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Full-Text Articles in Engineering Mechanics

An Analytical Framework For Quantifying Urban And Community Resilience To Natural Hazards From Cell-Phone Gps-Location And Traffic-Flow Data, Georgios Chatzikyriakidis May 2026

An Analytical Framework For Quantifying Urban And Community Resilience To Natural Hazards From Cell-Phone Gps-Location And Traffic-Flow Data, Georgios Chatzikyriakidis

Civil and Environmental Engineering Theses and Dissertations

Urban areas are increasingly exposed to natural hazards while accommodating a growing share of the global population, yet a consistent science-based framework for quantifying urban and community resilience remains lacking. This dissertation develops a physics-based analytical framework grounded in statistical mechanics and the quantitative theory of Brownian motion. A city is conceptualized as a complex medium in which citizens move analogously to Brownian particles within a viscoelastic environment, influenced by socioeconomic interactions and infrastructure functionality.

A central premise is that urban resilience, interpreted as engineering resilience (an outcome), can be quantified through a single metric: the mean-square displacement MSD=⟨r²(t)⟩, of …


Blade Design And Validation Of Hydrokinetic Turbine To Harvest Water Current Energy, Setare Sadeqi May 2024

Blade Design And Validation Of Hydrokinetic Turbine To Harvest Water Current Energy, Setare Sadeqi

LSU New Orleans Theses and Dissertations

The innovative aspect of this research lies in the careful integration of cutting-edge technologies throughout the entire process of designing, fabricating, and testing the carbon fiber propeller for the 3-bladed horizontal axis ocean current turbine (OCT). SolidWorks software played a pivotal role in the initial design phase, enabling a meticulous and precise modeling of the propeller's geometry. The utilization of SolidWorks allowed for a detailed exploration of various design parameters, ensuring that the propeller's structure and form were optimized for performance in ocean current conditions. Moving beyond the realm of virtual design, the choice of carbon fiber as the fabrication …


Towing Tank Trials Of Hydrokinetic Turbine Scale Model To Support Marine Energy System Verification, Shahab Rouhi May 2024

Towing Tank Trials Of Hydrokinetic Turbine Scale Model To Support Marine Energy System Verification, Shahab Rouhi

LSU New Orleans Theses and Dissertations

In response to the escalating demand for sustainable energy solutions and the critical reevaluation of conventional fossil fuels due to environmental concerns, this dissertation embarks on a comprehensive exploration of hydrokinetic energy as a promising alternative. The study delves into the underexplored domain of hydrokinetic energy, leveraging innovative methodologies for effective utilization and harnessing, particularly through the development and investigation of hydrokinetic turbines.

In the realm of hydrokinetic energy conversion, our research has exclusively concentrated on horizontal-axis turbines, distinct from other turbine configurations. Noteworthy is the adaptation of a conventional horizontal-axis wind turbine for water currents, revealing enhanced performance through …


Numerical Simulation Of Laser Induced Elastic Waves In Response To Short And Ultrashort Laser Pulses., Alireza Zarei May 2024

Numerical Simulation Of Laser Induced Elastic Waves In Response To Short And Ultrashort Laser Pulses., Alireza Zarei

All Dissertations

In an era of intensified market competition, the demand for cost-effective, high-quality, high-performance, and reliable products continues to rise. Meeting this demand necessitates the mass production of premium products through the integration of cutting-edge technologies and advanced materials while ensuring their integrity and safety. In this context, Nondestructive Testing (NDT) techniques emerge as indispensable tools for guaranteeing the integrity, reliability, and safety of products across diverse industries.

Various NDT techniques, including ultrasonic testing, computed tomography, thermography, and acoustic emissions, have long served as cornerstones for inspecting materials and structures. Among these, ultrasonic testing stands out as the most prevalent method, …


Simulation Of Wave Propagation In Granular Particles Using A Discrete Element Model, Syed Tahmid Hussan Jan 2024

Simulation Of Wave Propagation In Granular Particles Using A Discrete Element Model, Syed Tahmid Hussan

College of Graduate Studies: Theses & Dissertations

The understanding of Bender Element mechanism and utilization of Particle Flow Code (PFC) to simulate the seismic wave behavior is important to test the dynamic behavior of soil particles. Both discrete and finite element methods can be used to simulate wave behavior. However, Discrete Element Method (DEM) is mostly suitable, as the micro scaled soil particle cannot be fully considered as continuous specimen like a piece of rod or aluminum. Recently DEM has been widely used to study mechanical properties of soils at particle level considering the particles as balls. This study represents a comparative analysis of Voigt and Best …


