Variable viscosity characteristics of two-layered isothermal calendering phenomena with non-Newtonian fluids

Abstract

Two-layer calendering using non-Newtonian fluids is used frequently in different industries to improve the working and the quality of the products. In this article, two-layer calendering of incompressible non-Newtonian fluids, that is, couple stress fluid and viscoplastic fluid in the upper and lower layer respectively is investigated. For the simplified governing equations, Lubrication approximation theory is applied. By the use of no-slip condition and non-dimensional variables, the exact solution for pressure gradient, pressure, and velocity profiles for both layers of fluid is calculated. The parameters are important from an engineering point of view like roll separating force, power delivered by rollers, torque acting by each roller, and adiabatic temperature are also calculated analytically. Additionally, the results of a couple stress parameter, viscoplastic Casson parameter, and viscosity ratios on all the parameters are also discussed and investigated in detail. Moreover, all the effects investigated are also verified with the literature results of single-layer calendering of non-Newtonian fluids by using large numeric values for the couple stress parameter and viscoplastic Casson parameter.

Introduction

Calendering is a process of making sheets of different thicknesses on an industrial scale by inserting a flowing material between two or more co-rotating calenders. Nowadays, this process is applied commonly in industries for making paper, leather, textile fabric, plastics, and polyvinyl sheets. Moreover, calendering is also used in foodstuff industries for making shapes of pasta, biscuits and breads, etc. In calendering, rolling calenders work to solidify and compress the induced material into desired thickness which can be uni-layered or multi-layered. The advantage of calendering is that the calendered material is obtained with the demanded surface texture, thickness, and finishing. So, the calendering process is of great importance and a matter of study for researchers due to its advantages.

Osswald and Menges ¹ overviewed in their book that Edwin M. Chaffee who worked in the rubber industry and Charles Goodyear were the first engineers who used calendering process for making a rubber sheet in 1836. First of all, Gaskell ² inspected that the roller’s width is much less than the roller’s radius. He applied this assumption to Newtonian and Bingham plastic fluid models. Many researchers worked on Gaskell’s model for different fluids and non-isothermal effects. Power law fluids in calendering were deeply studied by McKelvey ³ and Brazinsky et al. ⁴ Middleman ⁵ collected all the work done on calendering till 1977 in his book. Polymer processing was discussed theoretically by Tadmor and Gogos ⁶ in their book. The calendering process using slip boundary conditions was discussed by Sofou and Mitsoulis. ⁷ Moreover, the numerical study of the calendering process for viscoelastic fluid was also studied by Mitsoulis and Sofou. ⁸ The effect of temperature on different parameters involved in the calendering process was discussed by Arcos et al. ⁹ Arcos et al. ¹⁰ further discussed the Newtonian fluid having viscosity dependent on temperature in the calendering process. Different non-Newtonian fluids in the calendering process were discussed by Ali et al., ¹¹ Sajid et al.,^12,13 and Javed et al. ¹⁴ For the viscous fluid model, the numerical solution for energy and the effect of temperature in calendering was studied by Abbas and Khaliq. ¹⁵ In no-isothermal calendering process, the consequences of heat conduction on third order fluid by applying the finite difference technique were also discussed by Khaliq and Abbas. ¹⁶ In non-isothermal calendering, the numerical study of exiting sheet thickness using micropolar Cason fluid was studied Abbas and Khaliq. ¹⁷ They discussed the effect of Cason and coupling number on different parameters involved using lubrication approximation theory.

Different types of fluid both Newtonian and non-Newtonian have been used in the calendering process by researchers till now. Non-Newtonian fluids have great importance as compared to Newtonian fluids due to their practical applications in industries. Couple stress fluid and viscoplastic Casson fluid are important non-Newtonian fluids. Casson fluid is a shear-thinning fluid that has a yield stress limit under which flow does not occur. Some real life examples of Casson fluid are human blood, soup, tomato sauces, honey, jellies, etc. The flow of Casson fluid in the calendering process was discussed by Eldabe et al. ¹⁸ Casson fluid flow for the oscillatory and steady blood flow was discussed by Boyd et al. ¹⁹ The model of Casson fluid is considered more appropriate to fit rheological data as compared to other viscoplastic fluid models.^20,21 The slip flow on MHD Casson fluid was discussed by Prasad et al. ²² They discussed the impact of magnetic field on different properties of the fluid. In the roll coating process, the micropolar Casson fluid and the effect of viscoplastic parameter on different variables were discussed by Abbas and Khaliq. ²³ Moreover, they ¹⁷ extended their work for non-isothermal calendering of micropolar Casson fluid and discussed the effect of micro rotation and viscoplastic parameter on various quantities of interest. Couple stress fluid is a unique kind of non-Newtonian fluid that is used to explain the behavior of different suspension fluids, lubricants and blood, etc. The initial work on a couple stress fluid was done by Stokes and Stokes. ²⁴ He discussed that the couple stress fluid has a couple stresses and body couples. Moreover, it has a large viscosity in general. Stokes stated that a couple stresses effects were significant in liquids with large molecules. The existence of couple stresses causes a decrease in the velocity and other parameters of interest which was discussed by Devakar et al., ²⁵ Ahmad et al., ²⁶ and Jangili et al. ²⁷ The hydromagnetic effect on the two-layer flow of couple stress and viscous fluids in an inclined channel was discussed by Abbas et al. ²⁸ They discussed the MHD and heat transfer effect on velocity and other parameters of interest. Ali et al. ²⁹ investigated the nonisothermal calendering using couple stress fluid. They discussed the effect of a couple stress parameter on velocity, pressure, and other variables. Zafar et al. ³⁰ discussed the roll coating process by using a couple stress fluid. Bitla and Sitotaw ³¹ discussed the effect of slip and magnetohydrodynamics on the flow of two immiscible fluids, that is, Jeffry fluid and Couple stress fluid in a porous channel.

