1-6stress-strain - 1 Notes on 1.63 Advanced Environmental...

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1 Notes on 1.63 Advanced Environmental Fluid Mechanics Instructor: C. C. Mei, 2001 [email protected], 1 617 253 2994 March 10, 2001 1-6stressstrain.tex, 1.6 Relations between stress and rate-of-strain tensors When the ° uid is at rest on a macroscopic scale, no tangential stress acts on a surface. There is only the normal stress, i.e., the pressure p δ ij which is thermodynamic in origin, and is maintained by molecular collisions. Denoting the additional stress by τ ij which is due to the relative motion on the continuum scale, σ ij = p δ ij + τ ij (1.6.1) The second part is called the viscous stress τ ij and must depend on gradients of velocity, q i x j , 2 q i x j x k • • • . 1.6.1 Newtonian ° uid For many ° uids in nature such as air and water, the relation between τ ij and q i / x i are linear under most circumstances. Such ° uids are called Newtonian. For second-rank tensors, the most general linear relation is, τ ij = C ij‘m q x m . (1.6.2) where C ij‘m is a coe cient tensor of rank 4. In principle there are 3 4 = 81 coe cients. It can be shown (Spain, Cartesian Tensors ) that in an isotropic ° uid the fourth-rank tensor is of the following form: C ij‘m = λδ ij δ ‘m + ° ( δ i‘ δ jm + δ im δ j‘ ) (1.6.3) Eighty one coe cients in C ij‘m reduce to two: λ and ° , and τ ij = ° ˆ q i x j + q j x i !
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