Impact Of Silicon Ion Irradiation On Aluminum Nitride-Transduced Microelectromechanical Resonators, David D. Lynes, Joshua Young, Eric Lang, Hengky Chandrahalim Nov 2023

Impact Of Silicon Ion Irradiation On Aluminum Nitride-Transduced Microelectromechanical Resonators, David D. Lynes, Joshua Young, Eric Lang, Hengky Chandrahalim

Faculty Publications

Microelectromechanical systems (MEMS) resonators use is widespread, from electronic filters and oscillators to physical sensors such as accelerometers and gyroscopes. These devices' ubiquity, small size, and low power consumption make them ideal for use in systems such as CubeSats, micro aerial vehicles, autonomous underwater vehicles, and micro-robots operating in radiation environments. Radiation's interaction with materials manifests as atomic displacement and ionization, resulting in mechanical and electronic property changes, photocurrents, and charge buildup. This study examines silicon (Si) ion irradiation's interaction with piezoelectrically transduced MEMS resonators. Furthermore, the effect of adding a dielectric silicon oxide (SiO2) thin film is …


Assessment Of Bridge Pier Response To Fire, Vehicle Impact, And Air Blast, Chen Fang, Qusai Alomari, Daniel G. Linzell May 2023

Assessment Of Bridge Pier Response To Fire, Vehicle Impact, And Air Blast, Chen Fang, Qusai Alomari, Daniel G. Linzell

Department of Civil and Environmental Engineering: Dissertations, Theses, and Student Research

Highway bridges exposed to intentional or unintentional fire followed by combined vehicle impact and air blast are at risk of significant damage and, possibly, collapse. Limited studies examining the complex effects of these extreme demands on bridge support elements and parametrizing their response and damage are found in the open literature. Research that is presented is part of an ongoing numerical investigation examining round, multi-column, reinforced concrete (RC), bridge pier behavior subject to multi-hazard scenarios involving fire, vehicle impact, and air blast. Detailed nonlinear finite element analysis models of single columns and multi-column piers supported by a pile foundation system …


Machine Learning-Based Data And Model Driven Bayesian Uncertanity Quantification Of Inverse Problems For Suspended Non-Structural System, Zhiyuan Qin May 2023

Machine Learning-Based Data And Model Driven Bayesian Uncertanity Quantification Of Inverse Problems For Suspended Non-Structural System, Zhiyuan Qin

All Dissertations

Inverse problems involve extracting the internal structure of a physical system from noisy measurement data. In many fields, the Bayesian inference is used to address the ill-conditioned nature of the inverse problem by incorporating prior information through an initial distribution. In the nonparametric Bayesian framework, surrogate models such as Gaussian Processes or Deep Neural Networks are used as flexible and effective probabilistic modeling tools to overcome the high-dimensional curse and reduce computational costs. In practical systems and computer models, uncertainties can be addressed through parameter calibration, sensitivity analysis, and uncertainty quantification, leading to improved reliability and robustness of decision and …


Crystal Growth And Property Tuning Of Layered Magnetic Topological Semimetals, Krishna Pandey May 2022

Crystal Growth And Property Tuning Of Layered Magnetic Topological Semimetals, Krishna Pandey

Graduate Theses and Dissertations

The demand for energy-efficient devices has been growing rapidly due to the need for data-driven technologies and the global energy crisis. As device size approaches the atomic scale, the miniaturization of electronic devices may stop in the near future unless fundamentally new materials or device concepts are developed. The emergent topological materials with exotic properties show remarkable robustness against crystal lattice defects, which are promising for next-generation technology. These materials host exotic properties such as high mobility, large magnetoresistance, chiral anomaly, and surface Fermi arcs, etc. Among various topological materials, the ZrSiS-family materials exhibit two types of Dirac states, which …


Securing Pint Glasses On Serving Trays With Magnetic Attachments, Jayden Commendatore Apr 2021

Securing Pint Glasses On Serving Trays With Magnetic Attachments, Jayden Commendatore

