a b s t r a c tAn exact solution of the three-dimensional incompressible Navier-Stokes equations with the continuity equation is produced in this work. The solution is proposed to be in the form V = ∇Φ + ∇ × Φ where Φ is a potential function that is defined as Φ = P (x, y, ξ ) R (y) S (ξ ), with the application of the coordinate transform ξ = kz − ς (t). The potential function is firstly substituted into the continuity equation to produce the solution for R and S. The resultant expression is used sequentially in the Navier-Stokes equations to reduce the problem to a class of nonlinear ordinary differential equations in P terms, in which the pressure term is presented in a general functional form. General solutions are obtained based on the particular solutions of P where the equation is reduced to the form of a linear differential equation. A method for finding closed form solutions for general linear differential equations is also proposed. The uniqueness of the solution is ensured because the proposed method reduces the original problem to a linear differential equation. Moreover, the solution is regularised for blow up cases with a controllable error. Further analysis shows that the energy rate is not zero for any nontrivial solution with respect to initial and boundary conditions. The solution being nontrivial represents the qualitative nature of turbulent flows.

Production logging in horizontal wells has been challenging due to phase stratification and complex flow regimes. New technologies were introduced to properly characterize such flow dynamics; however, adding a new complex completion strategy, such as a slotted liner, raises the challenge to a new level. When a slotted liner is not equipped with a hanger, two flow paths become possible—one inside the slotted liner and the other outside it—with possible fluid exchange between the two paths. Production logging tools can only see the inner flow. The lack of information about the outer flow raises uncertainties about the real producing zone(s) and may affect the success of future well intervention operations. We describe a new methodology that combines multiphase production logging measurements with distributed temperature survey data. Both sets of data were acquired simultaneously during conveyance of the production logging tool on coiled tubing. The production logging data offers insights into phase distribution and enables production profiling inside the slotted liner; distributed temperature data solves for total formation production. Combining the two interpretations enables quantification of flow in each of the two paths and helps identify the fluid exchange between them, which would otherwise be misinterpreted as fluid entry from the formation. The result of this integrated interpretation methodology is accurate determination of zonal contributions.

a b s t r a c tThe three-dimensional incompressible Navier-Stokes equations with the continuity equation are solved analytically in this work. The spatial and temporal coordinates are transformed into a single coordinate ξ . The solution is proposed to be in the form V = ∇Φ + ∇ × Φ where Φ is a potential function that is defined as Φ = P(x, ξ )R(ξ ). The potential function is firstly substituted into the continuity equation to produce the solution for R and the resultant expression is used sequentially in the Navier-Stokes equations to reduce the problem to the class of nonlinear ordinary differential equations in P terms. Here, more general solutions are also obtained based on the particular solutions of P. Explicit analytical solutions are found to be mathematically similar for the cases of zero and constant pressure gradient. Two examples are given to illustrate the applicability of the method. It is also concluded that the selection of variables for the potential function can be interchanged from the beginning, resulting in similar explicit solutions.

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In this paper, we introduced a new Complex linear operator in the terms of the Mittag-Leffler type functions. Furthermore, several subordination, superordination and sandwich-type outcomes for certain analytic functions related to this new operator are introduced and discussed. Relevant connections of the outcomes are also presented.