The multi-layer flow of Newtonian and non-Newtonian fluids using different geometries is of great importance for researchers due to their applications. Multi-layer flow has great importance in process industry and nuclear industry. Specifically, two-layer calendering finds its application in polymer processing, the textile industry, the paper and packaging industry, the medical industry, the food industry, and for making of thin film transistors. It helps to produce high-quality fabrics, paper with smooth finishing, and multi-layer coatings for the packaging of food which can have many properties like waterproof, incombustible and free of moisture, etc. In two-layer calendering, separate extruders are used to extrude different types of fluid into a single sheet which is then used to produce a multi-layer sheet of homogeneous thickness. Then this sheet is inserted among two rotating rollers to produce a sheet of homogeneous thickness. Different studies have been made for two-layer flow in different channels. Hannachi and Mitsoulis ³² were the first researchers who discussed multi-layer flow in the calendering process using LAT. They discussed the different numeric consequences by using various assumptions and arrangements of fluids. The multi-layer calendering process using metal strips for Newtonian and power law models was further discussed by Calcagno et al. ³³ Ilyas et al.^34,35 discussed the two-layer isothermal calendering using Newtonian and non-Newtonian fluids with different viscosities. The effect of different viscosity ratios of different fluids on different operating parameters was discussed in detail and mathematical expressions were computed analytically.

Despite of these few above-mentioned results, no research is made in two-layered calendering using non-Newtonian fluids. The main aim of this article is to extend the study of the multi-layer calendering process using non-Newtonian fluids, that is, couple stress fluid and viscoplastic Casson fluid in the first and second layers respectively having different viscosities. LAT is applied for simplification and the effects of viscosity ratios, viscoplastic fluid parameter, and couple stress fluid parameter are investigated on different operating variables of engineering importance like pressure gradient, maximum pressure, roll separating force, power input, the torque acting on both rollers and adiabatic temperature, etc.

Governing equations

The fluid flow occurs at extremely low Reynolds numbers in the calendering process. Due to this creeping flow approximation is applied and inertial forces are neglected as compared to viscous forces. The flow is considered as steady. By ignoring body forces and temperature effect, the governing equations for the incompressible fluid with the above assumptions are given by Ilyas et al.^34,35 as

$$\nabla ·\mathbf{V}=0,$$

(1)

$$0=-\nabla p+\mathrm{div}\mathbf{\tau },$$

(2)

where $\mathbf{V}=\mathbf{V}(u,v,w)$ is the velocity field, $p$ the pressure, and $\mathbf{\tau }$ is the extra stress tensor for non-Newtonian fluids which are Couple stress ³¹ and Casson ³⁴ fluids respectively given as

$${\mathbf{\tau }}_1=({\mu }_1\frac{\partial u_1}{\partial y}-\eta \frac{{\partial }^{3}u}{\partial y^3}){\mathbf{D}}_1,$$

(3)

$${\mathbf{\tau }}_2={\mu }_2(1+\frac{1}{\beta })\frac{\partial u_2}{\partial y}{\mathbf{D}}_2,$$

(4)

where ${\mu }_1$ and ${\mu }_2$ are the viscosities of couple stress and viscoplastic fluids, $\eta$ is a material constant of the couple stress fluid, and $\beta =\frac{{\mu }_B\sqrt{2{\pi }_c}}{p_y}$ is the constant of the viscoplastic Casson fluid with ${\mu }_B=$ plastic dynamic viscosity, ${\pi }_c=$ critical value of the product of deformation rate and $p_y=$ fluid yield stress. The first Rivlin-Erickson Tensor is ${\mathbf{D}}_1=\left(\mathrm{grad}\mathbf{V}\right)+{\left(\mathrm{grad}\mathbf{V}\right)}^t.$

Problem formulation

Suppose a steady, laminar, incompressible, immiscible, and viscous flow of two isothermal non-Newtonian fluids having distinct viscosities moves among two identical rolling cylinders with the same radius to make a multi-layer sheet of uniform thickness (see Figure 1).

Figure 1.

Representation of two-layer flow of non-Newtonian fluids in the nip region between the two rolling calenders.

