Tracking Multiphase Flows through Steep Reservoirs with External Constraint

Abstract

Problem statement: The study offers theoretical formulations for high-viscosity particulate flows in inclined reservoirs, taking into account the presence of homogeneous spheroidal particles of various types to produce discrete two-phase suspensions. Purpose: The primary objective of this analytical and comparative study is to identify the most dependable nanoparticles among hafnium and crystal metals that are suspended in an Eyring–Powell fluid through an inclined channel while being subjected to external magnetic and gravitational forces. Solution methodology: The flow dynamics of multiphase flows are formulated utilizing the stress tensor of the base fluid. The regular perturbation method (RPM) is employed to attain a more closed-form solution. The perturbation method is frequently employed in engineering problems to obtain an approximated solution, even when demonstrating the convergence of the solution is challenging. The rough solution is also validated through a thorough parametric analysis that shows the role of relevant parameters that contribute to the multiphase flow. Results: A concise parametric study is carried out against some of the most pertinent parameters and reveals that additional particles have promising effects on the momentum of each multiphase flow, whereas Eyring–Powell multiphase suspensions lessen in momentum due to strong internal viscous forces. The velocity of fluid and particle phases diminish with Hartmann number M and Froude number Fr. The second-order material constant B and concentration of nanoparticles C boost the motion of the fluid. The velocities of the particulate phase are quicker than the fluid phase. The hafnium particle is more reliable than crystal particles. Solution benchmark: Numerical and graphical findings have also been compared with the existing literature for the limiting case and found to be fully in agreement. Applications: This study’s findings provide a wider understanding of subterranean flows, specifically within the petroleum sector, with a focus on multiphase flows. Originality: The current study represents the authors’ original work and has not been previously submitted or published elsewhere.

Applications of lubrication approximation theory in the analysis of the roll‐coating using a tangent hyperbolic fluid model

“…The differential equation obtained in Equation ( 24) is highly non-linear, so its solution cannot be calculated easily, therefore the semi-analytical (i.e., perturbation technique) [40][41][42][43] is used for small 𝑊𝑒 << 1. We define the following expressions:…”

“…The differential equation obtained in Equation () is highly non‐linear, so its solution cannot be calculated easily, therefore the semi‐analytical (i.e., perturbation technique) [40–43] is used for small $$We &lt; &lt; 1.$$ We define the following expressions: $$$$\begin{equation}u = {u}_0 + We{u}_1 + W{e}^2{u}_2 + O(W{e}^3),\end{equation}$$$$ $$$$\begin{equation}\frac{{dp}}{{dx}} = \frac{{d{p}_0}}{{dx}} + We\frac{{d{p}_1}}{{dx}} + W{e}^2\frac{{d{p}_2}}{{dx}} + O\left( {W{e}^3} \right),\end{equation}$$$$ $$$$\begin{equation}\lambda = {\lambda }_0 + We{\lambda }_1 + W{e}^2{\lambda }_2 + O\left( {W{e}^3} \right),\end{equation}$$$$ $$$$\begin{equation}{x}_s = {x}_0 + We{x}_1 + W{e}^2{x}_2 + O\left( {W{e}^3} \right),\end{equation}$$$$ …”