Exact Mathematical Solution Research Articles

Impact of induced electric field on the motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time

The classical and quantum motion of a charged particle in a uniform constant magnetic field is widely studied in the scientific literature since it is a problem that allows an exact mathematical analytical solution. On the other hand the cases of a non-uniform, time-dependent or both non-uniform and time-dependent magnetic field lead to mathematical problems that do not have general exact analytical solutions except for a few very specific cases. We present in this work, the results for the two-dimensional motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time for a certain finite interval and then stabilizes to a constant value afterwards. The work draws attention to certain interesting features observed for the two-dimensional motion of the charged particle as a result of the interplay between the effects of an induced electric field and the time varying magnetic field.

Channel flow dynamics of fractional viscoelastic nanofluids in molybdenum disulphide grease: A case study

Nanoscopic fluids especially viscoelastic Nanofluids are very useful in engineering and industrial problems. This research is to determine the open channel flow of a viscoelastic nanofluid (namely Oldroyd-B (OBNF)). Oldroyd-B fluid (OBF) was used as the base fluid and molybdenum disulphide nanopatrials were added in the fluid to form desired OBNF. To convert the mathematical model to a fractional model from a classical order partial differential equation PDE, fractional type derivative named Caputo-Fabrizio (CF) was used. The main objective of this study is to find the exact mathematical solution for the temperature, concentration and velocity distributions by using integral transformation technique. Final results are discussed graphically for the influence of different parameters on calculated temperature, concentration and velocity. Skin friction of the said fluid and engineering related dimensionless numbers including Reynolds number (Re), Prandtl number (Pr), Grashof number (Gr) and Schmidt number (Sc) are also discussed. At the end a comparison is illustrated in graphical form between current studied fluid (i.e. OBNF), another non-Newtonian fluid (i.e. Maxwell Nanofluid (MWNF)) and Newtonian fluid. As we know speed of Newtonian fluid is greater than non-Newtonian fluid, the same statement is validated by our solution. It is also noted that adding molybdenum disulphide nanoparticles to grease, heat transmission increased to 19.11% and mass transmission decreased to 2.51%.

A study of the non-linear properties and wave generation of the multi-layer non-hydrostatic wave model SWASH

Despite the increasing use of non-hydrostatic models in the study of wave processes in coastal regions, there is still limited understanding of the non-linear properties of the governing equations and how they improve with increased vertical resolution. In this study, the governing equations of the non-hydrostatic wave model SWASH are analysed and the linear and non-linear solutions up to third-order of all dependent variables are derived, considering one to four vertical layers. The analysis concludes that the model can achieve excellent non-linear properties with respect to the Stokes theory over a large range of water depths using only a few vertical layers. Furthermore, deriving solutions for all variables enables the formulation of improved wave generation and absorption boundary conditions for non-hydrostatic models. A well-known issue of non-linear wave models is related to the generation and propagation of spurious free waves, resulting to non-homogeneous wave fields. In this study, it is proven that by imposing the derived exact mathematical solutions of the governing equations at the model’s boundaries, the target first- and second-order wave profiles can be generated with high accuracy, while the spurious waves can be entirely eliminated.

A pulse-decay method for low (matrix) permeability analyses of granular rock media

Abstract. Nanodarcy level permeability measurements of porous media, such as nano-porous mudrocks, are frequently conducted with gas invasion methods into granular-sized samples with short diffusion lengths and thereby reduced experimental duration; however, these methods lack rigorous solutions and standardized experimental procedures. For the first time, we resolve this by providing an integrated technique (termed gas permeability technique, GPT) with coupled theoretical development, experimental procedures, and data interpretation workflow. Three exact mathematical solutions for transient and slightly compressible spherical flow, along with their asymptotic solutions, are developed for early- and late-time responses. Critically, one late-time solution is for an ultra-small gas-invadable volume, important for a wide range of practical usages. Developed to be applicable to different sample characteristics (permeability, porosity, and mass) in relation to the storage capacity of experimental systems, these three solutions are evaluated from essential considerations of error difference between exact and approximate solutions, optimal experimental conditions, and experimental demonstration of mudrocks and molecular-sieve samples. Moreover, a practical workflow of solution selection and data reduction to determine permeability is presented by considering samples with different permeability and porosity under various granular sizes. Overall, this work establishes a rigorous, theory-based, rapid, and versatile gas permeability measurement technique for tight media at sub-nanodarcy levels.