Senior Theses

It is extremely common for servers in the restaurant industry to have pint glasses slip off their serving tray, wasting both time and money for the restaurant. I have designed a sleeve attachment that will clip on to the bottom of a pint glass. The metal sleeve attachment will be attracted to a magnet placed into the serving tray. This increases the normal force of the pint glass, resulting in an increase in the friction force between the tray and sleeve attachment, and helping secure the glass to the tray. In order to optimize the design, the value of the …


Fluid-Wall Interactions In Pseudopotential Lattice Boltzmann Models, Cheng Peng, Luis F. Ayala, Orlando M. Ayala Jan 2021

Fluid-Wall Interactions In Pseudopotential Lattice Boltzmann Models, Cheng Peng, Luis F. Ayala, Orlando M. Ayala

Engineering Technology Faculty Publications

Designing proper fluid-wall interaction forces to achieve proper wetting conditions is an important area of interest in pseudopotential lattice Boltzmann models. In this paper, we propose a modified fluid-wall interaction force that applies for pseudopotential models of both single-component fluids and partially miscible multicomponent fluids, such as hydrocarbon mixtures. A reliable correlation that predicts the resulting liquid contact angle on a flat solid surface is also proposed. This correlation works well over a wide variety of pseudopotential lattice Boltzmann models and thermodynamic conditions.


Mechani-Kits Senior Design Project, Jake Utley, Sophie Carson, Vincent Seguin Jun 2020

Mechani-Kits Senior Design Project, Jake Utley, Sophie Carson, Vincent Seguin

Mechanical Engineering

Studies suggest that when designed and executed well, hands-on activities can enhance student understanding of key mechanics concepts. Current products are expensive and typically not designed to meet a variety of learning objectives. Through the Mechanics of Inclusion and Inclusivity in Mechanics grant, the Cal Poly Physics and Engineering Departments are seeking to incorporate new hands-on activities into their courses. Our team has designed three inexpensive ”MechaniKits” to be used in physics, statics and dynamics courses [1]. This Final Design Review outlines our findings, objectives, and final designs for this project. It also explains our manufacturing and design verification plans. …


Dual-Axis Solar Tracker, Bryan Kennedy Jan 2020

Dual-Axis Solar Tracker, Bryan Kennedy

All Undergraduate Projects

Renewable energies, and fuels that are not fossil fuel-based, are one of the prolific topics of debate in modern society. With climate change now becoming a primary focus for scientists and innovators of today, one of the areas for the largest amount of potential and growth is that of the capturing and utilization of Solar Energy. This method involves using a mechanical system to track the progression of the sun as it traverses the sky throughout the day. A dual-axis solar tracker such as the one designed and built for this project, can follow the sun both azimuthally and in …


Investigation Of Fundamental Principles Of Rigid Body Impact Mechanics, Khalid Alluhydan Jul 2019

Investigation Of Fundamental Principles Of Rigid Body Impact Mechanics, Khalid Alluhydan

Mechanical Engineering Research Theses and Dissertations

In impact mechanics, the collision between two or more bodies is a common, yet a very challenging problem. Producing analytical solutions that can predict the post-collision motion of the colliding bodies require consistent modeling of the dynamics of the colliding bodies. This dissertation presents a new method for solving the two and multibody impact problems that can be used to predict the post-collision motion of the colliding bodies. Also, we solve the rigid body collision problem of planar kinematic chains with multiple contacts with external surfaces.

In the first part of this dissertation, we study planar collisions of Balls and …


Transferring Power Through A Magnetic Couple, Nickolas Cruz Villalobos Jr. May 2019

Transferring Power Through A Magnetic Couple, Nickolas Cruz Villalobos Jr.

Senior Theses

Properties of several working magnetic coupled rotors have been measured and their performance compared to theoretical models. Axial magnetic couplers allow rotors to work within harsh environments, without the need for seals, proper alignment, or overload protection on a motor. The influence of geometrical parameters, such as distance from the center of the rotors, polarity arrangement, and the number of dipole pairs were experimentally tested. These results can be used to improve rotor designs, to increase strength and efficiency.