The linear velocity $U=\omega R$ is the same for both rollers where ω represents the angular velocity. The upper roller rotates counterclockwise while the lower roller rotates in clockwise direction. The minimum gap between the rollers is 2 $H_0$ and this is the point $x_0$ where the fluid comes in connection with the rollers while $\kappa$ is the exiting sheet thickness where the gap between the rollers is $2H.$ The radius of the rollers is much greater than the distance between the rollers, that is, Assume that the upper half of the space between the rollers is filled by the couple stress fluid while the lower half is filled by the viscoplastic Casson fluid. Moreover, suppose that the fluids do not stretch out of the space between the rollers. So the flow is considered two-dimensional parallel to the x-axis and y-axis is orthogonal to the flow. The velocity profile becomes

$$\mathbf{V}=[u(x,y),v(x,y),0].$$

(5)

According to the above suppositions, equations (1) and (2) transformed as follow

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(6)

$$-\frac{\partial p}{\partial x}+\frac{\partial {\tau }_{\mathrm{xx}}}{\partial x}+\frac{\partial {\tau }_{\mathrm{xy}}}{\partial y}=0,$$

(7)

$$-\frac{\partial p}{\partial y}+\frac{\partial {\tau }_{\mathrm{xx}}}{\partial x}+\frac{\partial {\tau }_{\mathrm{xy}}}{\partial y}=0,$$

(8)

where ${\tau }_{\mathrm{xx}},{\tau }_{\mathrm{xy}},{\tau }_{\mathrm{yx}}$ and ${\mathbf{\tau }}_{\mathrm{yy}}$ represent the components of $\mathbf{\tau }.$

The creation of pressure and chain orientation in the gap between the rollers has a significant effect on polymer processing and the calendering process. The space between the rollers is known as the nip region. In the nip region and by going to any side of this with a very small distance $x_0,$ the roller surfaces are almost parallel and $H_0<<R.$ So lubrication approximation theory (LAT) can be applied as

$$\begin{array}{c}\frac{\partial }{\partial x}<<\frac{\partial }{\partial y},\hfill \\ v<<u,p=p(x).\hfill \end{array}$$

(9)

By using equation (9), equations (6)–(8) become

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(10)

$$-\frac{\mathrm{dp}}{\mathrm{dx}}+\frac{\partial {\tau }_{\mathrm{xy}}}{\partial y}=0.$$

(11)

So the momentum equations for both fluids take the form

$${\mu }_1\frac{{\partial }^2{u}_1}{\partial y^2}-\eta \frac{{\partial }^{4}u}{\partial y^4}=\frac{\mathrm{dp}}{\mathrm{dx}},$$

(12)

$${\mu }_2(1+\frac{1}{\beta })\frac{{\partial }^2{u}_2}{\partial y^2}=\frac{\mathrm{dp}}{\mathrm{dx}},$$

(13)

where the boundary conditions involved in this model are given as

$$\begin{array}{c}{u}_1=u_2\mathrm{at}y=0,\hfill \\ u_1=U\mathrm{at}y=h(x),\hfill \\ u_2=U\mathrm{at}y=-h(x),\hfill \\ \frac{{\partial }^2{u}_1}{\partial y^2}=0\mathrm{at}y=h(x),\hfill \\ \frac{{\partial }^2{u}_1}{\partial y^2}=0\mathrm{at}y=0,\hfill \\ {\mu }_1\frac{\partial u_1}{\partial y}-\eta \frac{{\partial }^3{u}_1}{\partial y^3}={\mu }_2(1+\frac{1}{\beta })\frac{\partial u_2}{\partial y}\mathrm{at}y=0.\hfill \end{array}\}$$

(14)

It is clear from the first boundary condition that due to no slip condition at the interface, both fluids are moving with same velocity while due to no slip condition at the surface of rollers, both fluids at the boundary move with the same velocity as that of rollers, that is, $U.$ Moreover, the shear stresses for both the fluids at the interface should be equal.

By applying integration on equations (12) and (13) and boundary conditions from equation (14), velocity profiles for both layers become

$$\begin{array}{c}{u}_1=U+\frac{1}{2{\mu }_1}\left(\frac{\mathrm{dp}}{\mathrm{dx}}\right)\hfill \\ \left[\begin{array}{c}{y}^2+h(x)(\frac{2{\mu }_1\beta }{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}-1)y-{\left[h\left(x\right)\right]}^2\left(\frac{2{\mu }_1\beta }{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right)\hfill \\ \multicolumn{1}{c}{+\frac{2\eta }{{\mu }_1}(1-\cosh \left(\sqrt{\frac{{\mu }_1}{\eta }}y\right)+\sinh \left(\sqrt{\frac{{\mu }_1}{\eta }}y\right)\tanh \left(\frac{h}{2}\sqrt{\frac{{\mu }_1}{\eta }}\right))}\end{array}\right],\hfill \end{array}$$

(15)

and

$$\begin{array}{c}\multicolumn{1}{c}{u_2=U+\frac{1}{2{\mu }_2}\left(\frac{\beta }{1+\beta }\right)\left(\frac{\mathrm{dp}}{\mathrm{dx}}\right)[y^2+h\left(x\right)\left(\frac{\beta ({\mu }_1-{\mu }_2)-{\mu }_2}{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right)y}\\ -\left(\frac{2{\mu }_2(1+\beta )}{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right){\left[h\left(x\right)\right]}^2],\hfill \end{array}$$

(16)

where $h\left(x\right)=H_0+R-\sqrt{R^2-x^2}.$

For the sake of simplification, the flow is considered to be limited as $x<<R.$ The approximated form of the above expression takes the form

$$h\left(x\right)=H_0(1+\frac{x^2}{2R{H}_0}).$$

(17)

