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Scientific Reports volume 14, Article number: 19709 (2024 ) Cite this article glass spray paint line
This article demonstrates a mathematical model and theoretical analysis of the Micropolar fluid in the reverse roll coating process. It is important because micropolar fluids account for the microstructure and microrotation of particles within the fluid. These characteristics are significant for accurately describing the behavior of complex fluids such as polymer solutions, biological fluids, and colloidal suspensions. First, we modeled the flow equations using basic laws of fluid dynamics. The flow equations are made modified using low Reynolds number theory. The simplified equations are solved analytically. The exact expression for velocity and pressure gradient are obtained, while pressure is calculated numerically using Simpson Rule. Graphical depictions are carried out to comprehend the impact of the newly emerged physical constraints. The influence of micropolar and microrotation parameters on the velocity, pressure and pressure gradient are elaborated with the help of different graphs.
Benkreira et al.13 The motion of the dynamic wetting line withinside the opposite roll coating become observed in a chain of experiments that gave the position, XD, angle, D and the essential metering roll speed, VM, cascade, for the begin of the waterfall. The statistics had been correlated with working situations and as compared with air infiltration charges of the diving belt. Hydrodynamic guide for wetting become found to be mentioned in a comparable way, however to a lesser degree, than for curtain lining. The correlated statistics for XD and θD additionally offer boundary situations that could override the assumptions normally made whilst simulating opposite roll coating flows. Rib instability, a totally not unusual place motive of choppy liquid movies in coating processes, is investigated each theoretically and experimentally. The Navier Stokes machine for two-dimensional go with the drift in symmetrical movie splitting in ahead roll coating is solved with the aid of using finite detail analysis. The balance of the go with the drift towards third-dimensional disturbances is tested with the aid of using making use of the principle of linear balance in a constant finite detail approach, the usage of the Fourier components with inside the transverse direction. The ensuing generalized uneven Eigen problem is solved for the boom charges of disturbances as a characteristic of wave quantity. The principle as it should be predicts the essential quantity of capillaries and the quantity of waves with inside the transition to nerves with massive amplitudes14. Malone15 provided 3 mathematical ways for the coating of meniscus rollers, where in a consistent glide of a Newtonian fluid happens with inside the slender area or line of touch among opposing rollers without bodily forces. In qualitative settlement with the observation, the Zero flow fashions are expecting a consistent strain gradient with inside the imperative center and eddies, every with an inner structure. The small glide version takes under consideration the small influx and makes use of the lubrication approximation to symbolize the fluid speed as an aggregate of the Couette and Poiseuille currents. A third version takes under consideration all of the consequences of curved menisci and the non-linear boundary situations of the unfastened floor. Pitts et al.16 argued that having a dynamic line of touch adjoining to the net at the pinnacle roll calls for the imposition of an obvious touch attitude and slip length. Numerical answers for speed and strain fields over the complete floor are received the usage of the finite detail method. The outcomes are steady with the experimental observations that the glide region includes big eddies and fluid switch jets or “snakes”. When the rollers, organized horizontally and aspect with the aid of using aspect in order that every is \(\frac{1}{2}\) of submerged in a field of liquid, rotate in contrary directions, the liquid flows thru the distance among them and splits on every curler to shape a layer. At low speeds, the sheets have a uniform thickness throughout the width of the rolls, however at better speeds they undulate regularly growing and lowering the thickness alternately. Bixler et al.17 The presented coating streams are small-scale viscous free surface streams through which a film of liquid, Newtonian or non-Newtonian, is continuously deposited on a flexible or rigid moving substrate. The liquid displaces the gas, usually air, from the substrate; where this occurs is called the wetting line. Preferred streams are stable and deposit a uniformly thin liquid film that is free from imperfections of any kind. Pearson et al.18 argued that when a thin film of viscous liquid is created by passing it through a small gap between a roller or scraper and a flat plate, it often has a corrugated or ridged surface. Here is an analysis of smear theory to show why in many cases a flow leading to a uniform film is unstable. Galerkin’s finite element method is successfully applied to flow in a relatively simple element of the roll liner; symmetrical film that splits at the line of contact between smooth, rigid and counter-rotating rollers. The calculated flow fields are solutions of the equations of moment and continuity together with the corresponding boundary conditions at the liquid / gas interface. The computer code for the analysis takes into account the effects of inertial rheology, gravity