Fluid Mechanics Course

Essential Foundation: Critical prerequisite for plasma physics (MHD), astrophysics, and engineering

Conservation Laws

Mass, momentum, and energy - the fundamental equations of fluid motion

Introduction

The motion of fluids is governed by conservation laws - mass, momentum, and energy cannot be created or destroyed, only transported and converted. These fundamental principles, combined with constitutive relations, give us the complete equations of fluid dynamics.

We present both integral forms (useful for control volume analysis) and differential forms (useful for detailed flow field solutions).

Conservation of Mass (Continuity)

Integral Form

The rate of change of mass in a control volume equals the net mass flux through the surface:

$$\frac{\partial}{\partial t}\int_{CV} \rho \, dV + \oint_{CS} \rho (\vec{v} \cdot \hat{n}) \, dA = 0$$

For steady flow: ∮ρ(v·n)dA = 0 → Mass in = Mass out

Differential Form

Using the divergence theorem:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$$

Or equivalently, using the material derivative:

$$\frac{D\rho}{Dt} + \rho (\nabla \cdot \vec{v}) = 0$$

Incompressible Flow

When Dρ/Dt = 0 (density of a fluid particle doesn't change), continuity simplifies to:

$$\nabla \cdot \vec{v} = 0$$

Valid for liquids and low-speed (Ma < 0.3) gas flows.

Conservation of Momentum

Integral Form (Control Volume)

Newton's second law applied to a control volume:

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

Forces include pressure, gravity, viscous stresses, and external forces.

Cauchy Momentum Equation

General differential form (valid for any continuous medium):

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

where τ is the deviatoric stress tensor (viscous stresses for Newtonian fluids).

Navier-Stokes Equations

For incompressible Newtonian fluids with constant viscosity:

$$\rho\left(\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v}\right) = -\nabla p + \mu \nabla^2 \vec{v} + \rho \vec{g}$$

∂v/∂t

Unsteady

(v·∇)v

Convection

μ∇²v

Diffusion

-∇p + ρg

Pressure + Gravity

Euler Equations (Inviscid)

When viscous effects are negligible (high Re, away from boundaries):

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

Conservation of Energy

First Law for a Control Volume

Energy balance including work and heat transfer:

$$\dot{Q} - \dot{W}_s = \frac{\partial}{\partial t}\int_{CV} e\rho \, dV + \oint_{CS} \left(e + \frac{p}{\rho}\right)\rho (\vec{v} \cdot \hat{n}) \, dA$$

where e = u + v²/2 + gz is specific energy (internal + kinetic + potential)

Differential Energy Equation

For a compressible fluid:

$$\rho \frac{De}{Dt} = -\nabla \cdot \vec{q} - p(\nabla \cdot \vec{v}) + \boldsymbol{\tau}:\nabla\vec{v}$$

Terms: heat conduction, pressure work, viscous dissipation

For Incompressible Flow

Often temperature can be treated separately from mechanics (weak coupling), and the energy equation simplifies to tracking temperature evolution via heat conduction and advection.

Bernoulli Equation

A special case of energy/momentum conservation for steady, inviscid, incompressible flow along a streamline:

$$p + \frac{1}{2}\rho v^2 + \rho g z = \text{constant along streamline}$$

p

Static pressure

½ρv²

Dynamic pressure

ρgz

Hydrostatic pressure

Extended Forms

Unsteady Bernoulli

$$\frac{p}{\rho} + \frac{v^2}{2} + gz + \int \frac{\partial v}{\partial t} ds = \text{constant}$$

With Head Loss (Engineering)

$$\frac{p_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{p_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_L$$

hL accounts for viscous losses (friction, minor losses)

Applications

Pitot Tube

Velocity from Bernoulli: v = √(2(p₀-p)/ρ) where p₀ is stagnation pressure

Venturi Meter

Flow rate from pressure difference using continuity + Bernoulli

Siphon

Flow driven by elevation difference - must stay above vapor pressure

Jet Propulsion

Momentum flux from nozzle provides thrust: F = ṁ(vexit - vinlet)

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