The linear stability of viscoelastic nanofluid layer is investigated. The rheological behavior of the viscoelastic fluid is described through the Walter's model. The normal modes analysis is utilized to treat the
equations of motion for stationary and oscillatory convection. The stability analysis resulted in a thirddegree dispersion equation with complex coefficients. The Routh–Hurwitz theory is employed to investigate the dispersion relation. The stability criteria divide the plane into several parts of stable/unstable
regions. This shows some analogy with the nonlinear stability theory. The relation between the elasticity and the longitudinal wave number is graphically analyzed. The numerical calculations show that
viscoelastic flows are more stable than those of the Newtonian ones.