Analysis of forced convection of Phan–Thien–Tanner fluid in slits and tubes of constant wall temperature with viscous dissipation

  • S. Z. DaghighiEmail author
  • M. Norouzi
Technical Paper


In the present work, analytical solutions are presented for thermal convection of the linear Phan–Thien–Tanner fluid (LPTT) in slits and tubes of constant wall temperature by taking account of the viscous dissipation term. Unlike the similar previous studies in which the advection term was neglected in the heat transfer equation, it is considered in this investigation. A continuous relation between the Nusselt number and the Brinkman number is obtained. Expressions for the temperature distribution are derived in closed form and in terms of a Frobenius series for the slit and tube flows, respectively. Based on these solutions, the effects of fluid elasticity and Brinkman number on thermal convection of LPTT fluid flows are studied in detail. It is shown that at negative Brinkman numbers (fluid cooling), increasing the Deborah number leads to a decrease in the Nusselt number, but an increase in the centerline temperature. Nonetheless, this trend is opposite for positive Brinkman numbers (fluid heating), i.e., an increase in the Nusselt number and a decrease in the centerline temperature. Also, there is a Brinkman number beyond which the Nusselt number is smaller than zero, meaning that there is weak heat convection in the flow. Also, the results confirm that the extensibility parameter affects the temperature profile in the same way as the Deborah number.


Viscoelastic fluid Phan–Thien–Tanner model Forced convection Constant wall temperature tube and slit Viscous dissipation 

List of symbols

\( A \)

Area of cross section, \( {\text{m}}^{2} \)

\( {\text{Br}} \)

Brinkman number, \( {\text{Br}} = \eta U^{2} /k(\tilde{T}_{\text{w}} - \tilde{T}_{\text{m}} ) \)

\( c_{\text{p}} \)

Specific heat, J kg−1 K−1

\( d_{\text{h}} \)

Hydraulic diameter, \( d_{\text{h}} = 2R \) for tube flow and \( d_{\text{h}} = 4H \) for slit flow, \( {\text{m}} \)

\( {\mathbf{D}} \)

Deformation rate tensor, Eq. (8)


Deborah number, defined as \( {\text{De}} = \,\lambda U/R \) for tube case and \( {\text{De}} = \,\lambda U/H \) for slit case


Dimensionless function

\( h \)

Heat transfer coefficient, W m−2 K−1

\( H \)

Half of the distance between two parallel plates, \( {\text{m}} \)


Equals 0 for slit and 1 for tube

\( k \)

Conductivity coefficient, W m−1 K−1

\( K \)

Equals 1.5 and 2 for slit and tube, respectively

\( {\text{Nu}} \)

Nusselt number, \( {\text{Nu}} = hd_{\text{h}} /k \)

\( p \)

Pressure, \( {\text{Pa}} \)

\( P \)

Perimeter of cross section, \( {\text{m}} \)

\( R \)

Tube radius, \( {\text{m}} \)

\( T \)

Fluid temperature, \( {\text{K}} \)

\( u \)

Axial velocity, \( {\text{ms}}^{ - 1} \)


Mean velocity, \( {\text{ms}}^{ - 1} \)

\( {\mathbf{V}} \)

Velocity vector

\( y \)

Radial (tube) or transverse (slit) direction, \( {\text{m}} \)

\( z \)

Axial direction, \( {\text{m}} \)

Greek symbols

\( \alpha \)

Equals 1 for pure entropy elasticity and 0 for pure energy elasticity

\( \alpha_{1} \)

\( \alpha_{1} { = }1.5_{{}} U_{\text{N}} /U \) for slit and \( \alpha_{1} { = }2\,U_{\text{N}} /U \) for tube

\( \delta \)

Constant, Eq. (12)

\( \varepsilon \)

Extensibility coefficient

\( \eta \)

Constant viscosity coefficient, \( {\text{Pa s}} \)

\( \lambda \)

Relaxation time, \( {\text{s}} \)

\( \rho \)

Density, Kg m−3

\( \mathop {\varvec{\uptau}}\limits_{{}}^{\nabla } \)

Upper-convected derivative of stress tensor, Eq. (9)

\( \phi \)

Viscous dissipation, Eq. (15)



