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| {{continuum mechanics|cTopic=Laws}}
| | Valentine's Day is around the corner again and congratulations, you may rack your brains to take into account what involving gift select for the one you love girlfriend. Truly it is really an uphill task to decide on an ideal gift since most ladies are picky and hard to please.<br><br>Blog can be a contraction of 'web firewood.' Basically, a blog is a log of thoughts, ideas, useful links, photos, videos, ugg news or scandal. Blogs are a series of posts assembled in chronological order, and a lot bloggers agree they're type of expression. Blogs, through the late 1990's were lists of links maintained by tech savvy females. But, in recent years, blogs have become personal observances, updated regularly, and many accommodate rants and findings.<br><br>The original ugg boots were popular among surfers, hikers and outdoor types. Although these are still some of UGG's customers, these days you discover all regarding people appreciating these comfortable sheepskin boots. UGG now has got a varied collecting styles. Their boots coming from useful to chic. The next are some designs of ugg boots that find pleasurable.<br><br>The obvious solution for this problem is to wear hunter boots. However, they must not necessarily cheap boots displayed discount stores, otherwise they'll wear out quickly as well as the joint belonging to the sole to the boot proper will be an entrance for not necessarily rocks and burs likewise for dirt and small, unpleasant pests. Quality boots are the only method to confirm they will not wear out and let things in through the seams.<br><br>If you hire a guide, she will provide you with a directory of the necessary equipment that you need to bring. Undoubtedly most likely provide much of your equipment, is additionally hired the guide through an expedition carrier. However, if you are required to collect your own list of gear, here's a few essential items that you'll need to pick up for a trip.<br><br>It always be summer or winter, girls never care; they wear what they want to utilize. And uggs your such footwear that girls can't stop wearing them. You can find girls using them not simply for casual wear but also for party wear significantly. Girls pair mainly because with skin tight denim skirts offering them a smashing and ravishing look and feel. Many girls love these boots but they just don't know to pick them love these boots (Great confusion!). If you would be the one that not for each other with these boots, when i suggest that wear them once with your lifetime to feel the structure and convenience these boots (I bet you will love wearing them again).<br><br>Here's more in regards to [http://horizonafrica.com/img/ ugg boots australia] review our own site. |
| The '''Clausius–Duhem inequality'''<ref>{{Citation |last=Truesdell |first=Clifford |authorlink=Clifford Truesdell |year=1952 |title=The Mechanical foundations of elasticity and fluid dynamics |journal=Journal of Rational Mechanics and Analysis |volume=1 |issue= |pages=125–300 |doi= }}.</ref><ref>{{Citation |last=Truesdell |first=Clifford |lastauthoramp=yes |first2=Richard |last2=Toupin |year=1960 |chapter=The
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| Classical Field Theories of Mechanics |title=Handbuch der Physik |volume=III |location=Berlin |publisher=Springer |isbn= }}.</ref> is a way of expressing the [[second law of thermodynamics]] that is used in [[continuum mechanics]]. This inequality is particularly useful in determining whether the [[constitutive relation]] of a material is thermodynamically allowable.<ref>{{Citation |last=Frémond |first=M. |year=2006 |chapter=The Clausius–Duhem Inequality, an Interesting and Productive Inequality |title=Nonsmooth Mechanics and Analysis |series=Advances in mechanics and mathematics |volume=12 |pages=107–118 |location=New York |publisher=Springer |doi=10.1007/0-387-29195-4_10 |isbn=0-387-29196-2 }}.</ref>
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| | |
| This inequality is a statement concerning the irreversibility of natural processes, especially when energy dissipation is involved. It was named after the German physicist [[Rudolf Clausius]] and French physicist [[Pierre Duhem]].
