1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Continuum Mechanics: Rigid Heat Conductor

  1. Dec 9, 2015 #1
    1. The problem statement, all variables and given/known data
    Determine the thermodynamic restrictions for a rigid heat conductor defined by the constitutive equations:
    [tex]\DeclareMathOperator{\grad}{grad}\psi = \hat{\psi}\left(\theta,\grad \theta, \grad \grad \theta\right) \\
    \eta = \hat{\eta}\left(\theta,\grad \theta, \grad \grad \theta\right)\\
    \textbf{q} = \hat{\textbf{q}}\left(\theta,\grad \theta, \grad \grad \theta\right)[/tex]
    2. Relevant equations

    [tex]\rho \left( \dot{\psi}+\dot{\theta} \eta \right)-\textbf{T}:\textbf{D}+\frac{\textbf{q}}{\theta}\cdot \grad\theta\leq0[/tex]

    3. The attempt at a solution
    I have already found the thermodynamic restrictions for [itex]\psi[/itex] because it's straightforward (take the material derivative and apply the chain rule), but I don't know where to start for [itex]\eta[/itex] or [itex]\textbf{q}[/itex]. We're using the Coleman-Noll approach and I understand the procedure once I have [itex]\psi = \hat{\psi}\left(\theta,\grad \theta, \grad \grad \theta\right)=something[/itex], but I'm struggling with finding a good starting place for [itex]\eta[/itex]. From what my professor has said, it seems like they can be arbitrary as long as they contain [itex]\theta,\grad \theta, \grad \grad \theta[/itex]. That doesn't seem like a very good way to go about this, though. A result of plugging in [itex]\dot{\psi}[/itex] into the relevant equation above (with [itex]\textbf{T}:\textbf{D}=0[/itex] because it's rigid) is [itex]\hat{\eta}=-\frac{\partial \hat{\psi(\theta)}}{\partial \theta}[/itex]. Can I just plug that into the relevant equation above while keeping [itex]\dot{\psi}[/itex] as [itex]\dot{\psi}[/itex] to find the thermodynamic restrictions for [itex]\eta[/itex]? The equation for [itex]\textbf{q}[/itex] is probably [itex]\textbf{q}=-\textbf{K}\textbf{g}[/itex].

    What do you think? Am I on the right track?
    Last edited: Dec 9, 2015
  2. jcsd
  3. Dec 10, 2015 #2
    I talked with my professor today and it turns out that I already finished the problem! Finding [itex]\dot{\psi}[/itex], plugging it in, and "solving" for the thermodynamic restrictions was the whole thing. Please consider this thread solved.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted