Global-in-time existence of solutions for incompressible magnetohydrodynamic fluid equations in a bounded domain Ω ⊂ ℝ³ with the boundary slip conditions is proved. The proof is based on the potential method. The existence is proved in a class of functions such that the velocity and the magnetic field belong to $W_p^{2,1}\left(\Omega \times (0,T)\right)$ and the pressure q satisfies $\nabla q\in L_p\left(\Omega \times (0,T)\right)$ for p ≥ 7/3.

This study deals with a numerical solution of a 2D flows of a compressible viscous fluids in a convergent channel for low inlet airflow velocity. Three governing systems – Full system, Adiabatic system, Iso-energetic system$basedontheNavier-Stokesequationsforlaminarflowaretested.Thenumericalsolutionisrealizedbyfinitevolumemethodandthepredictor-correctorMacCormackschemewithJamesonartificialviscosityusingagridofquadrilateralcells.TheunsteadygridofquadrilateralcellsisconsideredintheformofconservationlawsusingArbitraryLagrangian-Eulerianmethod.Thenumericalresults,acquiredfromadevelopedprogram,arepresentedforinletvelocity$u=4.12 ms-1$andReynoldsnumberRe=$4 103$.$

Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space ℝ³. Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved $L^q$-estimates of second order derivatives uniformly in the angular and translational velocities,...

We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $\epsilon \to 0$, the Froude number proportional to $\sqrt{\epsilon }$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $\epsilon \to 0$. The limit velocity field is solenoidal and satisfies the incompressible Oberbeck–Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we...

The existence and uniqueness of solutions to the Navier-Stokes equations in a cylinder Ω and with boundary slip conditions is proved. Assuming that the azimuthal derivative of cylindrical coordinates and azimuthal coordinate of the initial velocity and the external force are sufficiently small we prove long time existence of regular solutions such that the velocity belongs to $W_{5/2}^{2,1}\left(\Omega \times (0,T)\right)$ and the gradient of the pressure to $L_{5/2}\left(\Omega \times (0,T)\right)$. We prove the existence of solutions without any restrictions on the...

The self-consistent chemotaxis-fluid system $$\left\{\begin{array}{cc}{n}_t+u·\nabla n=\Delta n-\nabla ·\left(n\nabla c\right)+\nabla ·\left(n\nabla \phi \right),\hfill & x\in \Omega ,t>0,\hfill \\ c_t+u·\nabla c=\Delta c-nc,\hfill & x\in \Omega ,t>0,\hfill \\ u_t+\kappa (u·\nabla )u+\nabla P=\Delta u-n\nabla \phi +n\nabla c,\hfill & x\in \Omega ,t>0,\hfill \\ \nabla ·u=0,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$$ is considered under no-flux boundary conditions for $n,c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $\Omega \subset {ℝ}^N$ $(N=2,3)$, $\kappa \in \{0,1\}$. The existence of global bounded classical solutions is proved under a smallness assumption on $\parallel c_0{\parallel }_{L^{\infty }\left(\Omega \right)}$. Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid...

Global existence of regular solutions to the Navier-Stokes equations for (v,p) coupled with the heat convection equation for θ is proved in the two-dimensional case in a bounded domain. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First an appropriate estimate is shown and next the existence is proved by the Leray-Schauder fixed point theorem. We prove the existence of solutions such that $v,\theta \in W_s^{2,1}\left({\Omega }^T\right)$, $\nabla p\in L_s\left({\Omega }^T\right)$, s>2.

It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function $G$ is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming $G$ to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial. ...

We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor ${\tau }^V\left(𝕖\right)=\tau \left(𝕖\right)-2{\mu }_1\Delta 𝕖$, where the nonlinear function $\tau \left(𝕖\right)$ satisfies ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge c{\left|𝕖\right|}^p$ or ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge {c\left(\right|𝕖|}^2+{\left|𝕖\right|}^p)$. First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p>1$ for both models. Then, under vanishing higher viscosity ${\mu }_1$, the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p>\frac{3n}{n+2}$, its uniqueness...

We study the non-existence of global classical solutions to 1D compressible heat-conducting micropolar fluid without viscosity. We first show that the life span of the classical solutions with decay at far fields must be finite for the 1D Cauchy problem if the initial momentum weight is positive. Then, we present several sufficient conditions for the non-existence of global classical solutions to the 1D initial-boundary value problem on $[0,1]$. To prove these results, some new average quantities...

It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function $G$ is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming $G$ to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial. ...

We formulate the Leray problem for inhomogeneous fluids in two dimensions and outline the proof of the existence of a solution. There are two kinds of results depending on whether the given value for the density is a continuous function or only an $L^{\infty }$ function. In the former case, the given densities are attained in the sense of uniform convergence and in the latter with respect to weak-* convergence.

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on $p$ and on the symmetric part of a gradient of $u$, namely, it is represented by a stress tensor $T(Du,p):=\nu (p,|D{|}^2)D$ which satisfies $r$-growth condition with $r\in (1,2]$. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for...