A kinetic analysis of morphing continuum theory for fluid flows

Abstract

To describe the behavior of a gas composed of spherical particles that rotate, the kinetic theory approach is presented. First-order approximations to the Boltzmann-Curtiss transport equation yield conservation equations that govern the translational velocity and rotation of the particles. The resulting equations match the form of the equations of morphing continuum theory (MCT), a theory derived from the principles of rational continuum thermomechanics. A direct comparison of corresponding terms provides expressions related to the new coefficients within MCT, showing a clear departure from classical expressions derived from a kinetic treatment of classical fluids. The identical expressions for the coefficients in the Cauchy stress and viscous diffusion terms in the kinetic linear momentum equation suggests that the coupling coefficient introduced by MCT outweighs the contribution of the classical kinematic viscosity. The kinetic theory equations reduce to the form of the Navier-Stokes equations when the local rotation is equated to the angular velocity, but the predominance of the coupling coefficient results in a viscous term that differs slightly from the classical expression derived using the Boltzmann distribution function. For simple cases of irrotational and incompressible flows, the kinetic equations mimic the form of the classical momentum equations derived from classical kinetic theory. This result is consistent with the fact that the difference between the two kinetic approaches is the local rotation of spherical particles. Preliminary numerical simulations of the MCT governing equations are discussed, with an emphasis on the importance of the new coupling coefficient. Turbulent incompressible profiles are achieved by setting dimensionless parameters to particular values. The key parameter involves the ratio of the coupling coefficient to the kinematic viscosity. The relationship between the coupling coefficient and kinematic viscosity is shown to be the driving force for the development of transitional and turbulent boundary layer profiles. Compressible turbulence results are generated using the same dimensionless parameter values that generated turbulence in the incompressible case. For supersonic flow over a cylinder, MCT displays an inverse energy cascade from small to large scales. In addition to visualizing turbulent processes, the results from MCT display the importance of coupling the linear and angular momenta equations, which is strengthened when the coupling coefficient increases. The expressions from kinetic theory coupled with the numerical results in MCT indicate that the physical phenomena driving a fluid composed of spherical particles depends heavily on the physical properties of the coupling coefficient.