Parallel superposition of oscillatory shearing on steady shear flow: Normal stresses

Of the many rheological material functions, the two most important are (i) steady shear flow and (ii) oscillatory shear flow. Another canonical rheological material function is constructed by superposing, in parallel, at small-amplitude, romanette (ii) upon (i). To this, complex fluids, including polymeric liquids, will respond with a complex viscosity that depends on both the steady shear rate of (i) and the angular frequency of (ii). Our recent work [Phys. Fluids 36(8), 083121 (2024)] uncovers the macromolecular origins of this complex viscosity dependence using rotarance theory. By rotarance, we mean at least involving the hydrodynamic resistances of the macromolecules to reorientation. However, to parallel superposition, complex fluids also respond with two normal stress differences. We devote this paper to uncovering the macromolecular origins of both of these normal stress differences, using rotarance theory. For both the first and second normal stress differences, we arrive at analytical expressions for the complex normal stress coefficients. We find that these increase with the lopsidedness of the macromolecular structure, be this lopsidedness prolate or oblate. We further find that, whereas the real and minus imaginary parts of the parts of the complex components of the primary normal stress difference are signed identically, the real and minus imaginary parts of the corresponding secondary are signed oppositely.