Predicting The Mechanical Properties Of Nanocomposites Reinforced With 1-D, 2-D And 3-D Nanomaterials, Scott Edward Muller May 2019

Predicting The Mechanical Properties Of Nanocomposites Reinforced With 1-D, 2-D And 3-D Nanomaterials, Scott Edward Muller

Graduate Theses and Dissertations

Materials with features at the nanoscale can provide unique mechanical properties and increased functionality when included as part of a nanocomposite. This dissertation utilizes computational methods at multiple scales, including molecular dynamics (MD) and density functional theory (DFT), and the coupled atomistic and discrete dislocation multiscale method (CADD), to predict the mechanical properties of nanocomposites possessing nanomaterials that are either 1-D (carbyne chains), 2-D (graphene sheets), or 3-D (Al/amorphous-Si core-shell nanorod).

The MD method is used to model Ni-graphene nanocomposites. The strength of a Ni-graphene nanocomposite is found to improve by increasing the gap between the graphene sheet and a …


Computational Aerothermodynamic Analysis Of Satellite Trans-Atmospheric Skip Entry Survivability, John J. Runco Mar 2019

Computational Aerothermodynamic Analysis Of Satellite Trans-Atmospheric Skip Entry Survivability, John J. Runco

Theses and Dissertations

Computational aerothermodynamic analysis is presented for a spacecraft in low Earth orbit performing an atmospheric skip entry maneuver. Typically, atmospheric reentry is a terminal operation signaling mission end-of-life and, in some instances, executed for spacecraft disposal. A variation on reentry – skip entry – is an aeroassisted trans-atmospheric maneuver in which a spacecraft utilizes the effects of aerodynamic drag in order to reduce energy prior to a terminal entry, pinpoint a targeted entry, or change orbital elements such as inclination. Spacecraft performing a skip entry enable new modes of maneuver to enhance operations in nominal or possibly contested mission environments. …


Energy Conversion System For Travelers (Ecost), Thipok Bovornratanaraks Jun 2018

Energy Conversion System For Travelers (Ecost), Thipok Bovornratanaraks

The International Student Science Fair 2018

We have innovated “The Energy Conversion System for Travelers” or the ECoST. With the fact that most travelers have wheeled cabin-bags, whilst walking, the wheels will rotate so why don’t we harvest electricity from this kinetic energy? We thus install our innovation, the ECoST, to the bag to generate electricity from the spinning wheels. The electricity is then kept in the storage unit and ready to charge your empty battery devices in an emergency case via a USB port. To make life easy, our ECoST was designed to replicate the power bank charging method; therefore, we can charge …


Formation Of Mound-Like Multiscale Surface Structures On Titanium By Femtosecond Laser Processing, Edwin Peng, Alfred Tsubaki, Craig A. Zuhlke, Ryan Bell, Meiyu Wang, Dennis R. Alexander, George Gogos, Jeffrey E. Shield Mar 2017

Formation Of Mound-Like Multiscale Surface Structures On Titanium By Femtosecond Laser Processing, Edwin Peng, Alfred Tsubaki, Craig A. Zuhlke, Ryan Bell, Meiyu Wang, Dennis R. Alexander, George Gogos, Jeffrey E. Shield

Department of Mechanical and Materials Engineering: Dissertations, Theses, and Student Research

Surface Functionalization Technique • Femtosecond Laser Surface Processing (FLSP) • Utilize high power, femtosecond (10-15 s) laser pulses • Produce self-organized, multiscale surface micro/nanostructures • Diverse range of applicable substrates: semiconductors, metals, polymers, & composites

Why? • What are the different types of FLSP structures on Ti? • Physical evidence needed for FLSP formation models • Optimize FLSP of Ti for biomedical & other applications

How? • Obtain evidence of mound growth processes by examining underlying microstructure • Utilize dual beam Scanning Electron Microscope-Focused Ion Beam instrument to cross section surface structures & fabricate transmission electron microscopy samples


Preface To Special Topic: A Tribute To John Lumley, Kiran Bhaganagar, Thomas B. Gatski, William K. George Feb 2017

Preface To Special Topic: A Tribute To John Lumley, Kiran Bhaganagar, Thomas B. Gatski, William K. George

CCPO Publications

This Special Topic Section is dedicated to the life and memory of John Leask Lumley(1930-2015), professor and scientist extraordinaire.


