text:
"Winter Wonderland"
I thought I'd try my hands at scratchbuilding some buildings for a temporary winter set. They are not great, but for a first attempt, its all about learning the do's and don'ts of scratchbuilding! The shed is made out of plasticard, with brick paper attached, the backscene of the Copley Hill carriage works is the wrong colour (should be whitewashed!) but the brick comes out better in the photographs. These are made from plywood, cut to shape and painted carefully on the roof, and not so carefully where the roof meets the front facade.
The snow is soap powder, stuck down with a very diluted PVA glue, and the track was then laid on top. This was done so that, while it looks less realistic, it enables the trains to run without problems, and with several little cousins visiting at christmas, a decent running train set for them to play on was a must!
Merry Christmas everyone - last blog before christmas!
"60163, Tornado..."
Seen above is my completed model of 60163, Tornado, depicted as she will look in 2011.
Here's how I built mine, for those of you who may want to build your own:
1. Bachmann Peppercorn A1 donor model (mine was Aberdonian - which needed some modifying)
2. Fox transfers nameplates and express passenger blue transfer lining
3. Black undercoat, Express Passenger Blue paint, Railmatch bufferbeam red paint, and a few other acrylic colours
4. Clear varnish sealant for paint and transfers
The model was taken apart to remove rivet detail (like Kestrel before her, the model of Tornado actually needs to be a darlington variant which requires a little alteration), rubbing down and a black primer being applied:
The cabside had its rivets removed and paint stripped.
To me, Bachmann's backhead is excellent for detail, but it only really comes out when painted - here, I have handpainted the backhead myself. Some further modifications will be made to the cab in the shape of cabinets under the driver's seat.
The tender tank needing rebuilding to better represent 60163's larger tender tank, and if you look carefully on the finished pictures, you'll see I modelled the cabinet of dials on the right hand side of the tender footplate as well!
The boiler had minor alterations - chimney shortened, dome lengthened - this is hardly noticeable on the model, but it's there! Tornado's safety valves, cab roof, dome and chimney were reprofiled to meet Network Rail's 13ft above rail height stipulation, and so has my model.
The model was then sprayed black using Gamesmaster's "Chaos Black" - I swear by this stuff, never had a problem with it. It forms a perfect undercoat for certain shades of blue (especially my specially mixed Express Passenger blue paint!), and dries evenly.
The nameplates were dealt with separately, the fantastic Fox transfer etched plates being attached:
After the undercoat was completed...
...the first coat of blue paint was applied. I have never liked either Hornby or Bachmann's shade of Express Passenger blue, and the railmatch equivalent doesn't float my boat either. I opted for mixing my own paint from various off-the-shelf acrylics, and came up with this shade, as shown on the tender body:
Be careful, however! This was my first attempt with a spray gun, and in retrospect while the coat is even, it is also too thick. I'll be more careful next time!
Next came the adding of transfers...it always helps to have another model around to check where the lining goes - you'll notice in the later photographs that my model is missing the lining on the footplate - there is a good reason for this. I forgot!
Here, my earlier model of 60130 helps with the lining:
Here is Tornado, halfway though finishing the lining:
The transfers are Fox Transfers - use lots of water, soak up the excess with a cloth or tissue, and use a cocktail stick to position it carefully!
Here's the finished article, I'm delighted with the colour and how the whole project has turned out:
There's more to do which shall be covered in a future post - adding the roller bearing axleboxes on the tender and cartazzi truck is the major one, and another aesthetic piece is the adding of spoked wheels on the tender (as on the real Tornado).
I'll also cover how I added the A4 chime whistle to the right hand deflector.
Posted by Simierski at 01:15 3 comments:
Labels: 60163, Bachmann, British, Hornby, Locomotive, Peppercorn A1, Railways, renumbering, Steam
"Return to Great Northern"
New bufferbeam added, and dome - and some pictures here!
A proper, detailed post entailing the kitbash will be posted later next week.
"Pacific Power Part Four"
60525 A.H Peppercorn stands outside the works, newly painted and finished, complete with etched name and numberplates!
There will be a seven engine lineup next update - six Peppercorn pacifics and one Gresley...!
"Pacific Power Part Three"
From the above to this below:
Nearly there! 60525 takes shape...
"Pacific Power Part Two"
This model of 60525, Peppercorn A2 A.H Peppercorn, has become this below:
And hopefully, will look something like this when finished:
Making a grand total of six Peppercorn designed pacifics in use on 56C (when finally built). More pics will be forthcoming, when it has been finished in the next few days, along with a further update entitled "North by Great Northern!"
