Continuum Mechanics

The force per area on a surface is called the traction \( \mathbf{t} \). The Cauchy stress tensor \( \boldsymbol{\sigma} \) is defined to satisfy \[ \mathbf{t} = \boldsymbol{\sigma}\hat{\mathbf{n}} \] where \( \hat{\textbf{n}} \) is the unit normal vector of the surface.

The stress is being applied to a control volume which is a infinitesimal inertial volume. The control volume is at a quasi-equalibrium, that is, the changes in tensile and shear stresses across the volume are infinitesimal and the forces and torques should sum to zero. This justifies the Cauchy stress tensor being symmetric.

In a coordinate system, \( \sigma_{ij} \) is the force in \( i \)th direction, when the surface is facing \( j \)th direction.

• Mean Normal Stress Tensor

The isotropic stress can change the volumn of the stressed body. \[ \pi \mathbf{I} := \frac{1}{3}\operatorname{tr} (\boldsymbol{\sigma}) \mathbf{I}. \] The three accounts for the three direction that the stress is applied.

In its full generality, \[ p := \zeta \nabla\cdot \mathbf{u} - \pi \] where \( \zeta \) is teh volume viscosity, \( \mathbf{u} \) is the flow velocity.

• \( \boldsymbol{\tau} \), \( s_{ij} \)
• Stress Deviator Tensor

Deviation of the Cauchy stress tensor from the isotropy. \[ \boldsymbol{\tau} := \boldsymbol{\sigma} - \pi \mathbf{I}. \]

Assuming

• The Cauchy stress tensor is Galilean invariant
• \( p \) is independent of the strain \( \boldsymbol{\varepsilon} \)
• The fluid is isotropic,

The relation (constitutive equation) between the strain and stress is given as: \[ \boldsymbol{\sigma} = -p\mathbf{I} + \lambda \operatorname{tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2\mu \boldsymbol{\varepsilon} \] where \( \lambda \) is the second viscosity.

The equation can also be written as: \[ \boldsymbol{\sigma} = -(p - \zeta \nabla\cdot \mathbf{u})\mathbf{I} + \mu \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^{\mathsf{T}} - \frac{2}{3}(\nabla \cdot \mathbf{u})\mathbf{I} \right) \] using the bulk viscosity.

\[ \boldsymbol{\tau} = \mu (\nabla \mathbf{u} + (\nabla \mathbf{u})^{\mathsf{T}} \] where \(\boldsymbol{\tau}\) is the deviatoric stress, \( \mu \) is the dynamic viscosity, and \(\mathbf{u}\) is the flow velocity.

Coordinate representation would be \[ \tau_{ij} = \mu \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right). \]

The fluid that satisfies the Newton's law of viscosity is called the Newtonian fluid.

• \(\mu\), \( \eta \)
• Shear Viscosity

The relational constant between stress and momentum change. It has the unit of \(\rm Pa\cdot s\) in SI, poise in cgs.

• Momentum Diffusivity

\[ \nu = \frac{\mu}{\rho}. \]

The stress occurs due to irreversible resistance, over the reversible resistance by ientropic bulk modulus.

• Volume Viscosity

\[ \zeta := \lambda + \frac{2}{3}\mu \] where \( \lambda \) is the second viscosity, and \( \mu \) is the dynamic viscosity.

For an object moving in a fluid with viscosity \( \mu \), if the size of the object is smaller than the mean free path of the fluid particles, the effective viscosity \( \mu_{\mathrm{eff}} \) deviates from the macroscopic viscosity \( \mu \). \[ \eta_{\mathrm{eff}} = \eta \frac{1}{1+\frac{b}{pr}} \] where \( p \) is the pressure, \( b = 8.22\times 10^{-3}\ \mathrm{Pa\cdot m}\) is a constant, and \( r \) is the radius of the particle.

This quantity is used when establishing the mechanical equalibrium using the Stokes' law: \[ 6\pi r \eta_{\mathrm{eff}} v = F. \] From the equalibrium, the radius \( r \) can be obtained. \[ r = \sqrt{\frac{9\eta_{\mathrm{eff}} v_t}{2g\rho}}. \] where \( v_t \) is the terminal velocity of free falling particle, \( g \) is the gravitational constant, \( \rho \) is the density of the fluid.

\[ \boldsymbol{\varepsilon} = \frac{1}{2}(\nabla \mathbf{u} + (\nabla \mathbf{u})^{\rm T}) \] This is the symmetric part of the gradient of the displacement field \( \mathbf{u} \).

\[ \mathbf{J} = -D\nabla \varphi \] where \(\mathbf{J}\) is the diffusion flux, amount of substance per area per time, and \(D\) is the diffusion coefficient of diffusivity, and \(\nabla\varphi\) is the concentration gradient.

• \(D\) has the unit of area per time.
• \(\varphi\) has the unit of concentration, the amount of substance per volume.

• \[ \frac{\partial \varphi}{\partial t} = D\nabla^{\cdot 2}\varphi. \]
• This is the same form as the heat equation.

• \[ \frac{\partial u}{\partial t} = \alpha \nabla^{\cdot 2} u \]
  • where \(\alpha\) is the thermal diffusivity.
• \[ \alpha = \frac{k}{\rho c} \]
  • where \(k\) is the thermal conductivity, \(\rho\) is the density of the material, \(c\) is the specific heat capacity of the material.

• Fundamental solution to the heat equation.

