scholarly journals Forward Roll Coating of a Williamson’s Material onto a Moving Web: A Theoretical Study

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
M. Zahid ◽  
I. Siddique ◽  
S. Saleem ◽  
A. Al-Zubaidi ◽  
M. A. Rana ◽  
...  
Keyword(s):  
Coating Thickness ◽  
Gradient Flow ◽  
Separation Point ◽  
Maximum Pressure ◽  
Perturbation Approach ◽  
Regular Perturbation ◽  
Roll Coating ◽  
Flow Equations ◽  
Operating Variables ◽  
Roll Separating Force

This paper presents a mathematical model for the thin film roll coating process of an incompressible Williamson material, passing through a closed passage between a rotating roll and a web. In light of lubrication approximation theory, the flow equations are nondimensionalized. The regular perturbation approach is used to provide solutions for the velocity profile, pressure gradient, flow rate per unit width, and shear stress at the roll surface. Important engineering quantities such as coating thickness, maximum pressure, separation point, roll/sheet separating force, and roll-transmitted power to the fluid are also obtained. The effects of several factors are graphically projected. The study shows that the material factors that are involved determine the operating variables. Coating thickness and separation point are controlled by Weissenberg’s number, therefore acting as a controlling parameter for the rate of flow, thickness in coating, power contribution, pressure, roll separating force, and separation point. In comparison to the existing results in the literature, the current results are broader and zero-order results are more accurate.

Roll coating analysis of a third grade fluid

2016 ◽  
Vol 33 (1) ◽  
pp. 72-91 ◽  
Author(s):  
M Zahid ◽  
T Haroon ◽  
MA Rana ◽  
AM Siddiqui
Keyword(s):  
Equations Of Motion ◽  
Gradient Flow ◽  
Material Constant ◽  
Maximum Pressure ◽  
Lubrication Approximation ◽  
Third Grade Fluid ◽  
Regular Perturbation ◽  
Roll Coating ◽  
Grade Fluid ◽  
The Web

This paper studies the roll-coating process of an incompressible viscoelastic fluid, where the roll and the web have equal velocities. The lubrication approximation theory is used to simplify the equations of motion. Solutions for velocity profile, pressure gradient, flow rate per unit width, and shear stress at the roll surface are obtained by using a regular perturbation method. Integrated quantities of engineering interest are also calculated. These include the maximum pressure, separation point, roll/sheet separating force, power transmitted to the fluid by the roll, and coating thickness. It is found that these quantities increase substantially and monotonically as the fluid’s material constant increases.

Mathematical Analysis of Roll Coating Process by Using Couple Stress Fluid

2019 ◽  
Vol 8 (8) ◽  
pp. 1683-1691 ◽  
Author(s):  
M. Zafar ◽  
M.A. Rana ◽  
M. Zahid ◽  
M.A. Malik ◽  
M.S. Lodhi
Keyword(s):  
Coating Thickness ◽  
Couple Stress ◽  
Flow Parameter ◽  
Couple Stress Fluid ◽  
Lubrication Approximation ◽  
Isothermal Model ◽  
Roll Coating ◽  
Flow Equations ◽  
Pressure Profiles ◽  
Controlling Parameter

In this article, an incompressible isothermal model of a couple stress fluid between two rotating rolls is developed. Lubrication approximation theory is applied to simplify the flow equations. Exact solutions for velocity and pressure profiles are derived. Parameters of an industrial interest like pressure, separating force, coating thickness, detachment point and power transmitted by the rolls to the fluid are computed numerically. It is observed that the flow parameter is a controlling parameter for an exiting coating thickness. As the problem is symmetric, only one half of the geometry is considered.

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

Dynamics of a non-spherical microcapsule with incompressible interface in shear flow

2011 ◽  
Vol 678 ◽  
pp. 221-247 ◽  
Author(s):  
P. M. VLAHOVSKA ◽  
Y.-N. YOUNG ◽  
G. DANKER ◽  
C. MISBAH
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Viscosity Ratio ◽  
Perturbation Expansion ◽  
Shear Plane ◽  
Creeping Flow ◽  
Regular Perturbation ◽  
Cell Dynamics ◽  
Initial Shape ◽  
Flow Equations

