2 where k$_{y}$ is the y component of the wave-vector as de- fined later in Section III. Such a constitutive equation is expected to be necessary for the description of flows in highly confined systems, due to the large change in the strain rate with position in the vicinity of the wall [8]. The best available theoretical predictions of the wave- vector dependent viscosity are based on mode-coupling theory and generalized Enskog theory [14–16]. However, these theories do not quantitatively agree with data ob- tained via computer simulations [12]. The theoretical predictions focus on the transverse momentum density autocorrelation function, which is found by an iterative numerical solution of a system of nonlinear equations. Consequently, the theories do not result in analytical ex- pressions for the correlation functions or the wave-vector dependent transport coefficients, which are the focus of the present study. More recently, a modified collective mode approach has been successfully applied by Omelyan et al. [17] to the TIP4 model of water. In contrast to other semi-phenomenological approaches used for in- stance in TIP4P and SPC/E models of water by Bertolini et al. [18] and Palmer [19], Omelyan et al. reproduced the reciprocal space kernel using a relatively small num- ber of modes. while the Fourier transform of the mass density is ˜ ρ$_{a}$ ( k , t ) = ∑ N$_{a}$ i $_{=1}$m$_{i}$ e i k · r i . We define the Fourier transform of a function f ( r ) as F [ f ( r )] = ˜ f ( k ) = ∫ ∞ $_{−∞}$e $^{ikr}$f ( r ) dr . The atomic representation of the momentum density for a molecular fluid can be written in real space as [29]: where ρ$_{a}$ ( r , t ) = ∑ N$_{a}$ i $_{=1}$m$_{i}$ δ ( r − r$_{i}$ ) is the mass density, m$_{i}$ , r$_{i}$ and v$_{i}$ are the mass, position and velocity of atom i . The summation runs over the number of atoms N$_{a}$ in the system. The Fourier transform of the momentum density is In this paper, we extend the work done by Hansen et al. [20] and focus on computing the spatially non-local viscosity kernel for monatomic and diatomic fluids over a wider range of wave-vectors, state points and potential energy functions. We are specifically interested in identi- fying functional forms that fit the reciprocal space kernel data. On the basis of these results, we are be able to assess the length scale (i.e. the width of the real space kernel) over which the governing generalized constitutive relation Eq. (3) must be used. We expect that non-local transport phenomena would be relevant in shock waves [12, 21–24], shear banding [25], flows of micellar solutions [26], suspensions of rigid fibers [27] and jammed or glassy systems [28]. where the mass density is defined as ρ$_{a}$ ( r , t ) = ∑ N$_{m}$ i =1 ∑ N$_{s}$ α $_{=1}$m$_{iα}$ δ ( r − r$_{iα}$ ). The inner summation extends over the N$_{s}$ mass points in a molecule and the outer sum- mation extends over the number of molecules N$_{m}$ in the system. In general, N$_{s}$ depends on the molecule index i for a multicomponent system, but in our systems N$_{s}$ is the same for all molecules and the interaction sites are assumed to have identical mass, namely m$_{iα}$ . The Fourier transform of the momentum density is This paper is structured as follows: In Section II A we give an overview of the general formulation and the expressions for the complex wave-vector and frequency dependent viscosity are given. In Section III we describe the simulation methodology and conditions. In Section IV we present our molecular dynamics simulation data and compare the results of our monatomic and diatomic viscosity kernels, particularly the shape of the kernels. Finally, we summarize and conclude our analysis in Sec- tion 5. where the transformed mass density is ˜ ρ$_{a}$ ( k , t ) = ∑ N$_{m}$ i =1 ∑ N$_{s}$ α $_{=1}$m$_{iα}$ e i k · r iα . For molecules composed of N$_{s}$ atoms we can define the mass of molecule i , M$_{i}$ = N$_{s}$ α $_{=1}$m$_{iα}$ , position of the molecular center of mass as r$_{i}$ = ∑ N$_{s}$ α $_{=1}$m$_{iα}$ r$_{iα}$ /M$_{i}$ , position of site α of molecule i relative to the center of mass of molecule i as R$_{iα}$ = r$_{iα}$ − r$_{i}$ , and center of mass momentum of the molecule as p$_{i}$ = ∑ N$_{s}$ α $_{=1}$p$_{iα}$ . This means that the atomic mass density can be written in k -space as ˜ ρ$_{a}$ ( k , t ) = ∑ N$_{m}$ i =1 ∑ N$_{s}$ α $_{=1}$m$_{iα}$ e i k · ( r $_{i}$+ R $^{$_{iα}$)}$. If we expand this rela- tion further we can express the atomic mass density in terms of molecular mass density in which we de- fine the mass density in the molecular representation as ˜ ρ$_{m}$ ( k , t ) = ∑ N$_{m}$ i $_{=1}$M$_{i}$ e i k · r i in reciprocal space and We shall briefly introduce the main conceptual back- ground used in this work, namely, the wave-vector de- pendent momentum density, stress and viscosity in the atomic and molecular formulations. A. Wave-vector dependent momentum density for atomic and molecular fluids II. METHODOLOGY ˜ J $^{A}$( k , t ) = N$_{m}$ ∑ i =1 N$_{s}$ ∑ α =1 m$_{iα}$ v$_{iα}$ ( t ) e i k · r iα (8) J $^{A}$( r , t ) = ρ$_{a}$ ( r , t ) v ( r , t ) = N$_{m}$ ∑ i =1 N$_{s}$ ∑ α =1 m$_{iα}$ v$_{iα}$ ( t ) δ ( r − r$_{iα}$ ) (7) ˜ J ( k , t ) = N$_{a}$ ∑ i =1 m$_{i}$ v$_{i}$ ( t ) e i k · r i (6) J ( r , t ) = ρ$_{a}$ ( r , t ) v ( r , t ) = N$_{a}$ ∑ i =1 m$_{i}$ v$_{i}$ ( t ) δ ( r − r$_{i}$ ) (5) For a single component atomic fluid the real space mi- croscopic momentum density is given by [13]