"Pacific Power Part One"
Here's a lineup of some of the pacifics that will grace Copley Hill over its building, inception, exhibition and possibly disrepair (1964). From left to right: 60114, 60114 (in blue), 60130, 60139, 60158 and the last one which won't appear, 4472...!
"Creative Writing"
LOUGHBOROUGH UNIVERSITYLOUGHBOROUGH UNIVERSITY
The London and North Eastern railway was one of the so-called
"Big Four"
companies, formed during the 1923 grouping act in Britain.
In this form, with engines from the North Eastern and Great Northern railways,
it lasted a mere twenty five years,
but left an ever lasting impression of Luxury, prestige and speed.
40 years after the end of British main line steam,
these are the stories they tell...
In April 2007, a group of English students were given their module titles for their first year at university. One of those modules was Language and Communication, and involved a film project.
After much discussion, it was decided to make a Children's series for Youtube. It had to be educational, historical and above all – fun to watch. Not only that, but it required a certain amount of views over a year, and to fulfil the conditions set down by the course.
But what to film? Many ideas were tried out – stop motion plasticine films about a dog and his bone, animated features about ocean liners and their trips around the world – all ideas that were put into the mix. Then one day, in a meeting at the home of group director, Simon Martin, the old Hornby train set was discovered...and thus, the first seed was sown.
Initially the series was researched with a certain little blue tank engine in mind. However, the more the team researched series similar in setup, the further away from their minds was the Reverend Awdry's creation.
You see, the series had to be historical, and by that, fully accurate with regards to liveries, time period, locomotives and rolling stock. The first trailer, which by then had a new title - "The British Railway Series: The End of the LNER" was broadcast on July 15th 2007, and featured an as yet, unnamed locomotive – a blue Peppercorn A1 pacific, without a face, and pulling a train of maroon Gresley coaches.
It was here that the first dilemma appeared. On the one hand, the series was going to be very similar in style and setup to the little blue tank engine – how to set it apart from that series? The solution came in the characters, and the scripts that followed. Between the July trailer, and the final, fifth trailer which arrived on August 20th, a group of five locomotives had been procured and given their names. It was to be a wholly more emotive series, with friends separated, reunited, rebuilt and even renamed along the way.
It is not unfair to say that the series became the brain child of its director, Simon Martin – more than anyone else on the series, he had worked long and hard in creating the characters, writing the scripts, and researching the time period and location "The British Railway Series" would be based in.
"My first character was Stephen – this one was very easy. I had an old Hornby B12 model, that had been named Stephen by my father and I for as long as I remember – the reason? The original designer of the real class was Stephen Holden, and my father and I being steam enthusiasts, gave it an appropriate name. The engine itself was quite elderly, so it was natural that the character of Stephen would be the oldest of the engines in the series. He first appeared in episode one, and has appeared in every episode since – and remains the only engine to do so".
By the time episode one had been scripted, then approved for filming, filmed and edited, it was clear that the series was not going to be a one-off episode. Around ten scripts had been written by then, and more, interesting, characters were popping up all the time.
Episode one, titled "A great problem goes west!" was the first story of the engines Allen (the young Peppercorn A1 Pacific), Stephen (the wise and elderly Holden B12) and Sir Ralph (the haughty A4 pacific). The story revolved around an old Great Western engine needing help getting back to the Western Region of British Railways. Allen was chosen to pull the train, and several mishaps along the way led to a happy conclusion (this episode was set in 1951, and the course of the first six episodes were spread, chronologically, over the years 1951-1952. Episode Seven is of course set a the end of 1952). There are several, deliberate, references to the Reverend Awdry's "Thomas' train" story, in which the plucky tank engine forgets to bring his train with him. The BRWS team (the name the students decided to give themselves) were all fans of the books when they were younger, and their first episode was in fact dedicated to the "The Railway Series" (from which the series gained its name).
The episode had a mixed reception. The faces, made from a special modelling clay, were a hit, but the filming and set were not up to scratch, and in fact many alterations were made over the course of the first six episodes.
By the time episode five had been released, the series then had Herbert (the nervous V2) and Nigel (the cricket loving V3). Both of these characters have proved popular, and have had some great moments in the series - "the bull was in the tunnel...the bull was in the tunnel?!" (Episode four: "Nigel, Herbert and the Cows").
