1 Introduction

It is crucial to understand clearly the physics of heat transfer in non-Newtonian flows because it affects the quality of products in polymer processing industries.

Forced heat convection of non-Newtonian fluids in ducts has been increasingly investigated. Some authors [1,2,3,4,5,6,7,8,9] have studied heat transfer of fluids through ducts by neglecting the viscous dissipation effect. One of the earliest studies of viscoelastic fluid flow obeying the Phan–Thien–Tanner (PTT) model was conducted by Oliveira and Pinho [10]. They presented hydrodynamic solutions to the problem of fully developed laminar flow of the PTT model through channels and pipes. Their results revealed that by increasing the dimensionless Deborah number, the velocity profiles were blunt in the center of the geometries. Pinho and Oliveira [11] investigated heat transfer of fully developed LPTT fluid in pipes and channels whose walls had a constant heat flux. They derived exact formulae for the temperature distribution in the presence of the viscous dissipation term. They reported that the heat transfer enhances by increasing either the Deborah number or the extensibility parameter. The problem of fully developed forced convection of the simplified PTT fluid in ducts with a constant wall temperature was analyzed semi-analytically by Coelho et al. [12]. In fact, they considered two cases in their study. The first case was the equilibrium between the axial convection and the radial conduction of thermal energy transfer equation, while the second case referred to the equilibrium between the radial conduction and the viscous dissipation terms. In the former case, Br = 0, their results showed that the Nusselt number increased monotonically when the fluid elasticity increased in the range of \( 0.001 \le \varepsilon {\text{We}}^{2} \le 100 \). They also showed that for all nonzero values of the Brinkman number, the Nusselt number increased with the fluid elasticity. Pinho and Coelho [13] studied the problem of fully developed heat transfer of the SPTT fluid flow with viscous dissipation in annuli of wall subjected to either a constant wall heat flux or a constant wall temperature. They showed that for the case of constant wall heat flux, the fluid elasticity would increase the heat transfer, especially by viscous dissipation effect. Nonetheless, for the constant wall temperature boundary condition, at low amounts of the Brinkman number, the fluid elasticity decreases the heat transfer. Norouzi [14] studied a similar problem for forced convection of PTT fluids through isothermal pipes, without the viscous dissipation effect, and derived exact solutions for the temperature distribution. Anand [15] investigated the heat transfer and entropy generation characteristics of a viscoelastic fluid flow modeled by the exponential formulation of LPTT model. By presenting an analytical model, Matias et al. [16] investigated the influence of Joule heating on the slip velocity of viscoelastic fluids.

Khan et al. [17] studied numerically the bioconvection Carreau nanofluid flow over a paraboloid surface of revolution to calculate the results of generalized Fick’s and Fourier’s laws on nanoscale. Arif et al. [18] studied the application of generalized Fourier heat conduction law on viscoelastic fluid flow over stretching surface. To investigate the behavior of transformed internal energy in a magnetohydrodynamic Maxwell nanofluid flow, Khan et al. [19] conducted a theoretical study. Exploring the temperature-dependent viscosity of Maxwell fluid flow, Khan et al. [20] concluded that some parameters such as the thermal stratification one control the temperature distributions inside a stretching surface. The problem of impact of generalized heat and mass flux models on Darcy–Forchheimer Williamson nanofluid flow has been recently worked out by Salahuddin et al. [21]. In their investigation, the viscosity was assumed a dependent function of temperature, and they explicated the effectiveness of some parameters like the Reynolds number and Prandtl number on dimensionless velocity and temperature profiles. Khan et al. [22] studied the effect of Darcy–Forchheimer on magnetohydrodynamic Carreau–Yasuda nanofluid flow. Their results showed that the Weissenberg number led to deceleration of the tangential and radial velocities, while the power law index accelerated them. Tanveer and Salahuddin [23] classified the simultaneous significances of complaint walls and emission of electromagnetic waves from walls which conduct Eyring–Powell fluid. They reported various solutions for dimensionless temperature distributions and velocity. Homogeneous–heterogeneous reaction effects in the flow of tangent hyperbolic fluid on a stretching cylinder were studied by Salahuddin et al. [24]. The variable fluid properties of a second-grade fluid by taking account of two different temperature-dependent viscosity models were investigated by Salahuddin et al. [25]. In their work, they derived the Nusselt number as well as the skin friction in the vicinity of a sheet surface. Haider et al. [26] analyzed the characteristics of Darcy–Forchheimer second-grade fluid flow surrounded by a deformable sheet by taking account of both variable thermal conductivity and magnetohydrodynamics.

Cruz et al. [27] presented analytical solutions for fully developed pipe and channel flows containing a Newtonian solvent described by the PTT or the finitely extensible nonlinear elastic followed by the Peterlin approximation (FENE-P) model. Norouzi et al. [28] investigated the heat transfer characteristics for a FENE-P fluid flow through straight ducts. Also, the effect of the viscous dissipation on the thermal entrance region of non-laminar, non-Newtonian fluids is studied by some authors [29, 30]. By considering the viscous dissipation, Oliveira et al. [31] studied the Graetz problem for the FENE-P fluid in a pipe and a channel. Their investigation showed that the viscous dissipation tends to lower the Nusselt number, while the elasticity increases it.

The viscous dissipation appeared in the energy equation has an effective role in heat transfer characteristics. The present study aims to examine the LPTT viscoelastic fluid flowing through constant wall temperature slits and tubes by taking account of the viscous dissipation effect. The temperature distribution is derived as a closed-form function for the slit flow and a Frobenius series for the tube flow. In addition, the effects of the Brinkman number (both positive and negative values) and the fluid elasticity on heat transfer characteristics are discussed in detail.

As mentioned earlier, Coelho et al. [12] investigated forced convection of the LPTT fluid for two cases. But, they neglected the axial convection (advection) term for the case in which the viscous dissipation term was involved. Consequently, their result in terms of the variation of the Nusselt number via the fluid elasticity is different compared to that of the present study. Actually, the present work could be considered as the generalization of solution of Coelho et al. [12] via considering the axial convection into the heat transfer equation.

