Generalized Einstein’s and Brinkman’s solutions for the effective viscosity of nanofluids

Abstract

In this paper, we derived closed form analytical solutions for the effective viscosity of the suspensions of solid spheres that take into account size effects. This result was obtained by using the solution for the effective shear modulus of particulate composites developed in the framework of the strain gradient elasticity theory. Assuming the incompressibility of the matrix and the rigid behavior of particles and using a mathematical analogy between the theories of elasticity and viscous fluids, we derived generalized Einstein’s formula for effective viscosity. Then, generalized Brinkman’s solution for the concentrated suspensions was derived using the differential method. The obtained solutions contain a single additional length scale parameter, which can be related to the interactions between the base liquid and solid particles in the suspensions. In the case of a large ratio between the diameter of the particles and the length scale parameter, the developed solutions were reduced to the classical solutions. However, for the small relative diameter of particles, an increase of the effective viscosity was predicted. It was shown that the developed models agree well with the known experimental data. Solutions for the fibrous suspensions were also derived and validated.