The stability of two-dimensional linear flows

A theoretical investigation is made of the linear stability of a viscous incompressible fluid undergoing a steady, unbounded two-dimensional flow in which the velocity field is a linear function of position. Such flows are approximately generated by a four-roll mill device which has many experimental applications, and can be characterized completely by a single parameter λ which ranges from λ=0 for simple shear flow to λ=1 for pure extensional flow. The linearized velocity disturbance equations are analyzed for an arbitrary spatially periodic initial disturbance to give the asymptotic behavior of the disturbance at large time for 0≤λ≤1. In addition, a complete analytical solution of the vorticity disturbance equation is obtained for the case λ=1. It is found that unbounded flows with 0<λ≤1 are unconditionally unstable. An instability criterion relating the initial disturbance wave vector α to the steady flow strain rate E, kinematic viscosity ν, and the parameter λ is obtained. This criterion shows that for all admissible values of E and ν, a wave vector α may be found which corresponds to disturbances that grow exponentially in time. The growth of these disturbances is accompanied by a growth of vorticity oriented along the principal axis of extensional strain in the case λ=1. The results of this investigation also confirm the established fact that simple shear flow (λ=0) is stable to all infinitesimal spatially periodic disturbances.