3.2 Conservation of momentum

Conservation of momentum

Conservation of momentum governs how forces, velocities, and pressures relate to each other in moving fluids. It provides the foundation for the Navier-Stokes equations and is the tool you'll reach for whenever you need to find forces on pipes, drag on objects, or thrust from propulsion systems.

Linear Momentum

Definition of linear momentum

Linear momentum is the product of mass and velocity:

$\vec{p} = m\vec{v}$

It's a vector quantity, so direction matters just as much as magnitude. The SI unit is kg·m/s.

Relationship between mass and velocity

Linear momentum scales directly with both mass and velocity. Double the mass at the same velocity, and momentum doubles. Double the velocity at the same mass, and momentum doubles again. This proportionality is straightforward, but keep it in mind when you move to fluid systems where mass is flowing continuously rather than sitting in a single object.

Angular Momentum

Definition of angular momentum

Angular momentum is the rotational counterpart of linear momentum:

$\vec{L} = I\vec{\omega}$

where $I$ is the moment of inertia and $\omega$ is the angular velocity. The SI unit is kg·m²/s.

Relationship between moment of inertia and angular velocity

Angular momentum depends on how mass is distributed (moment of inertia) and how fast the object rotates. An object with mass concentrated far from the rotation axis has a larger $I$ and therefore greater angular momentum at the same $\omega$. This is why a figure skater spins faster when pulling their arms in: $I$ decreases, so $\omega$ must increase to keep $L$ constant.

Momentum Conservation Laws

Conservation of linear momentum

When no external forces act on a system, total linear momentum stays constant:

$\sum_{i=1}^{n} m_i \vec{v}_i = \text{constant}$

During any interaction between parts of the system, momentum can shift between them, but the total doesn't change. In fluid dynamics, "external forces" include things like pressure from surroundings, gravity, and wall reactions, so you need to account for these carefully when drawing your control volume.

Conservation of angular momentum

Similarly, when no external torques act on a system, total angular momentum is conserved:

$\sum_{i=1}^{n} I_i \omega_i = \text{constant}$

This governs rotating flows like vortices and turbomachinery. A classic fluid example: as flow spirals inward toward a drain, the radius decreases, so the tangential velocity increases to conserve angular momentum.

Applications of Momentum Conservation

Collisions and impacts

Momentum conservation lets you predict outcomes of collisions. In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, momentum is still conserved but some kinetic energy converts to heat or deformation. Water hammer in pipes is a fluid-mechanics example: when a valve closes suddenly, the momentum of the moving fluid must go somewhere, producing large pressure spikes.

Jet propulsion

Jet engines and rockets work by ejecting mass at high velocity in one direction, which generates thrust in the opposite direction. The thrust depends on two factors:

• Mass flow rate ($\dot{m}$): how much mass is ejected per second
• Exhaust velocity ($v_e$): how fast that mass leaves

Thrust is approximately $F = \dot{m} \, v_e$. Higher exhaust velocity or higher mass flow rate means more thrust.

Fluid flow in pipes and channels

When fluid flows through a pipe bend or a nozzle, the change in momentum between inlet and outlet tells you the net force the fluid exerts on the pipe walls. This is one of the most common applications of the momentum equation in engineering. You select a control volume around the pipe section, evaluate momentum fluxes at each opening, and solve for the reaction forces.

Momentum Flux

Definition and units

Momentum flux is the rate at which momentum crosses a surface per unit area. For a fluid with density $\rho$ and velocity $v$, the momentum flux through a surface with normal velocity component $v_n$ is:

$\rho \, v \, v_n$

The units work out to kg/(m·s²), which is equivalent to N/m² (same units as pressure). This isn't a coincidence: pressure itself is a form of momentum flux at the molecular level.

Momentum flux in fluids

The full description of momentum transport in a fluid uses the momentum flux tensor (also called the stress tensor), which captures both:

• Normal components: momentum transferred perpendicular to a surface (related to pressure and normal viscous stresses)
• Shear components: momentum transferred parallel to a surface (related to viscous shear stresses)

Understanding momentum flux is how you calculate drag on a submerged object or the force a jet of water exerts on a plate.

Momentum Transfer

Mechanisms of momentum transfer

Momentum moves through a fluid by three mechanisms:

1. Advection: the bulk flow carries momentum from one place to another. If fluid is moving to the right, it carries its rightward momentum with it.
2. Molecular diffusion: random molecular motion transfers momentum between adjacent fluid layers. This is what causes viscous shear stress, and it's governed by the fluid's viscosity.
3. Turbulent mixing: in turbulent flows, chaotic eddies transport momentum far more effectively than molecular diffusion alone.

