Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow

Sabeeh Khaliq; Nida Shaheen; Moin-ud-Din Junjua; Mehmood Ahmad; Zaheer Abbas

doi:10.1515/phys-2025-0173

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Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow

Sabeeh Khaliq , Nida Shaheen , Moin-ud-Din Junjua , Mehmood Ahmad and Zaheer Abbas

Published/Copyright: July 17, 2025

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From the journal Open Physics Volume 23 Issue 1

Abstract

Coating techniques are broadly used in the production of sticky tapes, plastic films, books, wallpapers and magazines, adhesive tapes, as well as the textile and metal preservation, packaging, material protection, and X-ray films. The rheology of the magneto-hydrodynamic (MHD) hybrid nanomaterial on the final coating thickness is studied by using the reverse roll coating technique. In the coating industry, the application of nanomaterials over the sheet/substrate has better non-Newtonian like rheological features as opposed to the ordinary fluid. The dimensionless governing equations are reduced using lubrication approximation theory, and the exact solutions for velocity and pressure gradient are obtained. To find the transition point, coating thickness, and other mechanical quantities, a numerical technique is utilized. Graphs and tables are used to study the impacts of hybrid nanoparticles and magnetic fields on the flow and engineering quantities in detail. The flow rate, transition point, and coating thickness decrease due to an increase in the nanoparticle volume fraction and MHD. Controlling these relevant parameters helps in engineering applications, including attaining an efficient coating process, protecting and prolonging the substrate life.

Graphical abstract

Keywords: hybrid nanomaterial; reverse roll coating process; MHD; coating thickness; lubrication theory

Nomenclature

B 0: applied magnetic field (T = kg/A m²)
σ: coating fluid electrical conductivity (S/m)
J: current density (A/m²)
λ: dimensionless flow rate
c t: final coating thickness (m)
V: fluid velocity ( m/s )
H f: forward roll film thickness (m)
ξ: geometric parameter
ρ hnf: hybrid nanofluid density (kg/m³)
μ hnf: hybrid nanofluid viscosity (N s m⁻²)
ϕ 1 , ϕ 2: hybrid nanoparticle volume fraction
B: magnetic field (T = kg/A m²)
2 H 0: nip gap separation (m)
H r: reverse roll film thickness (m)
x t: separation point (m)
∇: gradient operator
K: velocity ratio
p: pressure (kg s²/m)
R: rolls radius (m)
Re: Reynolds number
u, v: fluid velocity in x ∗ - and y ∗ -axis ( m/s )
x, y: horizontal and vertical directions ( m )

1 Introduction

The one roll-to-roll coating technique for wet coatings is called reverse roll coating. It is different from other roll coating techniques because it has two reverse-running nips. With an exact spacing between them, the measuring roll and the applicator roll rotate in opposite directions. The applicator roll’s surface is loaded with extra coating before the metering nip, ensuring that its surface exits the metering nip precisely coated to the gap’s width. When the applicator roll reaches the application nip, it wipes all of this coating onto the substrate by moving in the opposite direction of the substrate’s movement.

An engineering technique comprising a thin, rigid liquid layer applied continuously to a movable web is called roll coating. Because this method is widely employed at the industrial level, it has gained a critical reputation in the coating sectors in recent years. Also, engineering professionals are likely aware of roll coating. The roll elements make it easy to apply fluid coating onto a web and have a major impact on controlling the coating layer thickness [1]. There is a broad use of coating in the production of sticky tapes, plastic films, books, wallpapers and magazines, adhesive tapes, as well as the preservation of textiles and metals, graphic movies, packaging, material protection and X-ray films. There are several techniques for consistently producing a fluid layer covering a substrate or a surface. As a result, a number of criteria impact the choice of project, including fluid rheology, surface element, the intended layer thickness, the type of liquid utilized, the consistency of the cover, the rate at which a substrate is surrounded, etc. [2,3].

The foremost monitoring factor for sheet thickness and homogeneity among rotating rolls is the flowing fluid. The procedure of coating thickness is primarily dependent on the distance between the rolls and also on the rotational speeds of the rolls. In most cases, the radii of both rotating rollers are greater than the nip distance between both rolls. There are three primary types of roll coating depending on the rollers: reverse roll coating (RRC), metering roll coating (MRC) and forward roll coating (FRC) [4,5]. Both rollers travel in the same direction in FRC, but the direction of rolls and web is opposite in the case of RRC [6,7].

