ME 310 Chap.7 Differential Analysis and the Navier-Stokes Equations
Differential Analysis and the Navier-Stokes Equations
From chunks to points
The syllabus describes two levels of analysis: control volumes, chunks of fluid, and differential analysis, points of fluid.
The October lectures take the second route, applying mass and momentum conservation to an infinitesimal element so that velocity and pressure can be predicted everywhere in the flow rather than only at inlets and outlets.
Continuity and the stream function
Mass conservation at a point says that the divergence of the velocity field is zero for an incompressible fluid.
Any proposed velocity field can be tested with that condition.
In two dimensions, defining the velocity components as derivatives of a stream function satisfies continuity automatically, makes lines of constant stream function into streamlines, and gives the flow rate between two streamlines as their difference.
Euler and Navier-Stokes
Momentum at a point balances density times material acceleration against pressure gradient, gravity and viscous stress.
Dropping the viscous term gives the Euler equation, whose integral along a streamline is Bernoulli. Keeping it for a Newtonian incompressible fluid gives the Navier-Stokes equations, covered over two lectures that include their form in other coordinate systems.
A further lecture compares Newtonian and non-Newtonian flow, using a handout, because the viscous term is only as good as the stress law behind it.
Exact solutions
For flows with straight parallel streamlines the nonlinear terms vanish and the equations can be solved exactly: flow between a sliding and a fixed plate, pressure-driven flow between plates or in a pipe, and a film draining down a slope.
The early homework already touches this: a film on an inclined surface with a given velocity profile, where the task is to find and interpret the shear stress, and a cylinder falling through oil in a thin gap.
The Navier-Stokes chapter explains where those profiles come from.
Why this unit matters later
The exact solutions become the laminar pipe-flow results of the pipe chapter, and the balance of inertia against viscous terms inside Navier-Stokes is what the Reynolds number measures in dimensional analysis and boundary layers.
What this chapter covers
- 01
Differential continuity equation
- 02
Testing velocity fields for incompressibility
- 03
Stream function and flow rate between streamlines
- 04
Vorticity and irrotational flow
- 05
Euler equation and its link to Bernoulli
- 06
Navier-Stokes equations for Newtonian fluids
- 07
Exact solutions: Couette, Poiseuille and falling films
Worked example · free
Checking Couette flow against Navier-Stokes
- 1Assume u = u(y) only, v = 0; continuity ∂u/∂x + ∂v/∂y = 0 is satisfied.
- 2The convective term u ∂u/∂x vanishes; with zero pressure gradient, x-momentum reduces to μ d²u/dy² = 0.
- 2Integrate twice with u(0) = 0 and u(h) = U: u = U y / h, a straight line.
- 1Shear stress τ = μU/h = 0.30 × 1.0 / 0.002 = 150 Pa, uniform across the gap.
Key terms
- Stream Function
- The stream function is a scalar whose derivatives give the velocity components of a two-dimensional incompressible flow.
- Vorticity
- Vorticity is the curl of the velocity field, equal to twice the local angular velocity of fluid particles.
- Navier-Stokes Equations
- The Navier-Stokes equations are the momentum equations for a Newtonian fluid, including pressure, gravity and viscous forces.
- Euler Equation
- The Euler equation is the momentum equation for an inviscid fluid, from which Bernoulli is obtained along a streamline.
- Couette Flow
- Couette flow is the flow between two parallel surfaces driven by the relative motion of one surface, with a linear velocity profile when no pressure gradient acts.
- Poiseuille Flow
- Poiseuille flow is the pressure-driven laminar flow in a channel or pipe, with a parabolic velocity profile.
Differential Analysis and the Navier-Stokes Equations FAQ
How do I know which terms of Navier-Stokes to cross out?
Use stated physical reasons: steady flow removes time derivatives, fully developed flow removes changes along the flow direction, and continuity then often removes the convective term. Write each reason beside the term you delete.
Why do Couette and Poiseuille flows count as exact solutions?
Their streamlines are straight and parallel, so the nonlinear convective acceleration is identically zero. What remains is a linear ordinary differential equation that can be integrated with the wall conditions.
What changes for a non-Newtonian fluid?
The viscous term in Navier-Stokes assumes stress proportional to rate of deformation. For shear-thinning, yield-stress or viscoelastic materials a different stress law must replace it, and familiar profiles such as the parabola change shape.
How do I test whether a velocity field is physically possible?
For an incompressible fluid, compute the divergence: the sum of the derivative of each velocity component with respect to its own coordinate. If it is zero everywhere the field conserves mass; if not, it cannot describe a liquid.
Exam move
Practise the five-step exact-solution routine on a new geometry each time: coordinates, deleted terms, integration, boundary conditions, engineering quantities.
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