Incompressible Equations Research Articles

Hemivariational inequality for contaminant reaction–diffusion model of recovered fracturing fluid in the wellbore of shale gas reservoir

The goal of this paper is to study the dynamic behavior of recovered fracturing fluid and the reaction–diffusion phenomenon of contaminants in the wellbore of shale gas reservoir during the early stage of fracturing fluid flowback. First, by applying various physical constitutive laws, a complicated recovered fracturing fluid model which consists of a stationary incompressible Stokes equation with a nonmonotone and multivalued friction law and a reaction–diffusion equation with the Neumann boundary condition, is established. Then, we use the variational techniques and the theory of multivalued analysis to formulate a new recovered fracturing fluid model. The latter is a variational system which consists of a hemivariational inequality with constraint driven by an elliptic variational equation. The system involves mutually coupled velocity and the concentration fields. Finally, under some mild assumptions, we employ a surjectivity theorem for pseudomonotone operators, monotonicity arguments and Schauder fixed point theorem to prove the existence of a weak solution to the recovered fracturing fluid model.

Axisymmetric solutions to the convective Brinkman-Forchheimer equations

In this paper, we prove the existence and uniqueness of axisymmetric solution to the three-dimensional incompressible convective Brinkman-Forchheimer equations{∂tu+(u⋅∇)u−μΔu+ν1u+ν2|u|β−1u+∇p=0,∇⋅u=0,u|t=0=u0, with 1≤β≤73, by developing the local-in-space estimate for solutions.

A Framework for Generating Radial and Surface-Oriented Regularized Stokeslets.

Error in the method of regularized Stokeslets is highly dependent on the choice of the blob or regularization function that is utilized to handle singularities in the flow. In this work, we develop a general framework to choose regularizations at the level of the vector potential via smoothing factors. We detail the derivation for radial smoothing factors and specify properties which ensure that the solution is a regularized flow satisfying the incompressible Stokes equations. Error analysis is completed for both the far-field flow (away from the location of the forces) as well as at the location of the forces, relating our newly derived smoothing factors to commonly used blob functions and moment conditions. When forces are on a surface, we extend the radial smoothing factor case to the case of non-radial regularizations that are surface-oriented. We illustrate the utility of this framework by computing the forward and inverse problems of a translating sphere using radial and surface-oriented regularizations.

Global well-posedness and optimal decay estimate for the incompressible porous medium equation near a nontrivial equilibrium

In this paper, we prove global well-posedness and optimal decay estimate of solution to the 2D inviscid incompressible porous medium equation near a nontrivial equilibrium x2. The obtained decay rate is optimal in sense that this rate coincides with that of the linear system. In particular, the obtained L∞ estimate of u2 improves the associated work in (T.M. Elgindi, Arch. Ration. Mech. Anal. 2017).

Global well-posedeness and large time asymptotic behavior of 2D density-dependent Boussinesq equations of Korteweg type with vacuum

In this paper, we are devoted to study the Cauchy problem of the incompressible density-dependent Boussinesq equations of Korteweg type with vacuum. We establish some key spatial weighted estimates and a priori decay-in-time rate of the strong solutions by using the energy method. Furthermore, we prove that there is a unique global strong solution for the 2D Cauchy problem when the spatial weighted norm of the initial density is suitably small. Finally, we also obtain the large time decay rates for the gradients of velocity, temperature and pressure.

Alternative formulation of weak magnetohydrodynamic turbulence theory

In a recent paper [P. H. Yoon and G. Choe, Phys. Plasmas 28, 082306 (2021)], the weak turbulence theory for incompressible magnetohydrodynamics is formulated by employing the method customarily applied in the context of kinetic weak plasma turbulence theory. Such an approach simplified certain mathematical procedures including achieving the closure relationship. The formulation in the above-cited paper starts from the equations of incompressible magnetohydrodynamic (MHD) theory expressed via Elsasser variables. The derivation of nonlinear wave kinetic equation therein is obtained via a truncated solution at the second-order of iteration following the standard practice. In the present paper, the weak MHD turbulence theory is alternatively formulated by employing the pristine form of incompressible MHD equation rather than that expressed in terms of Elsasser fields. The perturbative expansion of the nonlinear momentum equation is carried out up to the third-order iteration rather than imposing the truncation at the second order. It is found that while the resulting wave kinetic equation is identical to that obtained in the previous paper cited above, the third-order nonlinear correction plays an essential role for properly calculating derived quantities such as the total and residual energies.