| |
| | |
| == Clausius–Duhem inequality in terms of the specific entropy ==
| |
| The Clausius–Duhem inequality can be expressed in [[integral]] form as
| |
| :<math>
| |
| \cfrac{d}{dt}\left(\int_\Omega \rho~\eta~\text{dV}\right) \ge
| |
| \int_{\partial \Omega} \rho~\eta~(u_n - \mathbf{v}\cdot\mathbf{n})~\text{dA} -
| |
| \int_{\partial \Omega} \cfrac{\mathbf{q}\cdot\mathbf{n}}{T}~\text{dA} +
| |
| \int_\Omega \cfrac{\rho~s}{T}~\text{dV}.
| |
| </math>
| |
| In this equation <math>t\,</math> is the time, <math>\Omega\,</math> represents a body and the [[integral|integration]] is over the volume of the body, <math>\partial \Omega\,</math> represents the surface of the body, <math>\rho\,</math> is the [[mass]] [[density]] of the body, <math>\eta\,</math> is the specific [[entropy]] (entropy per unit mass), <math>u_n\,</math> is the [[normal]] velocity of <math>\partial \Omega\,</math>, <math>\mathbf{v}</math> is the [[velocity]] of particles inside <math>\Omega\,</math>, <math>\mathbf{n}</math> is the unit normal to the surface, <math>\mathbf{q}</math> is the [[heat]] [[flux]] vector, <math>s\,</math> is an [[energy]] source per unit mass, and <math>T\,</math> is the absolute [[temperature]]. All the variables are functions of a material point at <math>\mathbf{x}</math> at time <math>t\,</math>.
| |
| | |
| In [[Vector calculus#Differential operations|differential]] form the Clausius–Duhem inequality can be written as
| |
| :<math>
| |
| \rho~\dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right)
| |
| + \cfrac{\rho~s}{T}
| |
| </math>
| |
| where <math>\dot{\eta}</math> is the time derivative of <math>\eta\,</math> and <math>\boldsymbol{\nabla} \cdot (\mathbf{a})</math> is the [[divergence]] of the [[Euclidean vector|vector]] <math>\mathbf{a}</math>.
| |
| | |
| {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
| |
| !Proof
| |
| |-
| |
| |Assume that <math>\Omega</math> is an arbitrary fixed [[control volume]]. Then
| |
| <math>u_n = 0</math> and the [[derivative]] can be taken inside the integral to give
| |
| :<math>
| |
| \int_\Omega \frac{\partial }{\partial t}(\rho~\eta)~\text{dV} \ge
| |
| -\int_{\partial \Omega} \rho~\eta~(\mathbf{v}\cdot\mathbf{n})~\text{dA} -
| |
| \int_{\partial \Omega} \cfrac{\mathbf{q}\cdot\mathbf{n}}{T}~\text{dA} +
| |
| \int_\Omega \cfrac{\rho~s}{T}~\text{dV}.
| |
| </math> | |
| Using the [[divergence theorem]], we get
| |
| :<math>
| |
| \int_\Omega \frac{\partial }{\partial t}(\rho~\eta)~\text{dV} \ge
| |
| -\int_\Omega \boldsymbol{\nabla} \cdot (\rho~\eta~\mathbf{v})~\text{dV} -
| |
| \int_\Omega \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right)~\text{dV} +
| |
| \int_\Omega \cfrac{\rho~s}{T}~\text{dV}.
| |
| </math>
| |
| Since <math>\Omega</math> is arbitrary, we must have
| |
| :<math>
| |
| \frac{\partial }{\partial t}(\rho~\eta) \ge
| |
| -\boldsymbol{\nabla} \cdot (\rho~\eta~\mathbf{v}) -
| |
| \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) +
| |
| \cfrac{\rho~s}{T}.
