Mathematical modeling on gas turbine blades/vanes under variable convective and radiative heat flux with tentative different laws of cooling

In the last twenty years the modeling of heat transfer on gas turbine cascades has been based on computational fluid dynamic and turbulence modeling at sonic transition. The method is called Conjugate Flow and Heat Transfer (CHT). The quest for higher Turbine Inlet Temperature (TIT) to increase electrical efficiency makes radiative transfer the more and more effective in the leading edge and suction/ pressure sides. Calculation of its amount and transfer towards surface are therefore needed. In this paper we decouple convection and radiation load, the first assumed from convective heat transfer data and the second by means of emissivity charts and analytical fits of heteropolar species as CO2 and H2O. Then we propose to solve the temperature profile in the blade through a quasi-two-dimensional power balance in the form of a second order partial differential equation which includes radiation and convection. Real cascades are cooled internally trough cool compressed air, so that we include in the power balance the effect of a heat sink or law of cooling that is up to the designer to test in order to reduce the thermal gradients and material temperature. The problem is numerically solved by means of the Finite Element Method (FEM) and, subsequently, some numerical simulations are also presented

Boundedness for a Fully Parabolic Keller–Segel Model with Sublinear Segregation and Superlinear Aggregation

This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem {ut=Δu−∇⋅(f(u)∇v) in Ω×(0,Tmax),vt=Δv−v+g(u) in Ω×(0,Tmax), where Ω is a bounded and smooth domain of Rn, for n≥ 2 , and f(u) and g(u) are reasonably regular functions generalizing, respectively, the prototypes f(u) = uα and g(u) = ul, with proper α, l> 0. After having shown that any sufficiently smooth u(x, 0) = u(x) ≥ 0 and v(x, 0) = v(x) ≥ 0 produce a unique classical and nonnegative solution (u, v) to problem (◊), which is defined on Ω × (0 , Tmax) with Tmax denoting the maximum time of existence, we establish that for any l∈(0,2n) and 2n≤α<1+1n−l2, Tmax= ∞ and u and v are actually uniformly bounded in time. The paper is in line with the contribution by Horstmann and Winkler (J. Differ. Equ. 215(1):52–107, 2005) and, moreover, extends the result by Liu and Tao (Appl. Math. J. Chin. Univ. Ser. B 31(4):379–388, 2016). Indeed, in the first work it is proved that for g(u) = u the value α=2n represents the critical blow-up exponent to the model, whereas in the second, for f(u) = u, corresponding to α= 1 , boundedness of solutions is shown under the assumption 0<2n

Improvements and generalizations of results concerning attraction-repulsion chemotaxis models

We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. More precisely, we are referring respectively to the papers “Boundedness in a nonlinear attraction-repulsion Keller–Segel system with production and consumption,” by S. Frassu, C. van der Mee and G. Viglialoro [J. Math. Anal. Appl. 504(2):125428, 2021] and “Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent,” by S. Frassu and G. Viglialoro [Nonlinear Anal. 213:112505, 2021]. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements, and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study

Uniform in time L∞ estimates for an attraction-repulsion chemotaxis model with double saturation

In this paper we focus on this attraction-repulsion chemotaxis model with consumed signals {ut = ?u - chi & nabla; middot (u & nabla;v) + xi & nabla; middot (u & nabla;w) in ? x (0, T-max),vt = ?v - uv in ? x (0, T-max),wt = ?w - uw in ? x (0, T-max), (?) formulated in a bounded and smooth domain ? of R-n, with n >= 2, for some positive real numbers chi, xi and with T-max is an element of (0, infinity]. Once equipped with appropriately smooth initial distributions u(x, 0) = u(0)(x) >= 0, v(x, 0) = v(0)(x) >= 0 and w(x, 0) = w(0)(x) >= 0, as well as Neumann boundary conditions, we establish sufficient assumptions on its data yielding global and bounded classical solutions; these are functions u, v and w, with zero normal derivative on & part;? x (0, T-max), satisfying pointwise the equations in problem (? ) with T-max = infinity. This is proved for any such initial data, whenever chi and xi belong to bounded and open intervals, depending respectively on Ilv(0)Il(L infinity()?) and Ilw0Il(L infinity)(?). Finally, we illustrate some aspects of the dynamics present within the chemotaxis system by means of numerical simulations