We study the motion and deformation of a liquid capsule enclosed by a surface-incompressible membrane as a model of red blood cell dynamics in shear flow. Considering a slightly ellipsoidal initial shape, an analytical solution to the creeping-flow equations is obtained as a regular perturbation expansion in the excess area. The analysis takes into account the membrane fluidity, area-incompressibility and resistance to bending. The theory captures the observed transition from tumbling to swinging as the shear rate increases and clarifies the effect of capsule deformability. Near the transition, intermittent behaviour (swinging periodically interrupted by a tumble) is found only if the capsule deforms in the shear plane and does not undergo stretching or compression along the vorticity direction; the intermittency disappears if deformation along the vorticity direction occurs, i.e. if the capsule ‘breathes’. We report the phase diagram of capsule motions as a function of viscosity ratio, non-sphericity and dimensionless shear rate.

scholarly journals Perturbation Approach in the Dynamic Buckling of a Model Structure with a Cubic-quintic Nonlinearity Subjected to an Explicitly Time Dependent Slowly Varying Load

2019 ◽  
pp. 1-19
Author(s):  
A. M. Ette ◽  
I. U. Udo-Akpan ◽  
J. U. Chukwuchekwa ◽  
A. C. Osuji ◽  
M. F. Noah
Keyword(s):  
Perturbation Technique ◽  
Dynamic Buckling ◽  
Buckling Load ◽  
Time Dependent ◽  
Initial Time ◽  
Perturbation Approach ◽  
Elastic Model ◽  
Model Structure ◽  
Regular Perturbation ◽  
Long Run

This investigation is concerned with analytically determining the dynamic buckling load of an imperfect cubic-quintic nonlinear elastic model structure struck by an explicitly time-dependent but slowly varying load that is continuously decreasing in magnitude. A multi-timing regular perturbation technique in asymptotic procedures is utilized to analyze the problem. The result shows that the dynamic buckling load depends, among other things, on the first derivative of the load function evaluated at the initial time. In the long run, the dynamic buckling load is related to its static equivalent, and that relationship is independent of the imperfection parameter. Thus, once any of the two buckling loads is known, then the other can easily be evaluated using this relationship.

scholarly journals A Flow Perspective on Nonlinear Least-Squares Problems

2020 ◽  
Vol 48 (4) ◽  
pp. 987-1003
Author(s):  
Hans Georg Bock ◽  
Jürgen Gutekunst ◽  
Andreas Potschka ◽  
María Elena Suaréz Garcés
Keyword(s):  
Least Squares ◽  
Newton Method ◽  
Large Scale ◽  
Gradient Flow ◽  
Nonlinear Least Squares ◽  
Bundle Adjustment ◽  
Least Squares Problems ◽  
Flow Equations ◽  
Backward Euler ◽  
Forward Euler

AbstractJust as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß–Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß–Newton flow equations. We highlight the advantages of the Gauß–Newton flow and the Gauß–Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg–Marquardt method, which are respectively equivalent to Newton flow forward Euler, gradient flow forward Euler, and gradient flow backward Euler. We finally show an unconditional descent property for a generalized Gauß–Newton flow, which is linked to Krylov–Gauß–Newton methods for large-scale nonlinear least squares problems. We provide numerical results for large-scale problems: An academic generalized Rosenbrock function and a real-world bundle adjustment problem from 3D reconstruction based on 2D images.

A theoretical analysis of the calendering of Ellis fluid

2016 ◽  
Vol 33 (2) ◽  
pp. 207-226 ◽  
Author(s):  
Muhammad Asif Javed ◽  
Nasir Ali ◽  
Muhammad Sajid
Keyword(s):  
Theoretical Analysis ◽  
Velocity Profile ◽  
Pressure Gradient ◽  
Pressure Distribution ◽  
Sheet Thickness ◽  
Power Input ◽  
Lubrication Approximation ◽  
Material Parameters ◽  
Operating Variables ◽  
Roll Separating Force

We present a theoretical analysis of calendering of Ellis fluid based on lubrication approximation. The equations governing the flow are nondimensionalized and solved to get closed form expressions of velocity and pressure gradient. Runge–Kutta algorithm is employed to compute the pressure distribution. The operating variables which are used in the calendering process, i.e. roll-separating force, power input to the rolls and exiting sheet thickness are calculated. The influence of the material parameters on the velocity profile, pressure gradient, pressure distribution and operating variables is shown graphically and discussed in detail.

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