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In this article, we investigate the effect of surface tension in the Rayleigh–Taylor (RT) problem of stratified incompressible viscoelastic fluids. We prove that there exists an unstable solution to the linearized stratified RT problem with a largest growth rate  Λ  under the instability condition (i.e., the surface tension coefficient  ϑ  is less than a threshold    $\vartheta _{c}$  ϑ c ). Moreover, for this instability condition, the largest growth rate    $\varLambda _{\vartheta }$  Λ ϑ decreases from a positive constant to 0, when  ϑ  increases from 0 to    $\vartheta _{c}$  ϑ c , which mathematically verifies that the internal surface tension can constrain the growth of the RT instability during the linear stage.

On effect of surface tension in the Rayleigh–Taylor problem of stratified viscoelastic fluids