A nonlinear attraction-repulsion Keller–Segel model with double sublinear absorptions: criteria toward boundedness

This paper deals with the zero-flux attraction-repulsion chemo-taxis model {u(t) = del center dot ((u + 1)(m1-1)del u-chi u(u + 1)(m2-1)del v in Omega x (0, T-max), +xi u(u + 1)(m3-1)del w) + h(u) (lozenge) v(t) = Delta v - f (u)v in Omega x (0, T-max), w(t) = Delta w - g(u)w in Omega x (0, T-max), in the unknown (u, v, w)= (u(x, t), v(x, t), w(x, t)). Here, x is an element of Omega, a bounded and smooth domain of R-n(n >= 1), t, chi, xi > 0, m(1), m(2), m(3) is an element of R, and f (u), g(u) and h(u) sufficiently regular functions generalizing the prototypes f(u) = K(1)u(alpha), g(u) = K2u(gamma) and h(u) = ku - mu u(beta), with K-1, K-2, mu > 0, k is an element of R, beta > 1 and suitable alpha, gamma > 0. Besides, further regular initial data u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), w(x, 0) = w(0)(x) >= 0 are given, whereas T-max is an element of (0, infinity] stands for the maximal instant of time up to which solutions to the system exist. We will derive relations between the parameters involved in (>) capable to warrant that u, v, w are global and uniformly bounded in time. The article generalizes and extends to the case of nonlinear effects and logistic perturbations some results recently developed in [3] where, for the linear counterpart and in the absence of logistics, criteria towards boundedness are established

Boundedness Through Nonlocal Dampening Effects in a Fully Parabolic Chemotaxis Model with Sub and Superquadratic Growth

This work deals with a chemotaxis model where an external source involving a sub and superquadratic growth effect contrasted by nonlocal dampening reaction influences the motion of a cell density attracted by a chemical signal. We study the mechanism of the two densities once their initial configurations are fixed in bounded impenetrable regions; in the specific, we establish that no gathering effect for the cells can appear in time provided that the dampening effect is strong enough. Mathematically, we are concerned with this problem {ut=Δu-χ∇·(u∇v)+auα-buα∫ΩuβinΩ×(0,Tmax),τvt=Δv-v+uinΩ×(0,Tmax),uν=vν=0on∂Ω×(0,Tmax),u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,x∈Ω ̄,◊ for τ= 1 , n∈ N , χ, a, b> 0 and α, β≥ 1 . Herein u stands for the population density, v for the chemical signal and Tmax for the maximal time of existence of any nonnegative classical solution (u, v) to system (◊). We prove that despite any large-mass initial data u , whenever (The subquadratic case) 1≤α[removed]n+42-α,(The superquadratic case) β>n2and2≤α<1+2βn, actually Tmax= ∞ and u and v are uniformly bounded. This paper is in line with the result in Bian et al. (Nonlinear Anal 176:178–191, 2018), where the same conclusion is established for the simplified parabolic-elliptic version of model (◊), corresponding to τ= 0 ; more exactly, this work extends the study to the fully parabolic case Bian et al. (Nonlinear Anal 176:178–191, 2018

Decay in chemotaxis systems with a logistic term

This paper is concerned with a general fully parabolic Keller-Segel system, defined in a convex bounded and smooth domain Ω of RN , for N ∈ {2, 3}, with coefficients depending on the chemical concentration, perturbed by a logistic source and endowed with homogeneous Neumann boundary conditions. For each space dimension, once a suitable energy function in terms of the solution is defined, we impose proper assumptions on the data and an exponential decay of such energies is established