Comment On "Roles Of Bulk Viscosity On Rayleigh-Taylor Instability: Non-Equilibrium Thermodynamics Due To Spatio-Temporal Pressure Fronts" Phys. Fluids 28, 094102 (2016), Robert L. Ash Feb 2017

Comment On "Roles Of Bulk Viscosity On Rayleigh-Taylor Instability: Non-Equilibrium Thermodynamics Due To Spatio-Temporal Pressure Fronts" Phys. Fluids 28, 094102 (2016), Robert L. Ash

Mechanical & Aerospace Engineering Faculty Publications

No abstract provided.


Experimental Building Demonstration Model With Viscous Fluid Dampers, Blake Thomas Reeve, Brianna Jean Kufa, Aden Malek Stepanians, Sophie Carmion Ratkovich Jun 2016

Experimental Building Demonstration Model With Viscous Fluid Dampers, Blake Thomas Reeve, Brianna Jean Kufa, Aden Malek Stepanians, Sophie Carmion Ratkovich

Architectural Engineering

The Architectural Engineering major places a heavy emphasis on structural dynamics and the role of wind and seismic loading in building analysis and design. Buildings of high importance that are critical to community function, such as hospitals, often utilize supplemental damping devices like supplemental viscous fluid dampers or base isolators to reduce the overall demands on the structural system. The design and analysis of these dampers are typically not taught at the undergraduate level, and is frequently performed by mechanical engineers, in lieu of structural engineers.

To better understand and research building behavior with supplemental damping devices, our multi-disciplinary team …


Secondary Flow Of Liquid-Liquid Two-Phase Fluids In A Pipe Bend, M. Ayala, P. Santos, G. Hamester, O. Ayala Jan 2016

Secondary Flow Of Liquid-Liquid Two-Phase Fluids In A Pipe Bend, M. Ayala, P. Santos, G. Hamester, O. Ayala

Engineering Technology Faculty Publications

A simulated study of oil and water in 90 degree bend was carried on COMSOL 5.1 to characterize flow pattern and analyze the secondary flow. The Euler-Euler k-e Reynolds Averaged Navier-Stokes model was used to represent the fluid motion. Changes in the Reynolds number, curvature ratio and direction of gravity were made to evaluate the effects in the intensity of the secondary flow. In the end, it was possible to see that the bend direction does not affect the formation of secondary flow for Reynolds above 100,000. It appears that the fluid behavior on the pipe bend is strongly related …


Effects Of Forcing Time Scale On The Simulated Turbulent Flows And Turbulent Collision Statistics Of Inertial Particles, B. Rosa, H. Parishani, O. Ayala, L.-P. Wang Jan 2015

Effects Of Forcing Time Scale On The Simulated Turbulent Flows And Turbulent Collision Statistics Of Inertial Particles, B. Rosa, H. Parishani, O. Ayala, L.-P. Wang

Engineering Technology Faculty Publications

In this paper, we study systematically the effects of forcing time scale in the large-scale stochastic forcing scheme of Eswaran and Pope ["An examination of forcing in direct numerical simulations of turbulence," Comput. Fluids 16, 257 (1988)] on the simulated flow structures and statistics of forced turbulence. Using direct numerical simulations, we find that the forcing time scale affects the flow dissipation rate and flow Reynolds number. Other flow statistics can be predicted using the altered flow dissipation rate and flow Reynolds number, except when the forcing time scale is made unrealistically large to yield a Taylor microscale flow Reynolds …


Effects Of Gravity On The Acceleration And Pair Statistics Of Inertial Particles In Homogeneous Isotropic Turbulence, H. Parishani, O. Ayala, B. Rosa, L.-P. Wang, W. W. Grabowski Jan 2015

Effects Of Gravity On The Acceleration And Pair Statistics Of Inertial Particles In Homogeneous Isotropic Turbulence, H. Parishani, O. Ayala, B. Rosa, L.-P. Wang, W. W. Grabowski

Engineering Technology Faculty Publications

Within the context of heavy particles suspended in a turbulent airflow, we study the effects of gravity on acceleration statistics and radial relative velocity (RRV) of inertial particles. The turbulent flow is simulated by direct numerical simulation (DNS) on a 2563 grid and the dynamics of O(106) inertial particles by the point-particle approach. For particles/droplets with radius from 10 to 60 µm, we found that the gravity plays an important role in particle acceleration statistics: (a) a peak value of particle acceleration variance appears in both the horizontal and vertical directions at a particle Stokes number …


