For our simulations we chose a particular realisation of Eq. (1) which after a scaling is given by the dimensionless equations x j ' = α j R j (Z 1, Z 2) - γ j x j, j = 1,...,3, where x j is the scaled version of y j, α3 ∈ ⟨0, 1⟩ and all γ j ∈ ⟨0, 1⟩, and Z j = x j p /(x j p + 1).
Before the dimensionless equations are presented we tidy up the presentation by introducing some dimensionless parameters.
The dimensionless equations obtained for the model have been solved numerically by employing the finite-volume method.
The relative error between the oil production calculated by the dimensionless IPR equation of horizontal wells in foamy heavy oil reservoirs and actual oil production is less than 13%.
The evolution of the fluid flow is thus given by the dimensionless Navier Stokes equation: begin{aligned} mathrm{E} left(!frac{partial vec u}{partial t} + vec u cdot vec nabla vec u !right) !&=! -vec nabla Pi+ E nabla^{2} vec u !- !2 hat z! times vec u + frac{mathrm{Ra E}}{text{Pr}} frac {vec r}{r_{text{cmb}}} T &quad+ frac 1 {text{Pm}} (vec nabla times vec B) times vec B, end{aligned} (2).
The values of the mass transfer coefficient were correlated well by a unified dimensionless equation in terms of the energy dissipation rate per unit mass of liquid to cover the liquid-solid and gas-liquid-solid fluidized systems.
Under steady-state conditions for the enzyme binding reactions the local dynamics of the PFK effectors ATP and ADP is described by the (dimensionless) set of equations [ 12] (3) d d t α = v − σ ϕ S, d d t γ = q σ ϕ S − γ (4) ϕ S = α γ n 1 + (1 + α ) γ n where ϕ S denotes the fractional saturation.
By deducing the dimensionless coupling equations of three exciters, the synchronization problem is reduced to an existence and stability problems of finding the zero solutions for the averaged differential equations.
The discrete variables at each grid point are calculated as a function of time by integrating the dimensionless differential equations (22)–(24) with the fourth-order Runge-Kutta method.