A modified Darcy’s law for viscoelastic flows of highly dilute polymer solutions through porous media

Viscoelastic flows of polymer solutions in complex geometries can generate a strong localization of stress within small regions of the fluid and the formation of birefringent strands. In porous media , these localized structures of stress drive preferential flow paths and increase global dissipation. Modeling the impact of such effects at Darcy or larger scales is a daunting task—one of the reasons being the lack of approaches using homogenization theories to help figure out both the correct form of the averaged transport equations and the relevant set of effective parameters. Here we homogenize the incompressible Oldroyd-B equations at zero Reynolds number to obtain a Darcy scale model that captures the effect of localized polymeric stress. This model consists of an advection—reaction transport equation for the average conformation tensor along with a form of Darcy’s law that contains an additional forcing term associated with structures of localized stress. The derivation is based upon a limit of high dilution, a regime where the Oldroyd-B model can be transformed into a sequence of linear problems using asymptotic developments. We validate our approach in test cases corresponding to flows in a channel and through arrays of circles. Besides providing a new model for viscoelastic flows in porous media , our work also shows that modeling viscoelastic flows through porous media is not simply a matter of determining an apparent permeability tensor—the homogenized model cannot be easily reduced to a simple form of Darcy’s law—but rather requires the development of specific homogenized models that capture the coupling between the transport of the polymeric stress and momentum. • The Oldroyd-B equations are homogenized using asymptotics and volume averaging. • We derive a Darcy-scale model for viscoelastic flow in porous media. • Our model couples a form of Darcy’s law with transport of the conformation tensor. • Our model captures the effect of stress localization at pore-scale on the average flow. • The notion of effective permeability alone seems to be insufficient for viscoelastic flows.

A stabilized formulation for the solution of the incompressible unsteady Stokes equations in the frequency domain

A stabilized finite element method is introduced for the simulation of time-periodic creeping flows. This new technique, which is formulated in the frequency rather than time domain, strictly uses real arithmetics and permits the use of similar interpolation functions for pressure and velocity for ease of implementation. It involves the addition of the Laplacian of pressure to the continuity equation with a complex-valued stabilization parameter that is derived systematically from the momentum equation. The numerical experiments show the excellent accuracy and robustness of the proposed method in simulating flows in complicated and canonical geometries for a wide range of conditions. The present method is mass conservative up to the tolerance of the underlying linear solver. It significantly outperforms its temporal counterpart, namely the standard pressure stabilized Petrov-Galerkin method, by lowering the cost and solution turnover time of the simulated cases by several orders of magnitude.

Global regularity for the 3D Hall-MHD equations with low regularity axisymmetric data

In this paper, we consider the global well-posedness of the incompressible Hall-MHD equations in \({\mathbb {R}}^3\). We prove that the solution of this system is globally regular if the initial data is axisymmetric and the swirl components of the velocity and magnetic vorticity are trivial. It should be pointed out that the initial data can be arbitrarily large and satisfy low regularity assumptions.

Random vortex dynamics via functional stochastic differential equations

In this paper, we present a novel, closed, three-dimensional random vortex dynamics system, which is equivalent to the Navier–Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled to a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the three-dimensional random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman–Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems.

Global well-posedness to the two-dimensional incompressible vorticity equation in the half plane

This paper is concerned with the global well-posedness of the two-dimensional incompressible vorticity equation in the half plane. Under the assumption that the initial vorticity ω0∈Wk,p(R+2) with k≥3 an integer and 1<p<2, it is shown that the two-dimensional incompressible vorticity equation admits a unique solution ω∈C([0,T];Wk,p(R+2)) for any T>0. An elementary and self-contained proof is presented and delicate estimates of the velocity and its derivatives are obtained in this paper. It should be emphasized that the uniform estimate on ∫0t‖∇u(τ)‖L∞(R+2)dτ is required to complete the global regularity of the solution. To do that, the double exponential growth in time of the gradient of the vorticity in the half plane is established and applied. This is different from the proof of global well-posedness of the Euler velocity equations in the Sobolev spaces, in which a Kato-type or logarithmic-type estimate of the gradient of the velocity is enough to close the energy estimates.