The fluid film’s thickness should mainly be constant and homogenous. According to researchers, a possible issue in the roll coating apparatus through coating is surface instabilities that are not enough and should be relinquished to accomplish a smooth coated layer in the case of Newtonian [8,9]. A numerical method was used by Chandio and Webster [10], which took into account the free surface calculation over time. They used a variable speed ratio to investigate the transitory instabilities and concluded that an increase in foil speed in place of roll speed could exacerbate flow instabilities. Coyle et al. [11] studied the flow instabilities, including ribbing and cascade.

For the experimental outcomes, they utilized the finite element technique to explain the RRC fluid dynamics. Jang and Chen [12] examined the fluid-free surface volume and finite-volume techniques in the RRC process to analyze the role of fluid properties. While analyzing the roll coating technique, they studied the power law fluid model, with a power-law index range of 0.95–1.05. They focused mostly on the impact of roll speed and coating thickness on rib-biting instabilities and concluded that when the film coating thickness is increased, the power law index also increases.

Greener and Middleman [13] conducted the roll coating ground-breaking work. Under the premise of a minor roll curvature, they developed a mathematical model for the situation where the sheet as well as the roll possessed equal linear speed. By using lubrication theory, they were able to determine the solutions numerically and analytically for both kinds of Newtonian and non-Newtonian fluids. A thorough investigation associated with film hypothesis and the process of roll coating was discussed by Middleman [14] as well as Kistler and Schweizer [15]. The viscocapillary representation using the lubrication approximation method was utilized by Carvalho and Scriven [16] to explore fluid flow between a rigid body and a counter-rotating roll that was deformable. Two models for determining the forward-roll coating sheet thickness were presented by Coyle et al. [17]. Two models were developed: the first model was based on short gap-to-roll diameter asymptotic expansion, and the second model on the finite element method of Galerkin for solving complete Navier–Stokes equations. Taylor and Zettlemoyer [18] used the lubrication approximation technique during printing press to explore the performance of ink flow operations and obtained the pressure and force distribution results. In the RRC method, Greener and Middleman [19] have also utilized lubrication approximation theory (LAT) to study viscoelastic liquids and viscous liquids. Recently, Sajid et al. [20] used the lubrication approximation technique by taking both planar and exponential coaters in blade coating in order to solve the third-grade fluid’s emergent equations. Sajid et al. [21] have also utilized the lubrication approximation technique to examine viscous fluids. They concluded that the magnetic field and slip parameter control the sheet velocity after taking into account magneto-hydrodynamic (MHD) and the calculation of the slip condition implied at the blade surface. Shahzad et al. [22] studied the model of Oldroyd’s four-constant fluid for blade coating. To clarify the governing partial differential equations that are dimensionless, they used LAT. They were able to determine that the coating’s quality and thickness depend on the pressure and load applied to the blade. In addition to the blade surface slip conditions, Wang et al. [23] used a viscous fluid model by supposing that the magnetic field and flow were normal to examine a flexible blade coater. Associated equations have been simplified through the application of lubrication theory. In the presence of a magnetic field and slip, they discovered that the fluid velocity and blade deflection are elements that may be controlled. Adopting the Adomian decomposition method, Bhatti et al. [24] studied the blade coating phenomena of non-Newtonian Carreau fluid by utilizing LAT. The impact on the flow and heat transfer mechanism was calculated using a numerical scheme with the help of graphical illustrations and tables.

Abbas and Khaliq [25] applied the micropolar-Casson model to theoretically investigate the roll coating method under an isothermal environment, and their outcomes indicate that both the liquid viscoplastic behavior along particle microrotation impact the uniform coating thickness on the moving sheet. Atif et al. [26] performed theoretical research on the roll coating procedure to determine the role of the rheological viscoelastic SPTT liquid in the final coating thickness and other engineering quantities. The implication of variable viscosity on the roll-over-web coating method was reported by Ali et al. [27] using the viscoelastic liquid model. Recently, the coating of a thin fluid layer of Sisko liquid was scrutinized by Hanif et al. [28] to report the heat and flow mechanism of the blade coating process under LAT. Atif et al. [29] applied the hyperbolic tangent fluid layer in an isothermal environment of the roll coating process and reported a thicker final coating layer in the case of a large Weissenberg number. Recently, a study of Garalleh et al. [30] focused on the blade coating technique by adopting the viscoelastic MHD nanofluid and verified the model with an artificial neural network framework. Some recent studies on the coating process are also reported in the literature for interested readers [31,32,33,34,35].

