Validity Of Analytical Solution Research Articles

Thermal performance analysis of non‐Newtonian fluid transport through turbine discs

AbstractIn this paper, the analytical investigation of non‐Newtonian fluid flow through a turbine disc with a heat generation effect is considered. The flow mechanics through the turbine disc is modeled using a coupled system of higher‐order partial equations, which are transformed into ordinary order utilizing proper similarity transforms. Governing models are analyzed using the variation iteration and homotopy perturbation analytical methods. As observed from the results, the study shows enhancement of heat generation for non‐Newtonian fluid flow through turbine disc, reveals abating thermal distribution. More so, the graphical expression shows that heat generation improves up till mid‐disc. Thereafter, an increasing trend toward the fluid injection surface is observed. Validation of analytical solutions with numerical analysis boosts confidence in this study, which may be used to provide further insights into the engineering science of non‐Newtonian flow of hydraulic/lube oil in practical applications, such as hydraulic couplings.

Modelling of III-Nitride Epitaxial Layers Grown on Silicon Substrates with Low Dislocation-Densities

Large lattice and thermal expansion coefficients mismatches between III-Nitride (III N) epitaxial layers and their substrates inevitably generate defects on the interfaces. Such defects as dislocations affect the reliability, life time, and performance of photovoltaic (PV) devices. High dislocation densities in epitaxial layer generate higher v-shaped pits densities on the layer top surface that also directly affect the device performance. Therefore, using an approach such as the embedded void approach (EVA) for defects reduction in the epitaxial layers is essential. EVA relies on the generation of high densities of embedded microvoids (~108/cm2), with ellipsoidal shapes. These tremendous number of microvoids are etched near the interface between the III N thin-film and its substrate where the dislocation densities present with higher values. This article used a 3-D constitutive model that accounts the crystal plasticity formulas and specialized finite element (FE) formulas to model the EVA in multi-junction PV and therefore to study the effect of the embedded void approach on the defects reduction. Mesh convergence and 2-D analytical solution validation is conducted with accounting thermal stresses. Several aspect and volume ratios of the embedded microvoids are used to optimize the microvoid dimensions.

Closed-form elastic solution for irregular frozen wall of inclined shaft considering the interaction with ground

An analytical solution is derived that provides a closed-form formulation for stresses and displacements of a deep inclined shaft with irregular frozen wall liner in an infinite elastic medium, subjected to a non-uniform stresses. This solution is based on: (i) the conformal mapping of a doubly connected domain of prescribed shape onto a circular ring by an appropriate numerical optimization scheme which is achieved by MATLAB program, (j) the complex variable method. For a particular case, the distributions of hoop stress and radial displacement inside frozen wall are presented with analytical solution. The accuracy of stress functions is verified by comparison of different number of coefficients in stress functions and it show that the number of coefficients k≥15 is advised. The validity of analytical solution is verified by a comparison between its result and that obtained from the finite element program ANSYS. Noting that the shaft excavation can be regarded as relaxation of initial ground stresses around shaft excavation boundary, this paper presents the mechanical problem as two separate problems: (i) initial stresses and displacements model where stresses keep constant and displacements equal zero around frozen wall and ground, (j) stresses relaxation model with relaxed normal and shear stresses acted on excavated boundary. The solution is worked out by the principle of superposition in elastic medium. The Young's modulus ratio (β) of the frozen wall to the ground reflects the influence of ground to frozen wall, and the influence decreases with the growth of β. The proposed solution can provide analytical basis for frozen wall design and be used as a quick-solver for back-analysis of in situ stresses and displacements.

Peristaltic transport of bi-viscosity fluids through a curved tube: A mathematical model for intestinal flow.

The human intestinal tract is a long, curved tube constituting the final section of the digestive system in which nutrients and water are mostly absorbed. Motivated by the dynamics of chyme in the intestine, a mathematical model is developed to simulate the associated transport phenomena via peristaltic transport. Rheology of chyme is modelled using the Nakamura-Sawada bi-viscosity non-Newtonian formulation. The intestinal tract is considered as a curved tube geometric model. Low Reynolds number (creeping hydrodynamics) and long wavelength approximations are taken into consideration. Analytical solutions of the moving boundary value problem are derived for velocity field, pressure gradient and pressure rise. Streamline flow visualization is achieved with Mathematica symbolic software. Peristaltic pumping phenomenon and trapping of the bolus are also examined. The influence of curvature parameter, apparent viscosity coefficient (rheological parameter) and volumetric flow rate on flow characteristics is described. Validation of analytical solutions is achieved with a MAPLE17 numerical quadrature algorithm. The work is relevant to improving understanding of gastric hydrodynamics and provides a benchmark for further computational fluid dynamic simulations.

