Stratified liquid-liquid flow through microchannels with grooved walls

Patterned surfaces have numerous applications in the microscale flow regime, such as heat transfer enhancement, mixing, and microfluidics devices. The present study analytically examines the pressure-driven flow of two immiscible Newtonian fluids through a grooved microchannel. The orientation of the channel is defined as the top and bottom walls being flat and wavy surfaces, respectively. A no-slip boundary condition is assumed at both walls. The present problem is investigated by invoking the Fourier theory for a flow along streamwise grooves at the Stokes flow limit. Flow rates and velocities of both fluids are determined analytically and numerically. A finite-element-based numerical study is conducted to understand the accuracy of the theoretical models. Results are generated to show the effects of viscosity ratio, wall undulation amplitude and wavelength of the patterned channel. For both fluids with a small wavelength, hydraulic permeability decreases with increasing the pattern amplitude at various viscosity ratios. Meanwhile, in the case of large wavelengths, hydraulic permeability increases with pattern amplitude at different values of viscosity ratio. This behaviour of permeability is identified for fluid 2 (which is in contact with the grooved surface). Roughness and confinement effects are captured with increasing pattern amplitude at different wavelengths and viscosity ratios. The present analytical model agrees well with numerical values. The findings provide a deeper understanding of the stratified flow through microchannels with undulating surfaces, with potential applications in electronic cooling, skin-friction drag, interfacial fluid dynamics, and enhancement and reduction of heat transfer.