The term nanofluid, devised by scientist Stephen U.S. Choi and Eastman [36], consists of nanosize particles in an engineered colloidal solution with a base fluid. Nanofluids generally can be utilized in several technologies as well as industrial applications, such as paints, biological solutions, glue, asphalts, and polymers. In the coating industry, the application of nanomaterials over the sheet/substrate has better non-Newtonian like rheological features as opposed to the ordinary fluid, which helps in protecting the sheet and improving efficiency. Researchers employed many kinds of nanoparticles, including carbon [37], metallic [38,39,40], and non-metallic nanoparticles [41,42]. Studies have previously reported the improvement in heat transportation as well as flow mechanisms in coating flows [43,44,45].

The relevant literature reporting the nanofluid rheological studies was limited to single (mono) nanofluid comprising unitary nanoparticles; until recently, engineers looked into the mixture of two nanoparticles in the same baseliquid termed as hybrid nanofluid, which simultaneously merges the chemical as well as physical features of distinct nanoparticles and produce sustainable mixture under homogeneous condition during the processes. Hybrid nanofluids are reported to have improved the mechanism of heat transfer, better rheological characteristics in liquid, viscosity modification, and thermal conductivity enhancement as related to usual unitary nanofluids. Scott et al. [46] focused on hybrid nanoliquid development and flow behavior in cavities. They noticed exceptional rheological and thermal features of the hybrid nanofluid because of the synergistic impact of distinct nanomaterials. Metallic nanoparticles were also considered by Prakash et al. [47], who reported the electroosmosis behavior in peristaltic microfluidic flow. Gandhi et al. [48] studied the drug delivery mechanism of nanoparticles by comparing the mono-nanoparticles (gold) efficiency with hybrid nanoparticles (magnetic Au−Al₂O₃/blood Au−Al₂O₃). Roy and Pop [49] reported the heat transportation with MHD flow of hybrid nanomaterials (Al₂O₃−Cu) between infinite parallel plates. Kumar et al. [50] studied the Casson hybrid nanofluid model to study the important Soret and Dufour impacts on flow dynamics over the curved Riga plate. Most recently, several research and review articles on the hybrid nanofluid flow are available for the interested readers [51,52,53,54,55,56].

A careful inspection of previous literature reveals that no study is present to discuss the nanofluid or hybrid nanofluid impact on the rheological flow of the RRC technique. The current study shows the rheological features of the hybrid nanofluid polymer and its influence on the existing coating thickness during the RRC process in isothermal enjoinment with applied MHD. In the following sections, the problem formulation, governing equations, problem simplification, problem solution, and the results are discussed with an extracted conclusion.

2 Problem formulation

The physical assumptions of the underlying problem are provided as:

We assume an incompressible, isothermal flow of a viscous, hybrid nanofluid polymer with applied MHD coated over a porous moving sheet.
A thin layer coating of hybrid nanomaterial ( Al 2 O 3 − Cu ) is performed by passing between two rollers revolving in opposite directions, having a radius R.
Suppose that the minimum gap in the middle of two rolls is 2 H 0 .
U f = R ω f and U r = R ω r are the forward (Applicator Roll) and reverse (Metering Roll) roll velocities.
ω is the angular roll velocity and the velocity ratio is K = U f U r .
Figure 1 shows two rolls centered in the plane where the two-dimensional coordinate system ( x ⁎ , y ⁎ ) is located, where u ⁎ represents the x ⁎ -directional component of the velocity and v ⁎ is its y ⁎ -directional component.

Figure 1

Systematic geometry of the RRC phenomena.