| |
| </math>
| |
| Expanding out
| |
| :<math>
| |
| \frac{\partial \rho}{\partial t}~\eta + \rho~\frac{\partial \eta}{\partial t} \ge
| |
| -\boldsymbol{\nabla} (\rho_\eta)\cdot\mathbf{v} - \rho~\eta~(\boldsymbol{\nabla} \cdot \mathbf{v}) -
| |
| \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) +
| |
| \cfrac{\rho~s}{T}
| |
| </math>
| |
| or,
| |
| :<math>
| |
| \frac{\partial \rho}{\partial t}~\eta + \rho~\frac{\partial \eta}{\partial t} \ge
| |
| -\eta~\boldsymbol{\nabla} \rho\cdot\mathbf{v} - \rho~\boldsymbol{\nabla} \eta\cdot\mathbf{v} -
| |
| \rho~\eta~(\boldsymbol{\nabla} \cdot \mathbf{v}) -
| |
| \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) +
| |
| \cfrac{\rho~s}{T}
| |
| </math>
| |
| or,
| |
| :<math>
| |
| \left(\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \rho\cdot\mathbf{v} + \rho~\boldsymbol{\nabla} \cdot \mathbf{v}\right)
| |
| ~\eta +
| |
| \rho~\left(\frac{\partial \eta}{\partial t} + \boldsymbol{\nabla} \eta\cdot\mathbf{v}\right)
| |
| \ge -\boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) +
| |
| \cfrac{\rho~s}{T}.
| |
| </math>
| |
| Now, the [[material time derivative]]s of <math>\rho</math> and <math>\eta</math> are given by
| |
| :<math>
| |
| \dot{\rho} = \frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \rho\cdot\mathbf{v} ~;~~
| |
| \dot{\eta} = \frac{\partial \eta}{\partial t} + \boldsymbol{\nabla} \eta\cdot\mathbf{v}.
| |
| </math>
| |
| Therefore,
| |
| :<math>
| |
| \left(\dot{\rho} + \rho~\boldsymbol{\nabla} \cdot \mathbf{v}\right)~\eta +
| |
| \rho~\dot{\eta}
| |
| \ge -\boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) +
| |
| \cfrac{\rho~s}{T}.
| |
| </math>
| |
| From the [[conservation of mass]] <math>\dot{\rho} + \rho~\boldsymbol{\nabla} \cdot \mathbf{v} = 0</math>. Hence,
| |
| :<math>
| |
| {
| |
| \rho~\dot{\eta} \ge -\boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right) +
| |
| \cfrac{\rho~s}{T}.
| |
| }
| |
| </math>
| |
| |}
| |
| | |
| == Clausius–Duhem inequality in terms of specific internal energy ==
| |
| The inequality can be expressed in terms of the [[internal energy]] as
| |
| :<math>
| |
| \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le
| |
| - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}
| |
| </math>
| |
| where <math>\dot{e}</math> is the time derivative of the specific internal energy <math>e\,</math> (the internal energy per unit mass), <math>\boldsymbol{\sigma}</math> is the [[stress (physics)|Cauchy stress]], and <math>\boldsymbol{\nabla}\mathbf{v}</math> is the [[gradient]] of the velocity. This inequality incorporates the [[conservation of energy|balance of energy]] and the [[conservation of momentum|balance of linear and angular momentum]] into the expression for the Clausius–Duhem inequality.
| |
| | |
| {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
| |
| !Proof
| |
| |-
| |
| |Using the identity
| |
| <math> \boldsymbol{\nabla} \cdot (\varphi~\mathbf{v}) = \varphi~\boldsymbol{\nabla} \cdot \mathbf{v} + \mathbf{v}\cdot\boldsymbol{\nabla} \varphi</math> | |
| in the Clausius–Duhem inequality, we get
| |
| :<math>
| |
| \rho~\dot{\eta} \ge - \boldsymbol{\nabla} \cdot \left(\cfrac{\mathbf{q}}{T}\right)
| |
| + \cfrac{\rho~s}{T} \qquad\text{or}\qquad
| |
| \rho~\dot{\eta} \ge - \cfrac{1}{T}~\boldsymbol{\nabla} \cdot \mathbf{q} -
| |
| \mathbf{q}\cdot\boldsymbol{\nabla} \left(\cfrac{1}{T}\right)
| |
| + \cfrac{\rho~s}{T}.
