Formulation of mesoscale models

Following a single particle, the rates of change of the mesoscale variables can be written in a Lagrangian form  

Note that the dependence of the mesoscale model on the moments of the NDF is used to introduce multi-particle effects through the mean-field variables such as the disperse-phase volume fraction. For this reason, the right-hand sides of Eq. (5.9) are different from the exact microscale models that were introduced in Chapter 4 (i.e. Eqs. (4. l)-(4.3) on page 103). Formally, given these dependences, the mesoscale models on the right-hand sides of Eq. (5.9) can be expressed explicitly (for example) as A p(t, X, ), where 

As mentioned above, the mesoscale model can contain stochastic processes, which lead to the diffusion terms appearing in Eq. (5.2). Using A p as an example, we can write 

In order to make the connection between the single-particle model and the mesoscale diffusion coefficients, we can rewrite Eq. (5.11) using vectors and matrices Z = Z + BW, where Z = [AJp, G , A j, Gjf and Z = [Afp, Gp, Apf, Gf] are column vectors of length TV, W = [Wpyp,Wpfp, Wpy,Wpfj is a column vector of length N, and B is a matrix of size N X N defined by 

The reader should note that the microscale model is used to determine the nonzero terms in B, and thus for the following discussion B can be assumed to be known. Using matrix notation and the properties of the Wiener process (Gardiner, 2004), a symmetric N x N diffusion matrix D can be defined by 

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