Document Type : Article

**Authors**

Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, P.O. Box 91775-1111, Iran

**Abstract**

The unsteady, viscous flow and mixed convection heat transfer of an incompressible fluid within a vertical concentric cylindrical annulus with time-dependent moving walls are investigated. Fluid is suctioned/injected through the cylinders' walls. Role of movement of the walls on flow and heat transfer inside the vertical annulus is sought. An exact solution of the Navier-Stokes and energy equations is obtained in this problem, for the first time. Here, the transfer of heat is from the hot cylinder walls with constant temperature to the cooler moving fluid. It is interesting to note that the results indicate that the time-dependency of the cylinder walls movement has no effect on the temperature profile. The results also indicate that, compared to other time functions, usage of inverse of time for velocity movement causes significant increase in velocity, stress tensor, and Nusselt number in the vicinity of the moving wall. Also, increasing the mixed convection and suction/injection parameters would increase the Nusselt number and decrease the stress tensor on the inner and outer cylinders no matter what the velocity function is chosen. Therefore, the flow and heat transfer is controllable using the velocity function of the wall movement and change of the other non-dimensional parameters.

**Keywords**

References

1. Wu, T.-H., Xu, Z., and Jackson, J. \Mixed convectionheat transfer to waterowing through a verticalpassage of annular cross section: part 2", Chemical Engineering Research and Design, 80(3), pp. 246{251(2002).2. Joshi, H.M. \Fully developed natural convection in an isothermal vertical annular duct", InternationalCommunications in Heat and Mass Transfer, 14(6), pp. 657{664 (1987).3. Chen, S., Tolke, J., and Krafczyk, M. \Numerical investigationof double-diusive (natural) convection in vertical annuluses with opposing temperatureandconcentrationgradients", International Journal of Heat and Fluid Flow, 31(2), pp. 217{226 (2010).4. Iannello, V., Suh, K.Y., and Todreas, N.E. \Mixed convection friction factors and Nusselt numbers invertical annular and subchannel geometries", International Journal of Heat and Mass Transfer, 31(10), pp.2175{2189 (1988).5. Malvandi, A., Moshizi, S., Soltani, E.G., and Ganji, D. \Modied Buongiorno's model for fully developedmixed convectionow of nanoflids in a verticalannular pipe", Computers & Fluids, 89, pp. 124{132 (2014).6. Avc, M. and Ayd, O. \Mixed convection in a verticalmicroannulus between two concentric microtubes", Journal of Heat Transfer, 131(1), p. 014502 (2009).7. Jha, B.K. and Aina, B. \Mathematical modelling and exact solution of steady fully developed mixedconvectionow in a vertical micro-porous-annulus", Afrika Matematika, 26(7{8), pp. 1199{1213 (2015).8. Jha, B.K., Aina, B., and Muhammad, S. \Combined effects of suction/injection and wall surface curvatureon natural convectionow in a vertical micro-porous annulus", Thermophysics and Aeromechanics, 22(2),pp. 217{228 (2015).9. Husain, S. and Siddiqui, M.A. \Experimental and numerical analysis of transient natural convection ofwater in a high aspect ratio narrow vertical annulus", Progress in Nuclear Energy, 106, pp. 1{10 (2018).10. Husain, S., Siddiqui, M.A., and Khan, S.A. \Eect of geometrical parameters on natural convection of waterin a narrow annulus", Progress in Nuclear Energy, 112, pp. 146{161 (2019).11. Husain, S. and Siddiqui, M.A. \Experimental andnumerical analyses of natural convectionw in a partially heated vertical annulus", Numerical HeatTransfer, Part A: Applications, 70(7), pp. 763{775(2016). 12. Husain, S. and Siddiqui, M.A. \Numerical and experimentalanalysis of natural convectionflow boiling of water in internally heated vertical annulus", NumericalHeat Transfer, Part A: Applications, 73(9), pp. 624{ 653 (2018).13. Mustafa, J., Husain, S., and Siddiqui, M.A.J.E.H.T.\Experimental studies on natural convection of water in a closed-loop vertical annulus", Experimental HeatTransfer an International Journal, 30(1), pp. 25{45 (2017).14. Huang, S. and Chun, C.H. \A numerical study of turbulentw and conjugate heat transfer in concentric annuli with moving inner rod", International Journalof Heat and Mass Transfer, 46(19), pp. 3707{3716 (2003).15. Lin, S. and Hsieh, D. \Heat transfer togeneralizedCouette ow of a non-Newtonianfluid in annuli with moving inner cylinder", Journal of Heat Transfer,102(4), pp. 786{789 (1980).16. Straord, K.N., Datta, P.K., and Coogan, C., Coatingsand Surface Treatment for Corrosion and Wear Resistance, Published for Institution of Corrosion Scienceand Technology by Ellis Horwood, Market Cross House, Cooper Street, Chichester, West Sussex PO 191 EB, England, 1984. 