Lubrication Approximation Theory Research Articles

Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials

Abstract The calendering process is pivotal in enhancing various materials’ surface properties and characteristics, making them indispensable for achieving desired product quality and performance. Also, this process holds significant relevance in various industrial applications, such as polymer processing, food production, and the manufacturing of composite materials. So, the aim of this study is to theoretically examine the calendering process applied to third-grade materials. It specifically explores how temperature variations impact material behavior during passage through two counter-rotating heated rolls. Particular consideration is given to the influence of temperature-dependent viscosity via Reynold’s model. The complexities of mass, momentum, and energy balance equations are reduced through the application of the Lubrication approximation theory. Solutions to these equations for variables such as velocity, flow rate, and temperature fields are accomplished by combining perturbation and numerical techniques. In relation to the calendering process, the thickness of the exiting sheet is specifically explored. Furthermore, this study quantifies substantial engineering parameters such as roll-separating force, pressure distribution, and power transferal from the rolls to the fluid. The governing equations belong to three key dimensionless parameters, namely, the Brinkman number, which is a product of Eckert number and Prandtl number, the temperature-dependent consistency index μ \mu , and a parameter η \eta correlating to non-Newtonian behavior. The outcomes of this study are presented both graphically and in tabular form. It has been observed that a rise in the third-grade parameter decreases detachment point and sheet thickness due to increased material rigidity. Furthermore, established results in the literature regarding the calendering of Newtonian fluids are validated.

Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena

Abstract The current study theoretically and computationally analyses the viscoelastic Sisko fluids during the non-isothermal rollover web phenomenon. The mathematical modeling produces a system of partial differential equations, which we further simplify into ordinary differential equations through appropriate transformations. We have formulated the problem based on the lubrication approximation theory. The solution has been obtained with the perturbation method, and the outcomes are found in mathematical, tabular, and graphical forms that highlight the influence of pertinent parameters on velocity profiles, pressure gradients, flow rates per unit width, Nusselt number, pressure profile, temperature distributions, and other significant engineering quantities. Further, A comparative analysis between analytic and numerical solutions, utilizing the middefer method in the Maple environment, demonstrates reasonable agreement. Also, we observe that the fluid parameter significantly influences both velocity and temperature profiles. Moreover, the determination of a separation point 2.5000, accompanied by the observation of a maximum coating thickness of 0.6960. The enhancement in fluid heat transfer rate is approximately 5% compared to non-Newtonian fluid parameter values, with potential for further improvement by increasing the non-Newtonian parameter values. This comprehensive investigation offers valuable insights for practical implementation and future scholarly endeavors, with zero-order findings showcasing enhanced precision.

Optimization and sensitivity analysis of heat transfer for Powell–Eyring fluid between rotating rolls with temperature-dependent viscosity: A mathematical modeling approach

This study explores the flow of a non-Newtonian fluid between two rolls that are counter-rotating at the same speed and of equal size. The fluid's viscosity depends on temperature, and we investigate its theoretical impact on the thickness of the sheet and other engineering parameters relevant to the process. We derive non-dimensional mass and momentum balance equations using suitable transformation and the lubrication approximation theory. The expressions for velocity distribution, pressure gradient, flow rate, temperature profile, and pressure fields have been obtained by utilizing the perturbation method. After obtaining these expressions, we compute engineering quantities such as the roll separation force, streamline, Nusselt number, and the power input required to drive both cylinders based on the system's kinematical and geometrical parameters. We also obtain numerical solutions using the finite difference method and built-in (BVP method) in Maple. Further, we use response surface methodology and analysis of variance to determine what the mathematical models mean and whether they are good enough for sensitivity and optimization analysis of the heat transmission and roll separation force. Using statistical tools such as the R2, we determine that our Nusselt number and roll separation force provide the best solution for the considered model. Additionally, it has been observed that as the Weissenberg number increases, velocity tends to rise; conversely, velocity decreases with a higher velocity ratio. Also, the temperature profile is notably influenced by the Brickman number and increases with the increase in the Brickman number. It has also been noted that as the values of velocities ratio increase, the separation points shift toward the nip region, while concurrently, the coating thickness decreases. Furthermore, we also demonstrate that compression between analytical and numerical solutions for the considered problem of fluid flow, which suggests that the results presented here are reasonable. Finally, we compare our work with published studies to validate our findings. Hence, these factors help in an efficient fluid coating process and improve the substrate life.