Many things were learnt over those early episodes, in that first few months. By the time episode six was being approved for filming, the set had been adjusted slightly, a new camera had been procured, which specialised in model filming, and the characters had started to settle into particular roles. They were by no means one dimensional characters – Sir Ralph, of course, is the best example of this. Originally a very haughty, arrogant engine, Sir Ralph's background and mishaps formed a story arc that started in episode one, and came to a head in episode ten, "The Legacy of Gadwall". Sir Ralph developed into a thoughtful, emotive locomotive with many character flaws – and of course, positives.
The series was developing into a complex look at the development of the Eastern Region of British Railways in the 1950s. Subjects like the "Modernisation Plan" of the British Railways board, scrapping of steam locomotives, and the onset of diesels appeared, giving the series a very real and historical perspective of British history.
One thing that made the series stand out was its very distinctive style and music from the outset. An orchestra was hired in to record a version of "Rule Brittania" for the intro, and a local student rock band were hired in to perform a remixed version of "The Dambusters March" for the end credits. The original ambient sounds and effects were donated by several members of the MTF forum, and their names appear in the credits of every episode as thanks.
By episode 8, released at the end of March 2008, the set had been re-designed, and the name of the yard – always referred to as "the sheds" or "the yard" was now revealed to be Copley Hill, a real life set of ex-LNER sheds in the Leeds area. The set was re-designed to better look like Copley Hill in real life, and the re-development of the set continues well into the next planned episodes over the course of 2008 and 2009.
The series had seen several guest engines appear from other regions of British Railways, to acknowledge the other regions and leave openings for the possibility of other series. These guest engines helped to cover up a few mishaps of the early set (like malfunctioning models and track work), and not only that, produced some very memorable episodes.
The series has a finite length, as per the original directive of being historical. The series will finish in the year 1967, when the last of the ex-LNER express locomotives leaves the rails for the scrapyard, and it's a certainty that only a few locomotives (including Scott the A3 pacific) will survive.
But that's a long way off, and of course, there are many, many stories still to tell...
"60163, Tornado's First Train"
What a mental week! Unfortunately university work, and my training have got in the way of any modelling or layout planning, but I did take time off from my busy schedule to see Tornado and travel behind her on her first public train.
Here's the link to my photographs:
She's alive!
Goodbye for now, I will make some sort of proper, modelling update to the blog this week - probably tomorrow!
Labels: 60163, Bachmann, British, Hornby, LNER, Locomotive, Peppercorn A1, Railways, Steam, The British Railway Stories, Tornado, trains
"North by Great Northern...Part 3"
Here's another pictorial update from the Copley Hill Works.
There will be a layout plan update this week, I am putting the finishing touches to the second layout plan as I type this.
Laters!
Labels: Bachmann, British, Great Northern, Hornby, Railways, renumbering, Steam
Another pictorial update:
Copley Hill Works
The tender is now a very nice shade of British Railways Expressenger Blue...
Here's hoping I can do a little more layout planning next, now that I'm waiting on some major components.
"North by Great Northern!"
I thought I'd give a pictorial update, and a link to the work in progress:
http://www.rmweb.co.uk/forum/viewtopic.php?f=8&t=15467&start=100#p445554
Great Northern is suddenly taking shape...
"Great Northern"
It's rather a brute of a machine, isn't it?
Seen here: http://www.lner.info/locos/A/a1_1.shtml. This database is a mine of information for anyone modelling the LNER and I highly reccomend it.
The lone A1/1 was an Edward Thompson rebuild of Sir Nigel Gresley's prototype pacific - Great Northern, which was to form the testbed for the Peppercorn A1 pacific. Suffice to say, not many locomotives have formed the prototype and test bed for two classes. A controversial rebuild, even today arguments break out over the locomotive's looks and even a question of should it have existed at all...
I was rather surprised to find in my research, how many times the unloved 60113, Great Northern, visited 56C. It was enough times, both in photographs (I have found no less than six photographs of the A1/1 at Copley Hill over a period of five years), that I decided - now that the Clan model is almost complete - to try to build another locomotive for the intended layout.
And why not? I love unusual machines, (RMwebbers will no doubt know of my passion for Brush Traction's Falcon... and, euh, be tired of my obssession with it, mind), and anything that adds interest to the shed scene will be well worth the effort.
However, after the agonising time put into a second hand Nu-Cast A2/1 kit (and failing so miserably to build it, that it's been sent away to be sorted out!), and the misery of failure, I decided on this occasion to kitbash, rather than kit build.