and shear thinning. The predicted positions of the film division line are consistent with available experimental data 19. The flow stability of reverse roll coating streams utilized in practical applications to coat thin liquid films on substrates of varied materials is the focus of Zheng et al.20. There are other coating processes, and several computer systems (for each approach) are developed to anticipate the set of parameters that allow for stable coating conditions. Siding windows are created using these programmers for each specific siding construction. The use of coating techniques with previous knowledge of the parameter settings for stable coatings is made possible by understanding these coating windows. Computer software to estimate coating windows for reverse roll coating are limited or non-existent, which was the impetus for the present research on the analytical factors detailed in this study. From a fluid mechanical standpoint, Coyle et al.21 explain the knife and roll coating process. It combines a critical analysis of what is known with some conjecture in the hopes of sparking new research avenues. The physical mechanics behind flow phenomena, as well as their practical ramifications, are examined. In the test portion of a laboratory reverse roll coater, the rheological behavior of the paper liner sheets is investigated. The aim is to combine the pressure readings in the measuring chamber and the torque on the gauge to calculate the viscosity of the process. Conventional rheological tests and numerical contact flow simulations are also required to complete the evaluation of the process viscosity. This viscosity is compared to that measured in step growth experiments in a rheometer22. Coyle et al.23 studied that non-Newtonian rheological conduct can strongly have an impact on both steady and steady flux and the instabilities to which it's far subject. The calculated answers of the applicable equations display that the flows of merely viscous drinks rarely fluctuate from the ones of Newtonian drinks with the identical range of capillaries, provided the dimensionless pressure courting is described with the viscosity on the corresponding shear price. Experiments with structurally viscous however surprisingly inelastic alignment answers affirm this, besides that the ribs are significantly exaggerated, in all likelihood because of the low elasticity of the fluid. Kang et al.24 mentioned the opposite roll coating procedure as an easy and bendy manner to deposit a skinny movie of liquid on a transferring substrate. Analysis of non-Newtonian nanofluid flow under radiation effects, thermophoresis, Brownian motion, chemical reactions and MHD through slender cylinder is investigated by Zaman25,26,27,28. Gaskell et al.29 provided a two-roll tool that was employed to investigate the targeted fluid dynamics of the meniscus roll coating in which the inlets were personal and the drift costs were low. The employment of optical sections in combination with dye injection and particle imaging methods is examined in the forward and opposite modes of operation (with counter-rotating and co-rotating rollers). The a part of the parameter area in which the meniscus coating takes place is recognized via way of means of various the distance among the rollers and the curler speeds, and hence the go with the drift price and the range of capillaries. Reverse roll coating might be the maximum generally used coating method. We exhibit via an extensive variety of experimental statistics that such an operation can produce very skinny and solid movies without ribs or cascading instabilities whilst the use of low viscosity fluids. In general, solid movie thicknesses of much less than five µm may be carried out at speeds of as much as one hundred fifty m/min the use of a rubber curler with a hundred µm gap with drinks with a viscosity of the order of 10, \(2 \text{hundred\; mPas}\) . MHD and heat transfer through disk is discussed by Nayak et al.30. Makinde et al.31,32 discussed bifurcation squeezed flow. This research was submitted by Fateh et al.33, and it details the construction of a mathematical model for reverse roll coating of a thin film for an incompressible non-isothermal magnetohydrodynamic (MHD) viscoplastic fluid flowing across a tiny gap between two counter-rotating rolls. The lubrication approximation theory is used to generate and simplify the equations of motion necessary for the fluid supplied to the route (LAT). The velocity profile, pressure gradient, and temperature distribution all yield analytical data. The trapezoidal rule or the regular disposition technique is used to determine pressure and flow distributions numerically. Zahid et al.34 present a mathematical model of the non-isothermal magnetohydrodynamic (MHD) flow of an incompressible Jeffrey fluid as it travels through a limited gap between two counter-rotating rollers; according to Suitable dimensionless parameters are used to obtain the dimensionless versions of the governing equations. Intelligent predictive networks for analysis of chaos in stochastic differential SIS epidemic model with various impact is discussed by raja et al.35,36,37.Radiative magnetodydrodynamic cross fluid thermophysical model passing on parabola surface with activation energy using MATLAB under various effects of eyeing powell and casson fluids are studied by Awais et al.38,39,40,41. The dimensionless flow equations are simplified using low Reynolds number theory. For velocity, pressure gradient, flow rate, Nusselt number, and temperature distribution, analytical solutions are offered.