Dimensionless variables

Now defining the non-dimensional variables as

$$\begin{array}{c}\multicolumn{1}{c}{x\prime =\frac{x}{\sqrt{2R{H}_0}},y\prime =\frac{y}{H_0},u{\prime }_1=\frac{u_1}{U},u{\prime }_2=\frac{u_2}{U},}\\ \multicolumn{1}{c}{h\prime (x\prime )=\frac{h\left(x\right)}{H_0}=1+x{\prime }^2,\alpha =\sqrt{\frac{{\mu }_1}{\eta }}h\prime ,}\\ \multicolumn{1}{c}{p{\prime }_1=\sqrt{\frac{H_0}{2R}}\frac{p{H}_0}{{\mu }_{1}U},p{\prime }_2=\sqrt{\frac{H_0}{2R}}\frac{p{H}_0}{{\mu }_{2}U},}\end{array}$$

(18)

Equations (15) and (16) and boundary conditions in dimensionless form by ignoring (’) sign become

$$\begin{array}{c}{u}_1=1+\frac{H_0}{2}\sqrt{\frac{H_0}{2R}}\frac{d{p}_1}{\mathrm{dx}}\hfill \\ \left[\begin{array}{c}{y}^2+(\frac{2{\mu }_1\beta }{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}-1)(1+x^2)y-\left(\frac{2{\mu }_1\beta }{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right){(1+x^2)}^2\hfill \\ +\frac{2{(1+x^2)}^2}{{\alpha }^2}(1-\cosh \left(\frac{y}{h}\alpha \right)+\sinh \left(\frac{y}{h}\alpha \right)\tanh \left(\frac{\alpha }{2}\right))\hfill \end{array}\right],\hfill \end{array}$$

(19)

$$\begin{array}{c}{u}_2=1+\frac{H_0}{2}\sqrt{\frac{H_0}{2R}}\frac{d{p}_2}{\mathrm{dx}}\left(\frac{\beta }{1+\beta }\right)[y^2+\left(\frac{\beta ({\mu }_1-{\mu }_2)-{\mu }_2}{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right)\hfill \\ (1+x^2)y-\left(\frac{2{\mu }_2(1+\beta )}{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right){(1+x^2)}^2],\hfill \end{array}$$

(20)

and

$$\begin{array}{c}{u}_1=u_2\mathrm{at}y=0,\hfill \\ u_1=1\mathrm{at}y=h=(1+x^2),\hfill \\ u_2=1\mathrm{at}y=-h=-(1+x^2),\hfill \\ \frac{{\partial }^2{u}_1}{\partial y^2}=0\mathrm{at}y=h=(1+x^2),\hfill \\ \frac{{\partial }^2{u}_1}{\partial y^2}=0\mathrm{at}y=0,\hfill \\ {\mu }_1\frac{\partial u_1}{\partial y}-\eta \frac{{\partial }^3{u}_1}{\partial y^3}={\mu }_2(1+\frac{1}{\beta })\frac{\partial u_2}{\partial y}\mathrm{at}y=0.\hfill \end{array}\}.$$

(21)

The flow rate $Q_1$ and $Q_2$ for the flow in both layers respectively are given as

$$\begin{array}{c}{Q}_1=\underset{0}{\overset{h\left(x\right)}{\int }}{u}_1\mathrm{dy},\hfill \\ Q_2=\underset{-h\left(x\right)}{\overset{0}{\int }}{u}_2\mathrm{dy}.\hfill \end{array}$$

(22)

To get the total flow rate, the flow rate for both layers is added. The final expression becomes

$$\begin{array}{c}Q=2\mathrm{Uh}\left(x\right)-\hfill \\ \left(\frac{\mathrm{dp}}{\mathrm{dx}}\right)[\frac{h^3\left(x\right)}{12}(\frac{1}{{\mu }_1}+\frac{\beta }{{\mu }_2(1+\beta )}+\frac{12\beta }{{\mu }_2+({\mu }_1+{\mu }_2)})+\frac{2{\eta }^{3/2}}{{\mu }_1^{5/2}}\tanh \left(\frac{h\left(x\right)}{2}\sqrt{\frac{{\mu }_1}{\eta }}\right)-\frac{\eta }{{\mu }_1^2}h\left(x\right)].\hfill \end{array}$$

(23)

Suppose that at the thickness $2H,$ the sheet detaches from the roller with speed $U.$ For this assumption, the total flow rate becomes

$$Q=2\mathrm{UH}.$$

(24)

The dimensionless form of the distance $h\left(x\right),$ for $x\prime =\kappa$ and $h\left(x\right)=H$ becomes

$$\frac{H}{H_0}=1+{\kappa }^2.$$

(25)

By putting this value, equation (24) takes the form as

$$Q=2U{H}_0(1+{\kappa }^2).$$

(26)

By substituting equations (26) into (23), the expression for pressure gradient becomes

$$\frac{\mathrm{dp}}{\mathrm{dx}}=\frac{24C{\mu }_1{\mu }_{2}U}{{[h(x)]}^2}\sqrt{\frac{2R}{H_0}}(1-\frac{H_0(1+{\kappa }^2)}{h\left(x\right)}),$$

