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Additional info for Fluid Dynamics : Theoretical and Computational Approaches, Third Edition

Sample text

In a field, the quantities are distributed according to certain physical laws that are peculiar to that field. In this section, we are interested in the distribution of velocity in a fluid medium, so as to uncover those kinematic quantities that play an important role in the structure of the field itself. Consider two neighboring points P and Q. What mathematical expression is required to predict the velocity at Q if we are given the velocity and its first partial derivatives at P? 3, let the coordinates of P be represented as r and those of Q as r + δr.

A translation 2. A deformation 3. 49 is called the angular velocity vector or the vorticity vector. At every point in the flow, the fluid particles rotate about an instantaneous axis, and the vector ω is represented by the direction of the instantaneous axis of rotation. Vortex Line A trajectory of the field of vorticity vectors is called a vortex line. In other words, a line in the flow whose tangents are directed along the local vorticity vectors forms a vortex line. 5. A crude example of a vortex line is the thread passing through the holes of a number of beads, the beads being treated as fluid particles.

Note that uI is the absolute velocity of the fluid particle with respect to an inertial frame whose origin coincides instantaneously with that of the rotating frame. (d) Use the time operator defined in (c) and show that d Ω dˆ Ω = dt dt and dˆ dΩ (Ω × r) = Ω × v + ×r dt dt (e) Using the results obtained in (a) to (d) show that, when reference is made to a Cartesian rotating frame, the acceleration is given by a= du dˆv d Ω =p+ + × r + 2 Ω × v + Ω × (Ω × r) dt dt dt Ω/dt) × r = acceleration due to the time variation of Ω.

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