The flow of a second-order fluid between two heated plates (Q2762838)

The authors study the motion of a second-order fluid between two parallel horizontal plates maintained at different temperatures, under constant pressure gradient. The second-order fluid is described by the relation \(T(x,t)=-pI+\mu A_1+\alpha_1A_2 +\alpha_2 A_1^2\), where \(A_1\) and \(A_2\) are the first two Rivlin-Ericksen tensors, \(p\) is hydrostatic pressure, and \(\mu\), \(\alpha_1\), \(\alpha_2\) are the constitutive moduli depending on the temperature \(\theta\) and on the invariants of the above tensors. The boundary value problem for fluid motion and heat transfer is written out in non-dimensional form and solved analytically by the transformation into a first-order system of differential equations. The system is linearized assuming that the unknown functions are obtained by small perturbations of unity: \(u=1+\widetilde u\), \(\theta=1+\widetilde \theta\), \(v=1+\widetilde v\), \(T=1+\widetilde T\). The authors also present some numerical solutions.