Rheological insight of wall slippage and microrotation on the coating thickness during non-isothermal forward roll coating phenomena of micropolar fluid

Abstract

Roll coating is important for coating industries, including wallpaper, magnetic records, photographic and plastic films, wrap**, adhesive tapes, magazines, books, etc. This theoretical study examines the non-isothermal, incompressible, steady flow of micropolar fluid inside a forward roll coating by using the lubrication approximation theory. Interesting engineering factors including pressure, roll-separating force, separation point, flow rate and power input are computed using a numerical technique, and further, the closed-form solutions for velocity, pressure gradient, temperature distribution and microrotation are also obtained. Graphs are used to show how the various parameters affect pressure, velocity, pressure gradient, shear stress and temperature distribution. Velocity profile, power input and separation points are the decreasing functions of the involved parameters. The fluid particle rotations increase the pressure distribution, leading to decreased coating thickness, which may help to improve the efficiency of coating process.

Appendix I

$${z}_{1}=3\mathrm{Br}\left(h-y\right){\varepsilon }^{2}\left(\begin{array}{c}64{A}_{2}{d}_{3}N\frac{dp}{dx}+2{{A}_{1}}^{5}{{a}_{3}}^{2}{{d}_{2}}^{2}h\left(1+N\right)y+{a}^{4}(-4{a}_{3}{{d}_{2}}^{2}hNy\\ +{{a}_{3}}^{2}(1+N)({{d}_{2}}^{2}-2{{d}_{3}}^{2}hy)-2{{d}_{2}}^{2}hy\varepsilon )\end{array}\right),$$

(56)

$${z}_{2}={{A}_{1}}^{2}\mathrm{Br}\left(h-y\right)\varepsilon \left(\begin{array}{c}8{d}_{1}h{N}^{2}y(3{a}_{2}{d}_{1}+2{a}_{1}\frac{\mathrm{d}p}{\mathrm{d}x}\left(h+y\right)+\\ +2N(-3{{d}_{2}}^{2}+6{{d}_{3}}^{2}hy+4{a}_{1}{A}_{2}h{\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)}^{2}y({h}^{2}+hy+{y}^{2})+\\ 4{a}_{2}{d}_{1}(-6{d}_{2}+{A}_{2}h\frac{\mathrm{d}p}{\mathrm{d}x}y(h+y)))\varepsilon -3{{d}_{3}}^{2}{\varepsilon }^{2}+\\ 6{a}_{3}(-8{d}_{1}{d}_{2}{N}^{2}+{d}_{3}({d}_{3}N-32{a}_{1}(1+N)\frac{\mathrm{d}p}{\mathrm{d}x})\varepsilon \end{array}\right),$$

(57)

$${z}_{3}=24y+\mathrm{Br}\left(h-y\right)\left(\begin{array}{c}3{{a}_{3}}^{2}{{d}_{3}}^{2}\left(1+N\right)+12{{a}_{2}}^{2}{{d}_{1}}^{2}h\left(1+N\right)y-\\ 6{a}_{3}N\left({{d}_{2}}^{2}-2{{d}_{3}}^{2}hy\right)-16{a}_{2}{d}_{1}\left(1+N\right)\\ (3{a}_{3}{d}_{2}-{a}_{1}h\frac{\mathrm{d}p}{\mathrm{d}x}y(h+y))+4hy(3{{d}_{2}}^{2}N+2{{a}_{1}}^{2}\\ (1+N){\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)}^{2}({h}^{2}+hy+{y}^{2}))+3({{d}_{2}}^{2}+2{{d}_{3}}^{2}hy)\varepsilon \end{array}\right),$$

(58)

$${z}_{4}=\left(h-y\right)\left(\begin{array}{c}12{{d}_{1}}^{2}h{N}^{3}y+8{d}_{1}{N}^{2}\left(-6{d}_{2}+{A}_{2}h\frac{dp}{dx}y\left(h+y\right)\right)\varepsilon +\\ {\varepsilon }^{2}(3{{d}_{3}}^{2}N+2{{A}_{2}}^{2}h{\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)}^{2}y(N({h}^{2}+hy+{y}^{2})+3\varepsilon )\\ -24{d}_{3}\frac{\mathrm{d}p}{\mathrm{d}x}(4{a}_{1}N+2{a}_{3}{A}_{2}N+{A}_{2}\varepsilon ))\end{array}\right),$$