Dimensional Tolerances in Mechanical Assemblies: A Cost-Based Optimization Approach

There is a widely accepted consensus that component manufacturing precision is directly correlated with improved functional performance. However, this increase in precision comes at the expense of higher manufacturing costs, resulting in a trade-off between quality and affordability. In light of this opposing behavior, low-cost products typically exhibit lower quality, whereas high-quality products tend to be more expensive. This study introduces a novel approach for optimizing the dimensional tolerances of mechanical assembly components, taking into account both their manufacturing requirements and the associated costs of non-quality. Furthermore, the method considers the functional constraints imposed by interrelated tolerance chains within the product. Instead of relying on an exact mathematical solution, the proposed solution employs a heuristic approach through a simple and flexible algorithm. This enables practical implementation, as different cost-tolerance functions can be selected based on specific requirements. To provide a comprehensive evaluation of the proposed method, a concise review of the relevant literature in the field was conducted, allowing a comparison with other state-of-the-art approaches.

Theoretical Study of the Effect of Permeability Tensor upon Drainage of Soils

the influence of permeability tensor upon drainage of anisotropic soils under ponded water and steady recharge (rainfall) is theoretically investigated. Tensorial permeability has led to the formulation of mixed type partial differential equations. Since there is no analytical solution to this problem, the formulation is therefore solved numerically by the method of finite elements. The finite element formulation is implemented into a computer model which can be applied to any problem of seepage under steady state conditions. Two different example problems representing two different flow conditions under full anisotropy have been studied. Results of the model for the isotropic case were checked against exact mathematical solutions derived analytically for isotropic soils and found to be accurate which indicates that using this model for anisotropic soils is safe and sound

Second Approximation of the Generalized Planetary Equation Based upon Golden Metric Tensors

In this paper, we consider the Post Einstein Planetary equation of motion. We succeeded in offering a solution using second approximation method, in which we obtained eight exact mathematical solutions that rebel amazing theoretical results. To the order of C-2, two of these exact solutions are reduced to the approximate solutions from the method of successive approximations.

STUDY OF CHANGES IN HYDRODYNAMIC PARAMETERS PATTERNS OF VISCOUS FLUID FLOW IN A FLAT DIFFUSER

The relevance. Diffusers, either as nozzles or constituent elements, are frequently used in many mechanisms and machines. In this regard, the study of viscous fluid flow in diffusers aims to discover patterns of changes in the flow's hydrodynamic parameters, allowing better understanding of the nature of flow as a function of Reynolds number. Following the results of the analysis of the study, conditions for the proper construction of the mechanism unit, ensuring its reliable and durable operation will be revealed. The main aim of this study is to determine the velocity profiles in the flat diffuser for a viscous incompressible fluid by integrating the simplified Navier–Stokes differential equations under the established initial and boundary conditions, as well as the bifurcation point's dependence on the opening angle and Reynolds number of the diffuser. Objects: a flat diffuser in which viscous incompressible fluid moves. At the same time, revealing the patterns of changes of the hydrodynamic parameters of the flow is of defining value when choosing the structural dimension of devices and mechanisms, the main part of which is the flat diffuser. Methods. To reveal the patterns of changes of the hydrodynamic parameters of the flow in a flat diffuser, the study is based on the fundamental nonlinear differential equations of viscous fluid mechanics, which in a general case are not subject to an exact mathematical solution. For integration in the nonlinear differential equations, due to the smallness, the nonlinear-convective terms are neglected, and the inertial terms are also partially simplified. Such a simplification is justified if the velocities are very small or if the dynamic coefficient of viscosity of the fluid is very large. A method for solving the boundary value problem was developed, and regularities for changing the flow parameters were obtained. According to the derived regularities, graphs of the change in velocity, pressure and shear stresses on the wall of the fixed channel were plotted and the coordinates of the separation point were determined. Results. Depending on the angle of the diffuser opening and the Reynolds number, a general solution of the approximating Navier–Stokes equations was given. In accordance with the nature of the motion, the boundary conditions of the problem were established and the boundary value problem was stated. A method for integrating a boundary value problem was developed, and regularities for the change in velocities along the length of the diffuser were obtained for a parabolic distribution of velocities in the inlet sections. Graphs of the change in radial velocities along the length and at a fixed value of the opening angle were constructed, a flow pattern and the transition of a single-mode flow to multimode operation were obtained. For a fixed opening angle and Reynolds number, the conditions for flow separation from a fixed wall were derived, where the flow velocity changes the sign.

