems.

At an interface between two viscous fluids, velocity is usually continuous when there is no interfacial slip. Tangential stress and normal-stress conditions must also be imposed, with surface tension included when the interface is curved or deforming.

4. Dimensional analysis and physical regimes

The Reynolds number compares inertial and viscous effects:

where U is a characteristic velocity, L a characteristic length, and ν = μ/ρ the kinematic viscosity in m² s⁻¹. Large Re indicates that inertia is relatively important; small Re indicates viscous domination. The numerical transition between laminar and turbulent flow depends on geometry, disturbances, and boundary conditions, so Re is not itself a universal transition criterion.

The unsteady-to-viscous balance is measured by the Womersley number,

where ω is angular frequency in rad s⁻¹. In oscillatory internal flow, small α indicates that viscous diffusion acts rapidly across the section, whereas large α indicates substantial phase lag and a thinner oscillatory boundary layer.

Non-dimensionalising the incompressible momentum equation using U and L gives

where starred variables are dimensionless. This form makes clear that exact viscous solutions are especially useful when Re is small or when geometry removes the nonlinear convective terms.

5. Exact solution I: plane Couette flow

Consider steady flow between two infinite parallel plates separated by distance h. The lower plate at y = 0 is stationary and the upper plate at y = h moves with constant velocity U in the x-direction. Assume:

• incompressible Newtonian fluid;
• constant μ and ρ;
• steady flow;
• fully developed unidirectional flow, 𝐮 = (u(y), 0, 0);
• no imposed pressure gradient;
• negligible body force in the x-direction.

Continuity is automatically satisfied. The x-momentum equation reduces to

Integrating twice,

The no-slip conditions are u(0) = 0 and u(h) = U, so

The velocity profile is linear. The wall shear stress is constant:

The volume flow rate per unit width is

There is no pressure gradient in ideal plane Couette flow. Energy is dissipated through viscous friction, and the viscous dissipation per unit volume is μ(U/h)².

6. Exact solution II: plane Poiseuille flow

Now consider pressure-driven flow between stationary parallel plates at y = ±h. The full gap is 2h. Let the pressure decrease in the positive x-direction, so dp/dx < 0. Under the same steady, fully developed, unidirectional assumptions,

Since dp/dx is constant, integration gives

using u(±h) = 0. The profile is parabolic, with maximum velocity at the centreline:

The mean velocity is

Thus u_max = 3U_mean/2. The flow rate per unit width is

The wall shear stress at y = h is

The solution demonstrates that increasing viscosity reduces velocity for a specified pressure gradient, while increasing the gap strongly increases flow rate.

7. Exact solution III: Hagen–Poiseuille flow in a circular pipe

For steady, incompressible, fully developed flow through a straight circular pipe of radius R, use cylindrical coordinates (r, θ, x) and assume 𝐮 = (u(r), 0, 0). Axisymmetry and full development eliminate dependence on x and θ. The axial momentum equation becomes

Regularity at r = 0 requires a finite derivative, and no slip gives u(R) = 0. Integration yields

The maximum velocity is

The volumetric flow rate is

For a pipe of length L and pressure drop ∆p = p_in − p_out,

This is the Hagen–Poiseuille law. Its hydraulic resistance is

The fourth-power dependence on radius is a major engineering result: a modest change in pipe radius can produce a very large change in flow rate. The mean velocity is U_mean = Q/(πR²), and u_max = 2U_mean.

The wall shear stress is

The formula is valid for laminar, fully developed flow of a Newtonian fluid in a straight circular tube. It is not a turbulent pipe-flow law and should not be used near entrances, fittings, strongly curved sections, or where properties vary substantially.

8. Start-up flow between parallel plates

Steady exact solutions do not describe the establishment of flow after a valve is opened. Consider a fluid between stationary plates at y = ±h. At t = 0, a constant pressure gradient is suddenly applied. The velocity satisfies

with u(±h,t) = 0 and u(y,0) = 0. The solution can be written as the steady Poiseuille profile minus a decaying series:

Here ν = μ/ρ. The series satisfies the initial and wall conditions and approaches the steady parabolic solution as t → ∞. The characteristic viscous diffusion time is

This is an order-of-magnitude estimate rather than an exact time to reach steady state. It shows that momentum diffuses across a distance h over a time proportional to h²/ν.

