Analysis of 2+1 diffusive-dispersive PDE arising in river braiding

We present local existence and uniqueness results for the following $2+1$ dispersive diffusive equation due to P. Hall arising in modeling of river braiding: $$u_{yyt} - \gamma u_{xxx} -\alpha u_{yyyy} - \beta u_{yy} + \left (u^2 \right)_{xyy} = 0$$ for $(x,y) \in [0, 2\pi] \times [0, \pi]$, $t> 0$, with boundary condition $u_{y}=0=u_{yyy}$ at $y=0$ and $y=\pi$ and $2\pi$ periodicity in $x$, using a contraction mapping argument in a Bourgain-type space $T_{s,b}$. We also show that the energy $\| u \|^2_{L^2}$ and cumulative dissipation $\int_0^t \| u_y \|_{L^2}^2 dt$ are globally controlled in time.Comment: 21 page

Analytic theory for the determination of velocity and stability of bubbles in a Hele-Shaw cell. Part 1: Velocity selection

An asymptotic theory is presented for the determination of velocity and linear stability of a steady symmetric bubble in a Hele-Shaw cell for small surface tension. In the first part, the bubble velocity U relative to the fluid velocity at infinity is determined for small surface tension T by determining transcendentally small correction to the asymptotic series solution. It is found that for any relative bubble velocity U in the interval (U(c),2), solutions exist at a countably infinite set of values of T (which has zero as its limit point) corresponding to the different branches of bubble solutions. U(c) decreases monotonically from 2 to 1 as the bubble area increases from 0 to infinity. However, for a bubble of arbitrarily given size, as T approaches 0, solution exists on any given branch with relative bubble velocity U satisfying the relation 2-U = cT to the 2/3 power, where c depends on the branch but is independent of the bubble area. The analytical evidence further suggests that there are no solutions for U greater than 2. These results are in agreement with earlier analytical results for a finger. In Part 2, an analytic theory is presented for the determination of the linear stability of the bubble in the limit of zero surface tension. Only the solution branch corresponding to the largest possible U for given surface tension is found to be stable, while all the others are unstable, in accordance with earlier numerical results

Micro electrical discharge machining

Experimental research in micromachining: A Case Study on LBMM-micro EDM based micro drillin

Microdrilling using laser-uEDM based sequential process

This presentation describes various analysis on the laser-uEDM based sequential micromachining process

A review on micro-patterning processes of vertically aligned carbon nanotubes array (VACNTs Array)

Vertically Aligned Carbon Nanotubes array which are also sometimes labeled as carbon nanotubes forests have many applications in several engineering fields for its remarkable mechanical, electrical, optical, and thermal properties. The Vertically Aligned Carbon Nanotubes array is often employed in developing microdevices such as pressure sensor, angle sensor, switches, etc. To successfully integrate carbon nanotubes forest to the micro-electro-mechanical systems based devices micropatterning of the carbon nanotubes forest is required. There are several methods available to realize micropatterning of Vertically Aligned Carbon Nanotubes array, from in-situ patterning during the growth process to post-patterning process. Each has its advantages and disadvantages. This paper will discuss elaborately different patterning processes of the carbon nanotubes forest and their different characteristics

Time-evolving bubbles in two-dimensional stokes flow

A general class of exact solutions is presented for a time evolving bubble in a two-dimensional slow viscous flow in the presence of surface tension. These solutions can describe a bubble in a linear shear flow as well as an expanding or contracting bubble in an otherwise quiescent flow. In the case of expanding bubbles, the solutions have a simple behavior in the sense that for essentially arbitrary initial shapes the bubble will asymptote an expanding circle. Contracting bubbles, on the other hand, can develop narrow structures ('near-cusps') on the interface and may undergo 'break up' before all the bubble-fluid is completely removed. The mathematical structure underlying the existence of these exact solutions is also investigated