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Register a new userStability of the flow due to a linear stretching sheet
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In this article we consider the linear stability of the two-dimensional flow induced by the linear stretching of a surface in the streamwise direction. The basic flow is a rare example of an exact analytical solution of the Navier-Stokes equations. Using results from a large Reynolds number asymptotic study and a highly accurate spectral numerical method we show that this flow is linearly unstable to disturbances in the form of Tollmien-Schlichting waves. Previous studies have shown this flow is linearly stable. However, our results show that this is only true for G{o}rtler-type disturbances.
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In homogeneous isotropic turbulence, slender rods are known to align with the Lagrangian stretching direction. However, how the degree of alignment depends on the aspect ratio of the rod is not understood. Moreover, many flows of practical interest are anisotropic and inhomogeneous. Here we study the alignment of rods with the Lagrangian stretching direction in a channel flow, which is approximately homogeneous and isotropic near the center but inhomogeneous and anisotropic near the walls. Our main question is how the distribution of relative angles between a rod and the Lagrangian stretching direction depends on the aspect ratio of the rod and upon the distance of the rod from the channel wall. We find that the distribution exhibits two regimes: a plateau at small angles that corresponds to random uncorrelated motion, and power-law tails that describe large excursions. The variance of the relative angle is described by the width of the plateau. We find that slender rods near the channel center align better with the Lagrangian stretching direction, compared to those near the channel wall. These observations are explained in terms of simple statistical models based on Jefferys equation, qualitatively near the channel center and quantitatively near the channel wall. Lastly we discuss the consequences of our results for the distribution of relative angles between the orientations of nearby rods (Zhao et al., Phys. Rev. Fluids, vol. 4, 2019, 054602).
In this chapter, we analyze the steady-state microscale fluid--structure interaction (FSI) between a generalized Newtonian fluid and a hyperelastic tube. Physiological flows, especially in hemodynamics, serve as primary examples of such FSI phenomena. The small scale of the physical system renders the flow field, under the power-law rheological model, amenable to a closed-form solution using the lubrication approximation. On the other hand, negligible shear stresses on the walls of a long vessel allow the structure to be treated as a pressure vessel. The constitutive equation for the microtube is prescribed via the strain energy functional for an incompressible, isotropic Mooney--Rivlin material. We employ both the thin- and thick-walled formulations of the pressure vessel theory, and derive the static relation between the pressure load and the deformation of the structure. We harness the latter to determine the flow rate--pressure drop relationship for non-Newtonian flow in thin- and thick-walled soft hyperelastic microtubes. Through illustrative examples, we discuss how a hyperelastic tube supports the same pressure load as a linearly elastic tube with smaller deformation, thus requiring a higher pressure drop across itself to maintain a fixed flow rate.
We revisit the somewhat classical problem of the linear stability of a rigidly rotating liquid column in this communication. Although literature pertaining to this problem dates back to 1959, the relation between inviscid and viscous stability criteria has not yet been clarified. While the viscous criterion for stability, given by $We = n^2+k^2-1$, is both necessary and sufficient, this relation has only been shown to be sufficient in the inviscid case. Here, $We = rho Omega^2 a^3/gamma$ is the Weber number and measures the relative magnitudes of the centrifugal and surface tension forces, with $Omega$ being the angular velocity of the rigidly rotating column, $a$ the column radius, $rho$ the density of the fluid, and $gamma$ the surface tension coefficient; $k$ and $n$ denote the axial and azimuthal wavenumbers of the imposed perturbation. We show that the subtle difference between the inviscid and viscous criteria arises from the surprisingly complicated picture of inviscid stability in the $We-k$ plane. For all $n >1$, the viscously unstable region, corresponding to $We > n^2+k^2-1$, contains an infinite hierarchy of inviscidly stable islands ending in cusps, with a dominant leading island. Only the dominant island, now infinite in extent along the $We$ axis, persists for $n= 1$. This picture may be understood, based on the underlying eigenspectrum, as arising from the cascade of coalescences between a retrograde mode, that is the continuation of the cograde surface-tension-driven mode across the zero Doppler frequency point, and successive retrograde Coriolis modes constituting an infinite hierarchy.
We study fluid-structure interactions (FSIs) in a long and shallow microchannel, conveying a non-Newtonian fluid, at steady state. The microchannel has a linearly elastic and compliant top wall, while its three other walls are rigid. The fluid flowing inside the microchannel has a shear-dependent viscosity described by the power-law rheological model. We employ lubrication theory to solve for the flow problem inside the long and shallow microchannel. For the structural problem, we employ two plate theories, namely Kirchhoff-Love theory of thin plates and Reissner-Mindlin first-order shear deformation theory. The hydrodynamic pressure couples the flow and deformation problem by acting as a distributed load onto the soft top wall. Within our perturbative (lubrication theory) approach, we determine the relationship between flow rate and the pressure gradient, which is a nonlinear first-order ordinary differential equation for the pressure. From the solution of this differential equation, all other quantities of interest in non-Newtonian microchannel FSIs follow. Through illustrative examples, we show the effect of FSI coupling strength and the plate thickness on the pressure drop across the microchannel. Through direct numerical simulation of non-Newtonian microchannel FSIs using commercial computational engineering tools, we benchmark the prediction from our mathematical prediction for the flow rate-pressure drop relation and the structural deformation profile of the top wall. In doing so, we also establish the limits of applicability of our perturbative theory.
Understanding ventilation strategy of a supercavity is important for designing high-speed underwater vehicles wherein an artificial gas pocket is created behind a flow separation device for drag reduction. Our study investigates the effect of flow unsteadiness on the ventilation requirements to form (CQf) and collapse (CQc) a supercavity. Imposing flow unsteadiness on the incoming flow has shown an increment in higher CQf at low free stream velocity and lower CQf at high free stream velocity. High-speed imaging reveals distinctly different behaviors in the recirculation region for low and high freestream velocity under unsteady flows. At low free stream velocities, the recirculation region formed downstream of a cavitator shifted vertically with flow unsteadiness, resulting in lower bubble collision and coalescence probability, which is critical for the supercavity formation process. The recirculation region negligibly changed with flow unsteadiness at high free stream velocity and less ventilation is required to form a supercavity compared to that of the steady incoming flow. Such a difference is attributed to the increased transverse Reynolds stress that aids bubble collision in a confined space of the recirculation region. CQc is found to heavily rely on the vertical component of the flow unsteadiness and the free stream velocity. Interfacial instability located upper rear of the supercavity develops noticeably with flow unsteadiness and additional bubbles formed by the distorted interface shed from the supercavity, resulting in an increased CQc. Further analysis on the quantification of such additional bubble leakage rate indicates that the development and amplitude of the interfacial instability accounts for the variation of CQc under a wide range of flow unsteadiness. Our study provides some insights on the design of a ventilation strategy for supercavitating vehicles in practice.
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