9. Stokes’ first and second problems

In Stokes’ first problem, an infinite plate suddenly starts moving at velocity U in its own plane, with fluid occupying y > 0. The fluid initially rests. The governing equation is

with u(0,t) = U, u(∞,t) = 0, and u(y,0) = 0. The solution is

where erfc is the complementary error function. The shear stress at the plate is

The idealised start produces a mathematically singular stress as t approaches zero. This does not imply infinite physical force in a real apparatus; it indicates that the instantaneous-start model neglects finite acceleration, molecular-scale effects, and apparatus compliance.

In Stokes’ second problem, a plate oscillates as U cos(ωt). After transients decay, the velocity amplitude decreases away from the plate and the phase lags. The Stokes-layer thickness is

This result is important in oscillating boundaries, acoustic streaming, turbomachinery, and biological flows.

10. Developing pipe flow and entrance effects

The Hagen–Poiseuille solution assumes fully developed flow. At a pipe entrance, the velocity profile is often approximately uniform, and viscous effects begin at the wall. Boundary layers grow inward until they meet at the centreline. A commonly used engineering estimate for laminar entrance length is

where D = 2R and Re_D = ρU_meanD/μ. The coefficient varies with inlet geometry and the definition of “fully developed,” so this relation is an estimate rather than a universal exact formula.

In the developing region, pressure loss includes both wall friction and entrance-region losses. Applying a fully developed pressure-drop equation too close to an inlet can therefore underpredict the required pressure difference.

11. Worked example: pressure-driven microchannel

A rectangular microchannel is approximated as two wide parallel plates separated by a gap of 0.80 mm. Its width is sufficiently large compared with the gap. A liquid has viscosity 0.002 Pa·s and density 1000 kg m⁻³. The pressure drop is 1200 Pa over a length of 0.30 m. Estimate the mean velocity and flow rate per unit width using plane Poiseuille flow.

Let h = 0.40 mm = 4.0 × 10⁻⁴ m, because the plates are located at y = ±h. The pressure gradient magnitude is

The mean velocity is

Substitution gives

The flow rate per unit width is

A characteristic Reynolds number based on the full gap is

The low Reynolds number supports a laminar model, although entrance effects and the finite channel width must still be assessed. The result is sensitive to the gap because q is proportional to h³ for a fixed pressure gradient and viscosity.

12. Applied case: diagnosing an unexpectedly low flow rate

Background

Consider an illustrative industrial cooling loop using a straight circular tube. The design calculation predicts a flow rate from the Hagen–Poiseuille relation, but the measured rate is approximately 35% lower.

Problem definition

The engineer must determine whether the discrepancy is caused by an incorrect fluid viscosity, an incorrect tube radius, additional fittings, entrance effects, or failure of the laminar fully developed assumption.

Analysis

The Hagen–Poiseuille relation predicts Q ∝ R⁴/μ for fixed pressure gradient and length. Consequently, a 10% reduction in radius produces approximately a 34% reduction in predicted flow because

This makes internal diameter measurement particularly important. The engineer should also calculate Re = ρUD/μ and compare it with the regime assumed in the model. If the flow is turbulent, the laminar law is invalid. If the tube contains valves, elbows, filters, or a developing entrance, minor and entrance losses must be included.

Approach and solution

The recommended diagnostic sequence is:

1. Verify actual temperature and viscosity, since viscosity may change strongly with temperature.
2. Measure internal diameter at several locations rather than relying only on nominal size.
3. Calculate Reynolds number using measured flow or an iterative estimate.
4. Add minor-loss terms, commonly represented as KρU²/2, where K is a dimensionless loss coefficient.
5. Check the pressure transducer calibration and whether pressure taps are located in fully developed regions.
6. Compare measured velocity profiles, if available, with the parabolic laminar prediction.

Suppose inspection finds that the actual radius is 0.90 times the design radius, while all other assumptions remain valid. The predicted flow becomes 65.61% of the original value, closely explaining the observed 35% reduction. This conclusion should still be confirmed experimentally because several errors may coexist.

Learning points

The case illustrates that an exact formula is not automatically a complete engineering model. Geometry, property variation, measurement location, entrance effects, and flow regime determine whether the assumptions are justified. It also demonstrates the value of sensitivity analysis: the fourth-power radius dependence makes dimensional inspection more influential than many small pressure-measurement errors.