In the current analysis, we assume that the flow is steady, incompressible and laminar. In this analysis, we differentiate the polymer melt by using the micropolar fluid model. The polymer melt is passing through the couple rotating rolls in opposite directions with their rotating velocities are \({U}_{f}=R{\omega }_{f},\text{ and }{U}_{r}=R{\omega }_{r}\) , in which \({\text{U}}_{\text{f}}\) and \({\text{U}}_{\text{r}}\) denote the forward and backward cylinders velocities respectively. The distance between these two cylinders is \(2{\text{H}}_{0}\) . Both cylinders having radius R and the velocity ratio of both cylinders is defined as \(k=\frac{{U}_{r}}{{U}_{f}}\) . In this study, we consider the velocity ratio is uniform throughout the analysis. We use Cartesians coordinates system, in which x-axis along the flow direction while the y-axis along the transverse to the flow direction. The flow geometry presented in Fig. 1.
For the analysis of the reverse roll coating, we use the conservation of the mass and momentum for incompressible fluid as
In the above equations,\(\overrightarrow{V}\) expresses the velocity vector, \(\rho\) symbolizes the density, \(\overrightarrow{q}\) represents the micro rotation vector, \(\overline{p }\) means the fluid pressure and \(j\) shows the micro-inertia parameter. It is interesting to note that microrotation becomes zero if \({\alpha }_{1}={\alpha }_{2}={\alpha }_{3}=0\) . If we set \(k=0\) then Eqs. (2) and (3) becomes uncoupled.
Eringen suggested that the following condition must satisfy.
In the reverse roll coating, we used two dimensional velocity field which is defined in the following way.
By utilizing Eqs. (5) and (6) into Eqs. (1)–(3), one can get
Introducing the dimensionless variables which are used to simplify the flow equations. \({u}^{*}=\frac{\overline{u}}{{U }_{f}} , {v}^{*}=\frac{\overline{v}}{\delta {U }_{f}} , {x}^{*}=\frac{\overline{x} }{{\left(R{H}_{0}\right)}^\frac{1}{2}} , {y}^{*}=\frac{\overline{y}}{{H }_{0}} , Q=\frac{\overline{Q}}{{2UH }_{0}}\)
By using these dimensionless variables, we made the Eqs. (7–10) in dimensionless form.
where \(N\) is the coupling number which indicates the ratio between the vortex viscosity coefficient and the shear viscosity coefficient \(\varepsilon\) denotes the micropolar parameter and \(\delta\) represents the geometric parameter, given as \(\delta =\sqrt{\frac{{H}_{0}}{2R}}\) , \(Re=\rho U{H}_{0}/\mu\) indicates the characteristic Reynolds number. The separation gap in the middle of the rolls is very small in comparison with the radii of the rolls i.e. \({H}_{0}\ll R\) , which implies that \(\delta \ll 1\) . Hence the above equations becomes
The dimensionless boundary condition is given by42
Simplify Eq. (15) in the following form
Now integrate Eq. (20) with respect to \(y\) ,we get
where \({c}_{1}\) shows the integration constant, which must be find.
To eliminate the \(u\) Eqs. (15) and (17) in the following form.
Multiplying Eq. (22) with \(\frac{1}{\epsilon },\) we get.