Insightful Facts on Peristalsis Flow of Water Conveying Multi-Walled Carbon Nanoparticles Through Elliptical Ducts With Ciliated Walls

In this research, a mathematical model is disclosed that elucidates the peristaltic flow of carbon nanotubes in an elliptic duct with ciliated walls. This novel topic of nanofluid flow is addressed for an elliptic domain for the very first time. The practical applications of current analysis include the customization of the mechanical peristaltic pumps, artificial cilia and their role in flow control, drug delivery and prime biological applications etc. The dimensional mathematical problem is transformed into its non-dimensional form by utilizing appropriate transformations and dimensionless parameters. Exact mathematical solutions are computed over the elliptic domain for the partial differential equations appearing in this convection heat transfer problem. A thorough graphical assessment is performed to discuss the prime results. The graphical visualization of the flow in this elliptic duct is obtained by plotting streamlines. The viscous effects are playing a vital role in the heat enhancement as compared to the molecular conduction. Since the incrementing Brinkman number results in a declined conduction due to viscous dissipation that eventually results in an enhanced temperature profile. This research first time elucidates the impacts of nanofluid flow on the peristaltic pumping through an elliptic domain having ciliated walls. Considering water as base fluid with multi-wall Carbon nanotubes for this ciliated elliptic domain having sinusoidal boundaries.

Effects of internal swelling on residual elasticity of a quasi-brittle material through a composite sphere model

This paper describes the development of a micromechanical-based constitutive model accounting for the effect of internal expansion on the residual elasticity of a Hashin composite material. This material is made of spherical inclusions that are subjected to gradual swelling within a quasi-brittle matrix. The main focus of this work is to describe and analyze the material mechanical response, with an additional focus on the internal swelling’s effect on the stress–strain response and residual elasticity. Microstructural features and parameters of major importance for the mechanical responses were identified. The innovative characteristics of the proposed approach are summarized as follows: (1) a full determination of the physics of a complete-damage problem throughout the whole process of inclusion swelling with upscaling techniques, which transfers the microcrack-related properties from the lower scale to upper scale; and (2) an evolution of the mechanical fields and the corresponding residual elasticity for various inclusion swelling levels. Concerning the matrix–inclusion composite, it was hypothesized that only the matrix was susceptible to cracking, with varied degrees of damage, whereas the inclusions behave elastically and the elastic modulus of the expanding inclusions remains constant. It is to emphasize that the gradual swelling of inclusions is modeled by an increasing strain in the current micromechanical-based constitutive model, and the microcracks are represented by a set of randomly oriented penny-shaped microcracks with identical radii (namely, closed cracks). The main contribution of the current research is to establish the exact mathematical solutions for the mechanical fields (stress, strain) caused by the swelling of the inclusions (mechanical loading) and derive the effective residual elasticity of a composite (structural response) subject to internal expansion and quasi-brittle damage. Based on the assumption of closed cracks that could be extended to open cracks in the upcoming work, the results proposed by the present paper help to understand the non-linear mechanical behavior of quasi-brittle materials subject to microcracking and provide a theoretical framework to be used as academical benchmark for numerical simulations.

Physical Survey of Thermally Heated Non-Newtonian Jeffrey Fluid in a Ciliated Conduit Having Heated Compressing and Expanding Walls

An analytical study is reported that highlights the physical aspects for a heated non-Newtonian Jeffrey liquid in a duct possessing sinusoidally moving ciliated walls. A comprehensive and specific convection analysis is conveyed for this ciliated elliptic duct problem by considering the viscous dissipation effects. The dimensional mathematical problem under consideration is transformed into its dimensionless form by means of appropriate and useful transformations. Then, velocity and temperature equations are exactly evaluated with given boundary conditions. The velocity profile is integrated over the elliptic cross-section and exact mathematical solution is obtained for the pressure gradient. Moreover, the distinct physical flow properties combined with the convection heat transfer phenomenon are discussed in detail through graphical outcomes. The illustrative streamline description shows an enhancing closed contour size with increasing Q (dimensionless flow rate).