Mean value


Maximum value


Newtonian case





Transpose operator


Dimensional parameter


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Barletta A, di Schio ER, Zanchini E (2003) Combined forced and free flow in a vertical rectangular duct with prescribed wall heat flux. Int J Heat Fluid Flow 24:874–887CrossRefGoogle Scholar
  2. 2.
    Chang SW, Yang TL, Huang RF, Sung KC (2007) Influence of channel-height on heat transfer in rectangular channels with skewed ribs at different bleed conditions. Int J Heat Mass Transf 50:4581–4599CrossRefGoogle Scholar
  3. 3.
    Sayed-Ahmed M, Kishk KM (2008) Heat transfer for Herschel–Bulkley fluids in the entrance region of a rectangular duct. Int Commun Heat Mass Transf 35:1007–1016CrossRefGoogle Scholar
  4. 4.
    Norouzi M, Kayhani M, Nobari M (2009) Mixed and forced convection of viscoelastic materials in straight duct with rectangular cross section. World Appl Sci J 7:285–296Google Scholar
  5. 5.
    Rao SS, Ramacharyulu NCP, Krishnamurty V (1969) Laminar forced convection in elliptic ducts. Appl Sci Res 21:185–193CrossRefGoogle Scholar
  6. 6.
    Bhatti M (1984) Heat transfer in the fully developed region of elliptical ducts with uniform wall heat flux. J Heat Transf 106:895–898CrossRefGoogle Scholar
  7. 7.
    Abdel-Wahed R, Attia A, Hifni M (1984) Experiments on laminar flow and heat transfer in an elliptical duct. Int J Heat Mass Transf 27:2397–2413CrossRefGoogle Scholar
  8. 8.
    Sakalis V, Hatzikonstantinou P, Kafousias N (2002) Thermally developing flow in elliptic ducts with axially variable wall temperature distribution. Int J Heat Mass Transf 45:25–35CrossRefGoogle Scholar
  9. 9.
    Maia CRM, Aparecido JB, Milanez LF (2006) Heat transfer in laminar flow of non-Newtonian fluids in ducts of elliptical section. Int J Therm Sci 45:1066–1072CrossRefGoogle Scholar
  10. 10.
    Oliveira PJ, Pinho FT (1999) Analytical solution for fully developed channel and pipe flow of Phan–Thien–Tanner fluids. J Fluid Mech 387:271–280MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pinho F, Oliveira P (2000) Analysis of forced convection in pipes and channels with the simplified Phan–Thien–Tanner fluid. Int J Heat Mass Transf 43:2273–2287CrossRefGoogle Scholar
  12. 12.
    Coelho P, Pinho F, Oliveira P (2002) Fully developed forced convection of the Phan–Thien–Tanner fluid in ducts with a constant wall temperature. Int J Heat Mass Transf 45:1413–1423CrossRefGoogle Scholar
  13. 13.
    Pinho F, Coelho P (2006) Fully-developed heat transfer in annuli for viscoelastic fluids with viscous dissipation. J Nonnewton Fluid Mech 138:7–21CrossRefGoogle Scholar
  14. 14.
    Norouzi M (2016) Analytical solution for the convection of Phan–Thien–Tanner fluids in isothermal pipes. Int J Therm Sci 108:165–173CrossRefGoogle Scholar
  15. 15.
    Anand V (2016) Effect of slip on heat transfer and entropy generation characteristics of simplified Phan–Thien–Tanner fluids with viscous dissipation under uniform heat flux boundary conditions: exponential formulation. Appl Therm Eng 98:455–473CrossRefGoogle Scholar
  16. 16.
    Matías A, Sánchez S, Méndez F, Bautista O (2015) Influence of slip wall effect on a non-isothermal electro-osmotic flow of a viscoelastic fluid. Int J Therm Sci 98:352–363CrossRefGoogle Scholar
  17. 17.
    Khan M, Hussain A, Malik M, Salahuddin T, Aly S (2019) Numerical analysis of Carreau fluid flow for generalized Fourier’s and Fick’s laws. Appl Numer Math 144:100–117MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hussain A, Malik MY, Khan M, Salahuddin T (2019) Application of generalized Fourier heat conduction law on MHD viscoinelastic fluid flow over stretching surface. Int J Numer Methods Heat Fluid Flow. CrossRefGoogle Scholar