For simplicity, the above equation can be expressed in the following form
The general solution of Eq. (24) can be calculated in the following way
Using Eqs. (27) and (28) into Eq. (26) we get
Now we substitute the Eq. (29) into (21), then we get
Integrate the Eq. (30), we obtain the velocity profile as
For simplicity, we define \({a}_{i}^{\prime}s\) in the following form
Using the values of \({a}_{i}^{\prime}s\) in Eq. (31),
When we use boundary condition \(u=1 at y= - \sigma\) .we get this solution.
Now we use boundary condition \(u=-k at y= \sigma\) , we get this solution.
The boundary conditions in Eq. (29) indicates that microrotation zero at the roll surface.
Using Eq. (36) into Eq. (29), we get
In the same we use Eq. (36) into Eq. (29), we get:
where \({c}_{1}, {c}_{2},{c}_{3}\) and \({c}_{4}\) are constants, which are unknown.
From Eqs. (34), (35), (38) and (39), we find the values of these unknown constants in the following manner.
To find the pressure gradient we use the dimensionless flow rate formula as
To find the pressure distribution and thickness of film we have to calculate the value of \(\lambda\) first. In order to calculate the value of \(\lambda\) , many researcher use Swift-Stieber condition on pressure and pressure gradient. According to this condition both pressure and pressure gradient are zero at transition point \(x= {x}_{t}\) . Apply this argument on the Eq. (48), we get
Unfortunately, we cannot find the exact expression for pressure; therefore, we only calculate pressure numerically with the condition on pressure \(p\to 0 as x\to -\infty .\)
The value of \(\lambda\) is calculated with the help of finding roots. Interesting Eq. (48) we get pressure as
In this section, we study the impact of the velocities ratio \((k)\) , coupling number \(\left(N\right),\) flow rate (\(\lambda )\) and \(\epsilon\) on the velocity distribution, pressure gradient and pressure distribution with different graphs. The effects of the velocities ratio \((k)\) on the velocity at three axial ppositions \(x=0,x=0.25\) and \(x=0.75\) are discussed in Figs. 2, 3, 4, 5 with fixed values of other parameters. It is noted in the figures that velocities increase near the reverse roll by increasing the values of \(k\) while it it decreases near the forward roll. The impact of different values of coupling parameter on velocity is shows in Fig. 2. Impact of different values of \(k\) on velocity is shows in Figs. 3, 4 and 5.
Impact of different values of coupling parameter on velocity for fixed \(\epsilon ,\lambda ,k on x=0.\)
Impact of different values of \(k\) on velocity for fixed \(\epsilon ,\lambda ,N,\) on \(x=0.\)
Impact of different values of k on velocity for fixed \(\epsilon ,\lambda ,N,\) on \(x=0.25.\)
Impact of different values of k on velocity for fixed \(\epsilon ,\lambda ,N,\) on \(x=0.75.\)
Figures 6, 7, 8 plotted to see the influences of the coupling number, \(\epsilon\) and \(\lambda\) on the pressure gradient. From these figures it is clear that graphical behavior of pressure gradient about \(x=0\) , symmetrical. The axial point \(x=0\) also called nip region.
Impact of different values of N on pressure gradient for fixed \(\epsilon ,\lambda ,k\) .
Impact of different values of \(\lambda\) on pressure gradient for fixed \(\epsilon ,N,k\) .
Impact of different values of \(\epsilon\) on pressure gradient for fixed \(\lambda ,N,k.\)
When we march on x-axis from left to right the pressure gradient zero at \(x=4.5\) and moving right on the x-axis, the pressure gradient increases and attained its maximum values at \(x\approx 1\) and then decreasing and reaches it minimum values at \(x=0.\) After this point pressure gradient again increases at reach maximum at \(x= -1\) and approaches to zero at \(x= -4.5.\) The figures reveals that the magnitude of pressure gradient increases with increases the matrial parameters value.
The effects of the matrial parameters (\(N,\epsilon )\) and velocity ratio on the pressure profile are presented in Figs. 9, 10, 11. Figure 9 shows the effects of flow rate on the pressure profile with fixed values of other parameters. It is noted from this figure, if we moving from left to right on the x-axis pressure increases and attained its maximum value after this pressure decreases continuously at reach zero for some cases otherwise show steady. Moreover the magnitude of pressure increases with increase the value of flow rate.