362 (1984).17. Shigechi, T. and Lee, Y. \An analysis on fully developedlaminar flidow and heat transfer in concentric annuli with moving cores", International Journalof Heat and Mass Transfer, 34(10), pp. 2593{2601 (1991).18. Chamkha, A.J. \Unsteady MHD convective heat andmass transfer past a semi-innite vertical permeable moving plate with heat absorption", InternationalJournal of Engineering Science, 42(2), pp. 217{230 (2004).19. Abedini, A. and Rahimi, A.B. \Numerical study ofmixed convection in an annulus between concentric rotating cylinders with time-dependent angular velocity",Iranian Journal of Science and Technology-Transactions of Mechanical Engineering, 36, pp. 165{ 180 (2012).20. Saleh, R. and Rahimi, A.B. \Axisymmetric stagnationpointflow and heat transfer of a viscousuid on a movingcylinder with time-dependent axial velocity and uniform transpiration", Journal of Fluids Engineering,126(6), pp. 997{1005 (2004).21. Jha, B., Joseph, S., and Ajibade, A. \Transient freeconvectiveow through a vertical porous annulus", Proceedings of the Institution of Mechanical Engineers,Part E: Journal of Process Mechanical Engineering,226(2), pp. 105{116 (2012). A. Shakiba and A.B. Rahimi/Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 1306{1317 1317 22. Jha, B., Samaila, A., Ajibade, A. \Unsteady/steadynatural convectionow of reactive viscousuid in a vertical annulus", International Journal of AppliedMechanics Engineering, 18(1), pp. 73{83 (2013). 23. Shakiba, A. and Rahimi, A.B. \Mixed convectiveowof electrically conductinguid in a vertical cylindrical annulus with moving walls adjacent to a radial magneticeld along with transpiration", Journal of Fluids Engineering, 141, pp. 1{11 (2019).24. Busedra, A. and Tavoularis, S. \Numerical simulationsof natural convective heat transfer in vertical concentric and eccentric annular channels", Journal of HeatTransfer, 140(10), p. 102502 (2018). 25. Hekmat, M.H. and Ziarati, K.K. \Eects of nanoparticlesvolume fraction and magneticeld gradient on the mixed convection of a ferrouid in the annulus betweenvertical concentric cylinders", Applied Thermal Engineering, 152, pp. 844{857 (2019).26. Jha, B. and Oni, M. \Natural convectionow in a vertical annulus with time-periodic thermal boundary conditions", Propulsion Power Research, 8(1), pp. 47{ 55 (2019). 27. Jha, B.K. and Malgwi, P.B. \Eects of Hall current and magnetic eld inclination on hydromagnetic natural convectionow in a micro-channel with asymmetric thermal boundary condition", Journal of Thermal Science Engineering Applications, 12(3) (2020). 28. Abbas, Z., Mehdi, I., Hasnain, J., and Aly, S. \Role of suction/injection on natural convectionow of magnetite Fe3O4 nanoparticles in vertical porous micro-annulus between two concentric tubes: A purely analytical approach", Arabian Journal for Science, 44(9), pp. 8113{8122 (2019). 29. Jha, B.K., Oni, M.O., and Aina, B. \Steady fully developed mixed convectionow in a vertical microconcentric- annulus with heat generating/absorbinguid: an exact solution", Ain Shams Engineering Journal, 9(4), pp. 1289{1301 (2018). 30. Wrobel, W., Fornalik-Wajs, E., and Szmyd, J.S. \Experimental and numerical analysis of thermo-magnetic convection in a vertical annular enclosure", International Journal of Heat Fluid Flow, 31(6), pp. 1019{ 1031 (2010). 31. Shakiba, A. and Rahimi, A.B. \Nanouidow and MHD mixed convection inside a vertical annulus with moving walls and transpiration consideringthe effect of Brownian motion and shape factor", Int. J. of Thermal Analysis and Calorimetry, 138, pp. 501{515 (2019). References 1. Wu, T.-H., Xu, Z., and Jackson, J. \Mixed convectionheat transfer to waterowing through a verticalpassage of annular cross section: part 2", Chemical Engineering Research and Design, 80(3), pp. 246{251(2002).2. Joshi, H.M. \Fully developed natural convection in an isothermal vertical annular duct", InternationalCommunications in Heat and Mass Transfer, 14(6), pp. 657{664 (1987).3. Chen, S., Tolke, J., and Krafczyk, M. \Numerical investigationof double-di