Numerical Analysis of the Blade Coating Process Using Non‐Newtonian Nanofluid with Magnetohydrodynamic (MHD) and Slip Effects

Numerical Analysis of the Blade Coating Process Using Non‐Newtonian Nanofluid with Magnetohydrodynamic (MHD) and Slip Effects

Applications of lubrication approximation theory in the analysis of the roll‐coating using a tangent hyperbolic fluid model

AbstractIn this study, the process of roll‐over‐web coating for a tangent hyperbolic fluid is theoretically examined. Fundamental laws of fluid mechanics are used to set up the flow equations and then simplified through lubrication approximation theory (LAT). The governing differential equations are solved with the help of the perturbation method in conjunction with the Runge–Kutta method. The outcomes of the study are presented through the graphs and tables. Study shows that enhancing the Weissenberg number upsurges the velocity and pressure while decreasing the pressure gradient. It is also noted that force shows an increasing trend while power is decreasing for bigger values . Results further depict that for a bigger value of the power law index , pressure gradient, pressure, force, and power decrease, and the coating thickness increases. It is worth mentioning here that for the shear thinning case, the tangent hyperbolic model predicts less pressure in the nip region when compared to the Newtonian model.

Thermal and mass exchange in a multiphase peristaltic flow with electric-debye-layer effects and chemical reactions using machine learning

The present investigation delves into the analysis of mass and thermal transmission during the peristalsis of a two phase flow of a Prandtl fluid, within a symmetric channel by considering entropy generation and minimization. The considered flow is examined by taking into account the impacts of electroosmotic effects through Electric Debye Layer (EDL) and chemical reactions followed by the lubrication theory approximation. Numerical solutions are acquired for the consequential differential equations through artificial neural networks (ANN) based fitness functions. The artificial neural networks (ANN) based fitness function optimized through hybrid evolutionary algorithms especially ABC (artificial bee colony) and NNA (neural network algorithm) like ANN-ABC-NNA. For the efficiency and validation of the algorithm we conduct the one hundred (100) independent runs. The obtained results through ANN-ABC-NNA are compared with the exact solutions for validation of results. In the present examination, the velocity profile is noticed to increase by the rise in the electroosmotic factor and solid particles concentration, while it decreases for the Prandtl fluid parameter. This research holds practical significance in diverse applications, particularly in the design and optimization of microfluidic devices.

Soft computing through supervised neural network based on Levenberg–Marquardt back propagation for numerical treatment of steady flow of Oldroyd 4-constant fluid between two rolls

This study presents a new intelligent computing method based on the Levenberg–Marquardt back-propagation-based supervised neural network (LMB-SNN) for the flow of Oldroyd 4-constant fluid in the forward roll coating process (OFC-FRCP). The mathematical formulations for OFC-FRCP are converted into ordinary differential equations using lubrication approximation theory (LAT) and suitable dimensionless parameters. The analytic expressions for velocity distribution, pressure and pressure gradients have been obtained using Adomian’s Decomposition Method (ADM) for several situations. The fluid’s separation point and coating thickness are then calculated using integration. The reference dataset for LMB-SNN has been obtained by ADM for multiple scenarios by adjusting various physical variables. The neural network’s training, testing and validation are carried out in parallel to minimize the mean-squared error function (MSEF). The proposed LMB-SNN’s effectiveness was confirmed by analyzing the MSEF, error histograms and regression plots. The precision of the method is verified by the close agreement between numerical outputs and dataset values. The intelligent numerical results computed using ADM and LMB-SNN are presented in various graphs and tables. The method’s accuracy is demonstrated by the numerical outputs closest to the built and dataset values at levels between [Formula: see text] and [Formula: see text].

Controlling factors of coating thickness of Sisko fluid in blade coating process

Blade coating is a technique for coating the moving substrate with a protective fluid layer through the blade as a common smoothing device. The heat transfer and slip effects of the rheological Sisko fluid model for plane coater are considered under blade coating process. This model combines the power-law and Newtonian fluids and taken as the limiting behavior of a blade coating in the stiff blade limit. The flow of greases is the most significant example of this type of fluid, which is easily found in nature and has many practical uses. To solve the non-linear expressions, LAT (Lubrication Approximation Theory) is applied, and a shooting technique is used for numerical solutions. The impacts of several parameters on flow characteristics, particularly the velocity ratio K, the Sisko fluid parameter b, the power-law index n, and the slip parameter α are investigated. The effects of both shear-thinning and shear-thickening behaviors are also investigated. The lubrication pressure and temperature increase with the Sisko fluid parameter b and decrease with the slip parameter α. As material parameter b increases from (0.001–1), 20.7% reduction in coating thickness and a 53.2% pressure reduction is noted as compared to the Newtonian case.