I've kitbashed one locomotive already: a Trix A2 became a "Trixby" A2 (that's a Trix A2 body on a Hornby Railroad Flying Scotsman chassis), which turned out okay - as 60525 A.H Peppercorn. I'm doing a similar conversion here, except I fully intend to modify the cylinders and valve gear to appear more Thompson-esque.
I have a plan - the components are coming together - and I really, really want to build this A1/1. It's almost as if it'll convince myself that I can actually model - and well enough to finally get on and do some proper, soldering kitbuilding, instead of kitbashing.
So next time, hopefully, I'll have some pics up of the work in progress...
"Doncaster or Darlington..."
"60130, Kestrel"
"Early Bird"...
"Suburban steam"
"56C or 37B?"
"Great Northern takes shape"
"These are the stories we tell..."


BS EN 343 specifies the requirements and test methods for materials and the seams of protective clothing that has been designed to give protection against precipitation (i.e. rain, snow), mist and ground moisture. There is also a separate test EN 14360 for assessing the waterproofness of the complete finished garments but this test is not included in EN 343.
This standard's icon shows an umbrella placed under two numbers that represent the X and Y values. The X value indicates the waterproofness and the Y value indicates the water vapour permeability (the garment's breathability).
This standard falls within the scope of 'self-certification'. This means that manufacturers are allowed to certify their own products, which therefore do not need to be assessed by an independent Notified Body. This is because the only risk to the wearer is that he/she might get wet, which is not a life-threatening hazard. If EN 343 is used in combination with other EN standards the whole combination of standards is assessed by the Notified Body.
The X value indicates the waterproofness of the garment. There are 3 Classes with 3 being the highest (i.e. the most waterproof) and 1 is the lowest. A specific quantity of pressurised water is applied to the fabric and the respective Class is derived from the amount of pressure the fabric can withstand.
The Y value indicates the breathability of the fabric/s (all the layers used in the article). There are also 3 classes for breathability with Class 1 being the lowest and Class 3 the highest. Wearing Class 1 garments longer than x minutes is not recommended (the length of time the article may be worn (x) is given in the standard in a table).


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redpajama_set_name: RedPajamaGithub


text:
\section{Introduction}

If colloidal particles get trapped at fluid-fluid interfaces they interact effectively via deformations of the interface. These so-called capillary interactions can easily be tuned by changing external fields and they depend sensitively on the shape of the particles. This makes them good candidates for designing self-assembling systems and provides a convenient experimental playground for studying basic issues of statistical mechanics in two dimensions in the presence of long-ranged interactions~\cite{Zahn2000,Dominguez2010}. On the other hand, recent experiments~\cite{Madivala2009} show that capillary forces between elongated particles floating on spherical interfaces can have important consequences for stabilizing so-called Pickering emulsions, which are formed by particle-covered droplets (e.g.,\ oil) in a solvent (e.g.,\ water).

Whereas considerable theoretical progress has been made in understanding capillary interactions at flat interfaces~\cite{Kralchevsky2000,Oettel2005,Dominguez2008a}, basic issues such as the balance of forces acting on the interface and the influence of the incompressibility of the liquid enclosed by spherical interfaces have not yet been fully resolved. Curved interfaces pose the additional difficulty~\cite{Wurger2006,Dominguez2007} that in the presence of external forces the condition of mechanical equilibrium demands to either fix the center of mass of the droplet by an external body force or to pin the droplet surface, e.g., to a solid plate. The experimentally relevant issues of the boundary conditions at the plate and of their influence on the pair potential between capillary \textit{monopoles} have been studied previously~\cite{Guzowski2010,Guzowski2011}. Here, we study \textit{higher} capillary multipoles and derive the corresponding effective pair potentials for pointlike particles in terms of an expansion in spherical harmonics. By using a finite element method~\cite{Brakke} we compare these analytical results with those obtained by the numerical minimization of the surface free energy of drops in the presence of ellipsoidal particles.

