In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.
Free convective of nanofluid inside dispersive porous medium adjacent to a vertical plate under the effects of the zero mass nanoparticles flux condition and the thermal and solutal dispersions is studied. Buongiorno's model revised is used considering Darcy and non Darcy laminar flows, and isothermal or convective flux outer the wall. Dimensionless governing equations formulated using velocity, temperature, concentration and nanoparticle volume fraction have been solved by finite difference method that implements the 3-stage Lobatto collocation formula. The numerical data obtained with semi or full dispersions cases are compared to predictions made using the non dispersive porous medium. Taking into account the dispersions, the influence of the zero mass nanoparticles flux condition is examined to test the validity of the control active nanoparticle assumption. It is found mainly that the thermal transfers can reach more than 100% in connection with the case where of a semi-dispersion of the porous medium is applied. Realistic condition, i.e. zero mass flux should be addressed for the heat transfer rate rather than the mass transfer rate, discovered markedly different to the active condition. This signifies the importance of considering the zero nanoparticles mass flux and dispersions in the performance characterization of nanofluid flow in porous media.
It is shown that the presence of zero mass particles makes the elements of the S-matrix divergent. We explain the physical origin of such divergences. We argue that they are due to the long range of the interactions which violate the assumptions we made when we derived the asymptotic conditions for scattering. We study these divergences in the particular case of QED at tree, as well as one-loop level and present the Bloch–Nordsieck solution. We show that the cancellation of infrared divergences among virtual exchanged and real emitted soft photons is true to all orders in the perturbation expansion and we obtain the Sudakov double logarithm formula for Coulomb scattering.
AbstractThis investigation aims to look at the thermal conductivity of dusty Micropolar nanoliquid with MHD and Cattaneo–Christov heat flux flow over an elongated sheet. The novelty of the envisioned mathematical model is augmented with the added impacts of the heat source/sink, chemical reaction with slip, convective heat, and zero mass flux boundary conditions. The salient feature of the existing problem is to discuss the whole scenario with liquid and dust phases. The graphical depiction is attained for arising pertinent parameters by using bvp4c a built-in MATLAB function. It is noticed that the thermal profile and velocity field increases for greater values of liquid particle interaction parameter in the case of the dust phase. An escalation in the thermal profile of both liquid and dust phases is noticed for the magnetic parameter. The rate of mass transfer amplifies for large estimates of the Schmidt number. The thickness of the boundary layer and the fluid velocity are decreased as the velocity slip parameter is augmented. In both dust and liquid phases, the thermal boundary layer thickness is lessened for growing estimates of thermal relaxation time. The attained results are verified when compared with a published result.
On the planar Schrödinger–Poisson system with zero mass potential