Protein aggregation rate is known to be influenced by shear flow in protein solutions. This has important physiological implications as many of the body functions involve shear flow. Fluid mechanical shear can affect interactions between protein molecules, initiate protein aggregation, and further affect their biological activity. The shear rate is therefore an important parameter either to determine or to influence the properties of the protein solution when it forms a nucleus or aggregates. For experiments, the number density of nuclei can be controlled by using an optimal shear rate and protein concentration. However, this requires theoretical information on the shear rate for the experimental conditions. With this motivation, we have designed an experiment in which we can effectively apply shear with flow characteristics that can be calculated. Specifically, in a small hemispherically-shaped bowl, 4 mm in diameter we place the protein solution and insert a rounded rod that can be vibrated rotationally or laterally, maintaining spherical symmetry in the liquid region. This system is particularly useful when only small quantities of expensive protein solutions can be used for experimentation. We have carried out the mathematical analysis of the time-dependent flow field between two concentric hemispheres by the perturbation method using ε = U0/ωa ≪ 1 as a small parameter where U0 is a characteristic velocity, ω is the oscillation frequency and a is a length scale based on the vessel dimensions (bowl radius). We have obtained an analytical solution for the velocity field, and the shear rate in the liquid. In addition, with the nonlinear interaction of the oscillatory flow, there is a nonzero time-independent mean flow (known as streaming). With the integrated effect of shear in the liquid region, this result will be useful for conducting aggregation experiment in which the effective shear rate can be correlated to the aggregation rate.

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