RESEARCH AND MATHEMATICAL MODELING OF FRACTIONAL-DIFFERENTIAL RHEOLOGICAL MODELS

Deformation processes in media with fractal structure have been studied. At present, research on the construction of mathematical methods and models of interconnected deformation-relaxation and heat-mass transfer processes in environments with a fractal structure is at an early stage. There are a number of unsolved problems, in particular, the problem of correct and physically meaningful setting of initial and boundary conditions for nonlocal mathematical models of nonequilibrium processes in environments with fractal structure remains unsolved. To develop adequate mathematical models of heat and mass transfer and viscoelastic deformation in environments with fractal structure, which are characterized by the effects of memory, self-organization and spatial nonlocality, deterministic chaos and variability of rheological properties of the material, it is necessary to use non-traditional approaches. -differential operators. The presence of a fractional derivative in differential equations over time characterizes the effects of memory (eridity) or non-marking of modeling processes. The implementation of mathematical models can be carried out by both analytical and numerical methods. In particular, in this paper the integral form of fractional-differential rheological models is obtained on the basis of using the properties of the non-integer integral-differentiation operator and the Laplace transform method. The obtained analytical solutions of mathematical models of deformation in viscoelastic fractal media made it possible to obtain thermodynamic functions, creep nuclei and fractal-type relaxation. Developed software to study the effect of fractional differentiation parameters on the rheological properties of viscoelastic media.

Application of a Drilling Simulator for Real-Time Drilling Hydraulics Training and Research

Challenging wells require an accurate hydraulic model to achieve maximum performance for drilling applications. This work was conducted with a simulator capable of recreating the actual drilling process, including on-the-fly adjustments of the drilling parameters. The paper focuses on the predictions of the drilling simulator's pressure losses inside the drill string and across the open-hole and casing annuli applying the most common rheological models. Comparison is then made with pressure losses from field data. Drilling data of vertical and deviated wells were acquired to recreate the actual drilling environment and wellbore design. Several sections with a variety of wellbore sizes were simulated in order to observe the response of the various rheological models. The simulator allows the input of wellbore and bottom-hole assembly (BHA) sizes, formation properties, drilling parameters, and drilling fluid properties. To assess the hydraulic model's performance during drilling, the user is required to input the drilling parameters based on field data and match the penetration rate. The resulting simulator hydraulic outputs are the equivalent circulation density (ECD) and standpipe pressure (SPP). The simulator's performance was assessed using separate simulations with different rheological models and compared with actual field data. Similarities, differences, and potential improvements were then reported. During the simulation, the most critical drilling parameters are displayed, emulating real-time measured values, combined with the pore pressure, wellbore pressure, and fracture pressure graphs. The simulation results show promise for application of real-time hydraulic operations. The simulated output parameters, ECD and SPP, have similar trends and values with the values from actual field data. The simulator's performance shows excellent matching for a simple BHA, with decreasing system's accuracy as the BHA design becomes more complex, an area of future improvement. The overall approach is valid for non-Newtonian drilling fluid pressure losses. The user can observe the output parameters, and by adding a benchmark safety value, the simulator gives a warning of a potential fracture of the formation or maximum pressure at the mud pumps. Thus, by simulating the drilling process, the user can be trained for the upcoming drilling campaign and reach the target depth safely and cost-effectively during actual drilling. The simulator allows emulation of real-time hydraulic operations when drilling vertical and directional wells, albeit with a simple BHA for the latter. The user can instantly observe the output results, which allows proper action to be taken if necessary. This is a step towards real-time hydraulic operations. The results also indicate that the simulator can be used as an excellent training tool for professionals and students by creating wellbore exercises that can cover different operating scenarios.

