The corrections for log law must be taken into account the presence of bubbles in the two phase turbulent boundary layer. In the present study, a logarithmic law for the wall based on the supposition of additional turbulent viscosity associated with bubble wakes in the boundary layer was proposed for bubbly flows. An empirical constant accounting both for shear induced turbulence interaction and for non-linearity of bubble was determined for the new wall law, this constant was deduced from experimental measurements. In the case of a turbulent boundary layer with millimetric bubbles developing on a vertical flat plate, the wall friction prediction achieved with the wall law was compared to the experiences. We obtained a good concordance between experimental and numerical result. This significant agreement for wall friction prediction was particularly important for the low void fraction when bubble induced turbulence have a considerable role.

‎In this paper‎, ‎for a singular perturbation problem consist of the Cauchy-Euler equation with local and non-local boundary conditions‎. ‎We investigate the condition of the self-adjoint and the non-self-adjoint‎, ‎also look for the formation or non-formation of boundary layers for local boundary conditions using the Frequent uniform limit method‎. ‎Also‎, ‎for the state of non-local conditions‎, ‎we convert the non-local boundary conditions into local conditions by finding the fundamental solution and then obtaining the necessary conditions with the help of the 4-step method‎. ‎Finally‎, ‎we determine the formation or non-formation of a boundary layer for non-local conditions such as local conditions‎.

This study develops a fuzzy sliding mode controller (FSMC) based on a variable boundary layer. A fuzzy inference mechanism has been used to tune the thickness of the boundary layers of the controller online. To lower the rate of calculation of the controller, a minimum rule base has been used. The aim of this paper is to design a controller which is able to remove the chattering and maintains the robustness of controller simultaneously. To prove the effectiveness of this method, a simulation has been done using MATLAB/SIMULINK. In this simulation, the results of 3 controllers, FSMC with auto-adjustable boundary layers, FSMC with fixed boundary layers, and FSMC with sign function are compared. The results of the simulation confirm that the performance of fuzzy sliding mode controller based on auto-adjustable boundary layer method is superior to the fixed boundary layer method.

In this paper, we investigate a singular perturbation problem including a fourth order O.D.E. with general linear boundary conditions. Firstly, we obtain the necessary conditions of solution of O.D.E. by making use of fundamental solution, then by compatibility of these conditions with boundary conditions, we determine that, for given perturbation problem, whether boundary layer is formed or not.

In this paper, an effective procedure based on coordinate stretching and radial ba-sis functions (RBFs) collocation method is applied to solve singularly perturbed differential-difference equations with layer behavior. It is well known that if the boundary layer is very small, for good resolution of the numerical solution at least one of the collocation points must lie in the boundary layer. In fact, a set of uniform centers is distributed in the computational domain, and then coordinate stretching based transform is used to move the centers, to the region with high gradients. In addition to the integrated multiquadric (MQ) collocation method is applied to solve the transformed equation. The effectiveness of our method is demonstrated on several examples with boundary layer in both cases, i.e., boundary layer on the left side as well as the right side.