arXiv subjects

Haithem Taha

Publications and source records attributed to Haithem Taha.

10 recordsLinked to original sources

arXiv2026

On the Mathematical Analysis and Physical Implications of the Principle of Minimum Pressure Gradient

In this paper, we establish a two-way equivalence between the incompressible Navier- Stokes equation (INSE) and the principle of minimum pressure gradient (PMPG). We prove that a candidate smooth flow field is a solution of the INSE if and only if its instantaneous evolution minimizes, at every instant, the norm of the pressure force, required to enforce incompressibility. We show that the PMPG is precisely the minimization formulation of the Leray-Helmholtz projection. Any admissible instantaneous evolution (e.g., onset of separation) resulting from the INSE necessarily minimizes the PMPG cost. Conversely, any other kinematically admissible evolution, requiring a strictly larger pressure force to ensure the same constraints, does not satisfy the INSE. Thus, the PMPG offers a variational perspective through which intricate incompressible flow behaviors may be interpreted. In a finite-dimensional setting with divergence-free modes, we show that the PMPG yields the same dynamics as classical Galerkin projection. Moreover, the PMPG provides a natural generalization of classical Galerkin projection beyond linear modal expansions, accommodating nonlinear and non-modal representations. We then examine the relation between instantaneous dynamical minimization and steady variational selection, including its connection to the variational theory of lift. Motivated by these observations, we formulate conjectures concerning necessary conditions for stability and the convergence of Navier-Stokes solutions to Euler's in the vanishing-viscosity limit.

physics.flu-dyn↗

arXiv2026

A Response to "Application of Gauss's Principle to the Classical Airfoil Lift Problem"

The classical theory of lift is confined to sharp edged airfoils. The search for a more general closure condition in potential flow remained elusive for over a century. Recently, a variational theory of lift, inspired by Gauss's principle of least constraint, was proposed as a remedy. The theory was shown to recover the Kutta condition as a special case for sharp-edged airfoils. However, recent criticism of the variational theory has asserted fundamental issues and discontinuities in its predictions. The present paper demonstrates that these assertions are incorrect and arise from inconsistencies with basic principles of analytical mechanics, the calculus of variations, and ideal-flow aerodynamics, as well as from misapplications of the variational theory itself. To resolve such misunderstandings, we review foundational concepts from analytical mechanics, including least action, Gauss's principle, and Hertz's principle; the definitions of impressed and constraint forces; and the distinction between actual work and virtual work. We then place these concepts in the context of incompressible fluid mechanics, utilizing the geometric interpretation of Helmholtz decomposition. In particular, we demonstrate that, for incompressible flows subject to the no-penetration boundary condition, the pressure force is orthogonal to the entire space of kinematically admissible flows and therefore performs no virtual work. The pressure force, thus, acts as the constraint force required to ensure the continuity constraint. From an aerodynamic perspective, we show that the classical and variational theories of lift, as well as any theory based on steady, irrotational motion, are necessarily reversible and therefore inapplicable to reversed-flow configurations.

physics.flu-dyn↗

arXiv2025

Variational Projection of Navier-Stokes: Fluid Mechanics as a Quadratic Programming Problem

Gauss's principle of least constraint transforms a dynamics problem into a pure minimization problem, where the total magnitude of the constraint force is the cost function, minimized at each instant. Newton's equation is the first-order necessary condition for minimizing the Gaussian cost, subject to the given kinematic constraints. The principle of minimum pressure gradient (PMPG) is to incompressible fluid mechanics what Gauss's principle is to particle mechanics. The PMPG asserts that an incompressible flow evolves from one instant to another by minimizing the L2-norm of the pressure gradient force. A candidate flow field whose evolution minimizes the pressure gradient cost at each instant is guaranteed to satisfy the Navier-Stokes equation. Consequently, the PMPG transforms the incompressible fluid mechanics problem into a pure minimization framework, allowing one to determine the evolution of the flow field by solely focusing on minimizing the cost. In this paper, we show that the resulting minimization problem is a convex Quadratic Programming (QP) problem-one of the most computationally tractable classes in nonlinear optimization. Moreover, leveraging tools from analytical mechanics and the Moore-Penrose theory of generalized inverses, we derive an analytical solution for this QP problem. As a result, we present an explicit formula for the projected dynamics of the spatially discretized Navier-Stokes equation on the space of divergence-free fields. The resulting ODE is ready for direct time integration, eliminating the need for solving the Poisson equation in pressure at each time step. It is typically an explicit nonlinear ODE with constant coefficients. This compact form is expected to be highly valuable for both simulation and theoretical studies, including stability analysis and flow control design. We demonstrate the framework on the lid-driven cavity problem.

physics.flu-dyn↗

arXiv2025

Casting Computational Fluid Mechanics into a Convex Quadratic Optimization Framework

We employ the principle of minimum pressure gradient to transform problems in unsteady computational fluid dynamics (CFD) into a convex optimization framework subject to linear constraints. This formulation permits solving, for the first time, CFD problems efficiently using well-established quadratic programming tools or using the well-known Karush-Kuhn-Tucker (KKT) condition. The proposed approach is demonstrated using three benchmark examples. In particular, it is shown through comparison with traditional CFD tools that the proposed framework is capable of predicting the flow field in a lid-driven cavity, in a uniform pipe (Poiseuille flow), and that past a backward facing step. The results highlight the potential of the method as a simple, robust, and potentially transformative alternative to traditional CFD approaches.

