We showed that various expressions of stress in different models of fluids and in different forms of applying Newton's Second Law can be chosen, however there is always one Newton's Second Law which restricts and provides flexibility in these expressions. When fluids are regarded as being made of cells from a microscopic point of view, temperature force (gradient of internal kinetic energy) can also drive them flow.
The behavior of flagellated bacteria swimming in non-Newtonian media remains an area with contradictory and conflicting results. We report on the behavior of wild-type and smooth-swimming E. coli in Newtonian, shear thinning and viscoelastic media, measuring their trajectories and swimming speed using a three dimensional real-time tracking microscope. We conclude that the speed enhancement in Methocel solution at higher concentration is due to shear-thinning and an analytical model is used to support our experimental result. We argue that shear-induced normal stresses reduce the wobbling behav…
In the present work we formulated the boundary-value-problem, comprising partial differential equations (PDEs) of steady flow for laminar/turbulent circular jet of a micropolar fluid. A new boundary layer-similarity transformation/solution was derived which is valid not only for the Newtonian fluids but also for the micropolar fluids. Through this transformation PDEs are transformed into the ordinary differential equations (ODEs). These ODEs were solved numerically by the finite-difference method. The obtained results were compared with existing results [9] for the Newtonian fluids. The compar…
In this study, it was formulated the boundary-value-problem (BVP), comprising partial differential equations (PDEs), of steady flow for plane, laminar jet of a micropolar fluid. A new similarity transformation/solution was derived which is valid not only for the Newtonian fluids but also for the micropolar fluids. Obviously, this transformation will be transformed the PDEs into the ordinary differential equations (ODEs). These ODEs were solved numerically by the finite difference method. The obtained results were compared with existing results [1, 12] for the Newtonian fluids. The comparison w…
The motion of incompressible fractional Oldroyd-B fluids between two parallel walls perpendicular to a plate that applies time-dependent shear stresses to the fluid is studied by means of integral transforms. In the special cases of Newtonian and second grade fluids, these shear stresses reduces to fH(t)sin(wt) or fH(t)cos(wt). General solutions for velocity are presented as a sum of Newtonian solutions and the corresponding non-Newtonian contributions. They reduce to the similar solutions corresponding to the motion over an infinite plate if the distance between walls tend to infinity and can…
We investigate the dynamics of a prolate spheroid in a shear flow of a shear-thinning Carreau fluid. The motion of a prolate particle is developed analytically for asymptotically weak shear thinning and then integrated numerically. We find that shear-thinning rheology does not lift the degeneracy of Jeffery orbits observed in Newtonian fluids but the instantaneous rate of rotation and trajectories of the orbits are modified. Qualitatively, shear thinning has a similar effect to elongating the particle in a Newtonian fluid. The period of rotation increases as the particle slows down more when a…
Self-propulsion at low Reynolds number is notoriously restricted, a concept that is commonly known as the "scallop theorem". Here we present a truly self-propelled swimmer (force- and torque- free) that, while unable to swim in a Newtonian fluid due to the scallop theorem, propels itself in a non-Newtonian fluid as a result of fluid elasticity. This propulsion mechanism is demonstrated using a robotic swimmer, comprised of a "head" sphere and a "tail" sphere, whose swimming speed is shown to have reasonable agreement with a microhydrodynamic asymptotic theory and numerical simulations. Schlier…
Microorganisms such as bacteria often swim in fluid environments that cannot be classified as Newtonian. Many biological fluids contain polymers or other heterogeneities which may yield complex rheology. For a given set of boundary conditions on a moving organism, flows can be substantially different in complex fluids, while non-Newtonian stresses can alter the gait of the microorganisms themselves. Heterogeneities in the fluid may also be characterized by length scales on the order of the organism itself leading to additional dynamic complexity. In this chapter we present a theoretical overvi…
Swimming microorganisms often self propel in fluids with complex rheology. While past theoretical work indicates that fluid viscoelasticity should hinder their locomotion, recent experiments on waving swimmers suggest a possible non-Newtonian enhancement of locomotion. We suggest a physical mechanism, based on fluid-structure interaction, leading to swimming in a viscoelastic fluid at a higher speed than in a Newtonian one. Using Taylor's two-dimensional swimming sheet model, we solve for the shape of an active swimmer as a balance between the external fluid stresses, the internal driving mome…
A port-Hamiltonian model for compressible Newtonian fluid dynamics is presented in entirely coordinate-independent geometric fashion. This is achieved by use of tensor-valued differential forms that allow to describe describe the interconnection of the power preserving structure which underlies the motion of perfect fluids to a dissipative port which encodes Newtonian constitutive relations of shear and bulk stresses. The relevant diffusion and the boundary terms characterizing the Navier-Stokes equations on a general Riemannian manifold arise naturally from the proposed construction.
Unlike the power-law model, the Ellis model describes the apparent viscosity of a shear-thinning fluid with no singularity in the limit of a vanishingly small shear stress. In particular, this model matches the Newtonian behaviour when the shear stresses are very small. The emergence of the Rayleigh-Bénard instability is studied when a horizontal pressure gradient, yielding a basic throughflow, is prescribed in a horizontal porous layer. The threshold conditions for the linear instability of this system are obtained both analytically and numerically. In the case of a negligible flow rate, the …
We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids, also called yield-stress fluids. In contrast to the conventional approaches relying on the non-linear viscosity concept of the Navier-Stokes theory and representation of the solid state as an infinitely rigid non-deformable solid, the solid state in our theory is deformable and the fluid state is considered rather as a "melted" solid via a certain procedure …
The dynamics of a single elastoviscoplastic (EVP) drop immersed in plane shear flow of a Newtonian fluid is studied by 3D direct numerical simulations using a finite-difference and level-set method combined with the Saramito model for the EVP fluid. This model gives rise to a yield stress behavior, where the unyielded state of the material is described as a Kelvin-Voigt viscoelastic solid and the yielded state as a viscoelastic Oldroyd-B fluid. Yielding of an initially solid drop of Carbopol is simulated under successively increasing shear rates. We proceed to examine the roles of nondimension…
We introduce a full-Lagrangian heterogeneous multiscale method (LHMM) to model complex fluids with microscopic features that can extend over large spatio-temporal scales, such as polymeric solutions and multiphasic systems. The proposed approach discretizes the fluctuating Navier-Stokes equations in a particle-based setting using Smoothed Dissipative Particle Dynamics (SDPD). This multiscale method uses microscopic information derived on-the-fly to provide the stress tensor of the momentum balance in a macroscale problem, therefore bypassing the need for approximate constitutive relations for …
In 1926, Murray proposed the first law for the optimal design of blood vessels. He minimized the power dissipation arising from the trade-off between fluid circulation and blood maintenance. The law, based on a constant fluid viscosity, states that in the optimal configuration the fluid flow rate inside the vessel is proportional to the cube of the vessel radius, implying that wall shear stress is not dependent on the vessel radius. Murray's law has been found to be true in blood macrocirculation, but not in microcirculation. In 2005, Alarcón et al took into account the non monotonous dependen…