Molecular vs. turbulent momentum transfer

Molecular transfer is driven by velocity gradients and is proportional to the fluid's dynamic viscosity $\mu$. It dominates in slow, laminar flows.

Turbulent transfer is caused by fluctuating velocity components in turbulent flows and is often modeled using an eddy viscosity $\mu_t$, which can be orders of magnitude larger than $\mu$. In most engineering applications (pipe flow at typical speeds, atmospheric flows, ocean currents), turbulent transfer dominates. The Reynolds number tells you which regime you're in: low Re means molecular transfer matters most, high Re means turbulence takes over.

Momentum Equations

Integral momentum equation

This form applies conservation of momentum to a finite control volume. It states:

$\sum \vec{F} = \frac{\partial}{\partial t} \int_{CV} \rho \vec{v} \, dV + \int_{CS} \rho \vec{v} (\vec{v} \cdot \hat{n}) \, dA$

In words: the net force on the control volume equals the rate of change of momentum stored inside it plus the net momentum flux leaving through its surfaces. For steady flow, the storage term drops out, which simplifies things considerably.

Differential momentum equation

Applying Newton's second law to an infinitesimal fluid element gives:

$\rho \frac{D\vec{v}}{Dt} = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \vec{g}$

where $\frac{D}{Dt}$ is the material derivative, $p$ is pressure, $\boldsymbol{\tau}$ is the viscous stress tensor, and $\vec{g}$ is gravitational acceleration. For a Newtonian fluid with constant viscosity, this becomes the Navier-Stokes equations, the central governing equations of fluid dynamics.

Boundary conditions for momentum

Solving momentum equations requires boundary conditions at every edge of your domain:

• No-slip: fluid velocity equals zero at a stationary solid wall (the most common wall condition)
• Free-slip: zero shear stress at the boundary (used for symmetry planes or idealized free surfaces)
• Inlet/outlet: specified velocity profile or pressure at flow entrances and exits

Choosing the wrong boundary condition will give you a mathematically valid but physically meaningless solution, so this step deserves careful thought.

Momentum Analysis

Control volume approach

The control volume method is the workhorse of applied momentum analysis. The steps are:

1. Select a control volume that encloses the region of interest (pipe bend, nozzle, turbine, etc.)
2. Identify all forces acting on the control volume: pressure at each opening, wall reaction forces, gravity
3. Evaluate momentum fluxes at every inlet and outlet using $\dot{m}\vec{v}$ or $\rho A v^2$ for each opening
4. Apply the integral momentum equation in each coordinate direction
5. Solve for the unknown forces or velocities

This approach works even when you don't know the detailed flow field inside the control volume, which is what makes it so powerful.

Forces and moments on fluid elements

By evaluating momentum fluxes and pressure distributions around an object, you can calculate net force and moment without needing to know every detail of the flow. This is the basis for determining:

• Drag: the force component parallel to the freestream flow
• Lift: the force component perpendicular to the freestream flow
• Torque: the moment about a reference point

These quantities are typically expressed as dimensionless coefficients ($C_D$, $C_L$) so they can be applied across different scales and speeds.

Examples of Momentum Conservation

Hydraulic jumps

A hydraulic jump occurs when fast, shallow flow (supercritical, Froude number $Fr > 1$) abruptly transitions to slow, deep flow (subcritical, $Fr < 1$). You can analyze the jump by applying momentum conservation between a section upstream and a section downstream. The result is the Bélanger equation, which relates the upstream and downstream depths. Significant energy is dissipated in the turbulent roller of the jump, which is why hydraulic jumps are deliberately used in spillway stilling basins to safely dissipate energy.

Flow around objects

For an object in a flow, you can draw a control volume around it and use momentum conservation to find drag and lift. The momentum deficit in the wake downstream tells you how much momentum the object extracted from the flow, which equals the drag force. Wind tunnel testing uses exactly this principle: measure velocity profiles upstream and downstream, compute the momentum change, and you have the drag.

Propulsion systems

Propellers, turbines, and rockets all rely on momentum conservation. A propeller accelerates air or water backward; the reaction force pushes the vehicle forward. For a rocket:

$F_{\text{thrust}} = \dot{m} \, v_e + (p_e - p_a) A_e$

where $p_e$ and $p_a$ are the exit and ambient pressures, and $A_e$ is the nozzle exit area. The first term is the momentum thrust; the second is the pressure thrust. Analyzing these momentum changes lets you predict thrust, efficiency, and fuel consumption.