2 Governing equations and formulation

In continuum mechanics, the principles of mass conservation, momentum conservation and energy conservation must be obeyed. The continuity and the momentum equations are presented in Eqs. (1) and (2) from literature [32]

$$ \tilde{\nabla }.\,{\tilde{\mathbf{V}}} = 0 $$
(1)
$$ \rho \,{\tilde{\mathbf{V}}}.\tilde{\nabla }{\tilde{\mathbf{V}}} = - \tilde{\nabla }\tilde{p} + \tilde{\nabla }.{\tilde{\mathbf{\tau }}} $$
(2)

where \( {\tilde{\mathbf{V}}} \) is the velocity vector, \( \tilde{p} \) is the pressure which is supposed to be a linear function, \( \rho \) is the density and \( {\tilde{\mathbf{\tau }}} \) is the stress tensor. The constitutive equation applied here for the PTT fluid [33] can be written as Eq. (3) in which f is the dimensionless function, \( \lambda \) is the relaxation time, \( \eta \) is the constant viscosity coefficient, \( {\tilde{\mathbf{D}}} \) represents the deformation rate tensor (\( {\tilde{\mathbf{D}}} = (\tilde{\nabla }{\tilde{\mathbf{V}}} + \tilde{\nabla }{\tilde{\mathbf{V}}}^{T} )/2 \)) and \( \mathop {{\tilde{\mathbf{\tau }}}}\limits^{\nabla } \) indicates the Oldroyd’s upper-convected derivative of the stress tensor shown in Eq. (4):

$$ f(tr({\tilde{\mathbf{\tau }}})){\tilde{\mathbf{\tau }}} + \lambda \mathop {{\tilde{\mathbf{\tau }}}}\limits^{\nabla } = 2\eta {\tilde{\mathbf{D}}} $$
(3)
$$ \mathop {{\tilde{\mathbf{\tau }}}}\limits^{\nabla } = \frac{{D{\tilde{\mathbf{\tau }}}}}{Dt} - \left( {\tilde{\nabla }{\tilde{\mathbf{V}}}^{T} .{\tilde{\mathbf{\tau }}} + {\tilde{\mathbf{\tau }}}.\tilde{\nabla }{\tilde{\mathbf{V}}}} \right) $$
(4)

Since there is no exact solution for the exponential type of the PTT fluid, just the linear form of the PTT fluid is adopted here in, and the corresponding dimensionless function will be

$$ f\left( {tr({\tilde{\mathbf{\tau }}})} \right) = 1 + \frac{\varepsilon \lambda }{\eta }tr({\tilde{\mathbf{\tau }}}) $$
(5)

where \( \varepsilon \) is the fluid extensibility coefficient limiting the fluid extensional viscosity. Indeed, nonzero values of \( \varepsilon \) preclude the possibility of infinite extensional viscosity in a stretching flow in contrast to the upper-convected Maxwell (UCM) model possessing \( \varepsilon = 0 \).

The underlying assumptions for the present study are summarized as follows. The model parameters are independent of temperature, the flow is incompressible, rectilinear, laminar and hydrodynamically and thermally fully developed, and the viscoelastic fluid is simulated with the LPTT constitutive equation in two dimensions. Since the flow is hydrodynamically fully developed, the velocity profile (along the longitudinal axis) is only dependent on the vertical side, and since the flow is thermally fully developed, the axial conduction is negligible relative to the radial conduction [32].

After some mathematical simplifications, the two following equations are derived from Eqs. (3) and (4) which are applicable for both the tube and slit flows.

$$ tr({\tilde{\mathbf{\tau }}}) = \tilde{\tau }_{yy} = \frac{{2\lambda \tilde{\tau }_{yz}^{2} }}{\eta } $$
(6)
$$ \frac{{{\text{d}}\tilde{u}_{z} }}{{{\text{d}}\tilde{y}}} = f\left( {\frac{{2\lambda \tilde{\tau }_{yz}^{2} }}{\eta }} \right)\frac{{\tilde{\tau }_{yz} }}{\eta } $$
(7)

Consequently, by considering the linear form of the PTT model, as shown in Eq. (5), the transverse (slit case) or radial (tube case) velocity gradient will be expressed as

$$ \frac{{{\text{d}}\tilde{u}_{z} }}{{{\text{d}}\tilde{y}}} = \left[ {1 + \frac{{2\varepsilon \lambda^{2} \tilde{\tau }_{yz}^{2} }}{{\eta^{2} }}} \right]\frac{{\tilde{\tau }_{yz} }}{\eta } $$
(8)

Finally, with the help of the momentum equation given in Eq. (2) and considering the component in the axial direction, we have,

$$ \tilde{\tau }_{yz} = \frac{{{\text{d}}\tilde{p}}}{{{\text{d}}\tilde{z}}}\frac{{\tilde{y}}}{(1 + j)} $$
(9)

where j equals 0 for the slit flow and 1 for the tube flow and \( {\text{d}}\tilde{p}/{\text{d}}\tilde{z} \) is the pressure gradient. Also, \( \tilde{y} \) is radial or transverse coordinate depending on the use of radial or Cartesian coordinate. By integrating Eq. (8) from an arbitrary position in the radial or transverse coordinate to the wall, and replacing the axial component of the stress tensor, in turn, the velocity distribution will be derived as [10]:

$$ \frac{{\tilde{u}(y)}}{U} = K\frac{{U_{\text{N}} }}{U}\left( {1 - y^{2} } \right)\left( {1 + 4K^{2} \varepsilon {\text{De}}^{2} \left( {\frac{{U_{\text{N}} }}{U}} \right)^{2} \left( {1 + y^{2} } \right)} \right) \, $$
(10)

where \( \tilde{u} \) is the main flow velocity (axial velocity), U is the mean velocity defined as the volumetric flow rate divided by the cross-sectional area, K takes the respective values of 1.5 and 2 for the slit and the tube flows and \( U_{\text{N}} \) is the corresponding mean velocity for the Newtonian flow which is \( U_{\text{N}} = - {\text{d}}\tilde{p}/{\text{d}}\tilde{z}(R^{2} /8\eta ) \) for the tube flow and \( U_{\text{N}} = - {\text{d}}\tilde{p}/{\text{d}}\tilde{z}(H^{2} /3\eta ) \) for the slit flow. In addition, \( y \) is the dimensionless variable defined as \( y = \tilde{y}/R \) for the tube flow and \( y = \tilde{y}/H \) for the slit flow (here \( H \) is half of width of the slit). The dimensionless Deborah number is defined as \( {\text{De}} = \,\lambda U/R \) for the tube flow and \( {\text{De}} = \,\lambda U/H \) for the slit flow and represents the level of the fluid elasticity. Expectedly, for \( {\text{De}} = 0 \) or \( \varepsilon = \,0 \), Eq. (10) reduces to the parabolic profile with a maximum velocity at the centerline equal to K. Oliveira and Pinho [10] showed that the velocity ratio (\( U_{\text{N}} /U \)) can be calculated from the following formulation in terms of the rheological properties:

$$ \frac{{U_{\text{N}} }}{U} = \frac{{432^{1/6} (\delta^{2/3} - 2^{2/3} )}}{{6b^{1/2} \delta^{1/3} }} $$
(11)

where \( \delta \) is:

$$ \delta = (4 + 27b)^{1/2} + 3^{3/2} b^{1/2} $$
(12)

In Eqs. (11) and (12), \( b \) is a constant, \( b = 54/5(\varepsilon {\text{De}}^{2} ) \) for the slit flow and \( b = 64/3(\varepsilon {\text{De}}^{2} ) \) for the tube flow. Now we turn into the energy equation which plays a crucial role in the present study.

