Exact analysis of heat convection of viscoelastic FENE-P fluids through isothermal slits and tubes

Abstract

In this article, two exact analytical solutions for heat convection in viscoelastic fluid flow through isothermal tubes and slits are presented for the first time. Herein, a Peterlin type of finitely extensible nonlinear elastic (FENE-P) model is used as the nonlinear constitutive equation for the viscoelastic fluid. Due to the eigenvalue form of the heat transfer equation, the modal analysis technique has been used to determine the physical temperature distributions. The closed form solution for the temperature profile is obtained in terms of a Heun Tri-confluent function for slit flow and then the Frobenius method is used to evaluate the temperature distribution for the tube flow. Based on these solutions, the effects of extensibility parameter and Deborah number on thermal convection in FENE-P fluid flow have been studied in detail. The fractional correlations for reduced Nusselt number in terms of material modulus are also derived. Here, it is shown that by increasing the Deborah number from 0 to 100, the Nusselt number is enhanced by 8.5 and 13.5% for slit and tube flow, respectively.

Appendices

Appendix A: Heun functions

Heun functions are one of the closed form solutions for particular ODEs in mathematics and there are four standard forms, namely HeunB, HeunC, HeunD and HeunT which correspond to Biconfluent, Confluent, Doubleconfluent and Triconfluent Heun equations. HeunT function is the solution for the linear differential equation of second order given by:

$$ \frac{{{\text{d}}^{2} }}{{{\text{d}}y^{2} }}f(y) - \left( {3{\mkern 1mu} y^{2} + \gamma } \right)\frac{\text{d}}{{{\text{d}}y}}f(y) - \left( {\left( { - \beta + 3} \right)y - \alpha } \right)f(y) = 0 $$

(29)

in which all of the four parameters, $\left( {\alpha ,\beta ,\gamma ,y} \right)$, are algebraic expressions. By solving the Eq. (29) the closed form solution (HeunT) would be derived as follows:

$$ f(y) = C_{1} HeunT\left( {\alpha ,\beta ,\gamma ,y} \right) + C_{2} {\mkern 1mu} HeunT\left( {\alpha , - \beta ,\gamma , - y} \right){\text{e}}^{{y(y^{2} + \gamma )}} . $$

(30)

Now based on the boundary conditions introduced in Eqs. (31) and (32), the second term in Eq. (30) will vanish, so that Eq. (30) will be simplified to the one given by Eq. (33).

$$ at\,\,\,\,\,y = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,f_{,y} = 0 $$

(31)

$$ at\,\,\,\,\,y = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,f = 0 $$

(32)

$$ f(y) = C_{1} HeunT\left( {\alpha ,\beta ,\gamma ,y} \right). $$

(33)

Furthermore, the HeunT function could be written in the series solution form. Since the single singularity is located at infinity, this series converges into the entire complex plane. The Handbook of Mathematical functions prepared by the National Institute of Standards and Technology (NIST), Maryland, USA, is an excellent reference for Heun functions.

Appendix B: The Nusselt number and constants of temperature profile

In this section, the polynomial equation of the Nusselt number of FENE-P flow in an isothermal tube has been presented. This polynomial is obtained by applying the boundary condition (14b) to Eq. (24) and considering C ₂  = 0. According to the scaling law in heat convection, the Nusselt number is the first order root of the following algebraic equation:

$$ \begin{aligned} - (\frac{{U_{N} }}{U})^{7} \left( \begin{aligned} \frac{1}{{3251404800{\mkern 1mu} {\kern 1pt} }} + \frac{1}{{29030400{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{2} + \frac{1}{{604800{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{4} + \frac{1}{{22680{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{6} \hfill \\ +\,\frac{{2{\mkern 1mu} {\kern 1pt} }}{{2835{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{8} + \frac{{32{\mkern 1mu} {\kern 1pt} }}{{4725{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{10} + \frac{{512{\mkern 1mu} }}{{14175{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{12} + \frac{{8192{\mkern 1mu} }}{{99225{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{14} \hfill \\ \end{aligned} \right)Nu^{7} . \hfill \\ +\,(\frac{{U_{N} }}{U})^{6} \left( \begin{aligned} - \frac{1}{{38707200{\mkern 1mu} }} - \frac{{23{\mkern 1mu} }}{{14515200{\mkern 1mu} }}(\frac{{DeU_{N} }}{LUa})^{2} - \frac{1}{{36288{\mkern 1mu} }}(\frac{{DeU_{N} }}{LUa})^{4} + \frac{1}{{5670{\mkern 1mu} {\kern 1pt} L^{6} U^{12} a^{6} }}(\frac{{DeU_{N} }}{LUa})^{6} + \hfill \\ \frac{{32{\mkern 1mu} {\kern 1pt} }}{{2835{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{8} + \frac{{1856{\mkern 1mu} {\kern 1pt} }}{{14175{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{10} + \frac{{1024{\mkern 1mu} {\kern 1pt} }}{{2025{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{12} \hfill \\ \end{aligned} \right)Nu^{6} \hfill \\ - (\frac{{U_{N} }}{U})^{5} \left( \begin{aligned} \frac{1}{{1382400{\mkern 1mu} {\kern 1pt} }} + \frac{{53{\mkern 1mu} {\kern 1pt} }}{{2419200{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{2} + \frac{{13{\mkern 1mu} {\kern 1pt} }}{{453600{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{4} - \frac{1}{{378{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{6} \hfill \\ +\,\frac{{8{\mkern 1mu} {\kern 1pt} }}{{525{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{8} + \frac{{1024{\mkern 1mu} {\kern 1pt} }}{{2835{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{10} \hfill \\ \end{aligned} \right)Nu^{5} \hfill \\ +\, (\frac{{U_{N} }}{U})^{4} \left( {\frac{{4631{\mkern 1mu} {\kern 1pt} }}{{203212800{\mkern 1mu} {\kern 1pt} }} + \frac{{51049{\mkern 1mu} {\kern 1pt} }}{{25401600{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{2} + \frac{{28789{\mkern 1mu} {\kern 1pt} }}{{529200{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{4} + \frac{{60064{\mkern 1mu} {\kern 1pt} }}{{99225{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{6} + \frac{{5056{\mkern 1mu} {\kern 1pt} }}{{2025{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{8} } \right)Nu^{4} \hfill \\-\, (\frac{{U_{N} }}{U})^{3} \left( {\frac{{59{\mkern 1mu} {\kern 1pt} }}{{51200{\mkern 1mu} {\kern 1pt} }} + \frac{{107003{\mkern 1mu} {\kern 1pt} }}{{1587600{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{2} + \frac{{26126{\mkern 1mu} {\kern 1pt} }}{{19845{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{4} + \frac{{837248{\mkern 1mu} {\kern 1pt} }}{{99225{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{6} } \right)Nu^{3} \hfill \\ +\,(\frac{{U_{N} }}{U})^{2} \left( {\frac{{73{\mkern 1mu} {\kern 1pt} }}{{2304{\mkern 1mu} {\kern 1pt} }} + \frac{{551{\mkern 1mu} {\kern 1pt} }}{{450{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{2} + \frac{{952{\mkern 1mu} {\kern 1pt} }}{{81{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{4} } \right)Nu^{2} - {\kern 1pt} \frac{{U_{N} }}{U}\left( {\frac{3}{8}{\mkern 1mu} + \frac{{64{\mkern 1mu} }}{{9{\mkern 1mu} {\kern 1pt} }}(\frac{{DeU_{N} }}{LUa})^{2} } \right)Nu + 1 = 0 \hfill \\ \end{aligned} $$

