Journal of Engineering Mathematics最新文献

We study the temperature evolution in the three-dimensional skin tissue exposed to an electromagnetic beam of millimeter wavelength. The skin absorption coefficient of the beam frequency determines how deep the electromagnetic energy penetrates into the skin tissue, which gives a sub-millimeter penetration depth for a 94 GHz wave. In contrast, in the lateral directions perpendicular to the depth, the beam size is usually much larger than the penetration depth. Based on this separation of length scales, we establish an asymptotic formulation in which each term has separable dependences on the depth coordinate and on the lateral coordinates. We solve it analytically to obtain a two-term asymptotic solution of the temperature distribution in the three-dimensional skin tissue. This closed-form analytical solution provides a practical and accurate way of predicting the temperature. When the beam size is moderately larger than the penetration depth (a ratio of 20), the effect of lateral heat conduction is well captured in the asymptotic solution with maximum error less than 0.0017 in the normalized temperature of magnitude well above 1.

One of the main roles of the lipid layer (LL) of the tear film (TF) is to help prevent evaporation of the aqueous layer (AL). The LL thickness, composition, and structure all contribute to its barrier function. It is believed that the lipid layer is primarily nonpolar with a layer of polar lipids at the LL/AL interface. There is evidence that the nonpolar region of the LL may have liquid crystalline characteristics. We investigate the structure and function of the LL via a model of the tear film with two layers, using extensional flow of a nematic liquid crystal for the LL and shear-dominated flow of a Newtonian AL. Evaporation is taken into account and is affected by the LL thickness, internal arrangement of its rod-like molecules, and external conditions. We conduct a detailed parameter study with a focus on the evaporative resistance parameter, the Marangoni number, and primary liquid crystal parameters including the Leslie viscosities and director angle. This new model responds similarly to previous Newtonian models in some respects; however, incorporating internal structure via the orientation of the liquid crystal molecules affects both evaporation and flow. As a result, we see new effects on TF dynamics and breakup.

In the machining process known as grinding, fluid is applied to regulate the temperature of the workpiece and reduce the risk of expensive thermal damage. The factors that influence the transport of this grinding fluid are not well understood; however, it is important to gain understanding in order to try to avoid the unnecessary cost incurred from its inefficient application. In this work, we use the method of matched asymptotic expansions to derive the multiscale system of equations that governs the flow. Under the lubrication approximation, we show that it is possible to calculate the flow rate through the grinding zone without having to solve for the flow far from the grinding zone. Additional empirically determined boundary conditions do not need to be imposed. With this lubrication model, we quantify the effect of experimental parameters on the flow field in the grinding zone and study how the flow regime responds to changes in these parameters.

We consider the equilibrium shape of a liquid drop on a hexagonal substrate as motivated by vapor–liquid growth of nanowires. We numerically determine the energy-minimizing liquid drop shape on a hexagonal base using the software Surface Evolver in conjunction with an efficient regridding algorithm and convergence monitoring. The drop shape depends on two nondimensional parameters, the drop volume, and the equilibrium contact angle. We show that sufficiently large drops are well approximated away from the base by a spherical cap drop with geometric parameters determined by the area of the hexagonal base. Notably, however, the drop/base contact region does not extend to the corners of the hexagonal base, even in the limit of large volume V. In particular, there is a self-similar structure to the dry corner region with a length scale proportional to (V^{-3/2}). Since steady-state growth of faceted hexagonal nanowires by vapor–liquid–solid growth requires the liquid drop to be commensurate with the underlying wire cross-section, our findings mean that steady-state growth of hexagonal wires is not strictly compatible with an equilibrium liquid drop acting as a catalyst.

The investigation focuses on the hydrodynamic instability of a fully developed pressure-driven flow within a bidisperse porous medium containing an electrically conducting fluid. The study explores this phenomenon using the Darcy theory for micropores and the Brinkman theory for macropores. The system involves an incompressible fluid under isothermal conditions confined in an infinite channel with a constant pressure gradient along its length. The fluid moves in a laminar fashion along the pressure gradient, resulting in a time-independent parabolic velocity profile. Two Chebyshev collocation techniques are employed to address the eigenvalue system, producing numerical results for evaluating instability. Our findings indicate that enhancing the values of the Hartmann numbers, permeability ratio, porous parameter, and interaction parameter contributes to an enhanced stability of the system. The spectral behavior of eigenvalues in the Orr-Sommerfeld problem for Poiseuille flow demonstrates noteworthy sensitivity, influenced by various factors, including the mathematical characteristics of the problem and the specific numerical techniques employed for approximation.