Determination of pressure from nonlinear equations of nonhomogeneous non-newtonian fluids

In this paper, the initial-boundary problem for a system of nonlinear equations modified by p-laplacian diffusion and nonlinear damping term, in which describe the motion of a non-homogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids is considered. Usually, a pressure does not include in the definition of weak generalized solution to equations of hydrodynamics for incompressible fluids, because weak solutions consider in the solenoidal space due to the incompressibility (continuity) equation. In the classical hydrodynamic equations, after determining a velocity and a density, a pressure can be determined by using the theorem of a representation of the space L2(Ω) into the direct sum of two orthogonal subspaces. The recovering of a pressure from the nonlinear equations of hydrodynamics modified by p-laplacian and nonlinear damping term requires a new approches. Here, under a suitable conditions on the data of the problem, a pressure has been uniquelly determined from the initial-boundary value problem for the system of nonlinear modified equations describing the motion of nonhomogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids. The main functional spaces and necessary axuillary conclusions are introduced. A space of generalized weak solutions is identified. The pressure uniquelly recovered by using the known de-Ram lemma.

Well-posedness of the surface wave problem for two dimensional micropolar fluids

We consider a viscous fluid in a finite-depth domain of two dimensions, with a free moving boundary and a fixed solid boundary. The fluid dynamics are governed by gravity-driven incompressible micropolar equations, and the surface tension is neglected on the free surface. The main result of this paper is to prove the global well-posedness of the surface wave problem when the fluid domain is horizontally periodic. And the solutions decay to equilibrium at an almost exponential rate. This is achieved by establishing delicate estimates when dealing with the strong coupling of fluid velocity and micro-rotation velocity. Our proof relies on two-tier nonlinear energy method developed by Guo and Tice [17–19].

Global existence of the strong solution to the 3D incompressible micropolar equations with fractional partial dissipation

Abstract In this paper, we considered the global strong solution to the 3D incompressible micropolar equations with fractional partial dissipation. Whether or not the classical solution to the 3D Navier–Stokes equations can develop finite-time singularity remains an outstanding open problem, so does the same issue on the 3D incompressible micropolar equations. We establish the global-in-time existence and uniqueness strong solutions to the 3D incompressible micropolar equations with fractional partial velocity dissipation and microrotation diffusion with the initial data $(\mathbf {u}_0,\ \mathbf {w}_0)\in H^1(\mathbb {R}^3)$ .

Assessment of new pressure-corrected design method for hypersonic internal waverider intake

The osculating axisymmetric flows concept is an extension of the concept of the osculating cones for designing waveriders which is also utilized to design the three-dimensional internal waverider intake, where the three-dimensional flowfield is generated using slices of locally two-dimensional flow. However, this design methodology neglects the effect of cross-flow between the osculating planes, which causes lateral pressure gradients. Therefore, this article introduces and assesses a new pressure-corrected osculating axisymmetric flows design approach for the three-dimensional internal waverider intake. The new design concept adds the lateral pressure gradient correction into the original design concept using cross-flow velocities calculated by Euler’s incompressible flow equation between the osculating planes. Thereafter, three-dimensional streamlines are formed instead of the osculating plane’s two-dimensional streamlines. The new design procedure is presented in detail and then verified using wavecatcher intake with a circular shape that contains zero lateral pressure gradients. No geometry changes are found when the pressure-corrected wavecatcher intake is compared with the uncorrected wavecatcher intake. Moreover, four hypersonic wavecatcher intakes of semi-rectangular and rectangular entrance and exit shapes are designed with a design point of Mach number 6.0 and numerically studied to assess the new design approach. The results revealed that the pressure-corrected wavecatcher intakes exhibited better total pressure recovery, flow uniformity, Mach number, and kinetic energy efficiency than uncorrected wavecatcher intakes. While the impinged shockwave is attached to the intake’s entrance, and the mass flow is completely captured in all designs. It is found that 38.49% and 30.36% of the exit area of the pressure-corrected wavecatcher intakes, and 35.68% and 28.37% of the exit area of the uncorrected wavecatcher intakes, have provided the combustor with air having a total pressure recovery above 0.7, respectively.