(34)

The nonzero constant of temperature distribution (C ₁ ) can be determined using the constraint presented in Eq. (15), for j = 1, as follows:

$$ \begin{aligned} C_{1}^{ - 1} & = T_{\hbox{max} }^{ - 1} = \frac{{4096{\mkern 1mu} {\kern 1pt} }}{{315{\mkern 1mu} {\kern 1pt} }}Nu^{4} \left( {\frac{De}{aL}} \right)^{10} \left( {\frac{{U_{N} }}{U}} \right)^{15} + \frac{704}{189}Nu^{4} \left( {\frac{De}{aL}} \right)^{8} \left( {\frac{{U_{N} }}{U}} \right)^{13} - \frac{1024}{27}Nu^{3} \left( {\frac{De}{aL}} \right)^{8} \left( {\frac{{U_{N} }}{U}} \right)^{12} \\ & \quad + \frac{80}{189}Nu^{4} \left( {\frac{De}{aL}} \right)^{6} \left( {\frac{{U_{N} }}{U}} \right)^{11} - \frac{{64{\mkern 1mu} {\kern 1pt} }}{{9{\mkern 1mu} {\kern 1pt} }}Nu^{3} \left( {\frac{De}{aL}} \right)^{6} \left( {\frac{{U_{N} }}{U}} \right)^{10} \\ & \quad + \left( {\frac{1}{42}{\mkern 1mu} {\kern 1pt} \left( {\frac{NuDe}{aL}} \right)^{4} + \frac{{18944{\mkern 1mu} {\kern 1pt} }}{{315{\mkern 1mu} {\kern 1pt} }}Nu^{2} \left( {\frac{De}{aL}} \right)^{6} } \right)\left( {\frac{{U_{N} }}{U}} \right)^{9} - \frac{13}{27}Nu^{3} \left( {\frac{De}{aL}} \right)^{4} \left( {\frac{{U_{N} }}{U}} \right)^{8} \\ & \quad + Nu^{2} \left( {\frac{1}{{1512{\mkern 1mu} {\kern 1pt} }}\left( {\frac{NuDe}{aL}} \right)^{2} + \frac{{312{\mkern 1mu} {\kern 1pt} }}{{35{\mkern 1mu} }}\left( {\frac{De}{aL}} \right)^{4} } \right)\left( {\frac{{U_{N} }}{U}} \right)^{7} + Nu{\mkern 1mu} \left( { - \frac{1}{{72{\mkern 1mu} }}\left( {\frac{NuDe}{aL}} \right)^{2} - \frac{1664}{27}\left( {\frac{De}{aL}} \right)^{4} } \right)\left( {\frac{{U_{N} }}{U}} \right)^{6} \\ & \quad + \left( {\frac{1}{138240}Nu^{4} + \frac{37}{84}\left( {\frac{NuDe}{aL}} \right)^{2} } \right)\left( {\frac{{U_{N} }}{U}} \right)^{5} + \left( { - \frac{1}{6912}Nu^{3} - \frac{{272{\kern 1pt} }}{45}Nu\left( {\frac{De}{aL}} \right)^{2} } \right)\left( {\frac{{U_{N} }}{U}} \right)^{4} \\ & \quad + \left( {\frac{83}{11520}Nu^{2} + \frac{{64{\kern 1pt} }}{3}\left( {\frac{De}{aL}} \right)^{2} } \right)\left( {\frac{{U_{N} }}{U}} \right)^{3} - \frac{{7{\mkern 1mu} {\kern 1pt} {\mkern 1mu} {\kern 1pt} }}{{48{\mkern 1mu} {\kern 1pt} }}Nu\left( {\frac{{U_{N} }}{U}} \right)^{2} + \frac{{U_{N} }}{U}. \\ \end{aligned} $$

(35)