More Like this

1. Stability of Fluid Flow through a Channel with Flexible Walls

   https://doi.org/10.1155/2021/8825677  

   Shubov, Marianna A. ; Edwards, Madeline M. ( March 2021 , International Journal of Mathematics and Mathematical Sciences)

   Solovjovs, Sergejs (Ed.)

   In the present paper, we summarize the results of the research devoted to the problem of stability of the fluid flow moving in a channel with flexible walls and interacting with the walls. The walls of the vessel are subject to traveling waves. Experimental data show that the energy of the flowing fluid can be transferred and consumed by the structure (the walls), inducing “traveling wave flutter.” The problem of stability of fluid-structure interaction splits into two parts: (a) stability of fluid flow in the channel with harmonically moving walls and (b) stability of solid structure participating in the energy exchange with the flow. Stability of fluid flow, the main focus of the research, is obtained by solving the initial boundary value problem for the stream function. The main findings of the paper are the following: (i) rigorous formulation of the initial boundary problem for the stream function, ψ x , y , t , induced by the fluid-structure interaction model, which takes into account the axisymmetric pattern of the flow and “no-slip” condition near the channel walls; (ii) application of a double integral transformation (the Fourier transformation and Laplace transformation) to both the equation and boundary and initial conditions, which reduces the original partial differential equation to a parameter-dependent ordinary differential equation; (iii) derivation of the explicit formula for the Fourier transform of the stream function, ψ ˜ k , y , t ; (iv) evaluation of the inverse Fourier transform of ψ ˜ k , y , t and proving that reconstruction of ψ x , y , t can be obtained through a limiting process in the complex k -plane, which allows us to use the Residue theorem and represent the solution in the form of an infinite series of residues. The result of this research is an analytical solution describing blood flowing through a channel with flexible walls that are being perturbed in the form of a traveling wave. 

   more » « less
2. Flow in oscillatory boundary layers over permeable beds

   https://doi.org/10.1063/5.0104305  

   Meza-Valle, Claudio ; Pujara, Nimish ( September 2022 , Physics of Fluids)

   In fluid dynamics applications that involve flow adjacent to a porous medium, there exists some ambiguity in how to model the interface. Despite different developments, there is no agreed upon boundary condition that should be applied at the interface. We present a new analytical solution for laminar boundary layers over permeable beds driven by oscillatory free stream motion where flow in the permeable region follows Darcy's law. We study the fluid boundary layer for two different boundary conditions at the interface between the fluid and a permeable bed that was first introduced in the context of steady flows: a mixed boundary condition proposed by Beavers and Joseph [“Boundary conditions at a naturally permeable bed,” J. Fluid Mech. 30, 197–207 (1967)] and the velocity continuity condition proposed by Le Bars and Worster [“Interfacial conditions between a pure fluid and a porous medium: Implications for binary alloy solidification,” J. Fluid Mech. 550, 149–173 (2006)]. Our analytical solution based on the velocity continuity condition agrees very well with numerical results using the mixed boundary condition, suggesting that the simpler velocity boundary condition is able to accurately capture the flow physics near the interface. Furthermore, we compare our solution against experimental data in an oscillatory boundary layer generated by water waves propagating over a permeable bed and find good agreement. Our results show the existence of a transition zone below the interface, where the boundary layer flow still dominates. The depth of this transition zone scales with the grain diameter of the porous medium and is proportional to an empirical parameter that we fit to the available data. 

   more » « less
3. Density Effects at a Concordant Bed Natural River Confluence

   https://doi.org/10.1029/2019WR026217  

   Horna‐Munoz, Daniel ; Constantinescu, George ; Rhoads, Bruce ; Lewis, Quinn ; Sukhodolov, Alexander ( April 2020 , Water Resources Research)

   Confluences are locations of complex hydrodynamic conditions within river systems. The effects on hydrodynamics and mixing of temperature‐induced density differences between incoming flows are investigated at a small‐size, concordant bed confluence. To evaluate density effects, results of eddy‐resolving simulations for a densimetric Froude numberFr= 4.9 (weak‐density‐effects cases) andFr= 1.6 (strong‐density‐effects cases) are compared to results of simulations in which the densities of the incoming flows do not differ (no‐density‐effects cases). Flow patterns predicted for both weak‐ and strong‐density‐effects cases show that secondary flow develops with increasing distance from the confluence apex. The pattern of secondary flow is characterized by denser fluid on one side of the confluence moving near the bed toward the side of the downstream channel corresponding to the less dense fluid and the less dense fluid moving near the free surface in the opposite direction. This pattern of fluid motion is similar to a spatially evolving lock‐exchange cross flow. In the strong‐density‐effects simulations, a cross‐stream cell of secondary flow develops at the density interface between the flows, similar to interfacial billows generated in classical lock‐exchange flows. Density effects increase global mixing with respect to corresponding no‐density‐effects cases regardless of whether the high‐momentum stream contains the higher‐density fluid or the lower‐density fluid. When density effects are weak, the lock‐exchange mechanism either reinforces the pattern of mixing associated with secondary flow induced by inertial forces, particularly helical motion, or opposes this pattern of mixing, depending on which tributary contains the denser fluid. When density effects are strong, flow from the upstream channel with the denser fluid moves under the flow from the upstream channel with the less dense fluid.

    

   more » « less
4. Stokes flow solutions in infinite and semi‐infinite porous channels

   https://doi.org/10.1111/sapm.12574  

   Bernardi, Francesca ; Chellam, Shankararaman ; Cogan, N. G. ; Moore, M. N. J. ( March 2023 , Studies in Applied Mathematics)

   We derive a class of exact solutions for Stokes flow in infinite and semi‐infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no‐slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi‐infinite channel domain, the ground‐state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane‐based water purification, heat and mass transfer, and fuel cells.

    

   more » « less
5. Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

   https://doi.org/10.1016/j.jfa.2023.110057  

   Koganemaru, Junichi ; Tice, Ian ( October 2023 , Journal of Functional Analysis)

   This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest. 

   more » « less