(27)

where C is a constant defined as

$$C=\left(\frac{{\alpha }^3(1+\beta )({\mu }_2+\beta ({\mu }_1+{\mu }_2))}{\left[\begin{array}{c}{\alpha }^3[{\mu }_2^2(1+2\beta )+{\beta }^2({\mu }_1^2+{\mu }_2^2)+12{\mu }_1{\mu }_2\beta (1+\beta )]+12{\mu }_2(1+\beta )(\tanh \left(\frac{\alpha }{2}\right)-\alpha )\hfill \\ \multicolumn{1}{c}{({\mu }_2+\beta ({\mu }_1+{\mu }_2))}\end{array}\right]}\right).$$

(28)

The non-dimensional forms of the pressure gradient for both layers become

$$\frac{d{p}_1}{\mathrm{dx}}=\frac{24C{\mu }_2}{H_0}\sqrt{\frac{2R}{H_0}}\left(\frac{(x^2-{\kappa }^2)}{{(1+x^2)}^3}\right),$$

(29)

and

$$\frac{d{p}_2}{\mathrm{dx}}=\frac{24C{\mu }_1}{H_0}\sqrt{\frac{2R}{H_0}}\left(\frac{(x^2-{\kappa }^2)}{{(1+x^2)}^3}\right).$$

(30)

Equations (29) and (30) show that the pressure gradient is zero for both the fluid layers at $x=\pm \kappa .$

$x=\kappa$ is the detachment point of the sheet from the rollers and $x=-\kappa$ is the point where the pressure has maximum value. By substituting the equations (29) and (30) into (19) and (20) and ignoring the (’), the final form of velocity expressions for couple stress fluid and Casson fluid becomes

$$u_1=1+12{\mu }_{2}C\left(\frac{(x^2-{\kappa }^2)}{{(1+x^2)}^3}\right)\left[\begin{array}{c}{y}^2+\left(\frac{\beta ({\mu }_1-{\mu }_2)-{\mu }_2}{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right)(1+x^2)y-\left(\frac{2{\mu }_1\beta }{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right){(1+x^2)}^2\hfill \\ +\frac{2{(1+x^2)}^2}{{\alpha }^2}(1-\cosh \left(\frac{y}{h}\alpha \right)+\sinh \left(\frac{y}{h}\alpha \right)\tanh \left(\frac{\alpha }{2}\right))\hfill \end{array}\right],$$

(31)

and

$$u_2=1+12{\mu }_{1}C\left(\frac{\beta }{1+\beta }\right)\left(\frac{(x^2-{\kappa }^2)}{{(1+x^2)}^3}\right)\left[\begin{array}{c}{y}^2+\left(\frac{\beta ({\mu }_1-{\mu }_2)-{\mu }_2}{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right)(1+x^2)y-\hfill \\ \left(\frac{2{\mu }_2(1+\beta )}{{\mu }_2+\beta ({\mu }_1+{\mu }_2)}\right){(1+x^2)}^2\hfill \end{array}\right].$$

(32)

Equations (29) and (30) are integrated and by using the conditions $p_1\left(\kappa \right)=0$ and $p_2\left(\kappa \right)=0,$ the pressure distribution among both layers of fluid is given as

$$p_1\left(x\right)=3C{\mu }_2\sqrt{\frac{2R}{H_0}}[\frac{x^2(1-3{\kappa }^2)-1-5{\kappa }^2}{{(1+x^2)}^2}x+\frac{1+3{\kappa }^2}{1+{\kappa }^2}\kappa +(1-3{\kappa }^2)({\tan }^{-1}x-{\tan }^{-1}\kappa )],$$

(33)

and

$$p_2\left(x\right)=3C{\mu }_1\sqrt{\frac{2R}{H_0}}[\frac{x^2(1-3{\kappa }^2)-1-5{\kappa }^2}{{(1+x^2)}^2}x+\frac{1+3{\kappa }^2}{1+{\kappa }^2}\kappa +(1-3{\kappa }^2)({\tan }^{-1}x-{\tan }^{-1}\kappa )].$$

(34)

The maximum pressure at $x=-\kappa$ for both fluids becomes

$$p_1(-\kappa )=6C{\mu }_2\sqrt{\frac{2R}{H_0}}(\frac{1+3{\kappa }^2}{1+{\kappa }^2}\kappa -(1-3{\kappa }^2){\tan }^{-1}\kappa ),$$

(35)

and

$$p_2(-\kappa )=6C{\mu }_1\sqrt{\frac{2R}{H_0}}(\frac{1+3{\kappa }^2}{1+{\kappa }^2}\kappa -(1-3{\kappa }^2){\tan }^{-1}\kappa ).$$

(36)

The limiting values of both pressures at $x\to -\infty$ are

$$p_1(x\to -\infty )=3C{\mu }_2\sqrt{\frac{2R}{H_0}}(\frac{1+3{\kappa }^2}{1+{\kappa }^2}\kappa -(1-3{\kappa }^2)(\frac{\pi }{2}+{\tan }^{-1}\kappa )),$$

(37)

and

$$p_2(x\to -\infty )=3C{\mu }_1\sqrt{\frac{2R}{H_0}}(\frac{1+3{\kappa }^2}{1+{\kappa }^2}\kappa -(1-3{\kappa }^2)(\frac{\pi }{2}+{\tan }^{-1}\kappa )).$$

(38)