Lightweight Uav Launcher, Ben Miller, Christian Valoria, Corinne Warnock, Jake Coutlee Jun 2014

Lightweight Uav Launcher, Ben Miller, Christian Valoria, Corinne Warnock, Jake Coutlee

Mechanical Engineering

This report discusses the design, construction, and testing of a lightweight, portable UAV launcher. There is a current need for a small team of soldiers to launch a US Marine Tier II UAV in a remote location without transport. Research was conducted into existing UAV launcher designs and the pros and cons of each were recorded. This research served as a basis for concept generation during the initial design development stage. It was required that the design weigh less than 110 lbs, occupy a smaller volume than 48" x 24" 18" in its collapsed state, be portable by a single …


Non-Equilibrium Pressure Control Of The Height Of A Large-Scale, Ground-Coupled, Rotating Fluid Column, R. L. Ash, I. R. Zardadhkan Jan 2013

Non-Equilibrium Pressure Control Of The Height Of A Large-Scale, Ground-Coupled, Rotating Fluid Column, R. L. Ash, I. R. Zardadhkan

Mechanical & Aerospace Engineering Faculty Publications

When a ground-coupled, rotating fluid column is modeled incorporating non-equilibrium pressure forces in the Navier-Stokes equations, a new exact solution results. The solution has been obtained in a similar manner to the classical equilibrium solution. Unlike the infinite-height, classical solution, the non-equilibrium pressure solution yields a ground-coupled rotating fluid column of finite height. A viscous, non-equilibrium Rankine vortex velocity distribution, developed previously, was used to demonstrate how the viscous and non-equilibrium pressure gradient forces, arising in the vicinity of the velocity gradient discontinuity that is present in the classical Rankine vortex model, effectively isolate the rotating central fluid column from …


Propagation Of Ultrasound Through Freshly Excised Human Calvarium, Armando Garcia Noguera Jul 2012

Propagation Of Ultrasound Through Freshly Excised Human Calvarium, Armando Garcia Noguera

Department of Engineering Mechanics: Dissertations, Theses, and Student Research

The propagation of ultrasound through complex biological media, such as the human calvarium, poses a great challenge for modern medicine. Several ultrasonic techniques commonly used for treatment and diagnosis in most of the human body are still difficult to apply to the human brain, in part, because of the properties of the skull. Moreover, an understanding of the biomechanics of transcranial ultrasound may provide needed insight into the problem of blast wave induced traumatic brain injury (TBI). In the present study, the spatial variability of ultrasonic properties was evaluated for relevant frequencies of 0.5, 1, and 2.25 MHz. A total …


The Influence Of Pressure Relaxation On The Structure Of An Axial Vortex, Robert L. Ash, Irfan Zardadkhan, Allan J. Zuckerwar Jan 2011

The Influence Of Pressure Relaxation On The Structure Of An Axial Vortex, Robert L. Ash, Irfan Zardadkhan, Allan J. Zuckerwar

Mechanical & Aerospace Engineering Faculty Publications

Governing equations including the effects of pressure relaxation have been utilized to study an incompressible, steady-state viscous axial vortex with specified far-field circulation. When sound generation is attributed to a velocity gradient tensor-pressure gradient product, the modified conservation of momentum equations that result yield an exact solution for a steady, incompressible axial vortex. The vortex velocity profile has been shown to closely approximate experimental vortex measurements in air and water over a wide range of circulation-based Reynolds numbers. The influence of temperature and humidity on the pressure relaxation coefficient in air has been examined using theoretical and empirical approaches, and …


Volume Viscosity In Fluids With Multiple Dissipative Processes, Allan J. Zuckerwar, Robert L. Ash Jan 2009

Volume Viscosity In Fluids With Multiple Dissipative Processes, Allan J. Zuckerwar, Robert L. Ash

Mechanical & Aerospace Engineering Faculty Publications

The variational principle of Hamilton is applied to derive the volume viscosity coefficients of a reacting fluid with multiple dissipative processes. The procedure, as in the case of a single dissipative process, yields two dissipative terms in the Navier-Stokes equation: The first is the traditional volume viscosity term, proportional to the dilatational component of the velocity; the second term is proportional to the material time derivative of the pressure gradient. Each dissipative process is assumed to be independent of the others. In a fluid comprising a single constituent with multiple relaxation processes, the relaxation times of the multiple processes are …