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Amjad Tuffaha

Publications and source records attributed to Amjad Tuffaha.

17 recordsLinked to original sources
arXiv2026

Euler Immersion

We address the Euler immersion problem, a fluid-structure interaction problem in which an elastic body is immersed in an incompressible inviscid fluid governed by the Euler equations. We show that the system exhibits a loss of one derivative and formulate the problem in analytic function spaces. We then prove local well-posedness in analytic spaces under velocity-matching boundary conditions. Finally, by means of an example, we show that existence fails in analytic spaces when both velocity and stress-matching boundary conditions are prescribed.

math.AP↗
arXiv2025

A geophysical free-boundary system modeling an ice-sheet interacting with an ocean

We consider a free-boundary model for the ice-sheet interacting with an ocean. The model captures the coupling between a viscous geophysical fluid and an elastic interface through kinematic and dynamic boundary conditions that account for hydrodynamic loading. Using the ALE formulation, we derive a system on a fixed reference domain and establish local-in-time a priori estimates for strong solutions with initial data in $H^2$. The main analytical difficulties arise from the nonlinear terms involving vertical derivatives and from high-order pressure contributions on the interface.

math.AP↗
arXiv2025

The Euler equations with variable coefficients

We establish local-in-time existence for the Euler equations on a bounded domain with space-time dependent variable coefficients, given initial data $v_0 \in H^r$ under the optimal regularity condition $r > 2.5$. In the case $r = 3$, we further prove a Beale-Kato-Majda criterion that relates blow-up in the $H^r$ norm to the BMO norm of the variable vorticity $\zeta$.

math.AP↗
arXiv2025

Optimization of a Nonlinear Acoustics -- Structure Interaction Model

In this paper, we consider a control/shape optimization problem of a nonlinear acoustics-structure interaction model of PDEs, whereby acoustic wave propagation in a chamber is governed by the Westervelt equation, and the motion of the elastic part of the boundary is governed by a 4th order Kirchoff equation. We consider a quadratic objective functional capturing the tracking of prescribed desired states, with three types of controls: 1) An excitation control represented by prescribed Neumann data for the pressure on the excitation part of the boundary 2) A mechanical control represented by a forcing function in the Kirchoff equations and 3) Shape of the excitation part of the boundary represented by a graph function. Our main result is the existence of solutions to the minimization problem, and the characterization of the optimal states through an adjoint system of PDEs derived from the first-order optimality conditions.

math.OC↗
arXiv2025

A free boundary inviscid model of flow-structure interaction

We address the existence and of solutions for the Euler-plate free-boundary system modeling an interaction of a three-dimensional inviscid fluid and an evolving plate. We prove the local existence and uniqueness of solutions for initial fluid and structural velocities belonging to $H^{2.5+}$ and $H^{2+}$, respectively. The results justify earlier a~priori estimates shown by two of the authors.

math.AP↗
arXiv2024

Inviscid fluid interacting with a nonlinear two-dimensional plate

We address a moving boundary problem that consists of a system of equations modeling an inviscid fluid interacting with a two-dimensional nonlinear Koiter plate at the boundary. We derive a priori estimates needed to prove the local-in-time existence of solutions. We use the Arbitrary Lagrange Euler (ALE) coordinates to fix the domain and obtain careful estimates for the nonlinear Koiter plate, ALE velocity, and pressure {without any viscoelastic smoothing}. For the nonlinear Koiter plate, higher order energy estimates are obtained, whereas estimates for the ALE pressure are obtained by setting up an elliptic problem. For the ALE velocity, the bounds are obtained through div-curl estimates by estimating the ALE vorticity. We then extend our results in two directions: (1) to include fractional Sobolev spaces and (2) to incorporate the normalized second fundamental form.

math.AP↗
arXiv2024

Global-in-time weak solutions for an inviscid free surface fluid-structure problem without damping

We consider the Cauchy problem for an inviscid irrotational fluid on a domain with a free boundary governed by a fourth order linear elasticity equation. We first derive the Craig-Sulem-Zakharov formulation of the problem and then establish the existence of a global weak solution in two space dimensions, in the general case without a damping term, for any initial data with finite energy.

