Large Deformation Problem of Bimodular Functionally-Graded Thin Circular Plates Subjected to Transversely Uniformly-Distributed Load: Perturbation Solution without Small-Rotation-Angle Assumption

In this study, the large deformation problem of a functionally-graded thin circular plate subjected to transversely uniformly-distributed load and with different moduli in tension and compression (bimodular property) is theoretically analyzed, in which the small-rotation-angle assumption, commonly used in the classical Föppl–von Kármán equations of large deflection problems, is abandoned. First, based on the mechanical model on the neutral layer, the bimodular functionally-graded property of materials is modeled as two different exponential functions in the tensile and compressive zones. Thus, the governing equations of the large deformation problem are established and improved, in which the equation of equilibrium is derived without the common small-rotation-angle assumption. Taking the central deflection as a perturbation parameter, the perturbation method is used to solve the governing equations, thus the perturbation solutions of deflection and stress are obtained under different boundary constraints and the regression of the solution is satisfied. Results indicate that the perturbation solutions presented in this study have higher computational accuracy in comparison with the existing perturbation solutions with small-rotation-angle assumption. Specially, the computational accuracies of external load and yield stress are improved by 17.22% and 28.79% at most, respectively, by the numerical examples. In addition, the small-rotation-angle assumption has a great influence on the yield stress at the center of the bimodular functionally-graded circular plate.

Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow. Article Highlights Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure. The predicted flowrate is accurate to within 5% for a wide range of permeability variations. If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

Numerical and perturbation solutions of cross flow of an Eyring-Powell fluid

This communication presents a comparative analysis of two-dimensional cross flow of non-Newtonian fluid with heat and mass transfer is presented in this article. Eyring-Powell fluid is chosen as the main carrier of heat and nano species through a uniform horizontal channel. Effects of suction are also taken into account by placing porous walls. Main source of the flow is the motion of upper plate that moves with a constant velocity in axial direction. Two different nano flows have been formulated by neglecting and, as well as, applying constant pressure gradient, respectively. In addition to this, the analytical solution is validated with the numerical solution. Perturbation technique is employed to obtain a sustainable solution for the highly nonlinear and coupled differential equations. Further, Range-Kutta method with shooting technique is employed to get an approximate solution. It if inferred that both numerical and series solutions display a complete agreement.

Entropy Analysis of an MHD Synthetic Cilia Assisted Transport in a Microchannel Enclosure with Velocity and Thermal Slippage Effects

The magnitude of shear stress at the ciliated wall is considered as the measure of efficiency of cilia beatings as it describes the momentum transfer between the medium and the cilia. Under high shear rate, some non-Newtonian fluids behave as visco-inelastic fluids. We consider here a ciliated channel coated with Prandtl fluid, a visco-inelastic fluid, with Hartmann layer under momentum and thermal slip effects. The flow in the channel is produced due to beatings of cilia that obey an elliptic path of motion in the flow direction. An entropy analysis of the flow is also conducted in wave frame. After introducing lubrication approximations in the governing equation, the perturbation solutions are calculated. The data for pressure rise per metachronal wavelength and frictional force at the ciliated wall are obtained by numerical integration. The analysis reveals that the higher values of cilia length and velocity slip parameters support fluid flow near the channel wall surface. Fluid temperature is an increasing function of thermal slip but a decreasing function of cilia length and slip parameters. Entropy in the channel can be minimized with an increase in cilia length and slip effect at the boundary. The magnitude of the heat transfer coefficient decreases by taking the substantial slippage and tiny cilia in length at the microchannel wall.