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This article is devoted to the mathematical analysis of a heat and mass transfer model for the pressure-induced flow of a viscous fluid through a plane channel subject to Navier's slip conditions on the channel walls. The important feature of our work is that the used model takes into account the effects of variable viscosity, thermal conductivity, and slip length, under the assumption that these quantities depend on temperature. Therefore, we arrive at a boundary value problem for strongly nonlinear ordinary differential equations. The existence and uniqueness of a solution to this problem is analyzed. Namely, using the Galerkin scheme, the generalized Borsuk theorem, and the compactness method, we proved the existence theorem for both weak and strong solutions in Sobolev spaces and derive some of their properties. Under extra conditions on the model data, the uniqueness of a solution is established. Moreover, we considered our model subject to some explicit formulae for temperature dependence of viscosity, which are applied in practice, and constructed corresponding exact solutions. Using these solutions, we successfully performed an extra verification of the algorithm for finding solutions that was applied by us to prove the existence theorem. [ABSTRACT FROM AUTHOR]

In this paper, we show that a weak solution for unsteady flows of 3D shear‐thinning fluids is strong under certain integrability assumptions about two components of the vorticity. In particular, if two components of the vorticity belong to the marginal critical space L1(0,T;BMO)$$ {L}^1\left(0,T; BMO\right) $$, then the solution is strong. [ABSTRACT FROM AUTHOR]

In this article, we investigate an optimal control problem for the coupled system of partial differential equations describing the steady-state flow of a corotational-type Oldroyd fluid through a bounded 3D (or 2D) domain. The control function is included in Dirichlet boundary conditions for the velocity field; in other words, we consider a model of inflow–outflow control. The main result is a theorem that states sufficient conditions for the solvability of the corresponding optimization problem in the set of admissible weak solutions. Namely, we establish the existence of a weak solution that minimizes the cost functional under given constraints on controls and states. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel mathematical model that describes steady‐state 3D flows of a non‐Newtonian fluid with shear‐dependent viscosity in a pipe network. Our approach is based on the rejection of averaging of the velocity field and the application of conjugation conditions that provide the mass balance for interior joints of the network. Using boundary conditions involving the pressure, we formulate the corresponding boundary value problem and introduce the concept of weak solutions by integral indentities. The main result of the work is an existence theorem in the class of weak solutions for large data. The proof of this result is performed with the help of the Galerkin procedure, methods of the topological degree theory, the technique of monotone operators as well as compactness arguments. [ABSTRACT FROM AUTHOR]

We consider a mathematical model that describes 3D steady ows of an incompressible viscoelastic uid of Oldroyd type in a bounded domain under mixed boundary conditions, including a threshold-slip boundary condition. Using the concept of weak solutions, we reduce the original slip problem to a coupled system of variational inequalities and equations for the velocity field and stresses. For arbitrary large data (forcing and boundary data) and suitable material constants, we prove the existence of weak solutions and establish some of their properties. [ABSTRACT FROM AUTHOR]

In this paper, we study the equations governing the steady motion of a class of second-grade fluids in a bounded domain of ℝn, n = 2, 3, with the nonlinear slip boundary condition. We prove the existence of a weak solution without assuming smallness of the data. Moreover, we give estimates for weak solutions and show that the solution set is sequentially weakly closed. [ABSTRACT FROM AUTHOR]

This paper deals with a 3D mathematical model for the non-isothermal steady-state flow of an incompressible fluid with temperature-dependent viscosity in a pipeline network. Using the pressure and heat flux boundary conditions, as well as the conjugation conditions to satisfy the mass balance in interior junctions of the network, we propose the weak formulation of the nonlinear boundary value problem that arises in the framework of this model. The main result of our work is an existence theorem (in the class of weak solutions) for large data. The proof of this theorem is based on a combination of the Galerkin approximation scheme with one result from the field of topological degrees for odd mappings defined on symmetric domains. [ABSTRACT FROM AUTHOR]

The Reynolds stress equations for two-dimensional and axisymmetric turbulent shear flows are simplified by invoking local equilibrium and boundary layer approximations in the near-wall region. These equations are made determinate by appropriately modelling the pressure–velocity correlation and dissipation rate terms and solved analytically to give a relation between the turbulent shear stress τ / ρ and the kinetic energy of turbulence (k = q 2 / 2) . This is derived without external body force present. The result is identical to that proposed by Nevzgljadov in A Phenomenological Theory of Turbulence, who formulated it through phenomenological arguments based on atmospheric boundary layer measurements. The analytical approach is extended to treat turbulent flows with external body forces. A general relation τ / ρ = a 1 1 − AF Ri F q 2 / 2 is obtained for these flows, where F Ri F is a function of the gradient Richardson number Ri F , and a 1 is found to depend on turbulence models and their assumed constants. One set of constants yields a 1 = 0.378, while another gives a 1 = 0.328. With no body force, F ≡ 1 and the relation reduces to the Nevzgljadov equation with a 1 determined to be either 0.378 or 0.328, depending on model constants set assumed. The present study suggests that 0.328 is in line with Nevzgljadov's proposal. Thus, the present approach provides a theoretical base to evaluate the turbulent shear stress for flows with external body forces. The result is used to reduce the k – ε model to a one-equation model that solves the k-equation, the shear stress and kinetic energy equation, and an ε evaluated by assuming isotropic eddy viscosity behavior. [ABSTRACT FROM AUTHOR]