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Modelling and analysis of the unsteady flow and heat transfer of immiscible micropolar and Newtonian fluids through a pipe of circular cross section

Abstract

This study deals with the unsteady flow and heat transfer of micropolar and Newtonian fluids, flowing immiscibly through a circular pipe. The micropolar and Newtonian fluids occupy core and peripheral regions, respectively. Initially, the pipe and fluids in both regions are at rest; after an instant of time, a constant pressure gradient is applied to generate the flow. The equations governing the flow are time dependent, coupled and nonlinear. The solutions for velocity, microrotation and temperature are acquired numerically employing Crank–Nicolson finite difference approach. Volume flow rate is also obtained numerically and presented in tabular form. At fluid–fluid interface, continuity of velocities, shear stresses, temperatures and heat fluxes are considered. The results for velocity, microrotation and temperature are displayed graphically. It is seen that the fluid velocities, microrotation and temperatures are increasing with time and attain a steady state after a particular time level. The micropolarity parameter has a decreasing effect on the fluid velocities and temperatures.

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Abbreviations

b :

Component of microrotation

\(C_{p}\) :

Ratio of specific heats

\({C_{p}}_{1}\) :

Specific heat of fluid in region-I

\({C_{p}}_{2}\) :

Specific heat of fluid in region-II

\(B_{R}\) :

Brinkman number

\({\bar{f}}\) :

Body force vector

\(G=-\frac{\partial p}{\partial z}\) :

Constant pressure gradient

h :

Step size in spatial direction

\(\hat{j}\) :

Dimensional Gyration parameter

\(J\) :

Non-dimensional Gyration parameter

k :

Step size in time

K :

Ratio of thermal conductivities

\(K_{1}\) :

Thermal conductivity of fluid in region-I

\(K_{2}\) :

Thermal conductivity of fluid in region-II

\({\bar{l}}\) :

Body couple vector

m :

Index representing interface

\(m_{1}\) :

Ratio of viscosities

\(m_{2}\) :

Ratio of densities

\(n_{1}\) :

Micropolarity parameter

p :

Fluid pressure

\(P_{R}\) :

Prandtl number

\({\bar{q}}\) :

Fluid velocity vector

R :

Radius of region-I

\(R_{0}\) :

Radius of pipe

Re :

Reynolds number

\((r,\theta ,z)\) :

Cylindrical polar co-ordinates

t :

Time

\(T_{1}\) :

Fluid temperature in region-I

\(T_{2}\) :

Fluid temperature in region-II

\(T_{w}\) :

Temperature of the boundary of pipe

\(T_{0}\) :

Initial temperature of fluids

W :

Average velocity in pipe

\(w_{1}\) :

Component of fluid velocity in region-I

\(w_{2}\) :

Component of fluid velocity in region-II

\(\alpha ,\beta ,\gamma\) :

Gyroviscosity coefficients

\(\delta _{ij}\) :

Kronecker delta symbol

\(\varepsilon _{ijk}\) :

Levi-Civita symbol

\(\lambda _{1},\mu\) :

Viscosity coefficients

\(\kappa\) :

Vortex viscosity of micropolar fluid

\(\mu _{1}\) :

Viscosity of fluid in region-I

\(\mu _{1}\) :

Viscosity of fluid in region-II

\({\bar{\upsilon }}\) :

Microrotation vector

\(\rho\) :

Density of the fluid

\(\rho _{1}\) :

Density of fluid in region-I

\(\rho _{1}\) :

Density of fluid in region-II

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Acknowledgements

The authors are grateful to National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India, for the financial support through the research project Ref. No.2/48(23)/2014/NBHM-R&D II/1083 dated 28-01-2015. Authors also thank anonymous reviewers for their constructive suggestions which led to the strengthening of the paper to a great extent.

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Correspondence to M. Devakar.

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Technical Editor: Cezar Negrao.