Passive control of nanoparticles in stagnation point flow of Oldroyd-B nanofluid with aspect of magnetic dipole

The present research focuses on nanoparticle suspensions and flow properties in the context of their applications. The application of these materials in biological rheological models has piqued the attention of many researchers. Magneto nanoparticles have an important function in controlling the viscoelastic physiognomies of ferrofluid flows. Having such substantial interest in the flow of ferroliquids our vision is to discuss the stagnation point flow of ferromagnetic Oldroyd-B nanofluid through a stretching sheet. The Buongiorno nanofluid model with Brownian motion and thermophoretic properties is examined. A chemical reaction effect and porous medium is also taken into account. Moreover, the modelled equations are changed to ordinary differential equations (ODEs) using suitable similarity transformations. Which are then solved using classical Runge-Kutta (RK) process with shooting technique. The solutions for the flow, thermal, concentration, skin friction, rate of heat and mass transfer features are attained numerically and presented graphically. The significant results of the current study are that, the growing values of ferromagnetic interaction parameter and porosity parameter declines the velocity profile. The rising values of chemical reaction rate parameter and Brownian motion parameter declines the mass transfer but inverse behaviour is seen for augmented values of thermophoresis parameter.

Characterization and Modeling of the Viscoelastic Behavior of Hydrocolloid-Based Films Using Classical and Fractional Rheological Models

Hydrocolloid-based films are a good alternative in the development of biodegradable films due to their properties, such as non-toxicity, functionality, and biodegradability, among others. In this work, films based on hydrocolloids (gellan gum, carrageenan, and guar gum) were formulated, evaluating their dynamic rheological behavior and creep and recovery. Maxwell's classical and fractional rheological models were implemented to describe its viscoelastic behavior, using the Vortex Search Algorithm for the estimation of the parameters. The hydrocolloid-based films showed a viscoelastic behavior, where the behavior of the storage modulus (G') and loss modulus (G'') indicated a greater elastic behavior (G' > G''). The Maxwell fractional model with two spring-pots showed an optimal fit of the experimental data of storage modulus (G') and loss modulus (G'') and a creep compliance (J) (Fmin < 0.1 and R2 > 0.98). This shows that fractional models are an excellent alternative for describing the dynamic rheological behavior and creep recovery of films. These results show the importance of estimating parameters that allow for the dynamic rheological and creep behaviors of hydrocolloid-based films for applications in the design of active films because they allow us to understand their behavior from a rheological point of view, which can contribute to the design and improvement of products such as food coatings, food packaging, or other applications containing biopolymers.