13. Limitations and research perspectives

Exact viscous solutions are generally available only for restricted geometries and idealised conditions. They may fail when:

• the flow is turbulent and fluctuating over many scales;
• the fluid is non-Newtonian, viscoelastic, multiphase, or reacting;
• density or viscosity varies strongly with temperature or composition;
• walls are rough, porous, moving, deformable, or permeable;
• geometry is three-dimensional, curved, branching, or time-dependent;
• compressibility, buoyancy, surface tension, or electromagnetic forces are important.

Nevertheless, exact solutions remain essential. They provide limiting cases, expose errors in discretisation and boundary conditions, support code verification, and guide reduced-order models. Modern research combines analytical solutions with direct numerical simulation, large-eddy simulation, experimental velocimetry, uncertainty quantification, and data-driven closure models. Artificial intelligence can assist surrogate modelling and parameter estimation, but it cannot replace conservation laws, dimensional analysis, validation, or physical interpretation. A model that fits data but violates mass conservation or produces dimensionally inconsistent results is not an acceptable fluid-mechanics solution.

Best practices for analysis

1. State geometry, coordinate system, fluid properties, and force directions before simplifying equations.
2. Apply continuity before deleting velocity components.
3. Distinguish fully developed flow, steady flow, and uniform flow; they are not synonymous.
4. Check every boundary condition, including regularity at an axis or centreline.
5. Verify dimensions and limiting behaviour, such as zero pressure gradient or vanishing viscosity.
6. Calculate an appropriate Reynolds number after obtaining a velocity scale.
7. Separate frictional pressure drop from local and entrance losses.
8. Use exact solutions to verify numerical codes, not to conceal invalid assumptions.
9. Report uncertainty in geometry, viscosity, pressure, and flow-rate measurements.

Chapter Summary

The Navier–Stokes equations result from conservation of mass and linear momentum combined with a constitutive law for viscous stress. For a constant-property incompressible Newtonian fluid, they consist of the divergence-free condition and a momentum equation containing unsteady inertia, convection, pressure, viscous diffusion, and body-force terms. Boundary and initial conditions are indispensable to obtain a unique solution.

Exact solutions arise when geometry and assumptions eliminate or simplify the nonlinear terms. Plane Couette flow produces a linear velocity profile through wall motion. Plane Poiseuille flow produces a parabolic profile through a pressure gradient. Hagen–Poiseuille flow gives the fourth-power radius dependence of laminar pipe flow. Unsteady diffusion solutions describe start-up, impulsively moving plates, and oscillatory boundary layers. These solutions clarify viscous momentum transport and serve as benchmarks for experiments and computation.

The validity of every solution depends on assumptions such as Newtonian behaviour, no slip, constant properties, laminar flow, full development, and ideal geometry. Engineering application therefore requires dimensional analysis, regime identification, sensitivity assessment, and validation against measurements.

Key Terms

Continuum hypothesis: Assumption that fluid properties vary continuously at the macroscopic scale.

Material derivative: Rate of change experienced by a fluid particle moving through a flow field.

Continuity equation: Differential expression of conservation of mass.

Incompressible flow: Flow satisfying ∇ · 𝐮 = 0, usually with negligible density variation.

Cauchy stress tensor: Tensor representing surface force per unit area.

Newtonian fluid: Fluid whose viscous stress is linearly related to rate of deformation.

Dynamic viscosity: Proportionality coefficient between viscous stress and deformation rate.

Kinematic viscosity: Ratio ν = μ/ρ, with units m² s⁻¹.

No-slip condition: Equality of fluid and wall tangential velocities at a conventional solid boundary.

Fully developed flow: Flow whose velocity profile no longer changes in the streamwise direction.

Plane Couette flow: Shear flow between parallel plates caused by wall motion.

Plane Poiseuille flow: Pressure-driven flow between parallel stationary plates.

Hagen–Poiseuille flow: Laminar pressure-driven flow in a circular pipe.

Reynolds number: Dimensionless ratio of inertial to viscous effects.

Womersley number: Dimensionless measure of unsteady inertia relative to viscous diffusion.