A computational study of the calendering processes using Oldroyd 8-constant fluid with slip effects

The analyses of the calendering processes using Oldroyd 8-constant fluid model with the non-linear slip condition effects are discussed in this paper. To model the flow equations for the study of calendering, we utilized the conservation of mass, momentum equations, and Oldroyd 8-constant model. The final equations are reduced into fourth order differential equations by utilizing lubrication approximation theory (LAT). MATLAB’s built-in function bvp4c is employed to calculate the stream function and the fluid velocity. Furthermore, pressure gradient, pressure, and mechanical quantities of calendering processes are obtained by fourth-order Runge-Kutta method. The effects of the slip and material parameters of Oldroyd 8-constant fluid on the physical quantities related to calendering are illustrated via graphical figures. The presence of slip is resistive to length of contract and pressure distribution. On the other hand the pressure showed the increasing trend with increase the value of material parameter ([Formula: see text]). The engineering quantity such as force function show decreasing with increase the values of the slip parameter K. As evident from the study, the material parameters play a crucial role in controlling the pressure, thereby influencing the final sheet thickness and separating force. With an increase in the material parameter [Formula: see text], both the sheet thickness and roll-separating force show a corresponding increase.

Finite coating thickness of Prandtl fluid in non-isothermal reverse roll coating process

PurposeThe coating process is broadly employed in the manufacturing of wallpapers, adhesive tapes, wrapping, protection of fabrics and metals, X-ray and photographic films, beautification, books and magazines, film foils, magnetic records, coated paper, etc.Design/methodology/approachIn this study, an incompressible flow of non-Newtonian fluid is modeled to inspect the rheological behavior of finite coating thickness in the reverse roll coating process. With the assistance of lubrication approximation theory (LAT), the dimensionless form of governing expressions is simplified. Exact solutions for distributions for velocity, flow rate, temperature and pressure gradient attained utilizing perturbation technique and their variation is presented as well as discussed in graphs. Meanwhile, some important factors from an engineering perspective including coating thickness and transition point were calculated mathematically and are displayed in a tabular manner. Also, streamlines are drawn to observe the flow pattern.FindingsPrandtl fluid parameters provide a controlling factor to regulate the flow rate, velocity, coating thickness, and pressure gradient leading to an efficient coating process. Moreover, the Brinkman number and Prandtl fluid parameters significantly improve the temperature distribution.Originality/valueIn the literature, this study fills a gap in the theoretical prediction of coating thickness rheologically influenced by Prandtl fluid in reverse roll coating process.

Comparative numerical and analytical computations for magnetized pseudoplastic materialwith roll-coating applications

The phenomenon of wire coating occurs in various applications in manufacturing systems, cables, drawing of sheets, polymer industries etc. The current investigation deals with the study of the reverse coating phenomenon of magnetized non-Newtonian fluid. A pseudoplastic material with applications of magnetic force is considered and the rheological impact based on the non-Newtonian model is investigated. The thin layer of pseudoplastic material is coated when it travels due to the small gap conveying by two reverse coating rollers. The formulation of problem is based on lubrication approximation theory. Both analytical and numerical simulations are performed, and the corresponding results are compared. The numerical assessment is based on the shooting technique while analytical outcomes are obtained using perturbation technique. The effects of the governing parameters on the flow profiles are depicted. The main parameters influence the flow are the Weissenberg number and velocity ratio parameter.

Analytical study of viscoelastic fluid during forward roll coating process under lubrication approximation theory

Roll coating is a popular method for applying a thin coating film to a continuous substrate, such as paper or foil. This study aims to investigate analytically the fluid flow through counter rotating rolls using a non-Newtonian model. The governing equation for the constitutive of the Oldroyd-B as a non-Newtonian fluid with momentum equation has been discovered. The nondimensionalizing of the fluid flow equations is done by using suitable dimensionless parameters. The equations are then simplified by using Lubrication Approximation Theory (LAT) to obtain the solution for various quantities. Analytical solutions for the velocity distribution, pressure and pressure gradients have been obtained by regular perturbation method. Numerical integration has been used to calculate separation point, coating thickness, separation force and power input of the fluid. The effect of involved parameter such as Weissenberg number We and viscosity on velocities, pressure gradient and pressure is presented in a variety of graphs and several other results are shown in tables. It has been observed that as the modified capillary number [Formula: see text] increases, the separation point moves toward the nip region, whereas the coating thickness declines. It has been witnessed that the We serves as the controlling parameter for flow rate, pressure distribution and power input, whereas for the coating thickness, the separation force increases, which are important for coating industries.

