Analysis of Rate of Deformation
The principal aim of the following two subsections is to derive a relationship between the stress and the
rate of strain to be used in the momentum Equation (3.2.25). The reader less familiar with tensor
notation may skip these sections, apart from noting some of the terms and quantities de?ned therein, and
. (proceed directly to Equations (3.2.38) or (3.2.39
where is the viscous part of the total stress and is called the deviatoric stress tensor, is the identity
tensor, and dij is the corresponding Kronecker delta (dij = 0 if i ¹ j; dij = 1 if i = j). We make further
assumptions that (1) the ?uid exhibits no preferred directions; (2) the stress is independent of any previous
history of distortion; and (3) that the stress depends only on the local thermodynamic state and the
kinematic state of the immediate neighborhood. Precisely, we assume that is linearly proportional to
the ?rst spatial derivatives of u, the coef?cient of proportionality depending only on the local thermodynamic
state. These assumptions and the relations below which follow from them are appropriate for
a Newtonian ?uid. Most common ?uids, such as air and water under most conditions, are Newtonian,
but there are many other ?uids, including many which arise in industrial applications, which exhibit socalled
non-Newtonian properties. The study of such non-Newtonian ?uids, such as viscoelastic ?uids,
is the subject of the ?eld of rheology.
With the Newtonian ?uid assumptions above, and the symmetry of which follows from the
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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