\section{Perturbation theory}
We consider a spherical, full, and spatially fixed droplet of radius $R_0$ with colloidal particles trapped at its surface with surface tension $\gamma$. We model the effect of particles in terms of an external surface pressure field $\Pi(\Omega)$ (see below) parameterized by spherical coordinates $\Omega=(\theta,\phi)$ on the unit sphere. The equilibrium shape of the droplet subjected to the pressure field $\Pi$ follows from minimizing the corresponding free energy functional $\mathcal{F}[\{v(\Omega)\}]$ expressed in terms of the dimensionless radial displacement of the interface $v(\Omega)=(r(\Omega)-R_0)/R_0$. In the limit of small interfacial gradients $|\nabla_a v|\ll 1$ one has~\cite{Wurger2006,Guzowski2010}
 \begin{equation}
  \dfrac{1}{\gamma R_0^2}\mathcal{F}[\{v(\Omega)\}]= \int\! d\Omega\, \left[\frac{1}{2}(\nabla_a v)^2 - v^2 - \big(\pi(\Omega)+\mu\big)v\right],
  \label{free_en_O2}
 \end{equation}
where $\nabla_a:=\boldsymbol{e}_{\theta}\partial_{\theta}+\dfrac{\boldsymbol{e}_{\phi}}{\sin\theta}\partial_{\phi}$ is the dimensionless \textit{a}ngular gradient on the unit sphere~\cite{comment_epsilon}. The first two terms in Eq.\ (\ref{free_en_O2}) represent the surface free energy, the third term corresponds to the work done by the dimensionless external pressure $\pi(\Omega)=\Pi(\Omega)R_0/\gamma$ in displacing the interface and the fourth term serves to implement the volume conservation. The value of the Lagrange multiplier $-\mu$ follows from imposing the volume constraint
 \begin{equation}
  \int\! d\Omega\, v = 0;
  \label{volume_constr2}
 \end{equation}
$\gamma\mu/R_0$ can be identified with the shift in the internal pressure of the droplet with respect to the Laplace pressure $2\gamma/R_0$ of an unperturbed, perfectly spherical droplet. The stationary condition $\delta \mathcal{F}/\delta v = 0$ leads to the Euler-Lagrange equation
 \begin{equation}
  -\big(\nabla_a^2+2\big)v(\Omega)=\pi(\Omega)+\mu.
  \label{helmholtz_pi}
 \end{equation}
We expand both the deformation $v(\Omega)$ and the pressure field $\pi(\Omega)$ in spherical harmonics $Y_{lm}(\Omega)$ so that Eq.\ (\ref{helmholtz_pi}) turns into an infinite set of algebraic equations~\cite{Morse1993}:
\begin{equation}
	[l(l+1)-2]v_{lm}=\pi_{lm}+\mu\delta_{l0},
	\label{helmholtz_multipole}
\end{equation}
where $l=0,1,\ldots$ and $m=-l,\ldots,l$, and the spherical multipoles are defined as $X_{lm}:=\int \!d\Omega'\, X(\Omega')Y_{lm}(\Omega')$ for $X(\Omega)=v(\Omega)$ or $X(\Omega)=\pi(\Omega)$. The volume constraint in Eq.\ (\ref{volume_constr2}) implies $v_{00}=0$ and thus $\mu = -\pi_{00}$, which means that the internal pressure shift counterbalances the external pressure. According to Eq.\ (\ref{helmholtz_multipole}) the $l=1$ components of the deformation $v$ are undefined. This is consistent with the fact that those components describe translations of the whole droplet without any change in shape which do not change the free energy. On the other hand all $l=1$ components of the external pressure $\pi$ must cancel, reflecting the condition of balance of forces acting on the droplet. Indeed, the multipoles $\pi_{1-1},\pi_{10},\pi_{11}$ are proportional to the Cartesian components $f_x,f_y,f_z$ of the total force $\boldsymbol{f}$ acting on the droplet. Hence the condition $\boldsymbol{f}=0$ is equivalent to $\pi_{1m}=0$ with $m=-1,0,1$.