usive (natural) convection in vertical annuluses with opposing temperature and concentrationgradients", International Journal of Heat and Fluid Flow, 31(2), pp. 217{226 (2010).4. Iannello, V., Suh, K.Y., and Todreas, N.E. \Mixed convection friction factors and Nusselt numbers invertical annular and subchannel geometries", International Journal of Heat and Mass Transfer, 31(10), pp.2175{2189 (1988).5. Malvandi, A., Moshizi, S., Soltani, E.G., and Ganji, D. \Modied Buongiorno's model for fully developedmixed convectionflow of nanofluids in a verticalannular pipe", Computers & Fluids, 89, pp. 124{132 (2014).6. Avc, M. and Ayd, O. \Mixed convection in a verticalmicroannulus between two concentric microtubes", Journal of Heat Transfer, 131(1), p. 014502 (2009).7. Jha, B.K. and Aina, B. \Mathematical modelling and exact solution of steady fully developed mixedconvection

usive (natural) convection in vertical annuluses with opposing temperature and concentrationgradients", International Journal of Heat and Fluid Flow, 31(2), pp. 217{226 (2010).4. Iannello, V., Suh, K.Y., and Todreas, N.E. \Mixed convection friction factors and Nusselt numbers invertical annular and subchannel geometries", International Journal of Heat and Mass Transfer, 31(10), pp.2175{2189 (1988).5. Malvandi, A., Moshizi, S., Soltani, E.G., and Ganji, D. \Modied Buongiorno's model for fully developedmixed convectionflow of nanofluids in a verticalannular pipe", Computers & Fluids, 89, pp. 124{132 (2014).6. Avc, M. and Ayd, O. \Mixed convection in a verticalmicroannulus between two concentric microtubes", Journal of Heat Transfer, 131(1), p. 014502 (2009).7. Jha, B.K. and Aina, B. \Mathematical modelling and exact solution of steady fully developed mixedconvection

ow in a vertical micro-porous-annulus", Afrika Matematika, 26(7{8), pp. 1199{1213 (2015).8. Jha, B.K., Aina, B., and Muhammad, S. \Combined e

ects of suction/injection and wall surface curvatureon natural convection

ects of suction/injection and wall surface curvatureon natural convection

ow in a vertical micro-porous annulus", Thermophysics and Aeromechanics, 22(2),pp. 217{228 (2015).9. Husain, S. and Siddiqui, M.A. \Experimental and numerical analysis of transient natural convection ofwater in a high aspect ratio narrow vertical annulus", Progress in Nuclear Energy, 106, pp. 1{10 (2018).10. Husain, S., Siddiqui, M.A., and Khan, S.A. \E

ect of geometrical parameters on natural convection of waterin a narrow annulus", Progress in Nuclear Energy, 112, pp. 146{161 (2019).11. Husain, S. and Siddiqui, M.A. \Experimental andnumerical analyses of natural convectionflow in a partially heated vertical annulus", Numerical HeatTransfer, Part A: Applications, 70(7), pp. 763{775(2016). 12. Husain, S. and Siddiqui, M.A. \Numerical and experimentalanalysis of natural convectionow boiling of water in internally heated vertical annulus", NumericalHeat Transfer, Part A: Applications, 73(9), pp. 624{ 653 (2018).13. Mustafa, J., Husain, S., and Siddiqui, M.A.J.E.H.T.\Experimental studies on natural convection of water in a closed-loop vertical annulus", Experimental HeatTransfer an International Journal, 30(1), pp. 25{45 (2017).14. Huang, S. and Chun, C.H. \A numerical study of turbulentflow and conjugate heat transfer in concentric annuli with moving inner rod", International Journalof Heat and Mass Transfer, 46(19), pp. 3707{3716 (2003).15. Lin, S. and Hsieh, D. \Heat transfer to generalizedCouetteow of a non-Newtonianuid in annuli with moving inner cylinder", Journal of Heat Transfer,102(4), pp. 786{789 (1980).16. Stra

