By A. J. Chorin, J. E. Marsden (auth.)

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44 We thus change our previous definition. Instead of assuming that { where ~ force on per unit area is the normal to { s S, force on we now assume that S per unit area where (J made. 1) is a matrix', about which some assumptions will have to be (Jon need not be parallel to n. 1) is somewhat ambiguous since component parallel to n. (Jon may contain a This issue will be resolved below when we give a more definite functional form to g. l). Thus we see that of momentum across the boundary of it approximates in a reasonable way Wt • ~ modifies the transport We will choose (J = so that the transport of momentum by molecular motion.

Our last main goal in this section is to develop the vorticity equation somewhat further for the important special case of incompressible flow. 12) where ~ a -- a ~(x " y t) v yu -x - the flow and u, v is the (scalar) vorticity field of are the components of u. Assume that the flow D with a fixed boundary is contained in some plane domain aD, with the boundary condition u·n where ~ = 0, on aD , is the unit outward normal to simply connected (Le. has no "holes"). 13) aD. 14) streamlines lie on level curves of be a streamline, so x' = u(x,y) and Then = -vu + uv = O.

Let 'i/u ayu "U x a v y a v z d w a w y a w z ['xu a v x Jacobian matrix of u. h + O(h 2 ) -- - where 'i/u(x)·h column vector. 2) (~1'~2'~3). 1). x D is a symmetric matrix, is the quadratic form associate with where ation tensor. is the inner product in ~3. e. We call D the deform- We now discuss its physical interpretation. is symmetric, there is for in which D ~ fixed, an orthonormal basis is diagonal: Since D 26 Keep of ~ fixed and consider the original vector field as a function The motion of the fluid is described by the equations ~.