math.AP↗
arXiv2023

On the interaction between compressible inviscid flow and an elastic plate

We address a free boundary model for the compressible Euler equations where the free boundary, which is elastic, evolves according to a weakly damped fourth order hyperbolic equation forced by the fluid pressure. This system captures the interaction of an inviscid fluid with an elastic plate. We establish a priori estimates on local-in-time solutions in low regularity Sobolev spaces, namely with velocity and density initial data v0,R0 in H3. The main new device is a variable coefficients space tangential-time differential operator of order 1 with non-homogeneous boundary conditions, which captures the hyperbolic nature of the compressible Euler equations as well as the coupling with the structural dynamics.

math.AP↗
arXiv2023

Well-posedness of a Nonlinear Acoustics -- Structure Interaction Model

We establish local-in-time and global in time well-posedness for small data, for a coupled system of nonlinear acoustic structure interactions. The model consists of the nonlinear Westervelt equation on a bounded domain with non homogeneous boundary conditions, coupled with a 4th order linear equation defined on a lower dimensional interface occupying part of the boundary of the domain, with transmission boundary conditions matching acoustic velocities and acoustic pressures. While the well-posedness of the Westervelt model has been well studied in the literature, there has been no works on the literature on the coupled structure acoustic interaction model involving the Westervelt equation. Another contribution of this work, is a novel variational weak formulation of the linearized system and a consideration of various boundary conditions.

math.AP↗
arXiv2023

Construction of the free-boundary 3D incompressible Euler flow under limited regularity

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We construct unique local-in-time solutions in the Lagrangian setting for $u_0 \in H^{2.5+\delta }$ such that the Rayleigh-Taylor condition holds and $\mathrm{curl}\,u_0 \in H^{2+\delta }$ in an arbitrarily small neighborhood of the free boundary. We show that the result is optimal in the sense that $H^{3+\delta}$ regularity of the Lagrangian deformation near the free boundary can be ensured if and only if initial vorticity has $H^{2+\delta}$ regularity of vorticity near the free boundary.

math.AP↗
arXiv2022

A free boundary inviscid model of flow-structure interaction

We obtain the local existence and uniqueness for a system describing interaction of an incompressible inviscid fluid, modeled by the Euler equations, and an elastic plate, represented by the fourth-order hyperbolic PDE. We provide a~priori estimates for the existence with the optimal regularity $H^{r}$, for $r>2.5$, on the fluid initial data and construct a unique solution of the system for initial data $u_0\in H^{r}$ for $r\geq3$. An important feature of the existence theorem is that the Taylor-Rayleigh instability does not occur.

math.AP↗
arXiv2017

Static Stability Analysis of a Thin Plate with a Fixed Trailing Edge in Axial Subsonic Flow: Possio Integral Equation Approach

In this work, the static stability of plates with fixed trailing edges in axial airflow is studied using the framework of Possio integral equation. First, we introduce a new derivation of a Possio integral equation that relates the pressure jump along thin plates to their downwash based on the linearization of the governing equations of an ideal compressible fluid. The steady state solution to the Possio equation is used to account for the aerodynamic forces in the steady state plate governing equation resulting in a singular differential-integral equation which is transformed to an integral equation. Next, we verify the solvability of the integral equation based on the Fredholm alternative for compact operators in Banach spaces and the contraction mapping theorem. Then, we derive explicit formulas for the characteristic equations of free-clamped and free-pinned plates. The minimum solutions to the characteristic equations are the divergence speeds which indicate when static instabilities start to occur. We show analytically that free-pinned plates are statically unstable. After that, we move to derive analytically flow speed intervals that correspond to static stability regions for free-clamped plates. We also resort to numerical computations to obtain an explicit formula for the divergence speed of free-clamped plates. Finally, we apply the obtained results on piezoelectric plates and we show that free-clamped piezoelectric plates are statically more stable than conventional free-clamped plates due to the piezoelectric coupling.

physics.flu-dyn↗
arXiv2017

Subsonic Flow over a Thin Airfoil in Ground Effect

In this paper, the problem of compressible flow over a thin airfoil located near the ground is studied. A singular integral equation, also known as Possio equation, that relates the pressure jump along the airfoil to its downwash is derived. The derivation of the equation utilizes Laplace transform, Fourier transform, method of images, and theory of Mikhlin multipliers. The existence and uniqueness of solution to the Possio equation is verified for the steady state case and an approximate solution is obtained. The aerodynamic loads are then calculated based on the approximate solution. Moreover, the divergence speed of a continuum wing structure located near the ground is obtained based on the derived expressions for the aerodynamic loads.

physics.flu-dyn↗