\section{Green's function and balance of forces}
As mentioned above we consider a pressure field in the form of the superposition of $N$ pointlike forces $f_i$ pointing into directions $\Omega_i$:
\begin{equation}
	\pi(\Omega)=\sum_{i=1}^{N}q_i\delta(\Omega,\Omega_i),
	\label{decomposition1}
\end{equation}
where $q_i:=f_i/(\gamma R_0)$ and $\delta(\Omega,\Omega_i)=\delta(\theta-\theta_i)\delta(\phi-\phi_i)/\sin\theta_i$ is the Dirac delta distribution expressed in terms of spherical coordinates. The pressure field as a whole must obey the condition of balance of forces which can be expressed as $\sum_i^{N}\pi_{i,1m}=0$ for $m=-1,0,1$. The corresponding interface deformation can be written as the superposition of single particle contributions:
\begin{equation}
	v(\Omega)=\sum_i^{N}q_iG(\Omega,\Omega_i),
	\label{v1}
\end{equation}
where $G(\Omega,\Omega')$ is Green's function describing the response of the interface in direction $\Omega$ to a point-force applied in direction $\Omega'$ and which, according to the constraints of constant liquid volume and fixed center of mass obeys the equation~\cite{Morse1993}
\begin{equation}
	-(\nabla_a^2+2)G(\Omega,\Omega')=\sum_{l\geq 2}\sum_{m=-l}^l Y^*_{lm}(\Omega)Y_{lm}(\Omega'),
	\label{greens_equation}
\end{equation}
where the right hand side is a modified Dirac delta function $\hat{\delta}(\Omega-\Omega')$ with the $l=0$ and $l=1$ components projected out. From integrating both sides of Eq.\ (\ref{greens_equation}) over the unit sphere it follows that the $l=0$ component of $G$ vanishes, which reflects the condition of constant volume $\int\! d\Omega\, G(\Omega,\Omega')=0$. The $l=1$ component of $G$ also vanishes, which can be seen by multiplying both sides of Eq.\ (\ref{greens_equation}) by a radial vector $\boldsymbol{e}_r$, integrating over the unit sphere and using the fact that $\boldsymbol{e}_r$ can be expressed in terms of spherical harmonics with $l=1$ only. This reflects the condition that the center of mass of the droplet is fixed in space, which can be formulated as $\int\!d\Omega\,\boldsymbol{e}_r G(\Omega,\Omega')=0$.
\begin{figure
	\centering
	\psfragscanon
		\psfrag{p1}[l][l][1]{$\phi_1$}
		\psfrag{b}[c][c][1]{$\bar{\theta}$}
		\psfrag{p2}[l][l][1]{$\phi_2$}
		\psfrag{o1}[r][l][1]{$\Omega_1=(0,0)$}
		\psfrag{o2}[c][c][1]{$\Omega_2$}
	  	\psfrag{l}[c][c][1]{$\Lambda_{12}$}
	  	\psfrag{x}[c][c][1]{$x$}
		\psfrag{y}[c][c][1]{$y$}
		\psfrag{z}[c][c][1]{$z$}
	  	\psfrag{x1}[c][c][1]{$x'$}
		\psfrag{y1}[c][c][1]{$y'$}
		\psfrag{z1}[c][c][1]{$z,z'$}
	  	\psfrag{x2}[c][c][1]{$x''$}
		\psfrag{y2}[c][c][1]{$y''$}
		\psfrag{z2}[c][c][1]{$z''$}
		\includegraphics[width=0.45\textwidth]{fig01}
	\psfragscanoff
		\caption{Angular configuration of the pressure distributions symbolically represented as gray patches centered at the directions $\Omega_1=(0,0)$ (north pole) and $\Omega_2$ on the unit sphere. The angles $\phi_1$ and $\phi_2$ represent orientations of the local coordinate frames $x'y'z'$ and $x''y''z''$ associated with the pressure distributions (see main text).}
	\label{sphere}
\end{figure}

According to the above considerations the pressure field due to a \textit{single} point-force violates the balance of forces acting on the droplet. Therefore, if one wants to cast the solution still into in the form given by Eq.\ (\ref{v1}) with $N=1$, there must be an additional pressure field counterbalancing the single point-force. Inserting  $v(\Omega)=G(\Omega,\Omega')$ into Eq.\ (\ref{helmholtz_pi}) and using Eq.\ (\ref{greens_equation}), one obtains the corresponding pressure field $\pi$ in the form
\begin{equation}
 \pi(\Omega) =\delta(\Omega,\Omega')+\pi_{CM}(\Omega,\Omega')
\label{delta2}
\end{equation}
with the effective pressure $\pi_{CM}(\Omega,\Omega')=-(3/(4\pi))\cos\bar{\theta}$, where $\bar{\theta}$ is the angle between the directions $\Omega$ and $\Omega'$. The latter pressure must emerge in order to cancel the $l=1$ component of the external pressure $\pi_{ext}(\Omega)=\delta(\Omega,\Omega')$. Therefore $\pi_{CM}$ corresponds to a body force fixing the center of mass ($CM$) of the droplet without contributing to the deformation field $v$.

The expression in Eq.\ (\ref{v1}) for the deformation under the action of point-forces can be generalized to the case of a pressure field $\pi_{ext}(\Omega)$ due to an arbitrary continuous force distribution:
\begin{equation}
	v(\Omega)=\int\!d\Omega'\, \pi_{ext}(\Omega')G(\Omega,\Omega').
