Hydrodynamic stability of round viscous fluid jets is considered within the framework of the non-modal approach. Both the jet fluid and surrounding gas are assumed to be incompressible and Newtonian; the effect of surface capillary pressure is taken into account. The linearized Navier-Stokes equations coupled with boundary conditions at the jet axis, interface and infinity are reduced to a system of four ordinary differential equations for the amplitudes of disturbances in the form of spatial normal modes. The eigenvalue problem is solved by using the orthonormalization method with Newton iterations and the system of least stable normal modes is found. Linear combinations of modes (optimal disturbances) leading to the maximum kinetic energy at a specified set of governing parameters are found. Parametric study of optimal disturbances is carried out for both an air jet and a liquid jet in air. For the velocity profiles under consideration, it is found that the non-modal instability mechanism is significant for non-axisymmetric disturbances. The maximum energy of the optimal disturbances to the jets at the Reynolds number of 1000 is found to be two orders of magnitude larger than that of the single mode. The largest growth is gained by the streamwise velocity component.

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This work is focused on the development of a dynamic criterion for the arching and bridging of spherical particles in a 3D suspension flow through a channel with plane walls. Elasticity of the particles and the channel walls are taken into account. The carrier fluid is viscous and incompressible. Bridging occurs under the balance of the hydrodynamic force exerted from the fluid on the particles and the friction force exerted from the walls on the particles. The 3D motion of particles in fluid is analyzed by means of direct numerical simulation. The bridging criterion is formulated as a domain on the plane in terms of the two nondimensional parameters: the particle size to channel width ratio and the flow velocity. For each scaled particle diameter there is a range of critical velocities, in which bridging occurs. Various geometrical configurations are considered: three and four particles across the slot. Stability of the bridge is studied. The dynamic bridging criterion is different from the earlier purely kinematic criteria, which were formulated in terms of the particle-to-channel width ratio only. The bridging criterion is implemented into the 2D width-averaged lubrication model of suspension flow through a plane channel, and illustrative simulations are conducted. Application is for proppant transport in hydraulic fractures.