Appendix

Appendix

\(Z=(z_{ij})_{(3m+1)\times (3m+1)}\), \(Y=(y_{ij})_{(3m+1)\times (3m+1)}\) and \({\bar{b}}={\bar{b_{j}}}_{(3m+1)\times 1}\)

in which

$$\begin{aligned} z_{ij}= & {} {\left\{ \begin{array}{ll} {\epsilon _{1}}_{i}, &{} {\text {for}} \,\, i+m+2=j\\ {\varPsi _{1}}_{i}, &{} {\text {for}} \,\, i+m+1=j\\ {\delta _{1}}_{i}, &{} {\text {for}} \,\, i+m=j\\ {\gamma _{1}}_{i},&{} {\text {for}} \,\, i+1=j\\ {\alpha _{1}}_{i},&{} {\text {for}} \,\,i=j\\ {\beta _{1}}_{i},&{} {\text {for}} \,\,i=j+1\\ {\varXi _{1}}_{i},&{} {\text {for}} \,\,i=j+m\\ {\varOmega _{1}}_{i},&{} {\text {for}} \,\,i=j+m+1\\ {\theta _{1}}_{i},&{} {\text {for}} \,\,i=j+m+2\\ 0,&{} {\text {otherwise}}\\ \end{array}\right. }\\ y_{ij}= & {} {\left\{ \begin{array}{ll} {\epsilon _{2}}_{i}, &{} {\text {for}} \,\, i+m+2=j\\ {\varPsi _{2}}_{i}, &{} {\text {for}} \,\, i+m+1=j\\ {\delta _{2}}_{i}, &{} {\text {for}} \,\, i+m=j\\ {\gamma _{2}}_{i},&{} {\text {for}} \,\, i+1=j\\ {\alpha _{2}}_{i},&{} {\text {for}} \,\,i=j\\ {\beta _{2}}_{i},&{} {\text {for}} \,\,i=j+1\\ {\varXi _{2}}_{i},&{} {\text {for}} \,\,i=j+m\\ {\theta _{2}}_{i},&{} {\text {for}} \,\,i=j+m+2\\ 0,&{} {\text {otherwise}}\\ \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\bar{b}}=\bar{b_{j}}= {\left\{ \begin{array}{ll} kG,&{} {\text {for}} \,\,j=1\,\, {\text {to}}\,\, m\\ 0, &{} {\text {for}}\,\, j=m+1 \,\,{\text {to}}\,\, 2m+2 \\ \frac{kG}{m_{2}} &{} {\text {for}}\,\,j= 2m+3\,\, {\text {to}}\,\, 3m+1\\ \end{array}\right. } \end{aligned}$$