The energy equation should be treated cautiously. For viscoelastic materials, as Peters and Baaijens [34] reported in their work, the mechanical energy is partly dissipated and partly stored as elastic energy. They presented the general energy equation in the form of Eq. (13).

$$ \rho c_{\text{p}} \dot{\tilde{T}} = \tilde{\nabla }.(k\tilde{\nabla }\tilde{T}) + \alpha \tilde{\tau } :\,\tilde{D} + (1 - \alpha )\frac{{tr(\tilde{\tau })}}{2\lambda }\left( {1 + \frac{\varepsilon \lambda }{\eta }tr(\tilde{\tau })} \right) $$
(13)

where \( c_{\text{p}} \) is the specific heat, \( \dot{\tilde{T}} \) is the temperature material derivative, k is the thermal conductivity, \( \tau :D \) is the viscous dissipation which is a contribution of the entropy elasticity and therefore contributes to temperature change. The last term in Eq. (13) denotes the energy elasticity representing the energy stored elastically and does not relate to temperature variation. The split of the mechanical energy into the entropic elasticity and the energy elasticity is measured by \( \alpha \). In the case of pure entropy elasticity we have \( \alpha = 1 \), while in pure elastic energy it is \( \alpha = 0 \).

In light of the assumptions mentioned earlier and by considering an isotropic thermal conductivity, Eq. (13) can be simplified for both the tube and the slit cases as follows:

$$ \rho c_{\text{p}} \tilde{u}_{z} \frac{{\partial \tilde{T}}}{{\partial \tilde{z}}} = k\left( {\frac{{\partial^{2} \tilde{T}}}{{\partial \tilde{y}^{2} }} + \frac{j}{{\tilde{y}}}\frac{{\partial \tilde{T}}}{{\partial \tilde{y}}}} \right) + \alpha \tilde{\phi } + (1 - \alpha )\frac{{\tilde{\tau }_{yy} }}{2\lambda }\left( {1 + \frac{\varepsilon \lambda }{\eta }\tilde{\tau }_{yy} } \right) $$
(14)

As it is available in the literature [32], for \( Pe_{D} \gg 1 \) the axial conduction is neglected in comparison with the radial conduction. Also, by considering the hypothesis of a fully developed flow in the axial direction where the velocity component is zero, the transverse convection term of the energy equation is neglected. In addition, the viscous dissipation term \( \tilde{\tau }:\tilde{D} \) is replaced by \( \tilde{\phi } \) and is resulted by tensor product of the velocity gradient into the stress tensor which for the present problem will be reduced to \( \tilde{\phi } = \tilde{\tau }_{yz} {\text{d}}\tilde{u}_{z} /{\text{d}}\tilde{y} \), and after replacing the velocity gradient by Eq. (8), the dimensional viscous dissipation will be

$$ \tilde{\phi } = \left( {1 + \frac{{2\varepsilon \lambda^{2} \tilde{\tau }_{yz}^{2} }}{{\eta^{2} }}} \right)\frac{{\tilde{\tau }_{yz}^{2} }}{\eta } $$
(15)

It is worthwhile to mention that the viscous dissipation term is not the same for the tube and the slit cases since they have a different shear stress (\( \tilde{\tau }_{yz} \)) as presented in Eq. (9). After some mathematical simplifications, the final energy equation will be derived as

$$ \tilde{u}_{z} \frac{{\partial \tilde{T}}}{{\partial \tilde{z}}} = \frac{k}{{\rho c_{\text{p}} }}\left( {\frac{{\partial^{2} \tilde{T}}}{{\partial \tilde{y}^{2} }} + \frac{j}{{\tilde{y}}}\frac{{\partial \tilde{T}}}{{\partial \tilde{y}}}} \right) + \frac{1}{{\rho c_{\text{p}} }}\left( {1 + \frac{{2\varepsilon \lambda^{2} \tilde{\tau }_{yz}^{2} }}{{\eta^{2} }}} \right)\frac{{\tilde{\tau }_{yz}^{2} }}{\eta } $$
(16)

As it can be seen from the above equation, the term \( \alpha \) has been diminished. Indeed, the terms involving \( \alpha \) cancel out and interestingly the final result, as shown in Eq. (16), is mathematically equivalent to setting \( \alpha = 1 \) in Eq. (14). In order to render the variables dimensionless, the following expressions are introduced.

$$ \begin{aligned} & u = \frac{{\tilde{u}}}{U},\,\,\,\,\,T = \frac{{\tilde{T} - \tilde{T}_{\text{w}} }}{{\tilde{T}_{\text{m}} - \tilde{T}_{\text{w}} }} \\ & \left\{ \begin{aligned} {\text{For}}\,{\text{slit}}\,{\text{flow:}}\,\,\,\,\,\,z = \frac{{\tilde{z}}}{H},\,\,\,\,y = \frac{{\tilde{y}}}{H}\,\,\,\,\,{\text{and}}\,\,\,\,\phi = \frac{{\tilde{\phi }}}{{3\eta U^{2} /H^{2} }} \hfill \\ {\text{For}}\,{\text{tube}}\,{\text{flow:}}\,\,\,\,z = \frac{{\tilde{z}}}{R},\,\,\,\,y = \frac{{\tilde{y}}}{R}\,\,\,\,\,{\text{and}}\,\,\,\,\phi = \frac{{\tilde{\phi }}}{{4\eta U^{2} /R^{2} }}\,\,\,\, \hfill \\ \end{aligned} \right. \\ \end{aligned} $$
(17)

where \( \tilde{T}_{\text{w}} \) is the wall temperature and \( \tilde{T}_{\text{m}} \) denotes the fluid mean temperature in the tube or the slit across the longitudinal coordinate. Consequently, the dimensionless form of the viscous dissipation term (Eq. (15)) will be

$$ \,\phi = (3 + j)(U_{\text{N}} /U)^{2} y^{2} \left( {1 + 2\alpha_{2} y^{2} } \right) $$
(18)

in which we have \( \alpha_{2} = 9\varepsilon {\text{De}}^{2} (U_{\text{N}} /U)^{2} \) for the slit flow and \( \alpha_{2} = 16\varepsilon {\text{De}}^{2} (U_{\text{N}} /U)^{2} \) for the tube flow. It is easily provable that the viscous dissipation term for De = 0, the Newtonian fluid, is \( \,\phi = (3 + j)y^{2} \).

