Conservation Laws and Constitutive Relations

TL;DR

Conservation laws tell you what quantities must be preserved in fluid flow (mass, momentum, energy). Constitutive relations describe how your specific fluid behaves under stress. Together, they give you the complete mathematical framework to derive the Navier-Stokes equations.

1. The Mental Model

Think of conservation laws as the universal accounting rules that every fluid must follow—nothing gets created or destroyed. Constitutive relations are like the personality traits of your particular fluid—how viscous it is, how it responds to shear. Conservation laws provide the skeleton; constitutive relations add the flesh.

2. The Core Material

2.1 The Three Conservation Laws

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Every fluid flow problem starts with three fundamental conservation principles that nature refuses to violate.

Mass Conservation (Continuity Equation)
The amount of fluid flowing into any volume must equal the amount flowing out, plus any accumulation inside. In mathematical form:

$$\frac{\partial \rho}{\partial t} + abla \cdot (\rho \mathbf{u}) = 0$$

where $\rho$ is density and $\mathbf{u}$ is velocity. For incompressible fluids (constant density), this simplifies beautifully to $
abla \cdot \mathbf{u} = 0$—the divergence of velocity is zero.

Momentum Conservation
Newton's second law applied to fluid elements. The rate of change of momentum equals the sum of all forces acting on the fluid:

$$\rho \frac{D\mathbf{u}}{Dt} = - abla p + abla \cdot \boldsymbol{\tau} + \rho \mathbf{g}$$

Here $\frac{D}{Dt}$ is the material derivative (following a fluid particle), $p$ is pressure, $\boldsymbol{\tau}$ is the stress tensor, and $\mathbf{g}$ represents body forces like gravity.

Energy Conservation
The first law of thermodynamics for moving fluids. Internal energy changes due to heat transfer, work done by pressure and viscous forces:

$$\rho \frac{DE}{Dt} = -p( abla \cdot \mathbf{u}) + \boldsymbol{\tau} : abla \mathbf{u} + abla \cdot (k abla T) + Q$$

where $E$ is specific internal energy, $k$ is thermal conductivity, $T$ is temperature, and $Q$ represents heat sources.

flowchart TD
    A["Conservation Laws<br/>(Universal Physics)"] --> B["Mass Conservation<br/>∂ρ/∂t + ∇·(ρu) = 0"]
    A --> C["Momentum Conservation<br/>ρ Du/Dt = -∇p + ∇·τ + ρg"]
    A --> D["Energy Conservation<br/>ρ DE/Dt = -p(∇·u) + τ:∇u + ∇·(k∇T)"]

    E["Constitutive Relations<br/>(Material Properties)"] --> F["Newtonian Viscosity<br/>τ = μ(∇u + (∇u)ᵀ) + λ(∇·u)I"]
    E --> G["Fourier Heat Conduction<br/>q = -k∇T"]
    E --> H["Equation of State<br/>ρ = ρ(p,T)"]

    B --> I["Complete System<br/>(Navier-Stokes Equations)"]
    C --> I
    D --> I
    F --> I
    G --> I
    H --> I

2.2 Constitutive Relations: Describing Fluid Behavior

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Conservation laws alone aren't enough—you need to specify how your particular fluid behaves. That's where constitutive relations come in.

Newtonian Viscosity Model
For most common fluids (water, air, oils), stress is proportional to the rate of strain:

$$\boldsymbol{\tau} = \mu [ abla \mathbf{u} + ( abla \mathbf{u})^T] + \lambda ( abla \cdot \mathbf{u}) \mathbf{I}$$

Here $\mu$ is dynamic viscosity and $\lambda$ is the second viscosity coefficient (often assumed to be $-\frac{2}{3}\mu$ for monatomic gases). The first term handles shear deformation, the second handles volumetric expansion/compression.

Fourier's Law of Heat Conduction
Heat flux is proportional to temperature gradient:

$$\mathbf{q} = -k abla T$$

The negative sign means heat flows from hot to cold regions. The thermal conductivity $k$ determines how efficiently your fluid conducts heat.

Equation of State
This relates density to pressure and temperature. For ideal gases:

$$p = \rho R T$$

where $R$ is the specific gas constant. For liquids, you might use more complex relations or simply assume incompressibility ($\rho$ = constant).