(59)

$${z}_{5}=-16y\left(\begin{array}{c}{A}_{1}(2-{A}_{1}{a}_{3}){d}_{1}{d}_{2}{N}^{2}+({{A}_{1}}^{2}{a}_{2}{d}_{1}{d}_{2}\left(-N+{A}_{1}{a}_{3}\left(1+N\right)\right)+\\ (-2{d}_{3}+{A}_{1}{d}_{2}h)(2{A}_{2}N-{A}_{1}(2{a}_{1}+{a}_{3}{A}_{2})N+2{{A}_{1}}^{2}{a}_{1}{a}_{3}(1+N))\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)\varepsilon \\ +{A}_{1}{A}_{2}{d}_{3}\frac{\mathrm{d}p}{\mathrm{d}x}{\varepsilon }^{2})cosh[\sqrt{{A}_{1}}h]\end{array}\right),$$

(60)

$${z}_{6}=y\varepsilon \left({A}_{1}{{d}_{2}}^{2}+{{d}_{3}}^{2}\right)\left(2N+{A}_{1}\left(-2{a}_{3}N+{A}_{1}{{a}_{3}}^{2}\left(1+N\right)+\varepsilon \right)\right)\mathrm{cosh}\left[2\sqrt{{A}_{1}}h\right],$$

(61)

$${z}_{7}=\left(\begin{array}{c}{A}_{1}\left(2-{A}_{1}{a}_{3}\right){d}_{1}{d}_{2}{N}^{2}+(-4{A}_{2}{d}_{3}N\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)+2{A}_{1}N\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)\left(2{a}_{1}{d}_{3}+{a}_{3}{A}_{2}{d}_{3}+{A}_{2}{d}_{2}y\right)\\ {{+A}_{1}}^{3}{a}_{3}{d}_{2}(1+N)({a}_{2}{d}_{1}+2{a}_{1}\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)y)-{{A}_{1}}^{2}({a}_{2}{d}_{1}{d}_{2}N+4{a}_{1}{a}_{3}{d}_{3}(1+N)\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)\\ +(2{a}_{1}+{a}_{3}{A}_{2}){d}_{2}N\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)y))\varepsilon +{A}_{1}{A}_{2}{d}_{3}\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right){\varepsilon }^{2})cosh[\sqrt{{A}_{1}}y]\end{array}\right),$$

(62)

$${z}_{8}=h\varepsilon \left({A}_{1}{{d}_{2}}^{2}+{{d}_{3}}^{2}\right)\left(2N+{A}_{1}\left(-2{a}_{3}N+{A}_{1}{{a}_{3}}^{2}\left(1+N\right)+\varepsilon \right)\right)\mathrm{cosh}\left[2\sqrt{{A}_{1}}h\right],$$

(63)

$${z}_{9}=8\left(\begin{array}{c}\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)\varepsilon \left(\left(2{d}_{2}-{d}_{3}h\right)\left(2{A}_{2}N-{A}_{1}\left(2{a}_{1}+{a}_{3}{A}_{2}\right)N+2{{A}_{1}}^{2}{a}_{1}{a}_{3}\left(1+N\right)\right)\right.\\ \left.-{A}_{1}{A}_{2}{d}_{2}\varepsilon \right)+{d}_{1}{d}_{3}((-2+{A}_{1}{a}_{3}){N}^{2}+{A}_{1}{a}_{2}(N-{A}_{1}{a}_{3}(1+N))\varepsilon ))sinh[\sqrt{{A}_{1}}h]\\ +{A}_{1}{d}_{2}{d}_{3}\varepsilon (2N+{A}_{1}(-2{a}_{3}N+{A}_{1}{{a}_{3}}^{2}(1+N)+\varepsilon ))sinh[2\sqrt{{A}_{1}}h]\end{array}\right),$$

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$${z}_{10}=\left(\begin{array}{c}\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)\varepsilon ((-2{A}_{2}N+{A}_{1}(2{a}_{1}+{a}_{3}{A}_{2})N-2{{A}_{1}}^{2}{a}_{1}{a}_{3}(1+N))(2{d}_{2}-{d}_{3}y)+{A}_{1}{A}_{2}{d}_{2}\varepsilon \\ +{d}_{1}{d}_{3}((2-{A}_{1}{a}_{3}){N}^{2}+{A}_{1}a2(-N+{A}_{1}{a}_{3}(1+N))\varepsilon ))sinh[\sqrt{{A}_{1}}y]\\ -2{{A}_{1}}^{3/2}{d}_{2}{d}_{3}h\varepsilon (2N+{A}_{1}(-2{a}_{3}N+{A}_{1}{{a}_{3}}^{2}(1+N)+\varepsilon ))sinh[2\sqrt{{A}_{1}}y]\end{array}\right).$$

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