\label{upi}
\end{equation}
The total pressure $ \pi$ acting on the interface again has an additional component $\pi_{CM}$ accomplishing the fixing of the center of mass, i.e., $\pi=\pi_{ext}+\pi_{CM}$. One can express the free energy in terms of Green's function by integrating by parts Eq.\ (\ref{free_en_O2}), by using Eq.\ (\ref{helmholtz_pi}) and the divergence theorem, and finally by inserting $v(\Omega)$ as given by Eq.\ (\ref{upi})~\cite{Morse1993}:
\begin{equation}
	\dfrac{1}{\gamma R_0^2}F = -\dfrac{1}{2}\int\!d\Omega\, \int\!d\Omega'\, \pi_{ext}(\Omega)G(\Omega,\Omega')\pi_{ext}(\Omega').
\label{free_en_sph}
\end{equation}

\section{Capillary interactions}
We study the pressure $\pi_{ext}$ localized around two different directions $\Omega_1$ and $\Omega_2$. To this end we introduce the following decomposition:
\begin{equation}
 \pi_{ext}(\Omega)=\pi_1(\hat{R}_1^{-1}\Omega)+\pi_2(\hat{R}_2^{-1}\Omega),
 \label{decomposition}
\end{equation}
where $\hat{R}_1$ ($\hat{R}_2$) denotes the rotation transforming the original coordinate frame $xyz$ into the coordinate frame $x'y'z'$ ($x''y''z''$), referred to as $O_1$ ($O_2$), associated with the pressure distribution $\pi_1$ ($\pi_2$) (see Fig.\ \ref{sphere}). We use the parameterization in terms of Euler angles, in which an arbitrary rotation $\hat{R}$ can be composed of a rotation around the $z$-axis by the angle $\alpha$, around the (rotated) $y$-axis by the angle $\beta$, and finally around the (rotated) $z$-axis by the angle $\gamma$.
Under the rotation $\hat{R}(\alpha,\beta,\gamma)$ of the coordinate frame the coordinates transform according to $\hat{R}^{-1}$, as indicated in Eq.\ (\ref{decomposition}).

The total free energy can be written as $F = F_{1,self} + F_{2,self} + \Delta F$, where $F_{i, self}= -[f_i^2/(4\pi\gamma)]\ln(R_0/a_i)+O(1)$ is the self-energy of particle $i$~\cite{Guzowski2010}, which does not depend on the relative position of the particles on the droplet but depends on the particle size $a_i$. For a particle located at the north pole this is defined as the smallest distance for which $\pi_i(\theta,\phi)=0$ for all $\theta>a_i/R_0$ and $\phi\in[0,2\pi)$; this corresponds to the solid angle circumscribing the particle-liquid interface. The interaction free energy $\Delta F$ is given by the cross-terms in Eq.\ (\ref{free_en_sph}) with $\pi$ from Eq.\ (\ref{decomposition}):
\begin{multline}
 \dfrac{1}{\gamma R_0^2}\Delta F = -\int \!d\Omega \int \!d\Omega'\, \pi_1(\hat{R}_1^{-1}\Omega)G(\Omega,\Omega')\times\\
\times\pi_2(\hat{R}_2^{-1}\Omega')
	= -\int \!d\Omega \int \!d\Omega'\, \sum_{l,m}\pi_{1,lm}Y^*_{lm}(\hat{R}_1^{-1}\Omega)\times\\
	\times\sum_{j,n} g_{j}Y^*_{jn}(\Omega)Y_{jn}(\Omega')\sum_{k,m'}\pi_{2,km'}Y^*_{km'}(\hat{R}_2^{-1}\Omega'),
\label{delta_f}
\end{multline}
where
\begin{equation}
 g_l=\left\{
 \begin{array}{lll}
 0			&\text{for} &l=0,1\\
 \dfrac{1}{l(l+1)-2}	&\text{for} &l\geq 2
 \end{array} \right.
 \label{gl}
\end{equation}
are the spherical harmonics coefficients of $G(\Omega,\Omega')$. Without loss of generality we can take $\Omega_1=0$ such that the reference frame $O_1$ coincides with the original reference frame $xyz$ (see Fig.\ \ref{sphere}), which implies that $\hat{R}_1=\mathbbm{1}$.