In the above,

$$\begin{aligned} {\alpha _{1}}_{i}= & {} {\left\{ \begin{array}{ll} 1+2M, &{} {\text {for}} \,\, i=1\\ 1+2A, &{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m\\ Q, &{} {\text {for}} \,\, i=m+1 \\ 1+2N, &{} {\text {for}} \,\, i=m+2 \\ 1+2E-H_{i-(m+2)}, &{} {\text {for}} \,\, i=m+3\,\,{\text {to}}\,\,2m+1 \\ 1,&{} {\text {for}} \,\,i=2m+2\\ 1+2K, &{} {\text {for}} \,\, i=2m+3\,\,{\text {to}}\,\, 3m+1\\ \end{array}\right. }\\ {\alpha _{2}}_{i}= & {} {\left\{ \begin{array}{ll} 1-2M, &{} {\text {for}} \,\, i=1\\ 1-2A, &{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m\\ 0, &{} {\text {for}} \,\, i=m+1 \\ 1-2N, &{} {\text {for}} \,\, i=m+2 \\ 1-2E+H_{i-(m+2)}, &{} {\text {for}} \,\, i=m+3\,\,{\text {to}}\,\,2m+1 \\ 0,&{} {\text {for}} \,\,i=2m+2\\ 1-2K, &{} {\text {for}} \,\, i=2m+3\,\,{\text {to}}\,\, 3m+1\\ \end{array}\right. }\\ {\beta _{1}}_{i}= & {} {\left\{ \begin{array}{ll} -(A-B_{i}), &{} {\text {for}} \,\, i=1\,\,{\text {to}}\,\,m-1 \\ -P, &{} {\text {for}} \,\, i=m\\ 0, &{} {\text {for}} \,\, i=m+1 \\ -(E-F_{i-(m+1)}), &{} {\text {for}} \,\, i=m+2 \,\,{\text {to}}\,\,2m\\ 0,&{} {\text {for}} \,\,i=2m+1 \,\,{\text {and}}\,\,2m+2 \\ -(K-L_{i-(m+1)}), &{} {\text {for}} \,\, i=2m+3\,\,{\text {to}}\,\, 3m\\ \end{array}\right. }\\ {\beta _{2}}_{i}= & {} {\left\{ \begin{array}{ll} (A-B_{i}), &{} {\text {for}} \,\, i=1\,\,{\text {to}}\,\,m-1 \\ 0, &{} {\text {for}} \,\, i=m\\ 0, &{} {\text {for}} \,\, i=m+1 \\ (E-F_{i-(m+1)}), &{} {\text {for}} \,\, i=m+2 \,\,{\text {to}}\,\,2m\\ 0,&{} {\text {for}} \,\,i=2m+1 \,\,{\text {and}}\,\,2m+2 \\ (K-L_{i-(m+1)}), &{} {\text {for}} \,\, i=2m+3\,\,{\text {to}}\,\, 3m\\ \end{array}\right. }\\ {\gamma _{1}}_{i}= & {} {\left\{ \begin{array}{ll} -2M, &{} {\text {for}} \,\, i=1 \\ -(A+B_{i-1}), &{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m\\ 0, &{} {\text {for}} \,\, i=m+1\\ -2N, &{} {\text {for}} \,\, i=m+2 \\ -(E+F_{i-(m+2)}), &{} {\text {for}} \,\, i=m+3\,\,{\text {to}}\,\,2m+1 \\ 0,&{} {\text {for}} \,\,i=2m+2\\ -(K+L_{i-(m+2)}), &{} {\text {for}} \,\, i=2m+3\,\,{\text {to}}\,\, 3m\\ \end{array}\right. }\\ \end{aligned}$$
$$\begin{aligned} {\gamma _{2}}_{i}= & {} {\left\{ \begin{array}{ll} 2M, &{} {\text {for}} \,\, i=1 \\ (A+B_{i-1}), &{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m\\ 0, &{} {\text {for}} \,\, i=m+1\\ 2N, &{} {\text {for}} \,\, i=m+2 \\ (E+F_{i-(m+2)}), &{} {\text {for}} \,\, i=m+3\,\,{\text {to}}\,\,2m+1 \\ 0,&{} {\text {for}} \,\,i=2m+2\\ (K+L_{i-(m+2)}), &{} {\text {for}} \,\, i=2m+3\,\,{\text {to}}\,\, 3m\\ \end{array}\right. } \\ {\varXi _{1}}_{i}= & {} {\left\{ \begin{array}{ll} 0, &{} {\text {for}} \,\, i=1\,\,{\text {and}}\,\,2\\ I, &{} {\text {for}} \,\, i=3 \,\,{\text {to}}\,\,m+1\\ 0,&{} {\text {for}} \,\,i=m+2 \,\,{\text {to}}\,\,2m+1\\ \end{array}\right. } \\ {\varXi _{2}}_{i}= & {} {\left\{ \begin{array}{ll} 0, &{} {\text {for}} \,\, i=1\,\,{\text {and}}\,\,2\\ -I, &{} {\text {for}} \,\, i=3 \,\,{\text {to}}\,\,m+1\\ 0,&{} {\text {for}} \,\,i=m+2 \,\,{\text {to}}\,\,2m+1\\ \end{array}\right. } \\ {\varOmega _{1}}_{i}= & {} {\left\{ \begin{array}{ll} S &{} {\text {for}} \,\, i=m+1\\ 0, &{} {\text {otherwise}} \end{array}\right. } \\ {\theta _{1}}_{i}= & {} {\left\{ \begin{array}{ll} -I, &{} {\text {for}} \,\, i=1\,\,{\text {to}}\,\,m-1\\ -S,&{} {\text {for}} \,\, i=m\\ -(K-L_{i}), &{} {\text {for}} \,\, i=m+1\\ 0,&{} {\text {for}} \,\, i=m+2 \,\,{\text {to}}\,\,2m-1 \end{array}\right. } \end{aligned}$$
$$\begin{aligned} {\theta _{2}}_{i}= & {} {\left\{ \begin{array}{ll} I, &{} {\text {for}} \,\, i=1\,\,{\text {to}}\,\,m-1\\ 0,&{} {\text {for}} \,\, i=m\\ (K-L_{i}), &{} {\text {for}} \,\, i=m+1\\ 0,&{} {\text {for}} \,\, i=m+2 \,\,{\text {to}}\,\,2m-1 \end{array}\right. }\\ {\delta _{1}}_{i}= & {} {\left\{ \begin{array}{ll} 0, &{} {\text {for}} \,\, i=1\\ I, &{} {\text {for}} \,\, i=2\,\,{\text {to}}\,\,m\\ 0,&{} {\text {for}} \,\,i=m+1\,\,{\text {to}}\,\,2m+1\\ \end{array}\right. }\\ {\delta _{2}}_{i}= & {} {\left\{ \begin{array}{ll} 0, &{} {\text {for}} \,\, i=1\\ -I, &{} {\text {for}} \,\, i=2\,\,{\text {to}}\,\,m\\ 0,&{} {\text {for}} \,\,i=m+1\,\,{\text {to}}\,\,2m+1\\ \end{array}\right. }\\ {\varPsi _{1}}_{i}= & {} {\left\{ \begin{array}{ll} 0, &{} {\text {for}} \,\, i=1\\ -D_{i-1},&{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m\\ 0, &{} {\text {for}} \,\, i=m+1\,\,{\text {to}}\,\,2m\\ \end{array}\right. }\\ {\varPsi _{2}}_{i}= & {} {\left\{ \begin{array}{ll} 0, &{} {\text {for}} \,\, i=1\\ D_{i-1},&{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m\\ 0, &{} {\text {for}} \,\, i=m+1\,\,{\text {to}}\,\,2m\\ \end{array}\right. }\\ {\epsilon _{1}}_{i}= & {} {\left\{ \begin{array}{ll} 0, &{} {\text {for}} \,\, i=1\\ -I, &{} {\text {for}} \,\, i=2\,\,{\text {to}}\,\,m\\ -R, &{} {\text {for}} \,\, i=m+1\\ 0, &{} {\text {for}} \,\, i=m+2 \,\,{\text {to}}\,\,2m-1\\ \end{array}\right. }\\ {\epsilon _{2}}_{i}= & {} {\left\{ \begin{array}{ll} 0, &{} {\text {for}} \,\, i=1\\ I, &{} {\text {for}} \,\, i=2\,\,{\text {to}}\,\,m\\ 0, &{} {\text {for}} \,\, i=m+1 \,\,{\text {to}}\,\,2m-1\\ \end{array}\right. } \end{aligned}$$