2.1 Governing equations

The incompressible, isothermal expressions representing the viscous hybrid nanofluid with applied MHD in the absence of body force are as follows [19,48]:

(1) ∇ ⋅ V ⁎ = 0 ,

(2) ρ hnf d V ⁎ d t = − ∇ p ⁎ + μ hnf ∇ V ⁎ 2 − J × B ,

where ρ hnf is the hybrid nanofluid density μ hnf is the hybrid nanofluid viscosity, B is the magnetic field, and J is the current density.

For small magnetic Reynolds number J × B , the Lorentz force changes to σ hnf ( V ⁎ × B ) × B , where the induced electric field is neglected, and the applied magnetic field is equal to J = σ ( V ⁎ × B ) where σ represents the fluid’s electric conductivity. In light of the flow scenario, the normal direction magnetic field and the velocity field are given as follows:

(3) B = [ 0 , B 0 , 0 ] , V ∗ = [ u ∗ ( x ∗ , y ∗ ) , v ∗ ( x ∗ , y ∗ ) , 0 ] .

The above Eqs. (1) and (2) can be written as [43,49] follows:

(4) ∂ u ∗ ∂ x ∗ + ∂ v ∗ ∂ y ∗ = 0 ,

(5) ρ hnf u ∗ ∂ u ∗ ∂ x ∗ + v ∗ ∂ u ∗ ∂ y ∗ = − ∂ p ∗ ∂ x ∗ + μ hnf ∂ 2 u ∗ ∂ x ∗ 2 + ∂ 2 u ∗ ∂ y ∗ 2 − σ hnf B 0 2 u ∗ ,

(6) ρ hnf u ∗ ∂ v ∗ ∂ x ∗ + v ∗ ∂ v ∗ ∂ y ∗ = − ∂ p ∗ ∂ y ∗ + μ hnf ∂ 2 v ∗ ∂ x ∗ 2 + ∂ 2 v ∗ ∂ y ∗ 2 .

Figure 1 shows the nip region as the most important region in terms of the occurrence of dynamical events, leading to a relation where H 0 (one dimension) is much less than the other (R = roll radius), triggering LAT. Therefore, a parallel flow seems likely in both the ± x directions, also directing to the relations v ∗ ≪ u ∗ and ∂ / ∂ x ≪ ∂ / ∂ y . We assume an isochoric flow with v ∗ = v 0 ∗ ( constant ) , where the injection velocity relates to v 0 ∗ > 0 and the suction velocity is related to v 0 ∗ < 0 .

Hence, Eqs. (4)–(6) become

(7) ∂ u ⁎ ∂ x ⁎ = 0 ,

(8) ρ hnf v 0 ∗ ∂ u ∗ ∂ y ∗ = − ∂ p ∗ ∂ x ∗ + μ hnf ∂ 2 u ∗ ∂ y ∗ 2 − σ hnf B 0 2 u ∗ ,

(9) ∂ p ∗ ∂ y ∗ = 0 .

Eq. (9) shows that pressure is a function of only x. We obtain the following equation:

(10) ρ hnf v 0 ∗ ∂ u ∗ ∂ y ∗ = − d p ∗ d x ∗ + μ hnf ∂ 2 u ∗ ∂ y ∗ 2 − σ hnf B 0 2 u ∗ .

Here, ρ hnf , μ hnf , and σ hnf are hybrid effective densities and can be defined as follows [48,49]:

(11) μ hnf = μ f ( 1 − ϕ 1 − ϕ 2 ) 2.5 ρ hnf = ρ 1 ϕ 1 + ρ 2 ϕ 2 + ρ f ( 1 − ϕ 1 − ϕ 2 ) σ hnf = ( 2 σ f + σ hp ) + 2 ϕ hnf ( σ hp − σ f ) ( 2 σ f + σ hp ) − 2 ϕ hnf ( σ hp − σ f ) σ f ϕ hnf = ϕ 1 + ϕ 2 ,

where ϕ 1 and ϕ 2 signify the nanoparticle volume fractions for the respective first and second nanomaterials. The thermophysical properties of the hybrid manomaterials ( Al 2 O 3 − Cu ) with the base fluid ( H 2 O ) are taken from Table 1.