Electroosmosis oriented flow of Jeffrey viscoelastic model through scraped surface heat exchanger

Scraped surface heat exchanger (SSHE) is vibrantly utilized in various industries. The heat exchanger devices play pivotal role in the energy transfer phenomenon between the fluids, and it improves the efficient use of energy. The steady isothermal flow of a rheological aqueous ionic solution in a narrow gap scraped surface heat exchanger is explored. The mathematical model is established with the aid of lubrication approximation theory and extracted analytical solution. Exact analytical expressions are gathered for velocities, volume fluxes, pressure gradients, pressure at the edges of the blades and stream functions in the various stations of SSHE. Strength of significant flow parameters on the velocities, volume fluxes, pressure gradients, pressure at the edges of the blades and stream functions is revealed with aid of various plots. Significant enhancement in the velocity profile is observed in main domain of the flow with increasing ratio of relaxation to retardation time parameter and Helmholtz Smoluchowski velocity. While, it is observed that electro-osmotic parameter creates impediments to the flow.

Heat transfer analysis of the non-Newtonian polymer in the calendering process with slip effects

In this paper, a non-isothermal study of the calendering processes is presented using Carreau–Yasuda model along with nonlinear slip condition introduced at the upper roll surface. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. The final equations are solved numerically using “bvp4c” to find stream function and velocity profiles, while the hybrid numerical method which is the combination of shooting and finite difference methods is used to solve the energy equation. Graphs show the impact of the concerned material parameters on various quantities of interest. The pressure distribution decreases with the increasing values of the slip parameter and Weissenberg number. The mechanical variables show an increasing trend with the increasing values of the slip parameter and Weissenberg number. The temperature distribution increases with an increase in the Brinkman number, while temperature shows declining trend near the roll surface with the increasing values of the slip parameter. The force separating the two rollers, total power input into both rolls, increase with the increasing values of the Weissenberg number and slip parameters. The results show that the Newtonian model predicts higher pressure in the nip zone than the Carreau–Yasuda model. It is interesting to note that for the case of shear thinning, the Carreau–Yasuda model predicts 30% less pressure in the nip region when compared to the Newtonian model.

Mathematical model for numerical simulations of thermal energy of nano-fluid in a complex peristaltic transport within a curved passage:Pharmacological and engineering biomedical application

The proposed scientific model is based on peristaltic movement of nanofluid within a curved channel of an electrically conducting material. To obtain a numerical solution for the electric potential function, the Debye-Hückel and Lubrication theory approximations are used. Stream function, electroosmotic, axial velocity, and curvature are all estimated using low zeta potentials. The behaviour of electroosmotic flow governs two dimensionless parameters: the Debye-Hückel parameter and the Helmholtz-Smoluchowski velocity. It is also investigated how curvature influences blood flow caused by electroosmosis. It was also investigated how the curvature parameter affects the nano-fluid flow induced by electroosmosis. Nano-biofluid evaluation with varied particle shapes such as cylinder, platelet, brick, and blade flowing in complex travelling wave. It can be stated that the velocity component of nanofluid is decreasing due to magnetic force, particle size, and wall compliance. It is also stated that nanoparticles considerably improve the thermal conductivity of the fluid.

Theoretical investigation of a fluid model in calendering process involving slip at the upper roll surface

AbstractIn this article, calendering analysis is presented using Johnson‐Segalman fluid model along with nonlinear slip condition introduced at the upper roll surface. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well‐known lubrication approximation theory (LAT). To eliminate the pressure gradient from the governed equations, stream function is introduced. The final equations are solved numerically using Matlab built‐in procedure “bvp4c” to get the stream function and pressure gradient. The pressure and engineering quantities such as power input function and roll‐separating force are calculated by Runge‐Kutta fourth order method. The impact of the Johnson‐Segalman parameter and slip parameter on pressure, velocity and engineering quantities are presented with the help of various graphs. The Newtonian model predicts higher pressure in the nip zone than the Johnson‐Segalman fluid model, according to our findings. In the presence of the slip parameter, the force and power functions exhibit 37% and 61.71% decreases from Newtonian values, respectively. As the slip parameter is increased, the pressure distribution diminishes.

Theoretical Analysis of Roll-over-web Coating of a Non-Newtonian Polymer Using Lubrication Approximation Theory

Roll-coating process plays an important role in many industries for its practical applications such as paint, PVC coated fabrics and plastic industries. There are several roll-to-roll coating methods including forward and reverse roll-coating. However, the roll-over-web coating study of the Rabinowitsch model is presented in this paper. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. We employ the regular perturbation technique to get analytical expressions for velocity, pressure, and pressure gradient. Engineering quantities such as power input function and roll-separating force are calculated by Runge-Kutta method. The dimensionless Rabinowitsch parameter effect on velocity, pressure, pressure gradient, load-carrying force, and power input are shown graphically. It is interesting to note that for the shear thickening case, the Rabinowitsch model predicts 35% higher pressure, while in the shear thinning case it predicts 29% less pressure in the nip region when compared to the Newtonian model. The force and power show a decreasing trend on increasing the dimensionless Rabinowitsch parameter a. Moreover, the separation point shifts right of its Newtonian value when fluid behaves like shear thickening and volumetric flow rate increases which causes the coating thickness to increase.