On heat and mass transfer analysis for the flow of a nanofluid between rotating parallel plates

In the present study, we have considered the three dimensional heat and mass transfer with magnetic effects for the flow of a nanofluid between two parallel plates in a rotating system. The reduced governing equations (in a system of ordinary differential equations) are solved by applying Homotopy Analysis Method (HAM). To see the validity of analytical solution, a numerical solution using RK-4 method coupled with shooting method has also been sought. Both the solutions are found to be in excellent agreement. To capture the effects of involved physical parameters graphical representation of the flow are included with comprehensive discussions. It has been observed that the thermophoresis and Brownian motion parameters are directly related to heat transfer but are inversely related to concentration profile. It is also recorded that the higher Coriolis forces decrease the temperature boundary layer thickness.

Multi-dimensional dual-phase-lag heat conduction in cylindrical coordinates: Analytical and numerical solutions

Abstract This work investigates the temperature distribution for the case of cylindrical geometry subjected to heat flux boundary condition on one of the bases and constant temperature on other boundary conditions considering dual-phase-lag (DPL) model of heat conduction. Two types of heat flux were considered: a continuously operating and exponential pulsed. Both exact analytical and numerical solutions were applied. For the analytical one, the separation of variables analytical method together with Duhamel’s theorem was employed. However, implicit finite difference method was used for the numerical technique. The DPL model of heat conduction renders the hyperbolic nature for heat transport, which is based on one correction made to the Cattaneo–Vernotte model of heat transfer. After comparing the results from both methods and confirmation regarding the validation of analytical solution, the effects of temperature gradient lags on the temperature distribution were analyzed.

Precracking and interfacial delamination in a bi-material structure: Static and dynamic loadings

The behavior of a precracked bi-material structure interface under given static and dynamic axial loading is an interest object in the present paper. Firstly, it is shown that the shear-lag model is a proper tool to analyze a delamination process in a precracked bi-material structure undergoing static loading. Secondly, the “shear-lag model” is applied to the structure under dynamic loading. To solve the problem for an interface delamination of the structure and to determine the debond length along the interface, our own 2D boundary element method (BEM) code is proposed in the case of static loading, and the shear-lag model together with the Laplace transforms and half-analytical calculations are used in the case of dynamic loading. The interface layer is assumed as a very thin plate compared with the other two. The parametric (geometric and elastic) analysis of the debond length and interface shear stress is done. The results from the 2D BEM code proved the validity of analytical solutions to the shear-lag model. In the dynamic case, the influence of loading characteristics, i.e., frequencies and amplitude fluctuations on the shear stress and the value of debond length for an interval of time, is discussed. The analysis of the obtained results is illustrated by an example of the modern ceramic-metal composite, namely cermet, and depicted in figures.

Experimental Validation of Analytical Solutions for Vertical Flat Plate of Finite Thickness Under Natural-Convection Cooling

The analytical solution for a vertical heated plate subjected to conjugate heat transfer due to natural convection at the surface and conduction below is presented. The heated surface is split into two regions; the uniform heat flux region toward upstream and remaining fraction as the uniform wall temperature region. The fractional areas under the two regions are considered variable. Adopting thermally thin wall regime approximation, the possible solutions were investigated and found to satisfactorily deal with longitudinal conduction and temperature variation in the transverse direction. A test setup was developed and the experiments were conducted to obtain relevant data for comparison with the analytical solutions. The ranges for Rayleigh number and heat conduction parameter (α) during various test conditions were 2×108–6×108, and 0.001–1, respectively. The limiting solutions for stipulated conditions are analyzed and compared with experimental data. Reasonable agreement is observed between the experimental and analytical results.

Numerical simulations of frequency shift and amplitude alteration of sound waves in random mass density and velocity fields

We investigate the effect of space-and time-dependent random mass density and velocity fields on frequencies and amplitudes of small sinusoidal acoustic waves. The dispersion relations and their approximate solutions, which are valid in the limit of weak random fields and long sound waves, are presented. These approximate solutions are verified by numerical simulations for the full set of hydrodynamic equations. The main findings are: (a) both the analytical and numerical results reveal that random fields affect the frequencies and amplitudes of sound waves. A space-dependent (time-dependent) random field leads to wave attenuation (amplification). Random mass density and time-dependent random velocity fields lift up frequencies of sound waves. A space-dependent random velocity field produces redshifts in frequencies; (b) in the limit of validity of analytical solutions the numerical results are close to the analytical data; (c) for a stronger random field and for shorter sound waves, numerical and analytical data depart from each other.