Table 1

Thermophysical properties of the nanomaterial and water [48,49]

Physical properties	Base fluid ( H 2 O )	Al 2 O 3 ( 1 )	Cu ( 2 )
ρ hnf ( kg/m 3 )	997.1	3,970	8,933
σ ( S/m − 1 )	0.05	3.69 × 10⁷	5.96 × 10⁷

Under physical conditions on both rolls [19]:

(12) u ⁎ = U f at y ⁎ = − h ⁎ ( x ⁎ ) ,

(13) u ⁎ = − U r at y ⁎ = h ⁎ ( x ⁎ ) .

The above conditions physically relate to the situation of the no-slip condition on rolls and both rolls move in a clockwise direction with different surface velocities U f and U r .

The variable height h ( x ) is defined as follows:

(14) h ⁎ ( x ⁎ ) = R ⁎ + H 0 − ( R ⁎ 2 − x ⁎ 2 ) 1 / 2 .

Suppose 1 ≫ H 0 R , so Eq. (14) becomes

(15) h ⁎ ( x ⁎ ) = 1 + x ⁎ 2 2 H 0 R ⁎ H 0 .

2.2 Dimensionless equations

Now, we consider the following dimensionless equations:

(16) y = y ∗ H 0 , p = H 0 R H 0 p ∗ U f μ f , x = x ∗ H 0 R , u = u ∗ U f , v 0 = R H 0 v 0 ∗ U f , h ( x ) = h ⁎ ( x ⁎ ) H 0 , M = H 0 B 0 σ f U f .

We obtain the dimensionless system:

(17) A 1 ζ Re v 0 ∂ u ∂ y = − ∂ p ∂ x + 1 A 2 ∂ 2 u ∂ y 2 − M 2 A 3 u ,

with:

(18) u = 1 at y = − h ( x ) ,

(19) u = − K at y = h ( x ) ,

where Re, M, and ζ are the Reynolds number Re = H 0 U f ρ f μ f , geometric parameter, and magnetic parameter. The ranges of the involved parameters in the RRC of the MHD hybrid nanofluid flow are given in Table 2. Also,

A 1 = ϕ 1 ρ 1 + ϕ 2 ρ 2 ρ f + ( 1 − ϕ 1 − ϕ 2 ) , A 2 = ( 1 − ϕ 1 − ϕ 2 ) , A 3 = ( 2 σ f + σ hp ) + 2 ϕ hnf ( σ hp − σ f ) ( 2 σ f + σ hp ) − 2 ϕ hnf ( σ hp − σ f ) .

Table 2

Dimensionless physical parameters with their respective ranges

Physical parameters	Range	Reference
Re (Reynolds number)	10 − 3 – 10	[2,14,15]
B (geometric parameter)	10 − 4 – 10 − 1	[2,14,15]
K	0.1–0.9	[2,19,14]
M (Hartmann number)	1–3.5	[30,33,48]
ϕ 1 , ϕ 2 (hybrid nanoparticle volume fraction)	0–0.06	[48,49,50]

2.3 Problem solution

Using the boundary conditions (18) and (19), we obtain the following solution:

(20) u ( x , y ) = 2 M A 2 ( − d 2 2 + d 1 2 ) A 3 × − d 1 2 d p d x h A 2 3 2 + A 2 d 2 2 d p d x h − d 1 ( 1 + K ) M A 3 − d 2 Csch [ d 2 h A 2 ] 2 d p d x ( Cosh [ d 2 h A 2 ] − Cosh [ d 1 h A 2 ] ) − ( ( 1 − K ) Cosh [ d 2 h A 2 ] + ( K − 1 ) Cosh [ d 1 h A 2 ] + ( 1 + K ) Sinh [ d 1 h A 2 ] A 3 ,

where

(21) d 1 = Re A 1 v 0 ξ and d 2 = d 1 2 A 2 M A 3 .

The flow rate can be computed from the following equation:

(22) λ = 1 2 ∫ − h h u d y .

To obtain the pressure gradient, Eq. (20) is integrated from − h to h and substituted in Eq. (22) to obtain

(23) d p d x = M A 3 ( d 2 Csch [ d 2 h A 2 ] ( ( − 1 + K ) Cosh [ d 2 h A 2 ] − ( − 1 + K ) Cosh [ d 1 h A 2 ] + ( 1 + K ) Sinh [ d 1 h A 2 ] ) − ( d 1 + d 1 K − d 2 2 λ ) A 2 − d 1 2 λ A 2 3 / 2 ) / ( 2 d 2 Coth [ d 2 h A 2 ] − 2 d 2 Cosh [ d 1 h A 2 ] Csch [ d 2 h A 2 ] − d 2 2 h A 2 + d 1 2 h A 2 3 / 2 ) . .