ord, K.N., Datta, P.K., and Coogan, C., Coatingsand Surface Treatment for Corrosion and Wear Resistance, Published for Institution of Corrosion Scienceand Technology by Ellis Horwood, Market Cross House, Cooper Street, Chichester, West Sussex PO 191 EB, England, 1984. 362 (1984).17. Shigechi, T. and Lee, Y. \An analysis on fully developedlaminarfluidflow and heat transfer in concentric annuli with moving cores", International Journalof Heat and Mass Transfer, 34(10), pp. 2593{2601 (1991).18. Chamkha, A.J. \Unsteady MHD convective heat andmass transfer past a semi-innite vertical permeable moving plate with heat absorption", InternationalJournal of Engineering Science, 42(2), pp. 217{230 (2004).19. Abedini, A. and Rahimi, A.B. \Numerical study ofmixed convection in an annulus between concentric rotating cylinders with time-dependent angular velocity",Iranian Journal of Science and Technology-Transactions of Mechanical Engineering, 36, pp. 165{ 180 (2012).20. Saleh, R. and Rahimi, A.B. \Axisymmetric stagnationpointow and heat transfer of a viscousuid on a movingcylinder with time-dependent axial velocity and uniform transpiration", Journal of FluidsEngineering,126(6), pp. 997{1005 (2004).21. Jha, B., Joseph, S., and Ajibade, A. \Transient freeconvective

ect of geometrical parameters on natural convection of waterin a narrow annulus", Progress in Nuclear Energy, 112, pp. 146{161 (2019).11. Husain, S. and Siddiqui, M.A. \Experimental andnumerical analyses of natural convectionflow in a partially heated vertical annulus", Numerical HeatTransfer, Part A: Applications, 70(7), pp. 763{775(2016). 12. Husain, S. and Siddiqui, M.A. \Numerical and experimentalanalysis of natural convectionow boiling of water in internally heated vertical annulus", NumericalHeat Transfer, Part A: Applications, 73(9), pp. 624{ 653 (2018).13. Mustafa, J., Husain, S., and Siddiqui, M.A.J.E.H.T.\Experimental studies on natural convection of water in a closed-loop vertical annulus", Experimental HeatTransfer an International Journal, 30(1), pp. 25{45 (2017).14. Huang, S. and Chun, C.H. \A numerical study of turbulentflow and conjugate heat transfer in concentric annuli with moving inner rod", International Journalof Heat and Mass Transfer, 46(19), pp. 3707{3716 (2003).15. Lin, S. and Hsieh, D. \Heat transfer to generalizedCouetteow of a non-Newtonianuid in annuli with moving inner cylinder", Journal of Heat Transfer,102(4), pp. 786{789 (1980).16. Stra

ord, K.N., Datta, P.K., and Coogan, C., Coatingsand Surface Treatment for Corrosion and Wear Resistance, Published for Institution of Corrosion Scienceand Technology by Ellis Horwood, Market Cross House, Cooper Street, Chichester, West Sussex PO 191 EB, England, 1984. 362 (1984).17. Shigechi, T. and Lee, Y. \An analysis on fully developedlaminarfluidflow and heat transfer in concentric annuli with moving cores", International Journalof Heat and Mass Transfer, 34(10), pp. 2593{2601 (1991).18. Chamkha, A.J. \Unsteady MHD convective heat andmass transfer past a semi-innite vertical permeable moving plate with heat absorption", InternationalJournal of Engineering Science, 42(2), pp. 217{230 (2004).19. Abedini, A. and Rahimi, A.B. \Numerical study ofmixed convection in an annulus between concentric rotating cylinders with time-dependent angular velocity",Iranian Journal of Science and Technology-Transactions of Mechanical Engineering, 36, pp. 165{ 180 (2012).20. Saleh, R. and Rahimi, A.B. \Axisymmetric stagnationpointow and heat transfer of a viscousuid on a movingcylinder with time-dependent axial velocity and uniform transpiration", Journal of FluidsEngineering,126(6), pp. 997{1005 (2004).21. Jha, B., Joseph, S., and Ajibade, A. \Transient freeconvective