• Material Coordinates

Lagrangian specification of the flow field represents individual fluid parcels. \[ \mathbf{X}(\mathbf{x}_0, t) \]

• Specification at fixed location.
• \[ \mathbf{u}(\mathbf{x}, t) \]

It is related to Mach number and Reynolds number by \[ \mathrm{Kn} = \frac{\mathrm{Ma}}{\mathrm{Re}} \sqrt{\frac{\gamma \pi}{2}}. \]

\[ \frac{D}{Dt} := \frac{\partial }{\partial t} + \mathbf{u}\cdot\nabla. \]

The first term describes the change due to time at a fixed position, and the second term describes the change due to movement in space at a fixed time, which combine to describe the rate of change while moving along with the flow.

For a system in which the velocity of the particles are not equilibrated, not even locally, the probability density function \( f(\mathbf{r}, \mathbf{p}, t) \) is a good description of the system.

It is the number density in the phase space, satisfying \[ dN = f(\mathbf{r}, \mathbf{p}, t)\, \mathrm{d}^3 \mathbf{r}\, \mathrm{d}^3 \mathbf{p} \] where \( \mathrm{d}N \) is the number of particles in a small chunk of the phase space \( \mathrm{d}^3\mathbf{r}\,\mathrm{d}^3\mathbf{p} \).

According to the Liouville's theorem, the density of state in the phase space does not change in a conservative system. When one is interested in the microscopic behavior, the force once called friction disappears and just molecular interactions remains, constituting a conservative system.

There's still collisions that can disturb the probability density function. So, \[ \frac{\mathrm{d}f}{\mathrm{d}t} = \left( \frac{\partial f}{\partial t} \right)_{\mathrm{coll}}. \] The change of \( f \) due to collision is swept under the simple looking collision term, and it is not an easy task to find the exact form of this term. The total derivative is taken along the trajectory of physical parcel in the phase space, therefore when expanded

\begin{align*} \frac{\partial f}{\partial t} &+ \frac{\mathrm{d}\mathbf{r}}{\mathrm{d} t} \cdot \frac{\partial f}{\partial \mathbf{r}} + \frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}\cdot \frac{\partial f}{\partial \mathbf{p}} = \left( \frac{\partial f}{\partial t} \right)_{\mathrm{coll}} \\ \implies \frac{\partial f}{\partial t} &+ \frac{\mathbf{p}}{m}\cdot \frac{\partial f}{\partial \mathbf{r}} + \mathbf{F}\cdot \frac{\partial f}{\partial \mathbf{p}} = \left( \frac{\partial f}{\partial t} \right)_{\mathrm{coll}}. \end{align*}

• Velocity \( \mathbf{v} := \mathbf{p}/m \)
• Flow Velocity \[ \mathbf{u} := \langle \mathbf{v} \rangle = \int \mathbf{v} f(\mathbf{r}, \mathbf{v}, t)\, \mathrm{d}^3 \mathbf{v}. \]
• Peculiar Velocity \[ \mathbf{w} := \mathbf{v} - \mathbf{u}. \]
• Pressure \[ p := \frac{1}{3} \rho \langle w^2 \rangle. \]
• Pressure Tensor \[ \mathbf{P} := -\boldsymbol{\sigma} = \rho \langle \mathbf{w} \mathbf{w}^{\mathsf{T}}\rangle \] where \( \boldsymbol{\sigma} \) is the stress tensor.
• Heat Flux \[ \mathbf{h} := \frac{1}{2} \rho \langle w^2 \mathbf{w} \rangle. \]

\[ \frac{\partial \rho}{\partial t} = -\nabla\cdot(\rho\mathbf{u}). \] It arises from the global mass conservation.

\[ \rho \frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t} = \vec{f}_m(\rho, \mathbf{u}, p, T) \] where \(D/Dt\) is the material derivative, the rate of change while moving along with the flow.

• The Eulerian specification is used, since the Lagrangian specification is unstable.
• One form of this is Navier-Stokes equation.

\[ \rho\frac{\mathrm{D}}{\mathrm{D}t}f(\mathbf{u}, T) = f_e(\rho, \mathbf{u}, p, T) \] where the \(f\) is the local energy function.

• \[ f_{s,1}(p, \rho, T) = 0 \]
• \[ f_{s,2}(\mu, T) = 0 \]

Assuming the fluid is isothermal, \[ \rho\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t}=\nabla\cdot\boldsymbol{\sigma}+\rho\mathbf{f} \] where \( \rho \) is the mass density, \( \mathbf{u} \) is the flow velocity, \( \mathrm{D}/\mathrm{D} t \) is the material derivative, \(\boldsymbol{\sigma}\) is Cauchy stress tensor, and \( \mathbf{f} \) is the body force.

We assume the linear stress. Substituting, we get: \[ \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = - \nabla p + \nabla \cdot \left[ \mu \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^{\mathsf{T}} - \frac{2}{3}(\nabla \cdot \mathbf{u})\mathbf{I} \right) \right] + \nabla (\zeta \nabla \cdot \mathbf{u}) + \rho \mathbf{f}. \]

For a incompressible fluid \( \nabla\cdot \mathbf{u} = 0 \), simplifying the equation: \[ \rho\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t}=-\nabla p+\mu\nabla^2\mathbf{u}+\rho\mathbf{f}. \]

Note that \(\nabla\cdot \boldsymbol{\tau} = \mu\nabla^{\cdot 2}\mathbf{u}\) here, and this fluid can be called Newtonian.

\[ \boldsymbol{\omega} = \nabla\times \mathbf{u}. \]

• The cross product of vorticity vector \(\boldsymbol{\omega}\) and velocity vector \(\mathbf{u}\) of the flow field: \[ \mathbf{l} = \mathbf{u}\times \boldsymbol{\omega}. \]
• This appears in the convective acceleration term of the material derivative in the Navier-Stokes equation.

• Flow in which the vorticity vector and the velocity vector are parallel.
• Flow in which the Lamb vector is zero.