In Eq. (22), λ can be associated with the simple material balancing [19]:

(24) 2 U f H 0 λ = H f U f − H r U r ,

(25) C t = 2 λ + K ,

where C t represents the coating thickness, H 0 is the half nip space separation, and H r and H f are the reverse and forward roll fluid thicknesses.

Here, the pressure profile can be computed under the condition p = 0 at x → − ∞ :

(26) p 1 = ∫ − ∞ x t M A 3 ( d 2 Csch [ d 2 h A 2 ] ( ( − 1 + K ) Cosh [ d 2 h A 2 ] − ( − 1 + K ) Cosh [ d 1 h A 2 ] + ( 1 + K ) Sinh [ d 1 h A 2 ] ) − ( d 1 + d 1 K − d 2 2 λ ) A 2 − d 1 2 λ A 2 3 / 2 ) / ( 2 d 2 Coth [ d 2 h A 2 ] − 2 d 2 Cosh [ d 1 h A 2 ] Csch [ d 2 h A 2 ] − d 2 2 h A 2 + d 1 2 h A 2 3 / 2 ) d x .

At this point, it must be noted that the pressure expression in (26) by keeping ϕ 1 = ϕ 2 = M = 0 is similar to the one predicted for the simple Newtonian case by Greener and Middleman [19]. The next step involves determining λ , which further gives pressure distribution and coating thickness. Under the Swift−Stieber condition, pressure and pressure gradient disappear at a transition point x = x t , so the theory holds. By putting d p / d x = 0 , we need an explicit relationship between λ 1 and the transition point. However, from Eq. (23), using the pressure gradient equal to zero at x = x t , we obtain an implicit relationship as follows:

(27) λ = 1 A 2 ( d 2 2 − d 1 2 A 2 ) × d 2 Coth d 2 1 + ( x t ) 2 2 A 2 − d 2 K Coth d 2 1 + ( x t ) 2 2 A 2 − d 2 Cosh d 1 1 + ( x t ) 2 2 A 2 Csch d 2 1 + ( x t ) 2 2 A 2 + d 2 K Cosh d 1 1 + ( x t ) 2 2 A 2 Csch d 2 1 + ( x t ) 2 2 A 2 − d 2 Csch d 2 1 + ( x t ) 2 2 A 2 Sinh d 1 1 + ( x t ) 2 2 A 2 − d 2 K Csch d 2 1 + ( x t ) 2 2 A 2 Sinh d 1 1 + ( x t ) 2 2 A 2 + d 1 A 2 + d 1 K A 2 .

It is difficult to find the explicit relationship of x t under this constitutive model. Hence, we used the data points from implicit Eq. (27) to interpolate the required 10-degree polynomial representing the explicit expression of x t in terms of λ . After using this polynomial in Eq. (23) and integrating the resultant expression, we obtain a transcendental expression in λ . We utilized the Newton−Raphson root finding algorithm to find the values of λ .

3 Results and discussion

This study performs the modeling of the RRC technique to report the rheological features of the viscous hybrid nanofluid. The influence of the magnetic parameter M , the hybrid nanoparticle volume fractions ϕ 1 , ϕ 2 and the velocity ratio parameter K on flow and engineering quantities is discussed in this section. Figure 2 shows the comparison of the pressure profile derived from Eq. (26) by keeping ϕ 1 = ϕ 2 = M = 0 with the previous literature of the Newtonian case [19]. We observe a strong agreement between the results, which further verifies the solution methodology adopted and outcomes of the present study.

$Figure 2 Comparison graph of the present study (limiting values of ϕ 1 = ϕ 2 = M = 0 {\phi }_{1}={\phi }_{2}=M=0 ) with the Newtonian case [19].$

Figure 2

Comparison graph of the present study (limiting values of ϕ 1 = ϕ 2 = M = 0 ) with the Newtonian case [19].

