The behaviour of an isolated nearly spherical drop in an ambient linear flow is examined analytically at small but finite Reynolds numbers, and thereby the first effects of inertia on the bulk stress in a dilute emulsion of neutrally buoyant drops are calculated. The Reynolds numbers, Re = a2ρ/μ and $\hat{\Rey} \,{=}\, \dot{\gamma}a^2\rho/\hat{\mu}$, are the relevant dimensionless measures of inertia in the continuous and disperse(drop) phases, respectively. Here, a is the drop radius, is the shear rate, ρ is the common density and and μ are, respectively, the viscosities of the drop and the suspending fluid. The assumption of nearly spherical drops implies the dominance of surface tension, and the analysis therefore corresponds to the limit of the capillary number(Ca) based on the viscosity of the suspending fluid being small but finite; in other words, Ca ≪ 1, where Ca = μa/T, T being the coefficient of interfacial tension. The bulk stress is determined to O(φRe) via two approaches. The first one is the familiar direct approach based on determining the force density associated with the disturbance velocity field on the surface of the drop; the latter is determined to O(Re) from a regular perturbation analysis. The second approach is based on a novel reciprocal theorem formulation and allows the calculation, to O(Re), of the drop stresslet, and hence the emulsion bulk stress, with knowledge of only the leading-order Stokes fields. The first approach is used to determine the bulk stress for linear flows without vortex stretching, while the reciprocal theorem approach allows one to generalize this result to any linear flow. For the case of simple shear flow, the inertial contributions to the bulk stress lead to normal stress differences(N1, N2) at O(φRe), where φ(≪1) is the volume fraction of the disperse phase. Inertia leads to negative and positive contributions, respectively, to N1 and N2 at O(φRe). The signs of the inertial contributions to the normal stress differences may be related to the O(ReCa) tilting of the drop towards the velocity gradient direction. These signs are, however, opposite to that of the normal stress differences in the creeping flow limit. The latter are O(φCa) and result from an O(Ca2) deformation of the drop acting to tilt it towards the flow axis. As a result, even a modest amount of inertia has a significant effect on the rheology of a dilute emulsion. In particular, both normal stress differences reverse sign at critical Reynolds numbers(Rec) of O(Ca) in the limit Ca ≪ 1. This criterion for the reversal in the signs of N1 and N2 is more conveniently expressed in terms of a critical Ohnesorge number(Oh) based on the viscosity of the suspending fluid, where Oh = μ/(ρaT)1/2. The critical Ohnesorge number for a sign reversal in N1 is found to be lower than that for N2, and the precise numerical value is a function of λ. In uniaxial extensional flow, the Trouton viscosity remains unaltered at O(φRe), the first effects of inertia now being restricted to O(φRe3/2). The analytical results for simple shear flow compare favourably with the recent numerical simulations of Li & Sarkar (J. Rheol., vol. 49, 2005, p. 1377).
Normal stress differences in dense suspensions