By Dimitris Drikakis, William Rider

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**Sample text**

8) depend on nine joint invariants of A1 and A2 , where trA1 = 0, and are usually determined experimentally. • Models of the rate type: These models consider that the stress tensor T(t) and the tensor of the history of the deformation gradient F(t) satisfy the diﬀerential equation ˙ . . , T(n−1) ; F, F, ˙ . . 60) T(n) = S(T, T, where S is a tensor function. The above equation deﬁnes media of the rate type of order n, the latter being determined by the order of the derivative of T. The Oldroyd constitutive equation [401, 402] for a viscoelastic ﬂuid falls in this category.

92) is more appropriate for low Reynolds numbers (Re → 0). For Re = 0 we obtain the Stokes ﬂow which is dominated solely by viscous eﬀects. 93) where u and l are the characteristic velocity and length scales of the largest eddies in the turbulent ﬂow; Rel is the corresponding Reynolds number. In engineering and geophysical applications, Rel is usually of the order of O(106 ) − O(108 ) and thus the number of grid points needed to perform simulations would be of the order of O(1013 ) − O(1018 ). Therefore, direct numerical simulation (DNS) of ﬂows at such high Reynolds numbers is beyond the projecting future capacity of parallel computers, unless there is a major breakthrough in computers technology.

Examples of ﬂows in which the Froude number can be important include the waves generated by a ship, open channel ﬂows, jet ﬂows. 5 Nondimensionalization of the Governing Equations 41 where σ is the coeﬃcient of surface tension (measured in SI units as Newtons per meter). The Weber number is the ratio of inertia forces to surface tension and is used when surface tension is important. Applications in which the Weber number may be important include droplets, capillary and free surface ﬂows. 86) Eu = ρU 2 where ∆p is a reference pressure diﬀerence.