Now it aims to derive a relationship for the dimensional axial gradient of temperature (\( \partial \tilde{T}/\partial \tilde{z} \)). In this regard, it is known that in the fully developed thermal condition, the axial gradient of the dimensionless temperature is zero [35]:

$$ \frac{\partial T}{\partial z} = \frac{\partial }{\partial z}\left( {\frac{{\tilde{T} - \tilde{T}_{\text{w}} }}{{\tilde{T}_{\text{m}} - \tilde{T}_{\text{w}} }}} \right) = 0 $$
(19)

By expanding the above equation and knowing that the tube wall temperature is constant, we have:

$$ \frac{{\partial \tilde{T}}}{{\partial \tilde{z}}} = T\frac{{{\text{d}}\tilde{T}_{\text{m}} }}{{{\text{d}}\tilde{z}}} $$
(20)

In addition, the dimensional axial gradient of the mean temperature appeared in the above equation, \( {\text{d}}\tilde{T}_{\text{m}} /{\text{d}}\tilde{z} \), could be obtained by considering a balance of energy on a differential control volume and taking account of the viscous dissipation for both the cases as follows:

(21)

where \( \tilde{P} \) and \( \tilde{A} \) are the perimeter and the area of the cross section, respectively. By applying Eq. (22) which is reported by Oliveira and Pinho [10], on Eq. (21), finally the axial gradient of the mean temperature can be expressed in the form of Eq. (23).

$$ \frac{{U_{\text{N}} }}{U}\left( {1 + b\left( {\frac{{U_{\text{N}} }}{U}} \right)^{2} } \right) = 1 $$
(22)
$$ \left\{ \begin{aligned} & {\text{For}}\,{\text{slit}}\,{\text{flow:}}\,\,\frac{{{\text{d}}\tilde{T}_{\text{m}} }}{{{\text{d}}\tilde{z}}} = \frac{{h(\tilde{T}_{\text{w}} - \tilde{T}_{\text{m}} )}}{{\rho HUc_{\text{p}} }} + \frac{{3\eta U_{\text{N}} }}{{\rho H^{2} c_{\text{p}} }} \\ & {\text{For}}\,{\text{tube}}\,{\text{flow:}}\,\,\frac{{{\text{d}}\tilde{T}_{\text{m}} }}{{{\text{d}}\tilde{z}}} = \frac{{2h(\tilde{T}_{\text{w}} - \tilde{T}_{\text{m}} )}}{{\rho RUc_{\text{p}} }} + \frac{{8\eta U_{\text{N}} }}{{\rho R^{2} c_{\text{p}} }} \\ \end{aligned} \right. $$
(23)

Finally, the dimensionless form of the energy equation for the LPTT flow in slits and tubes of constant wall temperature is derived as

$$ \begin{aligned} \frac{{{\text{d}}^{2} T}}{{{\text{d}}y^{2} }} + \frac{j}{y}\frac{{{\text{d}}T}}{{{\text{d}}y}} + \alpha_{1} \left( {1 - y^{2} } \right)\left( {1 + \alpha_{2} \left( {1 + y^{2} } \right)} \right)\left( {\frac{1}{4 - 3j}{\text{Nu}} + 2(j + 1)\alpha_{1} {\text{Br}}} \right)T \hfill \\ \quad - \,4\alpha_{1}^{2} {\text{Br}}\left( {1 + 2\alpha_{2} y^{2} } \right)y^{2} = 0 \hfill \\ \end{aligned} $$
(24)

where we have \( \alpha_{1} { = }1.5_{{}} U_{\text{N}} /U \) and \( \alpha_{1} { = }2\,U_{\text{N}} /U \) for the slit and tube flows, respectively. Moreover, there are two well-known dimensionless numbers, namely the Nusselt number and the Brinkman number, which are defined as \( {\text{Nu}} = hd_{\text{h}} /k \) and \( {\text{Br}} = \eta U^{2} /(k(\tilde{T}_{\text{w}} - \tilde{T}_{\text{m}} )) \), respectively. The parameter h is the convective heat transfer coefficient and \( d_{\text{h}} \) is the hydraulic diameter (\( d_{\text{h}} = 2R \) for the tube flow and \( d_{\text{h}} = 4H \) for the slit flow).

3 Analytical solution

As mentioned before, the motive of the present study is to obtain the dimensionless temperature field, T(y), in constant wall temperature slits and tubes. To this aim, the temperature distribution is determined by solving the second-order non-homogeneous differential equation [Eq. (24)] by applying an analytical method. Besides, there are two boundary conditions consisting of the symmetry condition at the centerline [Eq. (25)] and the constant wall temperature [Eq. (26)]. Since the number of unknown parameters exceeds the number of the boundary conditions, a physical constrain will be introduced for each case [Eqs. (29) and (32)].

$$ {\text{at}}\,\,\,\,\,y = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,\frac{{{\text{d}}T}}{{{\text{d}}y}} = 0 $$
(25)
$$ {\text{at}}\,\,\,\,\,y = 1\,\,\,\,\, \Rightarrow \,\,\,\,T = 0 $$
(26)

3.1 Slit flow

By substituting j = 0 into Eq. (24) and then solving it, the temperature distribution for the LPTT fluid flow in a slit of constant wall temperature will be determined as follows:

$$ \begin{aligned} & T\left( y \right) \\ & \quad = C_{1} {\text{HeunT}}\left( \begin{aligned} & 0.13{\mkern 1mu} \frac{{\alpha_{1}^{2} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)^{2} \left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)} \right)^{4/3} }},0, \\ & 0.7211\frac{{\alpha_{1} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} \alpha_{1} B + {\text{Nu}}} \right)} \right)^{2/3} }}, - 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ & \quad \times {\text{e}}^{{\frac{1}{12}{\kern 1pt} \frac{{\alpha_{1} \left( {8{\kern 1pt} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)y\left( {2{\kern 1pt} \alpha_{2} y^{2} + 3} \right)}}{{\sqrt {\alpha_{1} \alpha_{2} \left( {8{\kern 1pt} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)} }}}} \\ & \quad + \,C_{2} {\text{HeunT}}\left( \begin{aligned} & 0.13{\mkern 1mu} \frac{{\alpha_{1}^{2} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)^{2} \left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)} \right)^{4/3} }},0, \\ & 0.7211\frac{{\alpha_{1} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} \alpha_{1} B + {\text{Nu}}} \right)} \right)^{2/3} }},0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ & \quad \times {\text{e}}^{{ - \frac{1}{12}{\kern 1pt} \frac{{\alpha_{1} \left( {8{\kern 1pt} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)y\left( {2{\kern 1pt} \alpha_{2} y^{2} + 3} \right)}}{{\sqrt {\alpha_{1} \alpha_{2} \left( {8{\kern 1pt} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)} }}}} \\ & \quad + \,\left[ \begin{aligned} & {\text{HeunT}}\left( \begin{aligned} & 0.13\frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)\left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2} \sqrt[3]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}}},0,0.7211{\mkern 1mu} \frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \\ & - 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ & {\text{e}}^{{\frac{1}{12}\frac{{\left( {2{\kern 1pt} y^{3} \alpha_{2} + 3{\kern 1pt} y} \right)\sqrt {\alpha_{1} \alpha_{2} \left( {8{\kern 1pt} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)} }}{{\alpha_{2} }}}} \int {\frac{A}{B}} {\text{d}}y \\ & - \,{\text{HeunT}}\left( \begin{aligned} 0.13\frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)\left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2} \sqrt[3]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}}},0,0.7211{\mkern 1mu} \frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \hfill \\ 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}y \hfill \\ \end{aligned} \right) \\ & {\text{e}}^{{ - \frac{1}{6}{\kern 1pt} \frac{{y\sqrt {\alpha_{1} \alpha_{2} \left( {8{\kern 1pt} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)} \left( {\alpha_{2} y^{2} + \frac{3}{2}} \right)}}{{\alpha_{2} }}}} \int {\frac{C}{B}} {\text{d}}y \\ \end{aligned} \right] \\ & \quad \times 24\alpha_{1}^{2} {\text{Br}}\sqrt {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} \alpha_{1} {\text{Br}} + {\text{Nu}}} \right)} \\ \end{aligned} $$
(27)