Tables 3–5 display the variation in λ , X t and C t as functions of ϕ 1 , ϕ 2 , M, and K. In Table 3, when the hybrid nanoparticle volume fractions ϕ 1 , ϕ 2 increase, the coating thickness C t and the transition point X t reduce as a result of the decreased flow rate λ with M = 1 and K = Re = v = ε = 0.1 . The higher volume fraction of the nanomaterials results in modified liquid viscosity, and hence, a greater pressure gradient is detected (Figure 3), which leads to a decrease in the final coating thickness. Similarly, an increase in M in Table 4 results in decreased λ , X t , and C t with ϕ 1 , ϕ 2 = 0.06 , and K = Re = v = ε = 0.1 . Enhancing the Hartmann number (M) results in invoking of the Lorentz force, which resists the coating flow, yielding a lower final coating thickness, which can positively influence the coating procedure efficiency. Meanwhile, Table 5 also shows a decreasing trend in all three quantities as a function of K.

Table 3

Influence of the hybrid nanoparticle volume fractions ( ϕ 1 , ϕ 2 )on the flow rate ( λ ), transition point ( X t ), and the final coating thickness ( C t )

ϕ 1 , ϕ 2	λ	X t	C t
0.0001	0.382332	0.355907	1.62933
0.01	0.381394	0.355124	1.62558
0.02	0.38068	0.354529	1.62272
0.03	0.380193	0.354122	1.62077
0.04	0.379926	0.3539	1.6197
0.05	0.379873	0.353856	1.61949
0.06	0.379743	0.352967	161,885

Table 4

Influence of the Hartmann number (M) on the flow rate ( λ ), transition point ( X t ), and the final coating thickness ( C t )

M	λ	X t	C t
1	0.38003	0.353987	1.62012
1.5	0.332184	0.314034	1.42873
2	0.297557	0.284723	1.29023
2.5	0.271241	0.261986	1.18496
3	0.250485	0.243681	1.10194
3.5	0.233636	0.228546	1.03454

Table 5

Influence of the velocity ratio (K) on the flow rate ( λ ), transition point ( X t ), and the final coating thickness ( C t )

K	λ	X t	C t
0.1	0.38003	0.353987	1.62012
0.6	0.171109	0.153026	1.28444
0.9	0.0435058	0.0381772	1.07402

$Figure 3 Influence of the hybrid nanoparticle volume fraction on the pressure gradient curve.$

Figure 3

Influence of the hybrid nanoparticle volume fraction on the pressure gradient curve.

The pressure gradient, represented against the axial direction x for increasing ϕ 1 , ϕ 2 , is shown in Figure 3. The plot is divided into three different regions, an upstream and downstream region of d p d x > 0 (indicating resistive flow nature) and the mid-nip region with d p d x < 0 (indicating assistive flow nature). Enhancement in the hybrid nanomaterial volume fraction gives a high assistance or resistance flow nature (high magnitude of dp/dx). The high volume fraction of the nanomaterials results in modified liquid viscosity, and hence, a greater pressure gradient is detected, indicating lower velocity, which leads to a reduced final coating thickness (Table 3). In Figure 4, it can be seen that with increasing values of the Hartmann number M , the resistance in flow is detected, which declines the pressure gradient (magnitude). Figure 5 shows the decrease in the pressure gradient curve due to an increment in the velocity ratio K.

Figure 4

Influence of the Hartmann number on the pressure gradient curve.

Figure 5

Influence of the velocity ratio on the pressure gradient curve.

In Figure 6, the pressure p is plotted against the axial direction x . When the hybrid nanoparticle volume fractions ϕ 1 , ϕ 2 increase, the pressure profile increases throughout the nip region. The high volume fraction of the nanomaterials results in modified liquid viscosity, and hence, a greater pressure profile is detected. This result is important in coating phenomena, which control the coating thickness. The behavior of M and K is opposite to that of ϕ 1 , ϕ 2 . As shown in Figures 7 and 8, the lower pressure is detected due to enhanced M and K, which also gives a lower flow rate and final coating thickness.

$Figure 6 Influence of the hybrid nanoparticle volume fraction on the pressure curve.$

Figure 6

Influence of the hybrid nanoparticle volume fraction on the pressure curve.

Figure 7

Influence of the Hartmann number on the pressure curve.

Figure 8

Influence of the velocity ratio on the pressure gradient curve.