The Effect of Length Scale on Flow Properties of Micro, Nano and Macro-Emulsions

<p>Flow properties of a complex fluid depend on not only the characterizations of the components that make up the system but also the interactions between the phases. One of the most significant factors that affect these interactions is the length scale of the dispersed phase. According to Stokes law, the root of complex fluid rheological models, the velocity of a moving particle in a fluid is a function of the viscosity of the fluid and also the size of the moving droplet. The main aim of this research is to understand the crucial elements that define and control the rheological behaviour of complex fluids and thereby provide evidence for proposed modifications of the available rheological models to include parameters that capture the deduced crucial elements. In particular, by adjusting different aspects of Stokes law. The modified models can then be applied to a wider range of complex fluid systems, including emulsions, regardless of the chemicals that form the system. The complex fluids used in this research to develop the above are emulsions with droplets ranging over four orders of magnitude, 10 nm to 100 µm. Within a single base chemical system microemulsions, nanoemulsions and macroemulsions could be formed. The length scale and flow properties of each group were examined and the effect of length scale on rheological properties was investigated. Critical elements there were identified include: • Use of the appropriate viscosity value for the fluid through which the dispersed phase diffuses. It is often assumed that the viscosity of the pure continuous phase fluid can be used as the reference viscosity in the Stokes equation. In a real system the viscosity of the continuous phase can be strongly affected, and thereby defined by, the presence of the dispersed phase itself and the interfacial layer. Hence it is paramount that the appropriate reference viscosity is used. It is noted that the standard assumption is often applicable for highly diluted suspensions that are composed of rigid spheres. However, the research undertaken here demonstrates that this assumption must be reconsidered for more concentrated systems and particularly for emulsions. We recommend that for such systems the viscosity of the pure continuous phase is replaced by the constant viscosity of the sample at a zero shear rate. • Consideration of structural factors that also affect the viscosity. In particular it is often assumed that: 1- the droplets/particles are spherical and non-deformable; and 2- the dispersed phase presents as a single length scale, i.e. the system is a monodisperse system. The inclusion of these assumptions limits dramatically the applicability of the available models to fit and describe the real flow behaviour and thereby does not allow for predictability of behaviours. Typically models have been modified by adding experimental factors rather than explicitly incorporating the above factors into the development of a model. In this work the deviation from these rheological models are explained and correlated to the deviation from spherical structure and monodispersity. • Defining the relative viscosity as the ratio between the sample viscosity and the reference viscosity is common practice in the application of most rheological models. The viscosity of water tends to be taken as the reference viscosity. This leads to no agreement between the well-known rheological models and the experimental data, especially when applied to analysis of microemulsion rheology. In this work, we show that by taking the viscosity of the relevant ternary surfactant solution as the reference viscosity, the existing models can be applicable to microemulsions. This work sheds light on the relationship between the non-Newtonian behaviour of nanoemulsions and their underlying thermodynamic instability. In these systems the Newtonian behaviour is not evident till a shear rate of 100/s is reached. On the other hand the Newtonian viscosity is observed in thermodynamically stable systems, e.g. surfactant solutions and microemulsions, beyond a shear rate of 5/s or less. The Newtonian region also was observed in normal emulsions with narrow size distributions, dilute monodisperse coarse emulsions or dilute normal emulsions prepared in a Warring blender while a short chain alcohol is added to the system. By adding the short chain alcohol to the system not only the densities of the two phases are made similar and the emulsification is eased but also the polydispersity of the final emulsion is decreased. Finally a single model to be applicable to different types of emulsions with droplet sizes over five orders of magnitude was proposed. However the relationship is applicable to the systems with a low degree of polydispersity and once polydispersity is introduced the flow behaviour becomes complicated and the proposed model is not applicable.</p>

The Effect of Length Scale on Flow Properties of Micro, Nano and Macro-Emulsions

<p>Flow properties of a complex fluid depend on not only the characterizations of the components that make up the system but also the interactions between the phases. One of the most significant factors that affect these interactions is the length scale of the dispersed phase. According to Stokes law, the root of complex fluid rheological models, the velocity of a moving particle in a fluid is a function of the viscosity of the fluid and also the size of the moving droplet. The main aim of this research is to understand the crucial elements that define and control the rheological behaviour of complex fluids and thereby provide evidence for proposed modifications of the available rheological models to include parameters that capture the deduced crucial elements. In particular, by adjusting different aspects of Stokes law. The modified models can then be applied to a wider range of complex fluid systems, including emulsions, regardless of the chemicals that form the system. The complex fluids used in this research to develop the above are emulsions with droplets ranging over four orders of magnitude, 10 nm to 100 µm. Within a single base chemical system microemulsions, nanoemulsions and macroemulsions could be formed. The length scale and flow properties of each group were examined and the effect of length scale on rheological properties was investigated. Critical elements there were identified include: • Use of the appropriate viscosity value for the fluid through which the dispersed phase diffuses. It is often assumed that the viscosity of the pure continuous phase fluid can be used as the reference viscosity in the Stokes equation. In a real system the viscosity of the continuous phase can be strongly affected, and thereby defined by, the presence of the dispersed phase itself and the interfacial layer. Hence it is paramount that the appropriate reference viscosity is used. It is noted that the standard assumption is often applicable for highly diluted suspensions that are composed of rigid spheres. However, the research undertaken here demonstrates that this assumption must be reconsidered for more concentrated systems and particularly for emulsions. We recommend that for such systems the viscosity of the pure continuous phase is replaced by the constant viscosity of the sample at a zero shear rate. • Consideration of structural factors that also affect the viscosity. In particular it is often assumed that: 1- the droplets/particles are spherical and non-deformable; and 2- the dispersed phase presents as a single length scale, i.e. the system is a monodisperse system. The inclusion of these assumptions limits dramatically the applicability of the available models to fit and describe the real flow behaviour and thereby does not allow for predictability of behaviours. Typically models have been modified by adding experimental factors rather than explicitly incorporating the above factors into the development of a model. In this work the deviation from these rheological models are explained and correlated to the deviation from spherical structure and monodispersity. • Defining the relative viscosity as the ratio between the sample viscosity and the reference viscosity is common practice in the application of most rheological models. The viscosity of water tends to be taken as the reference viscosity. This leads to no agreement between the well-known rheological models and the experimental data, especially when applied to analysis of microemulsion rheology. In this work, we show that by taking the viscosity of the relevant ternary surfactant solution as the reference viscosity, the existing models can be applicable to microemulsions. This work sheds light on the relationship between the non-Newtonian behaviour of nanoemulsions and their underlying thermodynamic instability. In these systems the Newtonian behaviour is not evident till a shear rate of 100/s is reached. On the other hand the Newtonian viscosity is observed in thermodynamically stable systems, e.g. surfactant solutions and microemulsions, beyond a shear rate of 5/s or less. The Newtonian region also was observed in normal emulsions with narrow size distributions, dilute monodisperse coarse emulsions or dilute normal emulsions prepared in a Warring blender while a short chain alcohol is added to the system. By adding the short chain alcohol to the system not only the densities of the two phases are made similar and the emulsification is eased but also the polydispersity of the final emulsion is decreased. Finally a single model to be applicable to different types of emulsions with droplet sizes over five orders of magnitude was proposed. However the relationship is applicable to the systems with a low degree of polydispersity and once polydispersity is introduced the flow behaviour becomes complicated and the proposed model is not applicable.</p>