where A, B and C are:

$$ \begin{aligned} A & = (2{\mkern 1mu} y^{4} \alpha_{2} + y^{2} ){\text{e}}^{{ - \frac{1}{6}{\kern 1pt} \frac{{y\sqrt {\alpha_{1} \alpha_{2} \left( {8{\kern 1pt} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)} \left( {\alpha_{2} y^{2} + \frac{3}{2}} \right)}}{{\alpha_{2} }}}} \\ & \quad \times {\text{HeunT}}\left( \begin{aligned} & 0.13\frac{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} \left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2}^{2} }},0,0.7211\frac{{\alpha_{1} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \\ & 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ B & = \left( \begin{aligned} & 48{\mkern 1mu} \left( {{\text{Br}}\alpha_{1} + \frac{\text{Nu}}{8}} \right)\left( {\alpha_{2} y^{2} + \frac{1}{2}} \right)\alpha_{1} \\ & {\text{HeunT}}\left( \begin{aligned} & 0.13\frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)\left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2} \sqrt[3]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}}},0,0.7211{\mkern 1mu} \frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \\ & 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ & - \,4.16{\mkern 1mu} \left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} Br\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} . \\ & {\text{HeunTPrime}}\left( \begin{aligned} & 0.13\frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)\left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2} \sqrt[3]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}}},0,0.7211{\mkern 1mu} \frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \\ & 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ \end{aligned} \right) \\ & \quad \times {\text{HeunT}}\left( \begin{aligned} & 0.13\frac{{\alpha_{1} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)\left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2} \sqrt[3]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)}}}},0,0.7211\frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \\ & - 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ & \quad - \,4.16{\mkern 1mu} \left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} \\ & \quad \times {\text{HeunTPrime}}\left( \begin{aligned} & 0.13\frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)\left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2} \sqrt[3]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}}},0,0.7211{\mkern 1mu} \frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \\ & - 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ & \quad \times {\text{HeunT}}\left( \begin{aligned} & 0.13\frac{{\alpha_{1} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)\left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2} \sqrt[3]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)}}}},0,0.7211\frac{{\alpha_{1} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \\ & 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ C & = (2{\mkern 1mu} y^{4} \alpha_{2} + y^{2} ){\text{e}}^{{\frac{1}{12}\frac{{\left( {2{\kern 1pt} y^{3} \alpha_{2} + 3{\kern 1pt} y} \right)\sqrt {\alpha_{1} \alpha_{2} \left( {8{\kern 1pt} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)} }}{{\alpha_{2} }}}} \\ & \quad \times {\text{HeunT}}\left( \begin{aligned} & 0.13\frac{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} \left( {2{\mkern 1mu} \alpha_{2} + 1} \right)^{2} }}{{\alpha_{2}^{2} }},0,0.7211\frac{{\alpha_{1} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)}}{{\left( {\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} B\alpha_{1} + {\text{Nu}}} \right)} \right)^{2/3} }}, \\ & - 0.6934\sqrt[6]{{\alpha_{1} \alpha_{2} \left( {8{\mkern 1mu} {\text{Br}}\alpha_{1} + {\text{Nu}}} \right)}}y \\ \end{aligned} \right) \\ \end{aligned} $$
(28)

As it can be seen, Eq. (27) represents a closed-form solution containing a special mathematical function, HeunT, for the slit case. Regarding Eq. (27), there are three unknown constant parameters in the slit temperature distribution, namely\( C_{1} \), \( C_{2} \) and Nu which one of them, \( C_{1} \), vanishes by applying Eq. (26) to Eq. (27). Unfortunately, there are no analytical solutions for the two integral expressions in Eq. (27). Therefore, these integrals have been solved numerically using the 1/3 Simpson rule. Because of the number of the unknown parameters, a physical constrain is presented in Eq. (29) which is the product of the dimensionless velocity profile into the dimensionless temperature distribution throughout the area of the geometry.

$$ \int\limits_{0}^{1} {u(y)T(y){\text{d}}y = 1} $$
(29)

Then, with applying Eqs. (25) and (29) to Eq. (27), a linear system of two equations with two variables (\( C_{2} \) and Nu) will appear by solving it; the slit temperature distribution will be completely determined.

3.2 Tube flow

The solution of forced heat convection of the LPTT fluid flow in a constant wall temperature tube can be obtained by solving Eq. (24) for j = 1. There is no closed-form solution for this case, and therefore, the temperature distribution has been solved by the Frobenius method. The corresponding temperature distribution for the tube flow is presented in Eq. (30).

