CHEN E4112 Y Spring 2023 Homework #2
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CHEN E4112 Y Spring 2023
Homework #2 - Due Feb. 15, 2023 11:40 AM EST
1. Recommended reading:
(a) Review of equilibrium thermodynamics: Venerus/Ottinger Chap-
ter 4 (sections 4.1-5).
(b) Kinematics and constitutive principles: Slattery_on_Consist_Funct_Fall_16 (on-line).
(c) Material frame indi§erence: Notes_Frame_Indi§erence_10 (on- line).
(d) Non-equilibrium thermodynamics (NET) and enforcing the second law: Muller_Thermodynamics_1987_on viscous áuids and Liu method (on-line); NET Rigid Media 2021 a (on-line); NET Vis- cous FLuids 2021 a (on-line).
2. Details of the mechanical energy balance: Venerus/Ottinger Exer- cise 5-12.
3. A candidate constitutive relation for q : The following expression has been proposed for the heat áux q
q + 入q(1) = ━kvT
where
(::)(1) = ╱ ━ vvi ·← (::)
is a "convected" time derivative and 入 and k are positive constants. What are the units of 入 and k ? Show that this model is frame indi§er- ent. Remember that corresponds to the Lagrangian time derivative, so that ╱ u、* = ╱ u、x(*) = ╱ u*、x* = u* ; where denotes
the derivative in the * frame and u is a vector Öeld; you may use results from Notes_Frame_Indi§erence_10 to solve this problem.
4. Details of the application of constitutive principles for viscous áuids: The on-line reading 疝ulldr T〉drmocyn_m/αs 69Aβ on v/sαous áu/cs _nc L/u mdt〉oc summarizes the application of constitutive prin- ciples for viscous áuids. Eqns 1.54 and 1.96 are the Önal results for compressible and incompressible viscous áuids, respectively.
(a) Rewrite Eqns 1.54 and 1.96 using the notion we use in class (also
used by Venerus and Ottinger, BSL).
(b) Summarize the steps used by Muller to arrive at Eqns 1.54.
(c) What are the main di§erences between the steps in Mullerís calcula- tion summarized in b). and that done in class for rigid media?
2023-03-16