## 2 Mathematical Model
### Governing Equations for Core-Fluid  
In dimensional terms, the governing mathematical equations for the viscoelastic core-fluid are derived from the conservation of mass, momentum, and energy, respectively, detailed in Equations (1)-(3):  
\[\
abla^{*}\cdot\mathbf{U}_{c}^{*}=0, \tag{1}\]  
Figure 1: Schematics of the model problem.  
\[\rho_{c}^{*}\bigg{(}\frac{\partial\mathbf{U}_{c}^{*}}{\partial t^{*}}+\mathbf{U}_{c}^{*} \cdot\
abla^{*}\mathbf{U}_{c}^{*}\bigg{)}=-\
abla^{*}p_{c}^{*}+\
abla^{*}\cdot\mathbf{ \sigma}_{c}^{*}+\mathbf{F}_{c}^{*}, \tag{2}\]  
\[\rho_{c}^{*}\mathbf{C}_{p_{c}}^{*}\bigg{(}\frac{\partial\mathbf{T}_{c}^{*}}{\partial t^ {*}}+T_{c}^{*}\cdot\
abla^{*}T_{c}^{*}\bigg{)}=\mathcal{Q}_{D_{c}}^{*}-\
abla^{ *}\cdot\mathbf{\Phi}_{q_{c}}+h_{c}^{*}(T_{w}^{*}-T_{c}^{*}). \tag{3}\]  
The asterisks \(*\) in Equations (1)-(3) represent dimensional variables, the subscript \(()_{c}\) denotes core-fluid, \(\
abla^{*}\) is the gradient operator, \(\mathbf{U}_{c}^{*}\) is the velocity field, \(\rho_{c}^{*}\) the density, \(t^{*}\) the time, \(p_{c}^{*}\) the pressure field, \(\mathbf{\sigma}_{c}^{*}\) the total stress tensor, \(\mathbf{F}_{c}^{*}\) represents body forces, \(\mathbf{C}_{p_{c}}^{*}\) is the specific heat at constant pressure, \(T_{c}^{*}\) is the temperature field of the core fluid, \(\mathcal{Q}_{D_{c}}^{*}\) the internal heat production, \(\mathbf{\Phi}_{q_{c}}^{*}\) the heat flux vector, \(T_{w}^{*}\) is the connecting wall temperature, and \(h_{c}^{*}\) and \(h_{c}^{*}\) are a convective heat-transfer parameters.  
The total stress tensor is given by:  
\[\mathbf{\sigma}_{c}^{*}=-p_{c}{}^{*}\mathbf{I}^{*}+\mathbf{\tau}_{c}^{*}+\eta_{c}^{*}\mathbf{S }_{c}^{*},\]  
where \(\mathbf{I}\) is the identity tensor, \(\mathbf{\tau}_{c}^{*}\) represents the polymer-stress tensor, and  
\[\mathbf{S}_{c}^{*}=\bigg{[}\
abla^{*}\mathbf{U}_{c}^{*}+(\
abla^{*}\mathbf{U}_{c}^{*})^{T }\bigg{]},\]  
is the rate of deformation tensor. The internal heat production is expressed as:  
\[\mathcal{Q}_{D_{c}}^{*}=\gamma\mathbf{\tau}^{*}:\mathbf{S}_{c}^{*}+(1-\gamma)\eta_{sol }^{*}\mathbf{S}_{c}^{*}:\mathbf{S}_{c}^{*},\]  
where \(\eta_{sol}^{*}\) is the solvent viscosity and \(\gamma\) is a viscoelastic non-isothermal parameter; for the full details, we refer the reader to [47; 48; 49]. The heat flux is mathematically expressed by [PERSON]'s law:  
\[\Phi_{q_{c}}^{*}=-K_{c}^{*}\
abla^{*}T_{c}^{*},\]  
where \(K_{c}^{*}\) is the thermal conductivity. The Giesekus viscoelastic constitutive equations are employed to model the polymeric stresses:  
\[\mathbf{\tau}^{*}+\varepsilon^{*}\mathbf{\tau}^{*2}+\lambda^{*}\mathbf{\tau}^{\
abla}=\eta _{p}^{*}\mathbf{S}_{c}^{*},\]  
where \(\varepsilon^{*}\) is the Giesekus nonlinear parameter, \(\lambda^{*}\) is the relaxation time, \(\eta_{p}^{*}\) is the polymer viscosity, and \(\mathbf{\tau}^{*}\) is the upper convected time derivative defined as:  
\[\mathbf{\tau}^{*}=\frac{\partial\mathbf{\tau}^{*}}{\partial t^{*}}+(\mathbf{U}_{c}^{*} \cdot\
abla^{*})\mathbf{\tau}^{*}-(\
abla^{*}\mathbf{U}_{c}^{*})\mathbf{\tau}^{*}-\mathbf{ \tau}^{*}(\
abla^{*}\mathbf{U}_{c}^{*})^{T}.\]  
The relaxation time and viscosity are assumed constant with the total viscosity given by:  
\[\eta_{c}^{*}=\eta_{sol}^{*}+\eta_{p}^{*}.\]