Rheology of three-phase suspensions determined via dam-break experiments

Three-phase suspensions, of liquid that suspends dispersed solid particles and gas bubbles, are common in both natural and industrial settings. Their rheology is poorly constrained, particularly for high total suspended fractions (≳0.5). We use a dam-break consistometer to characterize the rheology of suspensions of (Newtonian) corn syrup, plastic particles and CO 2 bubbles. The study is motivated by a desire to understand the rheology of magma and lava. Our experiments are scaled to the volcanic system: they are conducted in the non-Brownian, non-inertial regime; bubble capillary number is varied across unity; and bubble and particle fractions are 0 ≤ ϕ gas ≤ 0.82 and 0 ≤ ϕ solid ≤ 0.37, respectively. We measure flow-front velocity and invert for a Herschel–Bulkley rheology model as a function of ϕ gas , ϕ solid , and the capillary number. We find a stronger increase in relative viscosity with increasing ϕ gas in the low to intermediate capillary number regime than predicted by existing theory, and find both shear-thinning and shear-thickening effects, depending on the capillary number. We apply our model to the existing community code for lava flow emplacement, PyFLOWGO, and predict increased viscosity and decreased velocity compared with current rheological models, suggesting existing models may not adequately account for the role of bubbles in stiffening lavas.

Rheological Study of an Extruded Fish Diet with the Addition of Hydrolyzed Protein Flour

The extrusion of food for human and animal consumption is a unit operation that includes mixing, shearing, and force to the materials related to the rheological properties of the materials in the extruder. The present work aims to study the rheological behavior of an extruded fish diet incorporating hydrolyzed protein flour (HPF) processed by extrusion. The measurement was carried out online with a slit die rheometer, defining the rheological models and parameters that fit the process. During the extrusion process, the raw materials used were hydrolyzed protein flour, fish meal, and cassava starch. For the results, the evaluated treatments were adjusted to the Power Law, where an increase in the shear rate decreases the viscosity of the material, corresponding to a pseudoplastic behavior. The incorporation of hydrolyzed protein flour presented a significant effect on the value of n and Klp, increasing the viscosity with the increase in the percentage of inclusion of HPF. The models obtained for the prediction of the viscosity are adjusted to the system’s changes in shear rate, temperature, and humidity. The importance of the study lies in the fact that the diet developed can be applied to feed fish, and production is currently being scaled up to pilot plants for direct use by some producer communities.