Dimensionless Groups For Understanding Free Surface Flows of Complex Fluids

Numerous processing operations of complex fluids involve free surface deformations; examples include spraying and atomization of fertilizers and pesticides, fiber-spinning operations, paint application, roll-coating of adhesives and food processing operations such as container-and bottle-filling. Systematically understanding such flows can be extremely difficult because of the large number of different forces that may be involved; including capillarity, viscosity, inertia, gravity as well as the additional stresses resulting from the extensional deformation of the microstructure within the fluid. Consequently many free-surface phenomena are described by heuristic and poorly-quantified words such as 'spinnability', 'tackiness' and 'stringiness'. Additional specialized terms used in other industries include 'pituity' in lubricious aqueous coatings, 'body' and 'length' in the printing ink business, 'ropiness' in yogurts and 'long/short textures' in starch processing. A good approach to systematically getting to grips with such problems is through the tools of dimensional analysis (Bridgman, 1963) and this short note is intended to review the physics behind some of the plethora of specialized dimensionless groups one encounters when reading the scientific literature. The dominant balance of forces controlling the dynamics of any process depends on the relative magnitudes of each underlying physical effect entering the set of governing equations. In many (but certainly not all) problems, the dynamics are controlled by no more than two or three different contributions; and results from disparate experimental observations or numerical studies which describe different processing regimes or domains of stable/unstable operation can often be conveniently assembled in terms of processing diagrams such as those sketched in Figures 1 & 2. For bulk flows of non-Newtonian fluids the relative importance of inertial effects and elastic stresses with respect to viscous stresses are characterized by the Reynolds