The $\kappa$ in terms of pressure can be obtained by applying pressure at upstream of the gap between the rollers, but such pressure is not applied in the calendering process. So, the appropriate supposition is

$$p_1(x\to -\infty )=0\mathrm{and}{p}_2(x\to -\infty )=0.$$

(39)

By using the above assumption, equations (37) and (38) give

$$\frac{1+3{\kappa }^2}{1+{\kappa }^2}\kappa -(1-3{\kappa }^2)(\frac{\pi }{2}+{\tan }^{-1}\kappa )=0.$$

(40)

It is clear that the above expression is non-linear in $\kappa$ . That’s why its exact value can be obtained with great difficulty. So, its numerical value can be obtained by applying the Regula-Falsi method with accuracy defined as ${10}^{-10}$ to get

$$\kappa \cong 0.4751.$$

(41)

At the entering point $x_0,$ the pressure for both fluids can be calculated by putting $x=x_0$ in equations (33) and (34). By using conditions, that is, $p_1\left(x_0\right)=0$ and $p_2\left(x_0\right)=0$ and random values of $\kappa ,$ the related entering points and the exiting sheet thickness ratios can be obtained as given in Table 1.

Table 1.

Dimensionless detachment point and related entrance location.

Detachment point ( $\kappa$ )	Entrance location ( $x_0$ )	Exiting sheet thickness ( $H/H_0$ )
0.4751	−22.8	1.22
0.45	−2.16	1.20
0.40	−1.30	1.16
0.35	−0.95	1.12
0.33	−0.87	1.10
0.30	−0.73	1.09
0.25	−0.56	1.06

Operating variables

All the important parameters of interest, that is, roll separating force, power input, torque at the upper and lower roller, and adiabatic temperature can be obtained from the expressions of velocities, pressure, and pressure gradient. A numerical Simpson 3/8 method is employed to obtain the following quantities.

Roll-separating force

The expression for roll separating force is obtained by

$$\frac{F}{W}=\underset{x_0}{\overset{\kappa }{\int }}p\left(x\right)\mathrm{dx},$$

(42)

where W represents the width of the roller.

By solving equations (32) and (33) in dimensionless form, the expression of force is given as

$$\begin{array}{c}\frac{F}{W}=\frac{6C{\mu }_1{\mu }_{2}UR}{H_0}[\frac{1+2{\kappa }^2+3{\kappa }^4}{1+{\kappa }^2}-\frac{1+{\kappa }^2}{1+x_0^2}-\frac{\kappa \left(1+3{\kappa }^2\right)}{1+{\kappa }^2}\hfill \\ x_0+x_0(1-3{\kappa }^2)({\tan }^{-1}\kappa -{\tan }^{-1}{x}_0)].\hfill \end{array}$$

(43)

Power input

Power input for both the fluid layers can be calculated by the following expression

$$p_{\mathrm{in}}=2\sqrt{2}\underset{x_0}{\overset{\kappa }{\int }}\frac{\mathrm{dp}}{\mathrm{dx}}(1+x^2)\mathrm{dx}$$

(44)

where $p_{\mathrm{in}}=\frac{{p\prime }_{\mathrm{in}}}{\mu W{U}^2}$ is the dimensionless power.

The final expression of power becomes

$$\begin{array}{c}\multicolumn{1}{c}{p_{\mathrm{in}}=24C{\mu }_1{\mu }_{2}W{U}^2\sqrt{\frac{2R}{H_0}}}\\ [(1-{\kappa }^2)({\tan }^{-1}\kappa -{\tan }^{-1}{x}_0)-\kappa +\frac{1+{\kappa }^2}{1+x_0^2}{x}_0].\hfill \end{array}$$

(45)

Torque

The torque on the upper and lower roller can be calculated as

$$T_U=W\underset{x_0}{\overset{\kappa }{\int }}{{\tau }_{\mathrm{xy}}|}_{Y_U}{R}_U\mathrm{dx},$$

(46)

and

$$T_L=W\underset{x_0}{\overset{\kappa }{\int }}{{\tau }_{\mathrm{xy}}|}_{Y_L}{R}_L\mathrm{dx}.$$

(47)

By using the above expressions, the final torque on both rollers is calculated as

$$\begin{array}{c}{T}_U=\hfill \\ \left[\begin{array}{c}6C{\mu }_1{\mu }_2\mathrm{WUR}((1-{\kappa }^2)({\tan }^{-1}\kappa -{\tan }^{-1}{x}_0)-\kappa +\frac{1+{\kappa }^2}{1+x_0^2}{x}_0)\hfill \\ \multicolumn{1}{c}{\left(\frac{\alpha \beta {\mu }_1+({\alpha }^4-1)({\mu }_2+\beta ({\mu }_1+{\mu }_2))\tanh \left(\frac{\alpha }{2}\right)}{\alpha ({\mu }_2+\beta ({\mu }_1+{\mu }_2))}\right)}\end{array}\right],\hfill \end{array}$$

(48)

and

$$\begin{array}{c}{T}_L=\hfill \\ \left[\begin{array}{c}6C{\mu }_1{\mu }_2\mathrm{WUR}((1-{\kappa }^2)({\tan }^{-1}\kappa -{\tan }^{-1}{x}_0)-\kappa +\frac{1+{\kappa }^2}{1+x_0^2}{x}_0)\hfill \\ \multicolumn{1}{c}{\left(\frac{{\mu }_1\beta +3{\mu }_2(1+\beta )}{({\mu }_2+\beta ({\mu }_1+{\mu }_2))}\right)}\end{array}\right].\hfill \end{array}$$

(49)

Adiabatic temperature

The temperature of both fluids is increased by the power which transmitted by both the rollers. This temperature can be calculated as

$$\mathrm{\Delta }T=\frac{p_{\mathrm{in}}}{\rho Q{C}_p},$$

(50)

where $\rho$ represents the density of the fluid, $Q$ represents the volumetric flow rate, and $C_p$ represents the heat capacity of the calendered material at a fixed pressure.

