arXiv subjects

Nastaran Naghshineh

Publications and source records attributed to Nastaran Naghshineh.

3 recordsLinked to original sources

arXiv2023

On the shape of an axisymmetric meniscus rising from a static liquid pool

We examine the classical problem of the height of a static liquid interface that forms on the outside of a solid vertical cylinder in an unbounded stagnant pool exposed to air. Gravitational and surface tension effects compete to affect the interface shape as characterized by the Bond number, $B = ρg R^{2}/σ$, where $ρ$ is fluid density, $g$ is the gravitational constant, $R$ is the radius of the cylinder, and $σ$ is surface tension. Here, we provide a convergent power series solution for interface shapes that rise above the horizontal pool as a function of Bond number. The power series solution is expressed in terms of a pre-factor that matches the large distance asymptotic behavior -- the modified Bessel function of zeroth order -- and an Euler transformation that moves the influence of convergence-limiting singularities out of the physical domain. The power series solution is validated through comparison with a numerical solution, and the $B\rightarrow0$ matched asymptotic solutions of Lo (1983, J. Fluid Mech., 132, p.65-78). For a benchmark static contact angle of $45$ degrees, the power series approach exceeds the accuracy of matched asymptotic solutions for $B>0.01$; this lower limit on $B$ arises from round-off error in the computation of the series coefficients in double precision arithmetic, and is reduced as the contact angle is increased.

physics.flu-dyn↗

arXiv2023

On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs

We examine the power series solutions of two classical nonlinear ordinary differential equations of fluid mechanics that are mathematically related by their large-distance asymptotic behaviors in semi-infinite domains. The first problem is that of the "Sakiadis" boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. The second problem is that of a static air-liquid meniscus with surface tension that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. For the latter problem, the exact analytic solution -- given as distance from the wall as function of meniscus height -- has long been known (Batchelor, 1967). Here, we provide an explicit solution as meniscus height vs. distance from the wall to elucidate structural similarities to the Sakiadis boundary layer. Although power series solutions are readily obtainable to the governing nonlinear ordinary differential equations, we show that -- in both problems -- the series diverge due to non-physical complex or negative real-valued singularities. In both cases, these singularities can be moved by expanding in exponential gauge functions motivated by their respective large distance asymptotic behaviors to enable series convergence over their full semi-infinite domains. For the Sakiadis problem, this not only provides a convergent Taylor series (and conjectured exact) solution to the ODE, but also a means to evaluate the wall shear parameter (and other properties) to within any desired precision. Although the nature of nonlinear ODEs precludes general conclusions, our results indicate that asymptotic behaviors can be useful when proposing variable transformations to overcome power series divergence.

physics.flu-dyn↗

arXiv2023

Asymptotically-consistent analytical solutions for the non-Newtonian Sakiadis boundary layer

The Sakiadis boundary layer induced by a moving wall in a semi-infinite fluid domain is a fundamental laminar flow field relevant to high speed coating processes. This work provides an analytical solution to the boundary layer problem for Ostwald-de Waele power law fluids via a power series expansion, and extends the approach taken for Newtonian fluids [Naghshineh et al., "On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs", IMA J. of Appl. Math., (2023)] in which variable substitutions (which naturally determine the gauge function in the power series) are chosen to be consistent with the large distance behavior away from the wall. Contrary to prior literature, the asymptotic behavior dictates that a solution only exists in the range of power law exponents, $α$, lying in the range $0.5 < α\leq 1$. An analytical solution is obtained in the range of approximately $0.74 \leq α< 1$, using a convergent power series with an asymptotically motivated gauge function. For power laws corresponding to $0.5 < α<0.74$, the gauge function becomes ill-defined over the full domain, and an approximate analytical solution is obtained using the method of asymptotic approximants [Barlow et al., "On the summation of divergent, truncated, and underspecified power series via asymptotic approximants", Q. J. Mech. Appl. Math., (2017)]. The approximant requires knowledge of two physical constants, which we compute a priori using a numerical shooting method on a finite domain. The utility of the power series solution is that it can be solved on the entire semi-infinite domain and--in contrast to a numerical solution--does not require a finite domain length approximation and subsequent domain length refinement.

physics.flu-dyn↗