$$ \begin{aligned} & T(y) = C_{1} \left( \begin{aligned} & 1 - \frac{1}{4}\alpha_{1} \left( {{\text{Nu}}(1 + \alpha_{2} ) + 4k_{1} (1 + \alpha_{2} ){\text{Br}}} \right)y^{2} \\ & + \,\frac{1}{64}\alpha_{1} \left( \begin{aligned} & {\text{Nu}}\left( {\alpha_{1} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{2} + 4} \right) \\ & + \,8\left( {\alpha_{1}^{2} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{2} + 2\alpha_{1} + 2\alpha_{1}^{3} \left( {1 + \alpha_{2} } \right)^{2} {\text{Br}}} \right){\text{Br}} \\ \end{aligned} \right)y^{4} \\ & - \,\frac{1}{2304}\alpha_{1} \left( {4\alpha_{1} {\text{Br}} + {\text{Nu}}} \right) \\ & \times \,\left( \begin{aligned} & \alpha_{1} {\text{Nu}}\left( {\alpha_{1} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{3} + 20\left( {1 + \alpha_{2} } \right)} \right) - 64\alpha_{2} + \\ & 8\left( \begin{aligned} & \alpha_{1}^{3} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{3} + 10\alpha_{1}^{2} \left( {1 + \alpha_{2} } \right) + \\ & 2\alpha_{1}^{4} \left( {1 + \alpha_{2} } \right)^{3} {\text{Br}} \\ \end{aligned} \right){\text{Br}} \\ \end{aligned} \right)y^{6} + O(y^{7} ) \\ \end{aligned} \right) \\ & \quad + \,C_{2} \left( \begin{aligned} & \ln (y)\left( \begin{aligned} & 1 - \frac{1}{4}\alpha_{1} \left( {{\text{Nu}}(1 + \alpha_{2} ) + 4\alpha_{1} (1 + \alpha_{2} )Br} \right)y^{2} \\ & + \,\frac{1}{64}\alpha_{1} \left( \begin{aligned} & {\text{Nu}}\left( {\alpha_{1} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{2} + 4} \right) + \\ 8\left( {\alpha_{1}^{2} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{2} + 2\alpha_{1} + 2\alpha_{1}^{3} \left( {1 + \alpha_{2} } \right)^{2} {\text{Br}}} \right){\text{Br}} \\ \end{aligned} \right)y^{4} \\ & - \,\frac{1}{2304}\alpha_{1} \left( {4\alpha_{1} {\text{Br}} + {\text{Nu}}} \right) \\ & \times \,\left( \begin{aligned} & \alpha_{1} {\text{Nu}}\left( {\alpha_{1} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{3} + 20\left( {1 + \alpha_{2} } \right)} \right) - 64\alpha_{2} + \\ & 8\left( \begin{aligned} \alpha_{1}^{3} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{3} + 10\alpha_{1}^{2} \left( {1 + \alpha_{2} } \right) + \hfill \\ 2\alpha_{1}^{4} \left( {1 + \alpha_{2} } \right)^{3} {\text{Br}} \hfill \\ \end{aligned} \right){\text{Br}} \\ \end{aligned} \right)y^{6} + O(y^{7} ) \\ \end{aligned} \right) \\ & + \,\frac{1}{4}\alpha_{1} \left( {{\text{Nu}}(1 + \alpha_{2} ) + 4\alpha_{1} (1 + \alpha_{2} ){\text{Br}}} \right)y^{2} \\ & + \,\left( \begin{aligned} & \frac{1}{16}\alpha_{1} \left( {{\text{Nu}} + 4\alpha_{1} {\text{Br}}} \right) \\ & - \,\frac{3}{128}\alpha_{1} \left( \begin{aligned} & \alpha_{1} {\text{Nu}}^{2} \left( {1 + \alpha_{2} } \right)^{2} + 4{\text{Nu}} \\ & + \,8\left( {\alpha_{1}^{2} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{2} + 2\alpha_{1} + 2{\text{Br}}\alpha_{1}^{3} \left( {1 + \alpha_{2} } \right)^{2} } \right){\text{Br}} \\ \end{aligned} \right) \\ \end{aligned} \right)y^{4} \\ & + \,\left( \begin{aligned} & - \frac{3}{576}\left( {{\text{Nu}} + 4\alpha_{1} {\text{Br}}} \right)\alpha_{1} \\ & \times \,\left( {\alpha_{1} {\text{Nu}}(1 + \alpha_{2} ) - 8\alpha_{2} + 4\alpha_{1}^{2} (1 + \alpha_{2} ){\text{Br}}} \right) \\ & + \,\frac{11}{13824}\left( {{\text{Nu}} + 4\alpha_{1} {\text{Br}}} \right)\alpha_{1} \\ & \times \,\left( \begin{aligned} & \alpha_{1}^{2} {\text{Nu}}^{2} \left( {1 + \alpha_{2} } \right)^{3} + 20{\text{Nu}}\alpha_{1} (1 + \alpha_{2} ) - 64\alpha_{2} \\ & 8\left( \begin{aligned} & \alpha_{1}^{3} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{3} + 10\alpha_{1}^{2} \left( {1 + \alpha_{2} } \right) + \\ & 2\alpha_{1}^{4} \left( {1 + \alpha_{2} } \right)^{3} {\text{Br}} \\ \end{aligned} \right){\text{Br}} \\ \end{aligned} \right) \\ \end{aligned} \right)y^{6} + O(y^{7} ) \\ \end{aligned} \right) \\ & \quad + \,{\text{Br}}\left( {\frac{1}{4}\alpha_{1}^{2} y^{4} + \left( {\frac{2}{9}\alpha_{1}^{2} \alpha_{2} - \frac{1}{144}\alpha_{1}^{2} \left( {\alpha_{1} {\text{Nu}}(1 + \alpha_{2} ) + 4\alpha_{1}^{2} (1 + 4\alpha_{2} ){\text{Br}}} \right)} \right)y^{6} } \right) + O(y^{7} ) \\ \end{aligned} $$
(30)

By applying the boundary condition given in Eq. (25) to Eq. (30) and considering the singularity of Eq. (30) at y = 0, the constant \( C_{2} \) vanishes. Suffice it to say that the temperature distribution is finite throughout the cross section. Finally, the temperature distribution, up to eight-order terms, for the tube flow is presented in Eq. (31) in which \( C_{1} \) is replaced by the maximum temperature located at the center of the tube.

$$ \begin{aligned} & T(y) = T_{c} \left( \begin{aligned} & 1 - \frac{1}{4}\alpha_{1} \left( {{\text{Nu}}(1 + \alpha_{2} ) + 4k_{1} (1 + \alpha_{2} ){\text{Br}}} \right)y^{2} \\ & + \,\frac{1}{64}\alpha_{1} \left( \begin{aligned} & {\text{Nu}}\left( {\alpha_{1} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{2} + 4} \right) \\ & + \,8\left( {\alpha_{1}^{2} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{2} + 2\alpha_{1} + 2\alpha_{1}^{3} \left( {1 + \alpha_{2} } \right)^{2} {\text{Br}}} \right){\text{Br}} \\ \end{aligned} \right)y^{4} \\ & - \,\frac{1}{2304}\left( {{\text{Nu}} + 4\alpha_{1} {\text{Br}}} \right)\alpha_{1} \\ & \times \left( \begin{aligned} & \alpha_{1} {\text{Nu}}\left( {\alpha_{1} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{3} + 20\left( {1 + \alpha_{2} } \right)} \right) - 64\alpha_{2} \\ & + \,8\left( \begin{aligned} & \alpha_{1}^{3} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{3} + 10\alpha_{1}^{2} \left( {1 + \alpha_{2} } \right) + \\ & 2\alpha_{1}^{4} \left( {1 + \alpha_{2} } \right)^{3} {\text{Br}} \\ \end{aligned} \right){\text{Br}} \\ \end{aligned} \right)y^{6} \\ & + \,\frac{1}{147456}\left( {{\text{Nu}} + 4\alpha_{1} {\text{Br}}} \right)\alpha_{1}^{2} \\ & \times \left( \begin{aligned} & \alpha_{1}^{2} {\text{Nu}}^{3} \left( {1 + \alpha_{2} } \right)^{4} + 56\alpha_{1} {\text{Nu}}^{2} \left( {1 + \alpha_{2} } \right)^{2} \\ & - \,640\alpha_{2} {\text{Nu}}\left( {1 + \alpha_{2} } \right) + 144{\text{Nu}} \\ & + \,\left( \begin{aligned} & 12\alpha_{1}^{3} {\text{Nu}}^{2} \left( {1 + \alpha_{2} } \right)^{4} {\text{Br}} + 48\alpha_{1}^{4} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{4} {\text{Br}} + \\ & 64\alpha_{1}^{5} \left( {1 + \alpha_{2} } \right)^{4} {\text{Br}}^{2} + 448\alpha_{1}^{2} {\text{Nu}}\left( {1 + \alpha_{2} } \right)^{2} - \\ & 2560\alpha_{1} \alpha_{2} \left( {1 + \alpha_{2} } \right) + 576\alpha_{1} + 896\alpha_{1}^{3} \left( {1 + \alpha_{2} } \right)^{2} {\text{Br}} \\ \end{aligned} \right){\text{Br}} \\ \end{aligned} \right)y^{8} \\ \end{aligned} \right) \\ & \quad + \,{\text{Br}}\left( \begin{aligned} & \frac{1}{4}\alpha_{1}^{2} y^{4} + \left( {\frac{2}{9}\alpha_{2} - \frac{1}{144}\alpha_{1} (1 + \alpha_{2} )\left( {{\text{Nu}} + 4\alpha_{1} {\text{Br}}} \right)} \right)\alpha_{1}^{2} y^{6} \\ & + \,\left( {\frac{1}{256} - \frac{{1 + \alpha_{2} }}{2304}\left( {8\alpha_{2} - \frac{1}{4}\alpha_{1} (1 + \alpha_{2} )\left( {{\text{Nu}} + 4\alpha_{1} {\text{Br}}} \right)} \right)} \right)\alpha_{1}^{3} \left( {{\text{Nu}} + 4\alpha_{1} {\text{Br}}} \right)y^{8} \\ \end{aligned} \right) + \cdots \\ \end{aligned} $$
(31)

