Exact Solutions for the Flow Spatiotemporal Linear Stability by Using a Research Paradigm Associated with Fractional Calculus

Abstract

In this research we studied the stability of certain types of liquid flows using mathematics. Specifically, we looked at flows that are slow to moderate in speed and have a property called viscoelasticity. We used a mathematical equation called the Fractional Upper Convicted Maxwell equation. They found that there are certain points in the flow where the behaviour changes, and wanted to identify these points. In this explanation, we show how the power in the equation that measures how a particle moves in a small scale follows a specific pattern. The exponent in this equation ranges from 0 to 1. This equation helps us understand how a particle moves in a small scale. Revis an abbreviation for Reverend, which is a title given to clergy members, such as priests or pastors, indicating their position in the religious community. Allow or permit. This text is about a mathematical equation from 1995. It discusses the relationship between a number and the derivative (which measures how a value changes) in a non-linear stress equation. The stability studies focus on two different conditions: how particles move in Rouse chain melts with a value of α equal to 1/2, and how particles move in Zimm chain solutions with a value of α equal to 2/3. The analysis of how stable something is over time shows that when the fractional derivative decreases, the stability also decreases. (a) the least stable mode becomes weaker (b) the highest point of the least stable mode moves to lower Re values (c) the highest point of the least stable mode in the Rouse model tends to approach the limit of Re → 0. Evanescent modes, complete and convective instabilities, and temporal stability are sorted using the Briggs idea of analytic continuation. When the fluid is flowing quickly, the spatiotemporal phase diagram reveals an unusual region of time stability. This shows that the surroundings may not be uniform, which is why the flow is being obstructed.