Results and discussion

In this article, the two-layered calendering is explored when both the layers contain non-Newtonian fluid, that is, viscoplastic and couple stress fluids. For the simplification, lubrication approximation theory is applied. The exact solution for velocity and pressure is evaluated. Some other parameters of engineering importance are also evaluated analytically. The impact of some parameters is shown by graphs and some are given in the form of tables.

Velocity profile

Dimensionless velocity profiles for both layers of non-Newtonian fluid at distinct values of viscosity ratios, viscoplastic parameter $\beta$ and couple stress parameter $\alpha$ are shown in Figures 2(a), (b), 3(a), (b), 4(a), (b), and 5(a), (b). Due to no slip boundary condition at the interface, both fluid layers have identical velocities there. It is investigated that with the increment in the value of the viscoplastic parameter, the velocity decreases for the lower layer of fluid and increases for the upper layer of fluid. As the Casson parameter increases, the yield stress and effective viscosity of viscoplastic fluids rise, requiring higher shear stress for the same shear rate, hence decreasing fluid velocity accordingly. Similarly, when the couple stress parameter increases, it indicates an increase in microstructural effects within the fluid, leading to higher resistance to deformation and thus causes a decrease in fluid velocity. Moreover, with the increase in the viscosity ratios, the velocity profile for the upper layer becomes more pressed at the entering point as compared to the lower layer of fluid. Additionally, with the variation in the couple stress parameter, the lower layer of fluid squeezed more by increasing both the viscosity ratios and couple stress parameter $\alpha .$ It is also noticed that velocity profiles of both the layers become the same near the entrance point.

Figure 2.

Effect of $\beta$ on velocity at x = −0.6 for: (a) $\frac{{\mu }_1}{{\mu }_2}=2$ and (b) $\frac{{\mu }_1}{{\mu }_2}=5.$

Figure 3.

Effect of $\beta$ on velocity at x = 0.4 for: (a) $\frac{{\mu }_1}{{\mu }_2}=2$ and (b) $\frac{{\mu }_1}{{\mu }_2}=5.$

Figure 4.

Effect of $\alpha$ on velocity at x = −0.6 for: (a) $\frac{{\mu }_1}{{\mu }_2}=2$ and (b) $\frac{{\mu }_1}{{\mu }_2}=5.$

Figure 5.

Effect of $\alpha$ on velocity at x = 0.4 for: (a) $\frac{{\mu }_1}{{\mu }_2}=2$ and (b) $\frac{{\mu }_1}{{\mu }_2}=5.$

Pressure distribution

Figure 6(a) and (b) show the non-dimensional pressure distribution for the couple stress fluid and Figure 7(a) and (b) show viscoplastic fluid depending on non-dimensional axial points $x$ with the variation in both couple stress and viscoplastic fluid parameters, that is, $\alpha$ and $\beta$ respectively. Moreover, comparison with the Newtonian fluid^34,35 for both the fluids is given. It is observed that that pressure is minimum for Newtonian case and increases with the increase in non-Newtonian effects for both fluid layers. Figures 6 and 7 show that pressure is maximum at $x=-\kappa$ and zero at $x=\kappa .$ It is also observed that maximum pressure decreases for both fluids with the increase in viscosity ratios. Moreover, overall maximum pressure occurs for both fluids when variation in viscoplastic parameter is considered. So it is clear that the viscoplastic Casson parameter controlled the maximum pressure for both the fluid’s layers.

Figure 6.

Variation in pressure distribution of couple stress fluid for: (a) $\beta$ and (b) $\alpha$ .

Figure 7.

Variation in pressure distribution of viscoplastic fluid at $\frac{{\mu }_1}{{\mu }_2}=2$ for: (a) $\beta$ and (b) $\alpha$ .

Roll separating force and power

Figure 8(a) and (b) show the variation in roll separating force and power with the change in viscoplastic Casson parameter and couple stress fluid parameter respectively. Figure 8(a) shows the dimensionless force depending on the viscosity ratios and Figure 8(b) shows the dimensionless power depending on the viscosity ratios. It is noticed from the figures that both increase with the increment in viscosity ratios. Moreover, with the increase in viscoplastic parameter $\beta$ and couple stress parameter $\alpha ,$ both roll separating force and power decreases because increase in both parameters causes an increase in fluid yield stress and internal resistance, respectively, requiring less force and power to achieve deformation, thus reducing roll separating force and power.

Figure 8.