Like the slit case, the Nusselt number and the maximum temperature have been calculated using the boundary condition shown in Eq. (26) as well as the physical constrain presented in Eq. (32).

$$ \int\limits_{0}^{1} {u(y)T(y)y{\text{d}}y = \frac{1}{2}} $$
(32)

4 Results and discussion

In this section, the results of forced convective heat transfer of the linear PTT fluid with viscous dissipation through constant wall temperature slits and tubes are discussed in detail. This study aims to clarify the effects of fluid elasticity (\( 0.1 \le {\text{De}} \le 100 \)), extensibility (\( 1 \le \varepsilon \le 10 \)) and the Brinkman number (\( - 10 \le {\text{Br}} \le 10 \)) on the heat transfer characteristics in a fully developed regime. The results for the tube flow are presented and discussed in detail, while those relating to the slit flow are summarized in tabular form (for the similarity). For estimating the dimensionless parameters that appear in this study, the typical values [36] of physical parameters are listed in Table 1. Also, the reliability of the present results is gained for the Newtonian fluid in tubes and slits in Sect. 4.1.

Table 1 Typical values of physical and geometrical parameters used for estimating the dimensionless parameters in the present study
Full size table

4.1 Validation

The authors believe that the existing studies on forced convection of the LPTT fluid flow do not match the present work and cannot be used for validation. The present results have been therefore validated only in the Newtonian case (\( {\text{De}} = 0 \) and \( \varepsilon = 0 \)) with those available in the literature. In the present work, the Nusselt number for a constant wall temperature tube flow has been calculated as \( {\text{Nu}} = 3.6571 \) which closely matches the value of \( {\text{Nu}} = 3.66 \) [32, 37]. For the slit case, the Nusselt number in a constant wall temperature slit is \( {\text{Nu}} = 7.541 \) [32], and this investigation revealed the value of \( {\text{Nu}} = 7.5407 \). It can be concluded that the present results are in good correspondence with those available in the literature. In Sect. 4.2, the effects of either negative or positive Brinkman numbers as well as the fluid elasticity on the heat transfer characteristics are examined.

4.2 Effects of Brinkman number and fluid elasticity

Figure 1 demonstrates the effect of negative Brinkman numbers, namely \( {\text{Br}} = - 10 \), − 1 and − 0.1, on the dimensionless temperature, considering different values of the extensibility parameter (\( \varepsilon \)) and the Deborah number (De). By considering the definition of the Brinkman number, given by \( {\text{Br}} = \eta U^{2} /k(\tilde{T}_{\text{w}} - \tilde{T}_{\text{m}} ) \), at negative Brinkman numbers there is fluid cooling adjacent to the wall. Figure 1 shows that at fixed values of \( \varepsilon \) and De, by increasing the Brinkman number, the flow temperature and its temperature gradient in the vicinity of the wall fall, leading to a smaller Nusselt number. As demonstrated in Fig. 1, at \( {\text{De}} = 0.1 \) when Br increases from \( {\text{Br}} = - 10 \) to \( {\text{Br}} = - 0.1 \), the respective Nusselt numbers decrease dramatically from \( {\text{Nu}} = 38.4616 \) to \( {\text{Nu}} = 4.1799 \). Indeed, higher values of Br strengthen the cooling process in the wall area, where the viscous dissipation effect is the strongest. Following a diametrically opposed trend, the centerline temperature, \( T_{\text{c}} \), increases by Br and the temperature profiles in the vicinity of the core region displace upward in the radial direction. According to Fig. 1, by increasing De the Nusselt number decreases and the centerline temperature increases. Consequently, the Deborah number intensifies the effect of the Brinkman number, meaning that the higher the Deborah number, the higher the Brinkman number, leading to lower Nusselt numbers, but higher centerline temperatures. Moreover, the increasing De causes a wider blunt velocity distribution in the core region as well as a lower value of centerline velocity, leading to weaker heat convection in the vicinity of the core. This is why the lowest value of the centerline temperature is recorded for \( {\text{De}} = 0.1 \), and the centerline temperature increases as De increases. Regarding the rheological aspect, when De increases the fluid shear-thinning behavior intensifies which results in a better heat distribution in the flow as shown in Fig. 1. Furthermore, as Br increases, the effect of De seems to be lessened. In other words, when the Brinkman number increases from \( {\text{Br}} = - 10 \) to \( {\text{Br}} = - 0.1 \), the deviation between the three temperature profiles nearly vanishes (for either the Nusselt number or the centerline temperature). For instance, at \( {\text{Br}} = - 10 \) we have \( {\text{Nu}} = 38.4616 \) at \( {\text{De}} = 0.1 \) and \( {\text{Nu}} = 5.1371 \) at \( {\text{De}} = 100 \), while the corresponding values at \( {\text{Br}} = - 0.1 \) are nearly the same, \( {\text{Nu}} = 4.1799 \) and \( {\text{Nu}} = 4.1760 \), respectively. And the same goes for the centerline temperature values.

Fig. 1
figure 1

Dimensionless temperature distribution for negative Brinkman numbers in constant wall temperature tubes a ε = 1 and b ε = 10

Full size image

Figure 1 also depicts the effect of the fluid extensibility parameter on the temperature distribution. It is found that at a certain Brinkman number, \( \varepsilon \) decreases the wall temperature gradient and therefore Nu, but it also increases the centerline temperature which is apparent by considering Fig. 1b. Indeed, the extensibility parameter affects the temperature profile in the same way as the Deborah number. For example, at \( {\text{Br}} = - 10 \) and \( {\text{De}} = 0.1 \) as demonstrated in Fig. 1b the Nusselt number is equal to \( {\text{Nu}} = 38.4616 \) at \( \varepsilon = 1 \), while it is decreased to \( {\text{Nu}} = 29.9092 \) at \( \varepsilon = 10 \).