2.3 Putting It All Together

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When you substitute the constitutive relations into the conservation equations, you get the complete Navier-Stokes system. For an incompressible, Newtonian fluid with constant properties, this becomes:

Continuity: $
abla \cdot \mathbf{u} = 0$

Momentum: $\rho \frac{D\mathbf{u}}{Dt} = -
abla p + \mu
abla^2 \mathbf{u} + \rho \mathbf{g}$

Energy: $\rho c_p \frac{DT}{Dt} = k
abla^2 T + \Phi$

where $c_p$ is specific heat and $\Phi$ is viscous dissipation.

These equations are coupled—velocity affects temperature through convection and viscous heating, while temperature affects density and viscosity. This coupling makes most real-world problems nonlinear and challenging to solve analytically.

3. Worked Example

Let's derive the momentum equation for steady, incompressible flow in a 2D channel to see how conservation and constitutive relations work together.

Given: Steady flow between parallel plates separated by distance $2h$. Flow is fully developed (velocity profile doesn't change with $x$), so $u = u(y)$ only, $v = 0$.

Step 1: Apply Mass Conservation
$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$

Since $u$ doesn't depend on $x$ and $v = 0$, continuity is automatically satisfied.

Step 2: Apply Momentum Conservation
For steady flow, $\frac{\partial}{\partial t} = 0$. The $x$-momentum equation becomes:
$$u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho}\frac{\partial p}{\partial x} + u \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)$$

Since $u = u(y)$ only and $v = 0$:
$$0 = -\frac{1}{\rho}\frac{\partial p}{\partial x} + u \frac{\partial^2 u}{\partial y^2}$$

Step 3: Recognize the Physics
The pressure gradient $\frac{\partial p}{\partial x}$ must be constant (independent of $y$) for this equation to have a solution. This makes physical sense—the pressure drop drives the flow.

Step 4: Solve with Constitutive Relation
Let $\frac{\partial p}{\partial x} = -G$ (constant). Then:
$$ u \frac{\partial^2 u}{\partial y^2} = \frac{G}{\rho}$$

Integrating twice:
$$u(y) = \frac{G}{2\mu} y^2 + C_1 y + C_2$$

Step 5: Apply Boundary Conditions
No-slip at walls: $u(-h) = u(h) = 0$

This gives us $C_1 = 0$ and $C_2 = -\frac{G h^2}{2\mu}$

Final Result:
$$u(y) = \frac{G}{2\mu}(h^2 - y^2)$$

This is the classic parabolic velocity profile for Poiseuille flow. Notice how we used mass conservation (continuity) to simplify the problem, momentum conservation to get the governing equation, and the Newtonian constitutive relation ($\tau = \mu \frac{du}{dy}$) to relate stress to velocity gradients.

4. Key Takeaways

4.1 Most Important Concepts

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• Conservation laws are universal—every fluid must conserve mass, momentum, and energy regardless of its material properties.

• Constitutive relations are material-specific—they encode how your particular fluid responds to stress, temperature gradients, and deformation.

• The material derivative $\frac{D}{Dt}$ follows fluid particles—it accounts for both local time changes and convective transport as fluid moves.

• Stress tensors capture all internal forces—pressure (normal stress) and viscous stresses (both shear and volumetric) acting on fluid elements.

• Coupling makes problems nonlinear—velocity affects temperature through convection, temperature affects viscosity, and viscosity affects velocity.

• Boundary conditions come from physics—no-slip at solid walls, specified pressures at inlets/outlets, heat transfer conditions at surfaces.

• Incompressible flow simplifies everything—constant density eliminates the need for energy equations in purely mechanical problems.

4.2 Common Misconceptions

• "Conservation laws and Navier-Stokes are the same thing" → Conservation laws are general principles; Navier-Stokes equations result from combining conservation laws with specific constitutive relations.

• "Viscosity always opposes motion" → Viscosity opposes relative motion between fluid layers, but can actually help maintain flow in some geometries through momentum transfer.

• "The material derivative is just a time derivative" → It includes convective transport; $\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{u} \cdot
abla$ accounts for fluid particle motion.

• "Incompressible means the fluid can't be compressed" → It means density changes are negligible for the flow problem at hand; even "incompressible" water can be compressed with enough pressure.

4.3 Compare & Contrast

5. Now Try It

Derive the energy equation for steady, incompressible flow over a heated flat plate. Start with the general energy conservation law, apply the constitutive relation for heat conduction (Fourier's law), and use the assumptions of constant properties and negligible viscous dissipation. Your final equation should relate temperature to velocity and thermal diffusivity. Then identify what boundary conditions you'd need at the plate surface and far from the plate.

Success looks like: You produce the advection-diffusion equation $u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha
abla^2 T$ and correctly identify that you need specified temperature or heat flux at the wall.