3.2 Equations of Motion and Potential Flow
Stanley A. Berger
Integral Relations for a Control Volume
Like most physical conservation laws those governing motion of a ?uid apply to material particles or
systems of such particles. This so-called Lagrangian viewpoint is generally not as useful in practical
?uid ?ows as an analysis through ?xed (or deformable) control volumes ? the Eulerian viewpoint. The
relationship between these two viewpoints can be deduced from the Reynolds transport theorem, from
which we also most readily derive the governing integral and differential equations of motion.
Reynolds Transport Theorem
The extensive quantity B, a scalar, vector, or tensor, is de?ned as any property of the ?uid (e.g.,
momentum, energy) and b as the corresponding value per unit mass (the intensive value). The Reynolds
transport theorem for a moving and arbitrarily deforming control volume CV, with boundary CS (see
where Bsystem is the total quantity of B in the system (any mass of ?xed identity), n is the outward normal
to the CS, Vr = V(r, t) ? VCS(r, t), the velocity of the ?uid particle, V(r, t), relative to that of the CS,
VCS(r, t), and d/dt on the left-hand side is the derivative following the ?uid particles, i.e., the ?uid mass
comprising the system. The theorem states that the time rate of change of the total B in the system is
equal to the rate of change within the CV plus the net ?ux of B through the CS. To distinguish between
the d/dt which appears on the two sides of Equation (3.2.1) but which have different interpretations, the
derivative on the left-hand side, following the system, is denoted by D/Dt and is called the material
derivative. This notation is used in what follows. For any function f(x, y, z, t),
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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