This paper reports the analytical results on the non-isothermal stationary fluid flow inside thin vertical annular region formed by two co-axial cylinders. The annulus is packed with the fluid-saturated sparsely packed porous medium which is cooled through the side wall. The flow is governed by the prescribed pressure drop between the top and bottom walls which are maintained at uniform, but different temperatures. The main objective of this work is to propose the approximate model describing the effective flow using rigorous asymptotic analysis with respect to the thickness of the annular region. Starting from the dimensionless Darcy-Brinkman-Boussinesq system endowed with the appropriate boundary conditions, we derive the explicit asymptotic approximation clearly showing the effects of the porous structure and thermal transfer. We also provide the theoretical error analysis in order to indicate the order of accuracy of the proposed model and justify its usage.

This paper describes the bearing performances of orifice and constant flow valve (CFV) restrictor-compensated conical journal bearings for variations in operating speed. The analytical studies for different cone angle bearings using FEM are represented in terms of different bearing characteristics such as maximum pressure, minimum fluid film thickness, fluid flow, stiffness, damping, and so on. The Reynolds equation for lubricant at the mating surfaces is solved to obtain these characteristics for the operating speed parameter range Ω = 0.1–1.0. The study reveals that the CFV conical journal bearing shows more significant results as the speed progresses than the orifice compensation. This will help the designer design bearings by considering varying speeds.

Water pollution is a critical global problem. The fixed- bed continuous adsorption column provides the most practical application in the industry for wastewater treatment. The mass transfer process in the column can be described using a mass balance differential equation, and a sorbate–adsorbent interaction rate equation. The objective of this work was to describe the mass transfer in an adsorption column, analyzing the differential equations of the process and their analytical solutions. A general rate equation with four parameters was proposed, adding a zero-order parameter. The general model was solved using Laplace Transform method. The model proposed was applied to describe the adsorption of hexavalent chromium on chitosan biopolymer. The theoretical solution found was satisfactory to estimate the experimental breakthrough curves, and the estimated parameters allowed to predict other curves with different operational conditions. The zero-order parameter added relates to the baseline height of the breakthrough curve. The general model proposed generalizes already known plug flow models based on a single rate equation. The present model uses the information obtained from the column and from the equilibrium batch isotherm, which constitutes a useful tool for describing the dynamic adsorption process and to make decisions on column design.

Despite several applications of the fractional advection–diffusion equation (fADE) in studying sediment transport in an open channel flow, its application is limited to apprehending the non-local movement of sediment particles in an ice-covered channel with a steady, uniform flow field. An unsteady fADE is considered where the space term is non-local with a non-integer order and the mathematical model with Caputo fractional derivative is able to estimate the variation of sediment concentration along a vertical as well as with time in the ice-covered channel. An eddy viscosity expression is used, which includes the variation in roughness between the channel bed and ice cover surface. The Chebyshev collocation method and the Euler backward method are used to solve the fADE with the initial and boundary conditions and the convergence of the methods is established. The temporal variation of concentration shows that for a zero initial condition, the concentration profile first increases and then becomes stable after a certain time; for a non-zero initial concentration, the profile decreases with an increase in time and eventually a steady state is achieved. The effect of the order of the fractional derivative on the vertical variation of concentration at different times for zero and non-zero initial concentrations is studied and it is found that the order of the fractional derivative has a greater impact at smaller times. The impact of several parameters on concentration profiles is studied at different times and the validation of the model is done by comparing it with experimental studies under restricted conditions.

Non-Newtonian fluid mechanics and computational rheology widely exploit elastic dumbbell models such as Oldroyd-B and FENE-P for a continuum description of viscoelastic fluid flows. However, these constitutive equations fail to accurately capture some characteristics of realistic polymers, such as the steady extension in simple shear and extensional flows, thus questioning the ability of continuum-level modeling to predict the hydrodynamic behavior of viscoelastic fluids in more complex flows. Here, we present seven elastic dumbbell models, which include different microstructurally inspired terms, i.e., (i) the finite polymer extensibility, (ii) the conformation-dependent friction coefficient, and (iii) the conformation-dependent non-affine deformation. We provide the expressions for the steady dumbbell extension in shear and extensional flows and the corresponding viscosities for various elastic dumbbell models incorporating different microscopic features. We show the necessity of including these microscopic features in a constitutive equation to reproduce the experimentally observed polymer extension in shear and extensional flows, highlighting their potential significance in accurately modeling viscoelastic channel flow with mixed kinematics.