Impact of different values of \(\lambda\) on pressure for fixed \(\epsilon ,N,k.\)
Impact of different values of \(N\) on pressure for fixed \(\epsilon ,\lambda ,k.\)
Impact of different values of \(k\) on pressure for fixed \(\epsilon ,N,\lambda .\)
Figure 10 plotted to see the influence of coupling number (\(N)\) on the pressure distribution.the diagram shows that the pressure distribution decreases with incrases the values of the coupling number. The length of the contant decreases with increases the values of the coupling number (\(N).\)
Similarly, the impact of the velocity ratio on the pressure distribution is present in Fig. 11. This figure indicates that pressure distribution decreases with increasing the value of the velocity ratio. Because the decrease in pressure distribution with increasing velocity ratio in reverse roll coating can be attributed to the following:
Higher shear rates near the reverse roll enhance viscous drag and align fluid motion with the reverse roll’s direction.
Increased fluid velocity near the reverse roll reduces pressure due to the Bernoulli principle.
Micropolar effects such as couple stresses and microrotation reduce resistance to fluid motion, lowering pressure.
Mathematical analysis and numerical integration confirm that higher velocity ratios lead to a steeper velocity gradient and lower pressure. It is noted from this figure that small variations in the velocity’s ratio impose a large impact on the pressure profile.
In this study, we present the theoretical analysis of reverse roll coating for micropolar fluid. We evaluate the impact of velocity ratio (\(k)\) , coupling number \((N),\) flow rate (\(\lambda )\) and \(\epsilon\) on velocity distribution, pressure distribution and pressure gradient. The key deductions of study are as follows.
Velocity increases as the value of \(k\) increases.
The maximum velocity occurs at the reverse roll surface. The maximum velocity occurring at the reverse roll surface in a reverse roll coating process is primarily due to the high relative motion between the reverse roll and the fluid, the opposing directions of the rolls, and the squeezing effect of the narrow gap. These factors combine to create a higher fluid velocity near the reverse roll compared to the forward roll, leading to the observed maximum velocity at the reverse roll surface.
Flow rate velocity decreases near the forward roll. The decrease in flow rate velocity near the forward roll in reverse roll coating is primarily due to the lower relative velocity of the forward roll, alignment of the roll’s motion with the fluid flow, and the redistribution of fluid velocity between the rolls.
Pressure gradient shows behavior symmetrical at \(x=0.\)
Behavior of pressure gradient changes in the interval (− 4.5, 4.5).
Maximum pressure gradient attained in the interval (− 1, 1).
Pressure distribution decreases with increase the values of coupling number (\(N)\) , velocity ratio (\(k).\)
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
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The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2024/01/921063).
Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
Department of Mathematics, University of Gujrat, Gujrat, Pakistan
Azad Hussain, Kaleem Ashraf, Lubna Sarwar & Faizan Hussain
Department of Computer Science, Prince Sattam Bin Abdulaziz University, 11942, Al-Kharj, Saudi Arabia
Faculty of Computing and Information Technology, King Abdulaziz University, 21589, Jeddah, Saudi Arabia
Neighbourhood of Akcaglan, Imarli Street, Number: 28/4, 26030, Eskisehir, Turkey
Department of Mathematics, College of Science, King Khalid University, P.O. Box: 9004, 61413, Abha, Saudi Arabia
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S.Z. wrote manuscript. A.H. did supervision, Formal analysis is done by K.A., Methdology is done by L.S., Validation is done by F.H., Funding is done by A.A. Editing is done by A.B. Revision is done by T.M.
Correspondence to Saquib Ul Zaman.
The authors declare no competing interests.
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Zaman, S., Hussain, A., Ashraf, K. et al. Mathematical analysis of isothermal study of reverse roll coating using Micropolar fluid. Sci Rep 14, 19709 (2024). https://doi.org/10.1038/s41598-024-70808-6
DOI: https://doi.org/10.1038/s41598-024-70808-6
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