\(\varTheta =(\varTheta _{ij})_{(2m)\times (2m)}\), \(\varXi =(\varXi _{ij})_{(2m)\times (2m)}\) and \(\varUpsilon ={\varUpsilon _{j}}_{(2m)\times 1}\)

in which

$$\begin{aligned} \varTheta _{ij}= & {} {\left\{ \begin{array}{ll} {\gamma _{3}}_{i},&{} {\text {for}} \,\, i+1=j\\ {\alpha _{3}}_{i},&{} {\text {for}} \,\,i=j\\ {\beta _{3}}_{i},&{} {\text {for}} \,\,i=j+1\\ 0,&{} {\text {otherwise}}\\ \end{array}\right. }\\ \varXi _{ij}= & {} {\left\{ \begin{array}{ll} {\gamma _{4}}_{i},&{} {\text {for}} \,\, i+1=j\\ {\alpha _{4}}_{i},&{} {\text {for}} \,\,i=j\\ {\beta _{4}}_{i},&{} {\text {for}} \,\,i=j+1\\ 0,&{} {\text {otherwise}}\\ \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} {\alpha _{3}}_{i}= & {} {\left\{ \begin{array}{ll} 1+2C_{15}, &{} {\text {for}} \,\, i=1\\ 1+2C_{5}, &{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m\\ 1+K, &{} {\text {for}} \,\, i=m+1 \\ 1+2C_{12}, &{}{\text {for}} \,\, i=m+2 \,\, {\text {to}}\,\,2m\\ \end{array}\right. }\\ {\alpha _{4}}_{i}= & {} {\left\{\begin{array}{ll} 1-2C_{15}, &{} {\text {for}} \,\, i=1\\ 1-2C_{5}, &{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m\\ 0, &{} {\text {for}} \,\, i=m+1 \\ 1-2C_{12}, &{}{\text {for}} \,\, i=m+2 \,\, {\text {to}}\,\,2m\\ \end{array}\right. }\\ {\beta _{3}}_{i}= & {} {\left\{\begin{array}{ll} -[C_{5}-{C_{6}}_{i}], &{} {\text {for}} \,\, i=1 \,\,{\text {to}}\,\,m-1\\ -1, &{} {\text {for}} \,\, i=m\\ -[C_{12}-{C_{13}}_{i}], &{} {\text {for}} \,\, i=m+1 \,\,{\text {to}}\,\,2m-1 \\ \end{array}\right. }\\ {\beta _{4}}_{i}= & {} {\left\{\begin{array}{ll} {[}C_{5}-{C_{6}}_{i}], &{} {\text {for}} \,\, i=1 \,\,{\text {to}}\,\,m-1\\ 0, &{} {\text {for}} \,\, i=m\\ {[}C_{12}-{C_{13}}_{i}], &{} {\text {for}} \,\, i=m+1 \,\,{\text {to}}\,\,2m-1 \\ \end{array}\right. }\\ {\gamma _{3}}_{i}= & {} {\left\{\begin{array}{ll} -2C_{15}, &{} {\text {for}} \,\, i=1 \\ -[C_{5}+{C_{6}}_{i-1}], &{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m \\ -K, &{} {\text {for}} \,\, i=m+1 \\ -[C_{12}+{C_{13}}_{i-1}], &{} {\text {for}} \,\, i=m+2 \,\,{\text {to}}\,\,2m-1 \\ \end{array}\right. }\\ {\gamma _{4}}_{i}= & {} {\left\{\begin{array}{ll} 2C_{15}, &{} {\text {for}} \,\, i=1 \\ {[}C_{5}+{C_{6}}_{i-1}], &{} {\text {for}} \,\, i=2 \,\,{\text {to}}\,\,m \\ 0, &{} {\text {for}} \,\, i=m+1 \\ {[}C_{12}+{C_{13}}_{i-1}], &{} {\text {for}} \,\, i=m+2 \,\,{\text {to}}\,\,2m-1 \\ \end{array}\right. } \end{aligned}$$

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Devakar, M., Raje, A. Modelling and analysis of the unsteady flow and heat transfer of immiscible micropolar and Newtonian fluids through a pipe of circular cross section. J Braz. Soc. Mech. Sci. Eng. 40, 325 (2018). https://doi.org/10.1007/s40430-018-1233-2

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Keywords

  • Micropolar fluid
  • Immiscible fluid
  • Unsteady flow
  • Circular pipe
  • Finite difference method