| |
| </math>
| |
| Now, using index notation with respect to a [[Cartesian coordinate system]] <math>\mathbf{e}_j</math>,
| |
| :<math>
| |
| \boldsymbol{\nabla} \left(\cfrac{1}{T}\right) =
| |
| \frac{\partial }{\partial x_j}\left(T^{-1}\right)~\mathbf{e}_j =
| |
| -\left(T^{-2}\right)~\frac{\partial T}{\partial x_j}~\mathbf{e}_j
| |
| = -\cfrac{1}{T^2}~\boldsymbol{\nabla} T.
| |
| </math>
| |
| Hence,
| |
| :<math>
| |
| \rho~\dot{\eta} \ge - \cfrac{1}{T}~\boldsymbol{\nabla} \cdot \mathbf{q} +
| |
| \cfrac{1}{T^2}~\mathbf{q}\cdot\boldsymbol{\nabla} T
| |
| + \cfrac{\rho~s}{T} \qquad\text{or}\qquad
| |
| \rho~\dot{\eta} \ge -\cfrac{1}{T}\left(\boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s\right) +
| |
| \cfrac{1}{T^2}~\mathbf{q}\cdot\boldsymbol{\nabla} T.
| |
| </math>
| |
| From the [[conservation of energy|balance of energy]]
| |
| :<math>
| |
| \rho~\dot{e} - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} + \boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s = 0
| |
| \qquad \implies \qquad
| |
| \rho~\dot{e} - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} = - (\boldsymbol{\nabla} \cdot \mathbf{q} - \rho~s).
| |
| </math>
| |
| Therefore,
| |
| :<math>
| |
| \rho~\dot{\eta} \ge \cfrac{1}{T}\left(\rho~\dot{e}-\boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v}\right) +
| |
| \cfrac{1}{T^2}~\mathbf{q}\cdot\boldsymbol{\nabla} T
| |
| \qquad \implies \qquad
| |
| \rho~\dot{\eta}~T \ge \rho~\dot{e}-\boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} +
| |
| \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T}.
| |
| </math>
| |
| Rearranging,
| |
| :<math>
| |
| {
| |
| \rho~(\dot{e} - T~\dot{\eta}) - \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v} \le
| |
| - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} \qquad \qquad \square
| |
| }
| |
| </math>
| |
| |}
| |
| | |
| == Dissipation ==
| |
| The quantity
| |
| :<math>
| |
| \mathcal{D} := \rho~(T~\dot{\eta}-\dot{e}) + \boldsymbol{\sigma}:\boldsymbol{\nabla}\mathbf{v}
| |
| - \cfrac{\mathbf{q}\cdot\boldsymbol{\nabla} T}{T} \ge 0
| |
| </math>
| |
| is called the [[dissipation]] which is defined as the rate of internal [[entropy]] production per unit volume times the [[absolute temperature]]. Hence the Clausius–Duhem inequality is also called the '''dissipation inequality'''. In a real material, the dissipation is always greater than zero.
| |
| | |
| == See also ==
| |
| * [[Entropy]]
| |
| * [[Second law of thermodynamics]]
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| | |
| == External links ==
| |
| * [http://www.mechanics.rutgers.edu/TruesdellMemories.pdf Memories of Clifford Truesdell] by Bernard D. Coleman, Journal of Elasticity, 2003.
| |
| * [http://www.math.cmu.edu/~wn0g/noll/Thoughts%20on%20Thermomechanics.pdf Thoughts on Thermomechanics] by [[Walter Noll]], 2008.
| |
| | |
| {{DEFAULTSORT:Clausius-Duhem inequality}}
| |
| [[Category:Continuum mechanics]]
| |
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