Nanoparticle enhanced evaporation of liquids: A case study of silicone oil and water

Evaporation is a fundamental physical phenomenon, of which many challenging questions remain unanswered. Enhanced evaporation of liquids in some occasions is of enormous practical significance. Here we report the enhanced evaporation of the nearly permanently stable silicone oil by dispersing with nanopariticles including CaTiO3, anatase and rutile TiO2. The results can inspire the research of atomistic mechanism for nanoparticle enhanced evaporation and exploration of evaporation control techniques for treatment of oil pollution and restoration of dirty water

Compactness of Riemann–Liouville fractional integral operators

We obtain results on compactness of two linear Hammerstein integral operators with singularities, and apply the results to give new proof that Riemann–Liouville fractional integral operators of order $\alpha\in (0,1)$ map $L^{p}(0,1)$ to $C[0,1]$ and are compact for each $p\in \bigl(\frac{1}{1-\alpha},\infty\bigr]$. We show that the spectral radii of the Riemann–Liouville fractional operators are zero

Theories of Fixed Point Index and Applications

This thesis is devoted to the study of theories of fixed point index for generalized and weakly inward maps of condensing type and weakly inward A-proper maps. In Chapter 1 we recall some basic concepts such as cones, wedges, measures of noncompactness and theories of fixed point index for compact and gamma-condensing self-maps. We also give some new results and provide new proofs for some known results. In Chapter 2 we study approximatively compact sets giving examples and proving new results. The concept of an approximatively compact set is of importance in defining our index for a generalized inward map since there exists upper semicontinuous multivalued metric projections onto the approximatively compact convex set. We also introduce the concept of an M1-set which will play an important role in defining our fixed point index for generalized inward maps of condensing type since there exists continuous single-valued metric projections onto an Ml-closed convex set. Many examples of M1-closed convex sets are given. Weakly inward sets and weakly inward maps are studied in detail. New properties and examples on such sets and maps are given. We also introduce the new concept of generalized inward sets and generalized inward maps. The class of generalized inward maps strictly contain the class of weakly inward maps. Several necessary and sufficient conditions for a map to be generalized inward and examples of generalized inward maps are given. In Chapter 3 we define a fixed point index for a generalized inward compact map defined on an approximatively compact convex set and obtain many new fixed point theorems and nonzero fixed point theorems. In particular, norm-type expansion and compression theorems for weakly inward continuous maps in finite dimensional Banach spaces are obtained, which have not been considered previously. In Chapter 4 we define a fixed point index for a generalized inward maps of condensing type defined on an M1-closed convex set and obtain many new fixed point theorems and nonzero fixed point theorems. We also apply the abstract theory to some perturbed Volterra equations. In Chapter 5 we define a fixed point index for weakly inward A-proper maps. We obtain new fixed point theorems, nonzero fixed point theorem and results on existence of eigenvalues. We also give an application of the abstract theory to the existence of nonzero positive solutions of boundary value problems for second order differential equations

“Buoyancy” in granular medium: how deep can an object sink in sand?

The behavior of granular matter is different from either fluids or solids. One may not be able to answer even a naive question such as how deep an object can sink in sand. Answers to the depth of footprints on sand beach and its dependence on grain size have never been seriously studied before and may deserve a closer look and better understanding. Laying a ball of fixed size onto granules, we have measured the sinking depth (SD) of the ball into granules of different sizes and studied the dependence of SD on the sizes of the ball and granules. We find that the SD is very sensitive to the size of granules and the variation of SD on granule size is not monotonic. The maximum SD occurs at r ≈ 1/20 R, where r and R are the radii of granules and the ball, respectively. This ratio does not depend on the density of the ball and the volume fraction of granules. An empirical formula of SD on densities and sizes of the ball and granules are obtained based on the experimental results

A new Bihari inequality and initial value problems of first order fractional differential equations

We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order α∈(0,1) . One novelty in this paper is that it is not assumed that f is continuous but that it satisfies an Lp -Carathéodory condition for some p>1α (detailed definitions are given in the paper). We prove existence on an interval [0, T] in cases where T can be arbitrarily large, called global solutions. The necessary a priori bounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist when f(t, u) grows at most linearly in u, and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature

GRANULAR FLOW IN THE PRESENCE OF AN ELECTRIC FIELD

The granular flow in a vertical pipe in the presence of electric field E is studied. Depending upon its initial state and the applied field voltage the controlled flow rate remains in two phases, dilute flow or dense flow. For dilute flow, the electric field has no effect on the flow rate until V reaches a critical value Vj. At V = V 1} the flow rate drops abruptly and a transition of the particulate from dilute to dense flow occurs. For dense flow, the flow rate decreases monotonically with increasing V. A two-dimensional computer simulation has been done and the results agree qualitatively well with the experimental measurements