The optimal design of two-phase flow pipes depends on various variables. Pressure drop and liquid hold-up are essential parameters in the appropriate design of pipes. These parameters can be estimated by numerical simulation of pipes. However, the simulation strongly depends on both phase frictions. Frictions are commonly calculated using friction factors, and various empirical correlations have been proposed to calculate friction factors, but most of these correlations are dependent on the experimental conditions. The present study proposes a semi-empirical friction factor model by modifying an existing model in the literature for stratified two-phase flows. Experimental data from various laboratories are taken to validate the interface shear stress, velocity profile and pressure gradient. Compared with previous models, the predicted velocity profiles and shear stresses are in better agreement with the experimental data.

In this study, the coupled Eulerian-Lagrangian approach MPPIC-VOF is adopted, which is capable of capturing solid particle motion as well as fluid phase development simultaneously. Prior to three-phase flow simulation, two-phase air-water flow in a specific range of gas and liquid superficial velocities with slug and plug flow patterns is simulated. Then, polypropylene particles are injected as the solid phase into the air-water flow and the effects of particles on pressure drop, void fraction, and flow patterns are studied. The pressure drop and void fraction results are validated for both two-phase and three-phase flows by different correlations and experimental results. As a result of particle injection at the low concentration of 0.5%, void fraction increases by 13.7% and pressure drop changes on average by 18%, respectively. However, there was no significant change in the flow patterns.

Rheology is central to the design and industrial manufacture of cosmetic products, which are multicomponent soft materials whose performance depends on formulation, microstructure, processing history, temperature, shear, and time. Although apparent viscosity is widely used for quality control, a single-point measurement cannot fully describe yield behavior, viscoelasticity, thixotropic recovery, wall slip, extensional response, crystallization, or sensory performance. This review presents an industrial framework for applying rheology across the cosmetic-product lifecycle, from formulation and raw-material selection to scale-up, manufacturing, filling, release, stability, quality assurance, and consumer experience. Steady-shear, oscillatory, transient, nonlinear, extensional, tribological, powder-flow, and slurry measurements are discussed together with key methodological limitations. Particular attention is given to emulsification, polymer hydration, pigment and powder incorporation, deaeration, cooling, crystallization, and scale-up criteria, including tip speed, power input, circulation, heat transfer, and energy history. Rheological fingerprints are also examined for batch comparability, process analytical technology, statistical process control, computational fluid dynamics, machine learning, and digital twins. The review highlights links among formulation, microstructure, rheology, processing, stability, package performance, sensory response, and consumer outcomes and proposes a stage-based testing framework and research priorities for more robust cosmetic manufacturing.