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Computational analysis of multiphase flow of non-Newtonian fluid through inclined channel: heat transfer analysis with perturbation method

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Abstract

The present investigation is based on the two-phase flow of a non-Newtonian fluid through a uniform channel with heat transfer. Stress tensor of third-grade fluid is taken into account to treat as non-Newtonian fluid. Two different types of viscous suspensions are formed with the tiny size Hafnium and crystal particles, respectively. Owing to the high magnetic susceptibility of the Hafnium metallic particles magnetic effects are applied, as well. Each magnetohydrodynamics bi-phase flow is caused, due to gravitational force. An asymptotic solution is obtained with the help of the “Regular perturbation method,” for the set nonlinear and coupled differential equations. A detailed parametric study is carried out to analyze the effective contribution of significant parameters and quantities. It is inferred that the strong magnetic effects and dominant viscous dissipation introduce additional thermal energy to the multiphase flow. Moreover, highly viscous multiphase suspensions are suitable in chemical industries to manufacture such paints and emulsions which contain small polymer particles.

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Abbreviations

\({\hat{\mathbf{T}}}\) :

Stress tensor of a 3rd grade fluid

\({\hat{\mathbf{V}}}_{{\text{f}}}\) :

The velocity profile of fluid phase

\({\hat{\mathbf{L}}}\) :

Velocity gradient

\({\hat{\mathbf{J}}}\) :

Current vector

\({\mathbf{S}}\) :

Drag force

M :

Hartmann number

\(\alpha\) :

Inclination of plates

\({\Lambda }_{1}\) :

Third-grade parameter

\({\hat{\text{u}}}_{{\text{f}}}\) :

Fluid’s velocity

\(\rho_{{\text{f}}}\) :

Fluid’s density

\(\hat{P}\) :

Pressure

\({\hat{\text{T}}}_{l}\) :

The temperature of the suspension at \({\Psi } = - {\text{L}}\)

\(\beta_{1} , \;\beta_{2} \& \beta_{3}\) :

Material constants

\(t\) :

Time

\(\sigma\) :

Electrical conductivity

\({\hat{\text{T}}}\) :

The temperature of the suspension

\({\hat{\mathbf{V}}}_{{\text{p}}}\) :

The velocity profile of particle phase

\({\text{L}}\) :

Separation between plates

\({\hat{\mathbf{B}}}\) :

Magnetic force

\({\hat{\mathbf{E}}}\) :

Electric force

Fr:

Froude number

\(C\) :

Density number

\({\Lambda }_{2}\) :

Brinkman number

\({\hat{\text{u}}}_{{\text{p}}}\) :

Particle’s velocity

\(\rho_{{\text{p}}}\) :

Particle’s density

\(\overline{P}\) :

Modified pressure

\({\hat{\text{T}}}_{u}\) :

The temperature of the suspension at \({\Psi } = {\text{L}}\)

\(\alpha_{1} \& \alpha_{2}\) :

Material constants

g :

Gravitational force

\(\frac{{\text{D}}}{{{\text{D}}t}}\) :

Material time derivative

p :

Particle

f :

Fluid

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Acknowledgment

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant No. R.G.P1/276/43.

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Authors and Affiliations

  1. Department of Mathematics, Institute of Arts and Sciences, Government College University, Faisalabad, Chiniot Campus, Chiniot, 35400, Pakistan

    Mubbashar Nazeer

  2. Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia

    Mohammed Z. Alqarni & S. Saleem

  3. Department of Mathematical Sciences (FABS) BUITEMS, Quetta, 87300, Pakistan

    Farooq Hussain

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  1. Mubbashar Nazeer
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Correspondence to Mubbashar Nazeer.

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Appendix

Appendix

\({A}_{1}=\frac{ M }{\sqrt{\left(1-C\right)}}\),\( {A}_{2}={m}_{1}+{m}_{3}\), \({A}_{3}={m}_{1}{m}_{3}\), \({A}_{4}=\frac{{m}_{1}+{m}_{3}}{{m}_{1}{m}_{3}}\), \({B}_{1}=-\frac{\mathrm{sech}\left({A}_{1}\eta \right)\left(\frac{{A}_{4}g\mathrm{sin}\alpha }{\left(1-C\right){\left(\mathrm{Fr}\right)}^{2}}-\frac{P}{\left(1-C\right)}\right)}{2{{A}_{1}}^{2}}\),

\({B}_{2}=-\frac{\mathrm{sech}\left({A}_{1}\eta \right)\left(\frac{{A}_{4}g\mathrm{sin}\alpha }{\left(1-C\right){\left(\mathrm{Fr}\right)}^{2}}-\frac{P}{\left(1-C\right)}\right)}{2{{A}_{1}}^{2}}\),

\({B}_{3}=-\mathrm{csch}\left({2A}_{1}\right){B}_{6}-{B}_{7}-\mathrm{coth}\left({2A}_{1}\right){B}_{8}+{B}_{9}-2\mathrm{cosh}\left({2A}_{1}\right){B}_{10}\),

\({B}_{4}=-{B}_{5}+\mathrm{coth}\left({2A}_{1}\right){B}_{6}+\mathrm{csch}\left({2A}_{1}\right){B}_{8}-2\mathrm{cosh}\left({2A}_{1}\right){B}_{9}+{B}_{10}\).