ow through a vertical porous annulus", Proceedings of the Institution of Mechanical Engineers,Part E: Journal of Process Mechanical Engineering,226(2), pp. 105{116 (2012). A. Shakiba and A.B. Rahimi/Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 1306{1317 1317 22. Jha, B., Samaila, A., Ajibade, A. \Unsteady/steadynatural convection

ow of reactive viscousuid in a vertical annulus", International Journal of AppliedMechanics Engineering, 18(1), pp. 73{83 (2013). 23. Shakiba, A. and Rahimi, A.B. \Mixed convectiveowof electrically conductinguid in a vertical cylindrical annulus with moving walls adjacent to a radial magneticeld along with transpiration", Journal of Fluids Engineering, 141, pp. 1{11 (2019).24. Busedra, A. and Tavoularis, S. \Numerical simulationsof natural convective heat transfer in vertical concentric and eccentric annular channels", Journal of HeatTransfer, 140(10), p. 102502 (2018). 25. Hekmat, M.H. and Ziarati, K.K. \Eects of nanoparticlesvolume fraction and magnetic eld gradient on the mixed convection of a ferrouid in the annulus betweenvertical concentric cylinders", Applied Thermal Engineering, 152, pp. 844{857 (2019).26. Jha, B. and Oni, M. \Natural convectionow in a vertical annulus with time-periodic thermal boundary conditions", Propulsion Power Research, 8(1), pp. 47{ 55 (2019). 27. Jha, B.K. and Malgwi, P.B. \E

ects of Hall current and magneticeld inclination on hydromagnetic natural convectionow in a micro-channel with asymmetric thermal boundary condition", Journal of Thermal Science Engineering Applications, 12(3) (2020). 28. Abbas, Z., Mehdi, I., Hasnain, J., and Aly, S. \Role of suction/injection on natural convectionow of magnetite Fe3O4 nanoparticles in vertical porous micro-annulus between two concentric tubes: A purely analytical approach", Arabian Journal for Science, 44(9), pp. 8113{8122 (2019). 29. Jha, B.K., Oni, M.O., and Aina, B. \Steady fully developed mixed convectionow in a vertical microconcentric- annulus withheatgenerating/absorbinguid: an exact solution", Ain Shams Engineering Journal, 9(4), pp. 1289{1301 (2018). 30. Wrobel, W., Fornalik-Wajs, E., and Szmyd, J.S. \Experimental and numerical analysis of thermo-magnetic convection in a vertical annular enclosure", International Journal of Heat Fluid Flow, 31(6), pp. 1019{ 1031 (2010). 31. Shakiba, A. and Rahimi, A.B. \Nanouidow and MHD mixed convection inside a vertical annulus with moving walls and transpiration considering theffect of Brownian motion and shape factor", Int. J. of Thermal Analysis and Calorimetry, 138, pp. 501{515 (2019).

ects of Hall current and magneticeld inclination on hydromagnetic natural convectionow in a micro-channel with asymmetric thermal boundary condition", Journal of Thermal Science Engineering Applications, 12(3) (2020). 28. Abbas, Z., Mehdi, I., Hasnain, J., and Aly, S. \Role of suction/injection on natural convectionow of magnetite Fe3O4 nanoparticles in vertical porous micro-annulus between two concentric tubes: A purely analytical approach", Arabian Journal for Science, 44(9), pp. 8113{8122 (2019). 29. Jha, B.K., Oni, M.O., and Aina, B. \Steady fully developed mixed convectionow in a vertical microconcentric- annulus withheatgenerating/absorbinguid: an exact solution", Ain Shams Engineering Journal, 9(4), pp. 1289{1301 (2018). 30. Wrobel, W., Fornalik-Wajs, E., and Szmyd, J.S. \Experimental and numerical analysis of thermo-magnetic convection in a vertical annular enclosure", International Journal of Heat Fluid Flow, 31(6), pp. 1019{ 1031 (2010). 31. Shakiba, A. and Rahimi, A.B. \Nanouidow and MHD mixed convection inside a vertical annulus with moving walls and transpiration considering theffect of Brownian motion and shape factor", Int. J. of Thermal Analysis and Calorimetry, 138, pp. 501{515 (2019).

Transactions on Mechanical Engineering (B)

May and June 2021Pages 1306-1317