The 2D inviscid Boussinesq equations with fractional diffusion in bounded domain

The incompressible Boussinesq equations serve as an important model in geophysics especially in the study of Rayleigh–Bénard convection. This paper focuses on the 2D incompressible inviscid Boussinesq equations with fractional diffusion (−Δ)βθ in bounded domain, equipping with the slip boundary condition for velocity vector field and Dirichlet boundary condition for temperature. We obtain the global existence and uniqueness of classical solutions in the range of β∈[3/4,1) and also show the local corresponding result for β∈[0,1). To the best of our knowledge, this is the first paper considering the Boussinesq equation with fractional diffusion in bounded domain. This result extends the work by K. Zhao (2010) to a fractional dissipation for temperature in bounded domain.

Singular geometry perturbation based method for shape-topology optimization in unsteady Stokes flow

This paper concerns the topological derivatives and its applications for solving shape and topology optimization problems in fluid mechanics. The fluid flow is governed by the unsteady incompressible Stokes equations in the two dimensional case. We derive a topological sensitivity analysis for this parabolic-type operator. The proposed approach is based on a preliminary estimate describing the variation of the velocity field caused by the presence of a small obstacle inside the fluid flow. We obtain a topological asymptotic expansion for the unsteady Stokes operator valid for a large class of shape functions and an arbitrarily shaped geometric perturbation. Then, the topological gradient is exploited for building an efficient and accurate topology optimization algorithm. Finally, we present some numerical investigations showing the efficiency of the proposed approach.

Global well‐posedness of 3D incompressible inhomogeneous magnetohydrodynamic equations

In this paper, we investigate the 3D inhomogeneous incompressible magnetohydrodynamic (MHD) system. By the assumption of the smallness of initial velocity and magnetic fluids in the critical Besov space, the local and global well‐posedness of 3D inhomogeneous incompressible equations is obtained. It improves some previous results of MHD equations by generalizing the range of exponent in Besov spaces with . Besides, the initial density belongs to the critical Besov space with , and it is removed the additional restriction of , which is an important condition in some previous results for both 3D inhomogeneous incompressible Navier–Stokes equations and MHD system.

An interface-tracking space–time hybridizable/embedded discontinuous Galerkin method for nonlinear free-surface flows

We present a compatible space–time hybridizable/embedded discontinuous Galerkin discretization for nonlinear free-surface waves. We pose this problem in a two-fluid (liquid and gas) domain and use a time-dependent level-set function to identify the sharp interface between the two fluids. The incompressible two-fluid equations are discretized by an exactly mass conserving space–time hybridizable discontinuous Galerkin method while the level-set equation is discretized by a space–time embedded discontinuous Galerkin method. Different from alternative discontinuous Galerkin methods is that the embedded discontinuous Galerkin method results in a continuous approximation of the interface. This, in combination with the space–time framework, results in an interface-tracking method without resorting to smoothing techniques or additional mesh stabilization terms.

General solutions of linear poro-viscoelastic materials in spherical coordinates

The cell cytoskeleton is a dynamic assembly of semi-flexible filaments and motor proteins. The cytoskeleton mechanics is a determining factor in many cellular processes, including cell division, cell motility and migration, mechanotransduction and intracellular transport. Mechanical properties of the cell, which are determined partly by its cytoskeleton, are also used as biomarkers for disease diagnosis and cell sorting. Experimental studies suggest that in whole cell scale, the cell cytoskeleton and its permeating cytosol may be modelled as a two-phase poro-viscoelastic (PVE) material composed of a viscoelastic (VE) network permeated by a viscous cytosol. We present the first general solution to this two-phase system in spherical coordinates, where we assume that both the fluid and network phases are in their linear response regime. Specifically, we use generalized linear incompressible and compressible VE constitutive equations to describe the stress in the fluid and network phases, respectively. We assume a constant permeability that couples the fluid and network displacements. We use these general solutions to study the motion of a rigid sphere moving under a constant force inside a two-phase system, composed of a linear elastic network and a Newtonian fluid. It is shown that the network compressibility introduces a slow relaxation of the sphere and non-monotonic network displacements with time along the direction of the applied force. Our results can be applied to particle-tracking microrheology to differentiate between PVE and VE materials, and to measure the fluid permeability as well as VE properties of the fluid and the network phases.