\({B}_{5}=-\frac{3}{2}{B}_{1}{B}_{2}^{2}\left({{A}_{1}}^{2}\right)\), \({B}_{6}=-3{B}_{1}{B}_{2}^{2}{{A}_{1}}^{6}\), \({B}_{7}=-\frac{3}{2}{B}_{1}^{2}{B}_{2}\left({{A}_{1}}^{2}\right)\), \({B}_{8}=3{B}_{1}^{2}{B}_{2}{{A}_{1}}^{6}\),

\({B}_{9}=-\frac{3}{4}{B}_{2}^{3}\left({{A}_{1}}^{2}\right)\), \({B}_{10}=-\frac{3}{4}{B}_{1}^{3}\left({{A}_{1}}^{2}\right)\), \({D}_{1}=\frac{1}{4}\mathrm{cosh}\left(2{A}_{1}\right){B}_{1}^{2}{\Lambda }_{2}+\frac{1}{4}\mathrm{cosh}\left(2{A}_{1}\right){B}_{2}^{2}{\Lambda }_{2}-{4B}_{1}{B}_{2}{A}_{1}^{2}{\Lambda }_{2}\), \({D}_{2}=\frac{1}{4}\mathrm{sinh}\left(2{A}_{1}\right){B}_{1}^{2}{\Lambda }_{2}-\mathrm{sinh}\left(2{A}_{1}\right){B}_{2}^{2}{\Lambda }_{2}\),

\({D}_{3}=\left.\begin{array}{l}\frac{1}{8}\overline{\lambda }(4Sinh\left(2{A}_{1}\right)(-{B}_{2}{B}_{6}+{B}_{1}{B}_{8})\\ +8{A}_{1}(6{A}_{1}^{3}{B}_{1}^{2}{B}_{2}^{2}+{B}_{1}{B}_{6}-{A}_{1}({B}_{1}\left({B}_{4}+{B}_{5}\right)\\ +{B}_{2}({B}_{3}+{B}_{7}))-{B}_{2}{B}_{8})-4Cosh\left(2{A}_{1}\right)\\ \left(\begin{array}{c}4{A}_{1}^{2}{B}_{1}{B}_{2}\left({B}_{1}^{2}+{B}_{2}^{2}\right)-{B}_{1}\left({B}_{3}+{B}_{7}-3{B}_{9}\right)\\ -{B}_{2}\left({B}_{4}+{B}_{5}-3{B}_{10}\right)\end{array}\right)\\ Cosh\left(2{A}_{1}\right)({A}_{1}^{2}({B}_{1}^{4}+{B}_{2}^{4})\\ +3({B}_{2}{B}_{9}+{B}_{1}{B}_{10}))){\Lambda }_{2}\end{array}\right\},\)

\({D}_{4}=\left.\begin{array}{l}\frac{1}{24}\overline{\lambda }(12Cosh\left(2{A}_{1}\right)\left({B}_{2}{B}_{6}+{B}_{1}{B}_{8}\right)\\ -8{A}_{1}^{2}\left({B}_{1}{B}_{6}+{B}_{2}{B}_{8}\right)\\ +3Sinh\left(4{A}_{1}\right)\left({A}_{1}^{2}\left({B}_{1}^{4}-{B}_{2}^{4}\right)\right.\\ \left.-3{B}_{2}{B}_{9}+3{B}_{1}{B}_{10}\right)-12Sinh\left(2{A}_{1}\right)\\ (4{A}_{1}^{2}{B}_{1}{B}_{2}({B}_{1}^{2}-{B}_{2}^{2})\\ -{B}_{1}({B}_{3}+{B}_{7}+3{B}_{9})+{B}_{2}\\ ({B}_{4}+{B}_{5}+3{B}_{10}))){\Lambda }_{2}\end{array}\right\}\).

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Nazeer, M., Alqarni, M.Z., Hussain, F. et al. Computational analysis of multiphase flow of non-Newtonian fluid through inclined channel: heat transfer analysis with perturbation method. Comp. Part. Mech. 10, 1371–1381 (2023). https://doi.org/10.1007/s40571-023-00569-y

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