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

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7. In the determinant of J, let Ai be the cofactor of ∂xi /∂Xα. 27) where δji is the Kronecker symbol. Writing either i = j = 1, or 2, or 3, gives the value of J. 5. 29 is dJ = Ju,ii dt where u,ii is the formula for div u in curvilinear coordinates. The other kinematic formula is due to Reynolds and is known as Reynolds’ transport theorem. Let F(r, t) be a physical property per unit volume. 30) which is a well-defined function of time. 31) where dν = dᐂ(t). 26, where dν0 = dᐂ(t0). 31 becomes * In this book, we have used the pseudosymbols dν for volume element and dS for surface element, with an occasional sub- or superscript in all integrals.

4, keeping t fixed, we have δr = (∂r/∂Xα)∂Xα, so that the edges of the parallelepiped are (∂r/∂X1)∂X1, (∂r/∂X2)∂X2, and (∂r/∂X3)∂X3. 7. 7. In the determinant of J, let Ai be the cofactor of ∂xi /∂Xα. 27) where δji is the Kronecker symbol. Writing either i = j = 1, or 2, or 3, gives the value of J. 5. 29 is dJ = Ju,ii dt where u,ii is the formula for div u in curvilinear coordinates. The other kinematic formula is due to Reynolds and is known as Reynolds’ transport theorem. Let F(r, t) be a physical property per unit volume.

33c. 3 Two closely spaced points , P and Q separ ated by the vector δr. 36) S*( t ) which is the required relation. Note that if c = u, then V*(t) = ᐂ(t), whereas if c = 0, then V* is a fixed volume. 36 means that the time differentiation is to be performed after the volume integral has been evaluated. 36 must be interpreted as the time rate of change of F within the control volume. It is then appropriate to use ∂/∂t in place of d/dt. , form a field of values. In a field, the quantities are distributed according to certain physical laws that are peculiar to that field.

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