In Ref. 1 , we derived a correction to the Navier-Stokes expression for the stress , in the one-dimensional case, for large values of the average velocity u . This correction has the form R(g) xu , where g xu is the longitudinal rate. The viscosity factor R(g) is a solution of a differential equation, subject to a certain initial condition Eq. 1 of Ref. 2 . This equation was not studied completely in Ref. 1 . In their Comment, Uribe and Pina indicated some interesting features of this equation. In particular, they asked what happens to the relevant solution at negative values of g? Let us denote the points of the (g ,R) plane as P (g ,R). The relevant solution R(g) emerges from point P0 (0, 4 3 ), and, as we demonstrate below, it can be unequally continued to arbitrary values of g , positive and negative. This solution can be constructed, for example, with the Taylor expansion used in Ref. 2 , and which is identical to the relevant subseries of the Chapman-Enskog expansion cf. Ref. 1 , Eq. 8 . However, the difficulty in constructing this solution numerically for g 0 originates from the fact that the same point P0 is the point of essential singularity of other irrelevant solutions to Eq. 1 . Indeed, for g 1, let us consider R(g) R(g) , where R(g) 4 3 8 9 ( 2)g is the relevant solution for small g , and (g) is a deviation. Neglecting in Eq. 1 all regular terms of the order g), and also neglecting g in comparison to , we derive the following equation: (1 )g(d /dg) 3 2 . The solution is (g) (g0)exp a(g 1 g0 ) , where a (3/2)(1 ) . The essential singularity at g 0 is apparent from this solution, unless (g0) 0 that is, no singularity exists except for the relevant solution R R). Let (g0) 0. If g 0, then →0, together with all its derivatives, as g→0. If g 0, the solution expands, as g→0. The complete picture for 1 is as follows: The lines g 0 and P (g ,g ) define the boundaries of the domain of attraction A A A , where A P g 0,R g 1 , and A P g 0,R g 1 . The graph G „g ,R(g)... of the relevant solution belongs to the closure of A , and goes through the points P0 (0, 4 3 ), P ( ,0), and P ( ,0). These points at the boundaries of A are the points of essential singularity of any other irrelevant solution with the initial conditions P A , and P A G . That is, if P A , P A G , the solution expands at P0, and is attracted to P . If P A , and P A G , the solution expands at P , and is attracted to P0. It is this latter case that was found numerically by Uribe and Pina. The above consideration is supported by our independent numerical study of Eq. 1 see Fig. 1, corresponding to the case of hard spheres, 1 2 ). The difficulty of numerical integration from P0 to negative values of g is quite clear: the integration then goes in a direction opposite to the direction of attraction of the irrelevant solutions. However, the same feature becomes an advantage for the integration to the positive values of g: because of the attraction of all irrelevant solutions to the relevant one, roundoff errors will be suppressed. This explains why no difficulty was encountered in this part of integration in Ref. 2 . One can utilize the attraction in the negative domain in
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Sep 11, 2024
Haukur Arnþórsson 
Open Access