physics.flu-dyn↗

arXiv2025

On the Separating Flow Behind a Cylinder: Insights from the Principle of Minimum Pressure Gradient

We study the separating flow over a circular cylinder with two objectives: (i) to demonstrate the validity of the condition of matching curvature, and (ii) to obtain a reasonable estimate of the separation angle in the subcritical regime (Re=10^4-10^5) without explicitly modeling the boundary layer. First, we study Roshko's free streamline model (1954); it is an ideal flow model with sheets of discontinuities that represent the separating shear layers in the near wake region. The model fails to predict the correct separation angle over a cylinder. Roshko attributed this discrepancy to the condition of matching curvature, which asserts that the curvature of the separating streamline at the separation point must match that of the cylinder. We show that such a condition is legitimate and is not the real culprit for the failure of Roshko's model in predicting separation. Second, we employ the principle of minimum pressure gradient (PMPG), which asserts that, an incompressible flow evolves by minimizing the total magnitude of the pressure gradient over the domain. Encouraged by the fact that the flow characteristics in the range Re=10^4-10^5 are fairly independent of Re, we aim to predict the separation angle in this regime without modeling the boundary layer -- a task that may seem impossible, though anticipated by Prandtl in his seminal paper (Prandtl 1904). Over the family of kinematically-admissible, equilibrium flows, we utilize the PMPG to single out the separating flow with the minimum pressure gradient cost. Interestingly, the obtained separation angles match experimental measurements over the regime Re=10^4-10^5.

physics.flu-dyn↗

arXiv2024

Minimizing Nature's Cost: Exploring Data-Free Physics-Informed Neural Network Solvers for Fluid Mechanics Applications

In this paper, we present a novel approach for fluid dynamic simulations by harnessing the capabilities of Physics-Informed Neural Networks (PINNs) guided by the newly unveiled principle of minimum pressure gradient (PMPG). In a PINN formulation, the physics problem is converted into a minimization problem (typically least squares). The PMPG asserts that for incompressible flows, the total magnitude of the pressure gradient over the domain must be minimum at every time instant, turning fluid mechanics into minimization problems, making it an excellent choice for PINNs formulation. Following the PMPG, the proposed PINN formulation seeks to construct a neural network for the flow field that minimizes Nature's cost function for incompressible flows in contrast to traditional PINNs that minimize the residuals of the Navier-Stokes equations. This technique eliminates the need to train a separate pressure model, thereby reducing training time and computational costs. We demonstrate the effectiveness of this approach through a case study of inviscid flow around a cylinder, showing its ability to capture the underlying physics, while reducing computational cost and training time. The proposed approach outperforms the traditional PINNs approach in terms of Root Mean Square Error, training time, convergence rate, and compliance with physical metrics. While demonstrated on a simple geometry, the methodology is extendable to more complex flow fields (e.g., Three-Dimensional, unsteady, viscous flows) within the incompressible realm, which is the region of applicability of the PMPG.

physics.flu-dyn↗

arXiv2022

Sea urchin sperm exploit extremum seeking control to find the egg

Sperm cells perform extremely demanding tasks with minimal capabilities. The cells must quickly navigate in a noisy environment to find an egg within a short time window for successful fertilization without any global positioning information. Many research efforts have been dedicated to deriving mathematical principles that explain their superb navigation strategy. Here we show that the navigation strategy of sea urchin sperm, also known as helical klinotaxis, is a natural implementation of a well-established adaptive control paradigm known as extremum seeking. This bridge between control theory and the biology of taxis in microorganisms is expected to deepen our understanding of the process. For example, the formulation leads to a coarse-grained model of the signaling pathway that offers new insights on the peculiar switching-like behavior between high and low gain steering modes observed in sea urchin sperm. Moreover, it may guide engineers in developing bio-inspired miniaturized robots with minimal sensors.

math.OC↗

arXiv2022

Singularly Perturbed Averaging with Application to Bio-Inspired 3D Source Seeking

We analyze a class of singularly perturbed high-amplitude, high-frequency oscillatory systems that arises in extremum seeking applications. We provide explicit formulas for averaging and establish the convergence of the trajectories of this class of systems to the trajectories of a suitably averaged reduced order system by combining the higher order averaging theorem with singular perturbation techniques. Finally, we propose a novel bio-inspired 3D source seeking algorithm and establish its singular practical stability.

math.OC↗

arXiv2022

Recursive Averaging with Application to Bio-Inspired 3D Source Seeking

We analyze a class of high-amplitude, high-frequency oscillatory systems in which periodicity occurs on two distinct time scales and establish the convergence of its trajectories to a suitably averaged system by recursively applying the averaging theorem. Moreover, we introduce a novel bio-inspired 3D source seeking algorithm for rigid bodies with a collocated sensor and prove its practical stability under typical assumptions on the source signal strength field by combining our averaging results with singular perturbation.

math.OC↗

arXiv2021

Lie Bracket Approximation-Based Extremum Seeking with Vanishing Input Oscillations

In recent years, an approach to extremum seeking control made it possible to design control vector fields that lead to asymptotic stability of the minimum point provided that the minimum value of the function is known a priori. In this work, we aim to relax that assumption. We propose an extremum-seeking control law that converges to the minimum point with vanishing control oscillations, without access to the minimum value of the cost function. We provide a numerical example to support our results.

math.OC↗