Figures 9–11 depict the shear stress variation in the nip region for the hybrid nanomaterial volume fractions ϕ 1 , ϕ 2 , Hartmann number (M), and velocity ratio (K), respectively. Generally, the trend suggests higher shear stress in the vicinity of the applicator roll ((y = −h)) as compared to the metering roll (y = h). As shown in Figure 9, enhancing the hybrid nanomaterial volume fraction results in increased shear stress in the nip region. A higher nanomaterial volume fraction results in modified polymer viscosity, which gives increased shear stress. This result helps in controlling the final coating thickness, as compared to the Newtonian case [19], where only the velocity ratio (geometrical parameter) is significant. Figure 10 suggests greater shear stress near the applicator roll due to an increase in M, whereas the opposite trend is detected near the metering roll. Figure 11 shows higher shear stress at a velocity ratio K. Figure 12 shows the 3D contour, which shows the velocity profile behavior of the hybrid nanomaterial polymer under applied MHD. Figure 13(a)−(f) shows the streamline pattern of the hybrid nanomaterial polymer. Streamlines project the imaginary suspended particle path of the hybrid nanomaterial polymer and transports along with it, which is a fixed path in the case of steady flow. Streamlines squeezing together or spreading out correspond to areas with higher or lower relative fluid velocities.

$Figure 9 Influence of the Hybrid nanoparticle volume fraction on the Shear stress.$

Figure 9

Influence of the Hybrid nanoparticle volume fraction on the Shear stress.

Figure 10

Influence of the Hartmann number on shear stress.

Figure 11

Influence of the velocity ratio on shear stress.

Figure 12

3D velocity profile contour of the hybrid nanomaterial polymer.

$Figure 13 Streamline visualization: (a) and (b) for the hybrid nanoparticle volume fraction, (c) and (d) Hartmann number, and (e) and (f) velocity ratio.$

Figure 13

Streamline visualization: (a) and (b) for the hybrid nanoparticle volume fraction, (c) and (d) Hartmann number, and (e) and (f) velocity ratio.

4 Conclusions

The RRC system is studied to determine the role of the MHD hybrid nanomaterial coating on a moving sheet. A mathematical representation is created and organized simply by using the lubrication approximation technique. The following main results are summarized as follows:

Both the pressure gradient and pressure decrease with an increase in the magnetic parameter M and velocity ratio K .
As the hybrid nanoparticle volume fractions ϕ 1 , ϕ 2 increase, the coating liquid viscosity changes, which in turn enhances the pressure gradient and pressure profile, especially in the nip region, leading to a lower velocity profile, which is important in controlling the final coating thickness.
Shear stress enhances in the case of the hybrid nanomaterial polymer with MHD and velocity ratio, which controls the final coating thickness as compared to the Newtonian case [19], where only the velocity ratio (geometrical parameter) is significant.
Coating thickness declines under the influence of the hybrid nanoparticle volume fraction and magnetic parameter, which may help in achieving an efficient coating process and protecting the substrate's life.
This investigation provides the theoretical framework role of MHD hybrid nanofluid polymer in the RRC process, where the adopted model delivers a suitable estimation of the nanomaterial properties. In the future, the work can be extended and verified with experimentations, and nanomaterial rheology can be studied in non-Newtonian fluids.
The theoretical RRC model of the hybrid nanomaterial with applied MHD presents precise control of the coating film thickness and nanoparticle distribution, helping in defect-free, uniform coatings, which are crucial in coating industry of electronics, thermal interfaces, and biomedical films. The model also assists in optimizing the roll speed and magnetic field for efficient, high-performance industrial coatings.

Funding information: The authors state no funding involved.
Author contributions: Sabeeh Khaliq: conceptualization, methodology, and writing – original draft preparation. Nida Shaheen: software, visualization, and writing – original draft preparation. Moin-ud-Din Junjua: investigation and writing – reviewing and editing. Mehmood Ahmad: software, methodology and writing – reviewing and editing. Zaheer Abbas: supervision and validation. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2025-03-18

Revised: 2025-05-11

Accepted: 2025-06-06

Published Online: 2025-07-17

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2025-0173

Keywords for this article

hybrid nanomaterial; reverse roll coating process; MHD; coating thickness; lubrication theory

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