  19. 19.
    Khan M, Salahuddin T, Malik M, Khan F (2019) Arrhenius activation in MHD radiative Maxwell nanoliquid flow along with transformed internal energy. Eur Phys J Plus 134:198CrossRefGoogle Scholar
  20. 20.
    Khan M, Malik M, Salahuddin T, Saleem S, Hussain A (2019) Change in viscosity of Maxwell fluid flow due to thermal and solutal stratifications. J Mol Liq 288:110970CrossRefGoogle Scholar
  21. 21.
    Salahuddin T, Muhammad S, Sakinder S (2019) Impact of generalized heat and mass flux models on Darcy–Forchheimer Williamson nanofluid flow with variable viscosity. Phys Scr 94:125201CrossRefGoogle Scholar
  22. 22.
    Khan M, Salahuddin T, Malik M (2019) Implementation of Darcy-Forchheimer effect on magnetohydrodynamic Carreau-Yasuda nanofluid flow: application of Von Kármán. Can J Phys 97:670–677CrossRefGoogle Scholar
  23. 23.
    Tanveer A, Salahuddin T (2019) Emission of electromagnetic waves from walls of esophagus under domination of wall slip effect. Chaos Solitons Fractals 127:110–117MathSciNetCrossRefGoogle Scholar
  24. 24.
    Salahuddin T, Tanveer A, Malik M (2019) Homogeneous-heterogeneous reaction effects in flow of tangent hyperbolic fluid on a stretching cylinder. Can J Phys. CrossRefGoogle Scholar
  25. 25.
    Salahuddin T, Arif A, Haider A, Malik M (2018) Variable fluid properties of a second-grade fluid using two different temperature-dependent viscosity models. J Braz Soc Mech Sci Eng 40:575CrossRefGoogle Scholar
  26. 26.
    Haider A, Salahuddin T, Malik M (2018) Change in conductivity of magnetohydrodynamic Darcy-Forchheimer second grade fluid flow due to variable thickness surface. Can J Phys 97(8):809–815CrossRefGoogle Scholar
  27. 27.
    Cruz D, Pinho FTD, Oliveira P (2005) Analytical solutions for fully developed laminar flow of some viscoelastic liquids with a Newtonian solvent contribution. J Non Newton Fluid Mech 132:28–35CrossRefGoogle Scholar
  28. 28.
    Norouzi M, Daghighi S, Bég OA (2018) Exact analysis of heat convection of viscoelastic FENE-P fluids through isothermal slits and tubes. Meccanica 53:817–831MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ou J-W, Cheng K (1974) Viscous dissipation effects on thermal entrance heat transfer in laminar and turbulent pipe flows with uniform wall temperature. In: AIAA/ASME 1974 thermophysics and heat transfer conferenceGoogle Scholar
  30. 30.
    Basu T, Roy D (1985) Laminar heat transfer in a tube with viscous dissipation. Int J Heat Mass Transf 28:699–701CrossRefGoogle Scholar
  31. 31.
    Oliveira P, Coelho P, Pinho F (2004) The Graetz problem with viscous dissipation for FENE-P fluids. J Non Newton Fluid Mech 121:69–72CrossRefGoogle Scholar
  32. 32.
    Bejan A (2013) Convection heat transfer. Wiley, New YorkCrossRefGoogle Scholar
  33. 33.
    Thien NP, Tanner RI (1977) A new constitutive equation derived from network theory. J Non Newton Fluid Mech 2:353–365CrossRefGoogle Scholar
  34. 34.
    Peters GW, Baaijens FP (1997) Modelling of non-isothermal viscoelastic flows. J Non Newton Fluid Mech 68:205–224CrossRefGoogle Scholar
  35. 35.
    Eckert ERG, Drake RM Jr (1987) Analysis of heat and mass transfer. Tata McGraw Hill, New DelhizbMATHGoogle Scholar
  36. 36.
    Nóbrega J, Pinho FTD, Oliveira P, Carneiro O (2004) Accounting for temperature-dependent properties in viscoelastic duct flows. Int J Heat Mass Transf 47:1141–1158CrossRefGoogle Scholar
  37. 37.
    Kays WM, Crawford ME, Weigand B (2012) Convective heat and mass transfer. Tata McGraw-Hill Education, New YorkGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringShahrood University of TechnologyShahroodIran

Personalised recommendations