Gas chromatograph calibration of gas mixtures using a versatile precision volumetric apparatus

A compact, volumetric apparatus was developed for accurate automated preparation of standard gas and gas-liquid mixtures for gas chromatograph detector calibration, with accuracies comparable to those from gravimetric or Coriolis flow methods. The method developed by the principal author is not an adaptation or extension of any other apparatus or technology, and measurements involve only stepper motor steps, temperature, and pressure ratios. Mixture preparation is accomplished via the displacement of gas between chambers in a highly uniform cylinder, separated by a movable piston. Piston movement, with piston end rods of exactly equal diameter, ensures that there is no change in interior volume, and the volume of gas displaced from the bottom into the top compartment is exactly proportional to the piston travel measured to within 10μm. The apparatus, operation, and previously unpublished measurements on common refinery gas mixtures over large concentration ranges are described in detail. These confirmed the accuracy and versatility of the apparatus and also the principle of no pressure change during mixture preparation, from sensitive pressure measurements. Conservative expanded uncertainties in prepared mixture mole fraction ranged from 0.001 to 0.002 over extended composition ranges. Absolute average deviations for the detector response factor for the mixtures ranged from 0.001 to 0.002. An exact new mathematical solution procedure permits the use of impure "pure" gases without loss of accuracy and can be applied to other procedures for standard gas mixture preparation. An exact expression is listed for determining "pure" gas purity.

Heat transfer analysis of MHD viscous fluid in a ciliated tube with entropy generation

This investigation aims to explain the study of heat transfer and entropy generation of Magnetohydrodynamics (MHD) viscous fluid flowing through a ciliated tube. Heat transfer study has massive importance in various biomedical and biological industry problems. The metachronal wave propagation is the leading cause behind this viscous creeping flow. A low Reynolds number is used as the inertial forces are weaker than viscous forces, and also, creeping flow limitations are fulfilled. For the cilia movement, a very large wavelength of a metachronal wave is taken into account. Entropy generation is used to examine the heat transfer through the flow. Numerical solutions are calculated by using MATHEMATICA. Exact mathematical solutions are calculated and analyzed with the help of graphs. Streamlines are also plotted. An axially symmetric flow as well as temperature profile is revealed through the graphical solutions. Both velocity and temperature profiles attain maximum value in the center of this ciliated tube that eventually declines toward the boundaries.

Influence of Membrane Vibration on Particles Rejection Using a Slotted Pore Membrane Microfiltration.

A new method is proposed to increase the rejection in microfiltration by applying membrane oscillation, using a new type of microfiltration membrane with slotted pores. The oscillations applied to the membrane surface result in reduced membrane fouling and increased separation efficiency. An exact mathematical solution of the flow in the surrounding solution outside the oscillating membrane is developed. The oscillation results in the appearance of a lift velocity, which moves oil particles away from the membrane. The latter results in both reduced membrane fouling and increased oil droplet rejection. This developed model was supported by the experimental results for oil water separation in the produced water treatment. It was proven that the oil droplet concentration was reduced notably in the permeate, due to the membrane oscillation, and that the applied shear rate caused by the membrane oscillation also reduced pore blockage. A four-times lower oil concentration was recorded in the permeate when the membrane vibration frequency was 25 Hz, compared to without membrane vibration. Newly generated microfiltration membranes with slotted pores were used in the experiments.

Entropy generation for the blood flow in an artery with multiple stenosis having a catheter

This article explicates the blood flow across an artery with multiple stenosis at outer wall and a thrombus at the centre. The symmetric multiple stenosis and non-symmetric multiple stenosis shapes are both considered in this study. Entropy analysis is also taken into account for a detailed study of irreversibility. The governing equations are interpreted with provided boundary conditions and exact mathematical solutions are developed. Further, these exact solutions are elucidated graphically. Streamlines show that the closed contours are symmetric in shape for symmetric multiple stenosis but non-symmetric in shape for non-symmetric multiple stenosis shape. The increase in multiple stenosis heights, increases the shear stress at wall having multiple stenosis.

Influence of a new‐bored neighbouring cavity on the seismic response of an existing tunnel under incident P‐ and SV‐waves

AbstractOne of the important problems regarding the tunnel design in seismically active areas is the assessment of the mutual effect of two closely running tunnels. With the propagation of seismic waves, the stresses in certain sections of a tunnel, owing to the presence of an adjacent tunnel, can exceed those occurring when considering the single‐tunnel case. In the focus of this study is the hitherto unexplored elastic wave diffraction problem, which considers the influence of the presence of a new‐bored neighbouring cavity on the seismic response of an existing tunnel, under incident P‐waves/SV‐waves. The wave function expansion method, the corresponding boundary conditions along the tunnel and cavity contours, and the translational addition theorem for bi‐cylindrical wave functions are jointly employed to develop an exact mathematical solution of the given problem in the form of infinite FourierBessel series, under the plane‐strain conditions. The dynamic effects of structures are presented in the form of a newly introduced dynamic stress alteration factor as a measure of the variation of dynamic hoop stress amplitudes in the lining and the surrounding medium of an existing tunnel affected by the presence of a closely running newly‐constructed tunnel opening. The results of this study are believed to be useful for interpretation of design rules, particularly considering that seismic standards for tunnels are still in the preliminary stage, where the aspect of the minimum seismically safe distance of a newly‐constructed tunnel opening in the vicinity of an existing tunnel should be of particular concern.