Rheology of peristaltic flow in couple stress fluid in an inclined tube: Heat and mass transfer analysis

The recent investigations ensure that the effect of an endoscope on the peristaltic flow is very important for medical diagnosis and it has many clinical applications such as gastric juice motion in the small intestine when an endoscope is inserted through it. Therefore, we have studied the problem of peristaltic flow with heat and mass transfer through the gap between coaxial inclined tubes where the inner tube is rigid and the outer tube has sinusoidal wave traveling down its wall. The inner tube fulfilled the slip condition while the outer tube has a no-slip condition. The lubrication approximation theory is utilized to simplify the normalized equations. The perturbation procedure is employed to concede the results for the pressure gradient and velocity field, whereas exact outcomes are established for the energy and concentration fields. Frictional forces and pressure rise per wavelength are calculated numerically. The impacts of embedded parameters are portrayed through graphs. It is found that fluid velocity reduces by increasing the couple stress parameter however the inverse effect is noted for the slip parameter. Furthermore, better pumping is viewed in the vertical tube as compared to the horizontal tube. A validity of perturbation solutions for velocity distribution is made with the finite element method and an admirable comparison is also noticed with previously published results.

Blade coating analysis of carreau fluid model with Magnetohydrodynamics and slip effects

Coating is a process that improves and enhances life, quality and efficiency of the web. Common industrial adhesive application to board and paper, metal foils polymer coatings, photographic film emulsions, recording films, textile fabrics, paints, coating on food items, medicine, catalogues are common examples. There are many fluids that are used as coatings. This paper analyses blade coating using both plane as well as exponential geometries along with slip and magnetohydrodynamics effects for a Carreau fluid model. To study the blade coating phenomenon, slip is introduced at the blade surface and a magnetic field is applied normal to the fluid flow. The LAT (Lubrication approximation theory) has been introduced to convert governing partial differential equations into a simpler form and a numerical solution is obtained by using shooting method. The significant physical quantities like volumetric flow rate, pressure, velocity, load, coated film thickness and pressure gradient are discussed in detail. The effects of power law index, slip parameter, Weissenberg number and Hartmann number on these physical quantities are presented. The observations are depicted in tabular data and in the graphical form as well. The Weissenberg number and Hartmann number play a very important role in controlling the load and coated film thickness.

Perturbation based analytical and numerical solutions of non-Newtonian differential equation during reverse roll coating process under lubrication approximation theory

Reverse roll coating process are utilized for the purpose of applying a coating to a substrate or web by utilising different rollers in order to get the required coated surface. As a result of this, reverse roll coating process have found applications in a wide variety of industries, such as those dealing with food and medicine, electronic components, optical and LCD devices, and optical products. The main objective of this paper is to develop the mathematical formulation for the coating of thin film for an incompressible isothermal viscoelastic fluid between two reversely rotating rolls. Through the use of an appropriate dimensionless parameter, Non-dimensional nonlinear ordinary differential equations (ODEs) are derived from governing partial differential equations (PDEs). LAT (lubrication approximation theory) simplifies the dimensionless equations of fluid motion. The expressions for velocity of flow, pressure gradient and flow rate is obtained analytically by using regular perturbation method, while numeric solution of some mechanical parameters such as power input, coating thickness, roll separation force and separation points are calculated. The numerical validation of the analytical solution of the fundamental model of nonlinear constitutive flow laws is done in the Maple environment using the numeric technique which is based on finite difference method. The influence of numerous non-Newtonian parameters such as velocities ratio and Weissenberg number on velocity profiles, pressure gradient, power input, roll separation force, separation points and coating thickness of a non-Newtonian Johnson–Segalman (JS) fluid are explored via graphically and in tabular form. The outcomes demonstrate that on increasing the Weisenberg number and velocities ratio, the coating thickness on web is decreases. For the numerous values of velocities ratio, it is important to note that separation points shifted towards the nip region. In addition, the non-Newtonain parameter have signifcant impact on power input and roll separating force. The maximum coating thickness that is 1.0927 has been observed at the value of separation poin 0.9620. Hence, these factors may help in an efficient coating process and improve the substrate life.