Similarly, Fig. 2 illustrates the effects of positive Brinkman numbers on temperature distribution in the LPTT fluid. It is worthwhile mentioning that, as a fact, the Nusselt number decreases continuously with the Brinkman number and thus it has lower values at positive Brinkman numbers. Generally, nearly all the trends presented at positive Brinkman numbers are opposite to those shown at negative Brinkman numbers. When the Brinkman number is greater than zero, flow is supposed to be heated, since the wall temperature is larger than the mean temperature. But it is revealed that at \( {\text{Br}} = 0.1 \) the flow is cooled further. Such small positive values of Br behave in the same way as \( {\text{Br}} = 0 \) does. At these values, the dominance of viscous dissipation is not strong enough to overcome fluid cooling, and therefore, the fluid in the vicinity of the wall becomes colder. As shown in Fig. 2, at \( {\text{Br}} = 0.1 \) and \( \varepsilon = 1 \), when \( {\text{De}} = 0.1 \), the fluid is cooled down and there is a negative temperature gradient next to the wall. When the Deborah number increases from \( {\text{De}} = 0.1 \) to \( {\text{De}} = 100 \), because of an improvement in the fluid shear-thinning behavior, both the Nusselt number and centerline temperature increase, and the temperature profiles move upward entirely. As Br increases, the viscous dissipation effect will be stronger and much more energy will be produced in the flow. As depicted in Fig. 2, for the case \( {\text{Br}} = 1 \), the viscous dissipation effect overcomes the cooling effect and the Nusselt number decreases to a negative number at \( {\text{De}} = 0.1 \). Indeed, at this moment there is the heating process in the flow which is borne out with the positive wall temperature gradient, leading to a negative Nusselt number (\( {\text{Nu}} = - 1.0723 \)). In this condition and at a constant Brinkman number, when the Deborah number increases, because of a larger value of velocity adjacent to the wall, the flow becomes warm faster than other areas, augmenting the heating process. As a consequence, the temperature profile in the vicinity of the wall moves upward which causes the temperature gradient to be positive, thereby resulting in a positive Nusselt number as well (\( {\text{Nu}} = 4.0686 \) at \( {\text{De}} = 100 \)). Like the cooling process, when De increases, the centerline velocity becomes lower, weakening the heat convection in the core region. For this reason, at higher De the centerline temperature follows a downward trend. In the following, through further growth in the Brinkman number up to \( {\text{Br}} = 10 \), a dramatic fall has been observed in the value of Nu, implying too weak heat convection in the flow (\( {\text{Nu}} = - 55.9386 \)). At this moment, the strength of the viscous dissipation is so much that causes difficulty in proper distribution of the heat generated as it can be seen in the case of \( {\text{De}} = 0.1 \). Since a huge amount of energy is accumulated next to the wall, where the viscous dissipation effect is the highest, the heat convection is too weak which causes the flow temperature to remain negative at radial distances more than \( y \approx 5 \). Conversely, the flow temperature in the core region is by far higher than other areas (\( T_{c} \approx 14 \)). Again, by increasing De there will be better heat distribution throughout the flow and at \( {\text{De}} = 100 \) it is much better where the temperature profile varies steadily in the radial coordinate (\( {\text{Nu}} = 3.1851 \)). In fluid heating, at higher Deborah numbers the value of the centerline temperature falls (\( T_{\text{c}} \approx 2 \)), so the flow in the core region will be heated slightly which is attributed to a lower centerline velocity.

Fig. 2
figure 2

Dimensionless temperature distribution for positive Brinkman numbers in constant wall temperature tubes a ε = 1 and b ε = 10

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As mentioned earlier, the extensibility parameter only intensifies the Deborah number effect. In comparison with Fig. 2a, the temperature profiles in Fig. 2b become closer to each other and there will be a lower deviation between them, leading to better heat distribution.

Figure 3a, b shows the impact of the fluid elasticity shown by \( \varepsilon {\text{De}}^{2} \) on the heat transfer characteristics. As explained before, it is clear from the line graphs that for the cases in which \( {\text{Br}} = 0.2 \) and \( {\text{Br}} = 1 \) there is fluid heating in the flow, while other cases are related to the cooling process.

Fig. 3
figure 3

Variations of a Nusselt number and b centerline temperature with fluid elasticity \( \varepsilon \text{De}^{\text{2}} \) for different Brinkman numbers in tube flow

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According to the Fig. 3a, the Nusselt number depends heavily on the Brinkman number and it is increases by decreasing the Brinkman number from positive to negative values.

Figures 4, 5 and 6 show the slit case which shows similar results in comparison with the tube case.

Fig. 4
figure 4

Dimensionless temperature distribution for negative Brinkman numbers in constant wall temperature slits a ε = 1 and b ε = 10

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Fig. 5
figure 5

Dimensionless temperature distribution for positive Brinkman numbers in constant wall temperature slits a ε = 1 and b ε = 10

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Fig. 6
figure 6

Variations of a Nusselt number and b centerline temperature with fluid elasticity \( \varepsilon \text{De}^{\text{2}} \) for different Brinkman numbers in slit flow

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In “Appendix,” the results of the Nusselt number and the centerline temperature are reported (Tables 2, 3, 4, 5, 6, 7, 8, 9) for both the tube and slit flows for various values of the Brinkman number, Deborah number and the extensibility parameter.

Table 2 Nusselt number and centerline temperature values for the LPTT fluid flow in a constant wall temperature tube for negative Brinkman numbers at different values of extensibility parameter and Deborah number
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Table 3 Nusselt number and centerline temperature values for LPTT fluid flow in a constant wall temperature tube for positive Brinkman numbers at different values of extensibility parameter and Deborah number
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Table 4 Nusselt number data as a function of \( \varepsilon {\text{De}}^{ 2} \) in a tube flow for different values of Brinkman number
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Table 5 Centerline temperature values as a function of \( \varepsilon {\text{De}}^{ 2} \) in tube flow for different values of Brinkman number
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Table 6 Nusselt number and centerline temperature values for LPTT fluid flow in a constant wall temperature slit for negative Brinkman numbers at different values of extensibility parameter and Deborah number
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Table 7 Nusselt number and centerline temperature values for LPTT fluid flow in a constant wall temperature slit for positive Brinkman numbers at different values of extensibility parameter and Deborah number
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Table 8 Nusselt number data as a function of \( \varepsilon {\text{De}}^{ 2} \) in a slit flow for different values of Brinkman number
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Table 9 Centerline temperature values as a function of \( \varepsilon {\text{De}}^{ 2} \) in slit flow for different values of Brinkman number
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5 Conclusions

In this article, exact analytical solutions for heat convection in the simplified PTT fluid flow through tubes and slits whose walls are subjected to a constant temperature have been presented for the first time. The viscous dissipation term is taken into account. Expressions for the temperature distribution are derived in closed form for the slit case and in terms of a Frobenius series for the tube case. It is shown that the Nusselt number is heavily dependent on the Brinkman number and decreases continuously with the Brinkman number. The results revealed that the variations in the cooling and heating processes are inversed. In the cooling process, the Nusselt number decreases when the Deborah number and the extensibility parameter increase, while the centerline temperature increases. However, during the heating process the Nusselt number increases and the centerline temperature decreases. In addition, they vary in two completely different trends when the Deborah number and the Brinkman number increase.