This paper presents a gradient theory of grade two based on an axiomatic conception of a nonlocal continuum theory for materials of grade n. The general classical model of linear momentum equation that is depending on the secondorder strain rate tensor and its gradients up to grade n À 1 is Cauchy's first law of motion. In special case, the secondgradient theory, the total stress tensor is a function of the rate of stress tensor and its first gradient. The objective of the present work is to study the turbulent flow through a pipe based on this theory. The linear momentum equation in this case is reduced to an ordinary differential equation of order four, which is solved by analytical method. The boundary conditions are generated by using two cases, slip and no-slip conditions. Finally, analytical results are compared with the experimental pieces of work and show good agreement with experimental data.

A parametric study involving the effects of some combinations of parameters, in particular, different combinations of the Rayleigh number, amplitude, temperature, and inclination angle of a two-dimensional long wavy-walled channel on a laminar incompressible fluid flow and natural convection within the channel is performed. The considered channel has an undulated wall as one side of the channel, and a parallel flat wall at a differentially different temperature as its counterpart. The channel sustains variable inclination angle, variable wavy wall amplitude, and variable temperaturedifference between its two walls. A perturbation technique in terms of the small waviness of the undulated wall is performed to obtain a set of non-linear ordinary differential equations for the main flow and its perturbations. Solving this set of equations determines the streamline and temperature profiles for the imposed varying parameters. The results reveal that the fluid velocity along the channel axis increases with increasing the Rayleigh number, but decreases with increasing the channel inclination angle. The results also show that eddies appeared due to the steep undulations intensify as the temperature-difference between the two channel walls increases. The veracity of the present work is demonstrated through comparing the obtained results with those available in the literature.

Purpose This study aims to introduce a new modified method for estimating steam turbine high pressure (HP)-intermediate pressure (IP) leakage flow based on the experimental data extracted from a 250 MW re-heat steam turbine. Design/methodology/approach Effects of measurement uncertainties on the test results are investigated and key parameters are specified via a new modified method to diminish the test uncertainties. The recommended method is based on a constant IP turbine pressure ratio at the same loads. Considering this assumption, it was found that the turbine pressure ratio can be achieved in the second and the third tests with a much longer duration. Findings The results showed that the cross-over temperature is a major parameter in the leakage flow estimation tests, whereas hot reheat and cross-over pressures are the next priorities. It was also observed that as the cross-over temperature varies by 1°C, the estimated leakage flow error significantly differs by up to 72.6 per cent. Originality/value It is concluded that the present modified HP-IP leakage flow estimation method seems to be more accurate in comparison with previously proposed methods in literature.