Effect of couple stress and viscoplastic parameter on: (a) roll-separating force and (b) power delivered by rollers at $\kappa =0.4571.$

Torque and temperature

Figure 9(a) and (b) represent the dimensionless form of torques $T_U$ and $T_L$ on both upper and lower rollers respectively and Figure 10 represents the adiabatic temperature as a function of viscosity ratios. It is observed that the torque on the upper roller decreased while the torque on the lower roller and temperature increased with the increase in viscosity ratios. Moreover, both torques and temperature decreased with the increase in viscoplastic parameter and couple stress parameter which shows that both non-Newtonian fluids change into Newtonian fluid at the large value of both the parameters.

Figure 9.

Effects of couple stress and viscoplastic parameters on: (a) torque on upper roller and (b) torque on lower roller at $\kappa =0.4571.$

Figure 10.

Effects of couple stress and viscoplastic parameters on adiabatic temperature at $\kappa =0.4571.$

Tables 2 and 3 show the numerical values of different important variables such as roll separating force, power, adiabatic temperature, and torques acting on upper and lower rollers for various values of viscosity ratios, viscoplastic parameter, and couple stress parameter respectively. It is noticed from Table 2 that the value of all the above-mentioned parameters is increased with the increase in viscosity ratio and decreased with the increase in the couple stress parameter. Moreover, it is clear from Table 3 that the effect of the viscoplastic parameter on all the parameters is the same as the effect of a couple stress parameter.

Table 2.

Effect of couple stress parameter and viscosity ratio on engineering parameters at $\beta =0.2.$

Viscosity ratio	$\alpha$	Roll separating force	Power input	Adiabatic temperature	Torque on rolls
					TU	TL
$\frac{{\mu }_1}{{\mu }_2}=2$	2	53.13	2.72	52.93	2.87	−1.20
	4	48.73	2.50	48.78	27.29	−1.10
	6	46.76	2.40	46.92	91.26	−1.06
	7	46.22	2.37	46.41	143.70	−1.04
$\frac{{\mu }_1}{{\mu }_2}=5$	2	73.11	3.75	86.23	4.09	−1.38
	4	69.65	3.57	83.32	39.14	−1.32
	6	68.02	3.49	81.91	132.87	−1.29
	7	67.56	3.46	81.52	210.15	−1.28

Table 3.

Effect of viscoplastic parameter and viscosity ratio on engineering parameters at $\alpha =6.$

Viscosity ratio	$\beta$	Roll separating force	Power input	Adiabatic temperature	Torque on rolls
					TU	TL
$\frac{{\mu }_1}{{\mu }_2}=2$	0.1	68.52	3.51	48.78	133.67	−1.67
	0.5	30.78	1.58	31.51	60.12	−0.61
	1	24.71	1.26	25.10	48.29	−0.44
	2	21.43	1.10	21.67	41.89	−0.36
	10	18.59	0.95	18.71	36.35	−0.28
$\frac{{\mu }_1}{{\mu }_2}=5$	0.1	96.46	4.95	121.55	188.32	−2.07
	0.5	48.32	2.48	57.57	94.48	−0.76
	1	40.55	2.08	47.49	79.32	−0.57
	2	36.02	1.84	41.61	70.47	−0.47
	10	31.70	1.62	36.03	62.03	−0.39

Conclusions

The problem under consideration deals with the two-layer calendering of incompressible non-Newtonian fluids having different viscosity ratios. The main aim of this study is to discuss the effect of non-Newtonian parameters on flow behavior in detail. The upper layer contains a couple stress fluid while the lower layer contains viscoplastic Casson fluid. Lubrication approximation theory is applied for simplified results. Different results are noticed during the process which are given as follows:

• By decreasing the radius of rollers $R$ or velocity or by increasing the gap between rollers, that is, $H_0,$ the maximum value of pressure can be decreased. Moreover, with the increase in viscoplastic parameter $\beta$ and couple stress fluid parameter $\alpha ,$ the maximum pressure decreased for both the fluid layers.

• It is noticed that with the increase in viscosity ratios, the maximum pressure for both fluids decreased.

• With the decrease in the speed of the roller, width of the roller $W$ , radius of the roller $R$ , and viscosity ratios, the roll separating force can be minimized. It can also be minimized with the increase in roller gap $H_0,$ $\beta$ and $\alpha .$

• The power input delivered by the rollers decreased with the decrease in speed of the roller and with the increase in the $\beta$ and $\alpha .$

• Power increased with the increase in the width of the roller $W$ , radius of the roller $R$ and with the decrease in the roller gap $H_0.$

• The torque acting on both rollers is decreased by decreasing the width of the roller $W,$ radius of the roller $R$ , speed of the roller, viscosity ratios and by increasing the roller gap $H_0$ , $\beta$ and $\alpha .$

• Adiabatic temperature for both fluid layers based on power, the density of the fluid, flow rate, and heat capacity of the fluid. It is noticed that adiabatic temperature is decreased by decreasing the power delivered by the rollers to the fluid and by increasing all the remaining factors.

• It is also noticed that the limit $\alpha \to \infty$ and $\beta \to \infty ,$ the effects of Newtonian fluid ³⁵ in the calendering process can be evaluated and proved.

This research work deals with only theoretical and mathematical formulation of two-layered calendering of non-Newtonian fluids, that is, viscoplastic fluid and couple stress fluid. Experimental verification of the final effects can be done in the future. Additionally, this work can be expanded for non-isothermal cases and other non-Newtonian fluids with slip boundary conditions and so on.