Note on semi‐analytical nonstationary solution for the rivulet flows of non‐Newtonian fluids

Abstract

We have presented in this analytical research the revisiting of approach for mathematical modeling the rivulet flows on inclined surface in terms of viscous‐plastic theory of two‐dimensional movements in a frame of (x, y)‐plane in Cartesian coordinates. A semi‐analytical solution has been obtained here for nonstationary creeping approximation of plane‐parallel flow slowly moving on inclined surface. The main motivation of the current development is that each additional enhancement of knowledge in regard to this phenomenon should obviously improve the accuracy of calculations for possible applications in technological or engineering areas of fluid dynamics researches. Even in such simple formulation, equations of motion that govern the dynamics of rivulet flows of non‐Newtonian fluids are hard to be solved analytically. Nevertheless, we have succeeded in obtaining analytical expressions for the components of velocity in {Ox, Oy}‐directions of motion for slowly moving the advancing front of rivulet flow (where Ox‐axis coincides to the initial main direction of slowly moving rivulet flow). Restrictions to the form of flow (stemming from the continuity equation) have been used accordingly.

Analytical Solutions to the Unsteady Poiseuille Flow of a Second Grade Fluid with Slip Boundary Conditions

“…which contradicts (28). Therefore, we introduce the concept of generalized solutions and then prove the corresponding existence and uniqueness theorem.…”

“…Analysis of the second grade hybrid nanofluid flow over a stretching flat plate is given in [27]. A semi-analytical approach for the investigation of rivulet flows of non-Newtonian fluids was developed in [28,29]. Finally, we mention mathematical studies devoted to the existence and uniqueness of solutions to PDEs describing second grade fluid flows [30][31][32][33][34][35][36] as well as to optimal flow control and controllability problems [37][38][39][40].…”

Optimal Dirichlet Boundary Control for the Corotational Oldroyd Model

“…The paper [32] is devoted to studying the flow of Oldroyd fluids with fractional derivatives over a plate that applies shear stress to the fluid. Semi-analytical solutions for rivulet flows of non-Newtonian fluids are obtained in [33,34].…”

Non-Newtonian Pressure-Governed Rivulet Flows on Inclined Surface

“…According to [17][18][19][20], the system of Equations ( 1)-( 3) that governs the dynamics for a particular family of viscoplastic rivulet flows (in general, a Bingham fluid if plasticity effect of flow is also taken into account) should be presented in a 2D case in the Cartesian coordinates, according to the approach suggested earlier in [17][18][19].…”

“…We will investigate here the novel family of solutions where non-zero plasticity effect for the flow field is also taken into account (more general solutions than those investigated previously in [17,19,20] for the case of absence of plasticity effect). This implies that we will consider the profiles of flow velocity in Equations ( 1)-( 3), which are given by formulae suggested in [19] for solutions (here and below we have chosen 8µ ρ = 1 in appropriate units of scaling in the physical sense), but with a novel type of pressure field p chosen for such solutions…”