Entrance length: Distance required for a developing internal flow to approach its fully developed profile.

Viscous diffusion: Transport of momentum through viscous stresses.

Hydraulic resistance: Ratio of pressure drop to flow rate in a specified flow system.

Review Questions

Short Answer Questions

1. What physical processes are represented by the local and convective terms in the material derivative?
2. State the incompressible continuity equation and explain its meaning.
3. What assumptions lead from the general momentum equation to the constant-property incompressible Navier–Stokes equation?
4. Why is the pressure gradient negative in the direction of flow in pressure-driven channel flow?
5. Define Reynolds number and explain what its magnitude indicates.

Long Answer Questions

1. Derive the velocity distribution and flow rate for plane Couette flow between two parallel plates.
2. Derive the Hagen–Poiseuille velocity profile and volumetric flow-rate relation, stating all boundary conditions.
3. Explain the physical meaning of every term in the incompressible Navier–Stokes equation.
4. Discuss the differences between fully developed flow, developing flow, steady flow, and unsteady flow.
5. Explain how exact viscous solutions support verification and validation of computational fluid-dynamics models.

Analytical / Application Questions

1. A pipe carries a Newtonian liquid under a fixed pressure gradient. Explain quantitatively how the flow rate changes if the radius is reduced by 15%, assuming laminar fully developed conditions.
2. For a suddenly imposed pressure gradient between parallel plates, identify the relevant diffusion time scale and explain how viscosity affects the time required to approach steady flow.
3. A measured flow rate is lower than the Hagen–Poiseuille prediction. Develop a systematic diagnostic procedure that considers viscosity, geometry, entrance effects, minor losses, pressure measurement, and flow regime.

Multiple Choice Questions with Answers

Question 1

Which equation expresses conservation of mass for a constant-density incompressible fluid?

A. ∇ · 𝐮 = 0

B. ∇ × 𝐮 = 0

C. ∇²𝐮 = 0

D. Dρ/Dt = ρ∇ · 𝐮

Correct Answer: A. ∇ · 𝐮 = 0

Question 2

For a constant-property incompressible Newtonian fluid, the viscous term in the momentum equation is:

A. μ∇p

B. μ∇²𝐮

C. ρ∇²p

D. μ∇ · 𝐮

Correct Answer: B. μ∇²𝐮

Question 3

In plane Couette flow with one plate moving at U and gap h, the velocity profile is:

A. u = U(1 − y/h)²

B. u = U y/h

C. u = U(1 − y²/h²)

D. u = U h/y

Correct Answer: B. u = U y/h

Question 4

For plane Poiseuille flow between plates at y = ±h, the centreline velocity is:

A. (−dp/dx)h²/(2μ)

B. (−dp/dx)h²/(3μ)

C. (−dp/dx)h²/(4μ)

D. (−dp/dx)h/(2μ)

Correct Answer: A. (−dp/dx)h²/(2μ)

Question 5

For laminar flow in a circular pipe, the Hagen–Poiseuille flow rate is proportional to:

A. R

B. R²

C. R³

D. R⁴

Correct Answer: D. R⁴

Question 6

The Reynolds number is defined as:

A. μ/(ρUL)

B. ρUL/μ

C. ρμ/(UL)

D. UL/(ρμ)

Correct Answer: B. ρUL/μ

Question 7

In fully developed circular-pipe flow, the maximum velocity is related to the mean velocity by:

A. u_max = U_mean/2

B. u_max = U_mean

C. u_max = 3U_mean/2

D. u_max = 2U_mean

Correct Answer: D. u_max = 2U_mean

Question 8

The characteristic time for viscous momentum diffusion across a distance h is of order:

A. hν

B. h/ν

C. h²/ν

D. ν/h²

Correct Answer: C. h²/ν

Question 9

The no-slip condition at a stationary solid wall requires:

A. Normal pressure equal to zero

B. Fluid velocity equal to zero at the wall

C. Zero density at the wall

D. Zero viscosity at the wall

Correct Answer: B. Fluid velocity equal to zero at the wall

Question 10

For an oscillating boundary, the Stokes-layer thickness is:

A. √(2ν/ω)

B. √(ω/2ν)

C. 2ν/ω

D. ω/(2ν)

Correct Answer: A. √(2ν/ω)