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In this paper, we consider the numerical approximations for a hydrodynamicalmodel of smectic-A liquid crystals. The model, derived from the variationalapproach of the modified Oseen-Frank energy, is a highly nonlinear system thatcouples the incompressible Navier-Stokes equations and a constitutive equationfor the layer variable. We develop two linear, second-order time-marchingschemes based on the "Invariant Energy Quadratization" method for nonlinearterms in the constitutive equation, the projection method for the Navier-Stokesequations, and some subtle implicit-explicit treatments for the convective andstress terms. Moreover, we prove the well-posedness of the linear system andtheir unconditionally energy stabilities rigorously. Various numericalexperiments are presented to demonstrate the stability and the accuracy of thenumerical schemes in simulating the dynamics under shear flow and the magneticfield.

Second Order, linear and unconditionally energy stable schemes for a hydrodynamic model of Smectic-A Liquid Crystals