Mathematical study of Electroosmotically driven peristaltic flow of Casson fluid inside a tube having systematically contracting and relaxing sinusoidal heated walls

The electro-osmotically driven flow of Casson fluid across a cylindrical geometry with systematically contracting and relaxing sinusoidal walls is mathematically interpreted. The phenomenon of heat transfer also has its worth in the present investigation. The Joule heating and viscous effects are also studied in detail. The mathematically formulated equations that govern the flow of Casson fluid are simplified in their dimensionless form by using lubrication approximation, i.e. (λ→∞). Finally, exact mathematical solutions are flourished for these dimensionless equations. The graphical results are presented to further explain the exact solutions of velocity, temperature, pressure rise, pressure gradient, and streamlines pattern. The major outcomes show that electro-osmosis is very helpful in controlling both flow and heat transfer.

On the performance of a novel multi-hop relay-assisted hybrid FSO / RF communication system with receive diversity

One of the main problems in mobile communication systems is long-range links in weather conditions that could disrupt Radio Frequency (RF) connection. This problem could be solved by use of diversity or consuming more power or adding processing complexity. However, a mobile device has limited size and a small battery, and does not afford multiple antennas, or higher power consumption, or more complexity. So it is better to implement these techniques on the receiver, which is the base station. In this paper, a novel multi-hop hybrid Free Space Optical (FSO) / RF system is presented as a solution for the proposed scenario (long-range links in bad weather condition). In the proposed structure, a mobile user is connected to the source Base Station with receive diversity, and the source Base Station is connected to the destination Base Station via a multi-hop hybrid parallel FSO / RF link. In order to have a favorable performance with the lowest possible complexity, selection combining as well as demodulate and forward relaying are used at all hops of the proposed structure. The proposed structure has advantages of FSO, RF, and relay-assisted systems at the same time. The innovations and contributions of this paper, which are first introduced in the multi-hop hybrid FSO/ RF hybrid structure, include the novelty of the proposed structure, the presentation of new exact and asymptotic mathematical solutions, use of demodulate and forward protocol, taking into account a wide range of atmospheric turbulence with the effect of pointing error, use of selection combining at each relay. In order to show the efficiency of the proposed structure, Bit Error Rate and Outage Probability of the proposed structure are investigated in a wide range of atmospheric turbulence from moderate to saturate. Considering these criteria, closed-form exact and asymptotic expressions are derived and verified by MATLAB simulations. Results indicate that the proposed structure shows independent performance at moderate to strong atmospheric turbulence regimes. Hence, it is not required to adaptively adjust system parameters in order to have a constant performance. Accordingly, this structure is economically affordable and particularly suitable for mobile communications that should deal with frequent changes in atmospheric turbulence in urban by a limited complexity and power consumption.

Mathematical computations for Peristaltic flow of heated non-Newtonian fluid inside a sinusoidal elliptic duct

This research work interprets the mathematical study of peristaltic flow of non-Newtonian fluid across an elliptical duct. The heat transfer mechanism for this elliptical duct problem is also considered in detail. The mathematical equations for Casson fluid model are developed and then by using appropriate transformations and long wavelength approximation, this mathematical problem is converted into its dimensionless form. After converting the problem in dimensionless form, we have obtained partial differential equations for both velocity and temperature profiles. These partial differential equations are solved subject to given boundary conditions over elliptical cross sections and exact mathematical solutions are obtained. The results are further discussed by plotting graphical results for velocity, pressure gradient, temperature, pressure rise and streamlines.

Analytical solution for partial heating on the exterior of the pipe with application to measuring fluid flow rate

An analytical model using Green's functions for partial external heating of a pipe is developed, which results in an exact mathematical solution for the radial and axial temperature distribution in the pipe wall. Partial heating consists of a constant heat flux function imposed over a small section of the exterior of a pipe and for a limited time duration. The solution comprises steady-state and transient parts, and an algebraic identity is used to decrease the number of summation terms in the slowly-converging steady-state part. Intrinsic verification principles are used to verify the solution.As an example application, this transient solution is applied toward the development of a simple, noninvasive method for in-field measurement of the flow rate in pipes. To simulate this application, a pulse of energy is imposed to the wall of the pipe, and the developed mathematical solution is used to find the flow rate of the fluid inside the pipe. An optimal experiment is designed to find the best measurement location and time.