Spherical harmonics transform under the representation of the group of rotations according to $Y_{lm}(\hat{R}^{-1}\Omega)=\sum_{m'=-l}^{l}D^{\, l}_{m',m}(\hat{R})Y_{lm'}(\Omega)$, where $D^{\, l}_{m',m}$ is the Wigner $D$-matrix and reads (adopting the convention used in Ref.~\cite{Brink_book})
\begin{equation}
 D^{\, l}_{m',m}(\hat{R})=D^{\, l}_{m',m}(\alpha,\beta,\gamma)=e^{-im'\alpha}d^{\, l}_{m',m}(\beta)e^{-im\gamma},
\end{equation}
where $d^{\, l}_{m',m}(\beta)$ is known as the Wigner (small) $d$-matrix. We use the parameterization in terms of the orientations $\phi_1$ and $\phi_2$ of the coordinate frames $O_1$ and $O_2$, respectively, relative to the great circle connecting the points $\Omega_1=0$ and $\Omega_2$ on the unit sphere (see Fig.\ \ref{sphere}). The rotation $\hat{R}_2$ is then parameterized by the triad of the Euler angles $(2\pi-\phi_1,\bar{\theta},\phi_2)$. By using the orthogonality  $\int \!d\Omega Y_{lm}(\Omega)Y_{l'm'}^*(\Omega)=\delta_{ll'}\delta_{mm'}$ of the spherical harmonics, and due to the identities $Y_{lm}^*(\Omega)=(-1)^mY_{l,-m}(\Omega)$ and ${D^{\, l}_{m',m}}^*(\alpha,\beta,\gamma)=(-1)^{m-m'}D^{\, l}_{-m',-m}(\alpha,\beta,\gamma)$ we finally obtain
\begin{multline}
 \dfrac{1}{\gamma R_0^2}\Delta F = -\sum_{l\geq2} \sum_{m=-l}^l\sum_{m'=-l}^l\pi_{1,lm}\,\pi_{2,lm'}\,(-1)^{m'}\,g_l\times\\
\times
d^{\, l}_{m,-m'}(\bar{\theta})\,e^{i(m\phi_1+m'\phi_2)}.
\label{multipole_expansion}
\end{multline}

\subsection{Limit of small particles}
Having in mind colloidal particles of radius $a$ as sources of the effective surface pressure $\pi$, we would like to relate the spherical multipoles $\pi_{lm}$, $l\geq0$, $m=-l,\ldots,l$ to prescribed forces, torques, and higher capillary multipoles defined for the case of a flat interface.
If $a\ll R_0$ the interface can be treated as to be locally flat in the close neighborhood $\Delta\Omega$ of angular extent $O(a/R_0)$ around each particle separately. Assuming for reasons of simplicity that $\Delta\Omega$ is centered at the $z$-axis and approximating it by a circular disk $D(a)$ of radius $a$ in the tangent plane we can use the asymptotic form of spherical harmonics~\cite{Edmonds_book}
\begin{equation}
	Y_{lm}(\theta,\phi) \xrightarrow[\theta\rightarrow0]{}i^{|m|+m}A_{l|m|}\theta^{|m|} e^{im\phi},
\end{equation}
where
\begin{equation}
	A_{ln}=\sqrt{\dfrac{2l+1}{4\pi}\dfrac{(l+n)!}{(l-n)!}}\dfrac{1}{2^{n}n!},\quad n>0,
	\label{Alm}
\end{equation}
in order to obtain
\begin{multline}
	\pi_{lm}=\int_{\Delta\Omega}\!d\Omega'\, \pi(\Omega') Y_{lm}(\Omega')\
	\xrightarrow[a/R_0\rightarrow0]{}\\
	i^{|m|+m}A_{l|m|}\left(\dfrac{a}{R_0}\right)^{|m|+1}Q_{|m|}e^{im\tilde{\phi}_{|m|}},
	\label{Qlm-Qm}
\end{multline}
where we have introduced the dimensionless moduli
\begin{equation}
	Q_{n}:=\left|\int_{D(a)}\!\dfrac{d^2x'}{a^2}\,\dfrac{a\Pi(z')}{\gamma} \left(\dfrac{z'}{a}\right)^n \right|
	\label{Qnsph}
\end{equation}
and the phase
\begin{equation}
	\tilde{\phi}_{n}:=\dfrac{1}{n}\,\text{arg}\int_{D(a)}\!\dfrac{d^2x'}{a^2}\,\dfrac{a\Pi(z')}{\gamma} \left(\dfrac{z'}{a}\right)^n,
	\label{phin}
\end{equation}
where $n\geq0$, $z'$ denotes a complex number, and $\Pi(\boldsymbol{x}')\equiv\gamma\pi(\Omega'(\boldsymbol{x}'))/R_0 $, with $\boldsymbol{x'}(\Omega')$ being a projection onto the plane tangent to the reference sphere at $\Omega_i$. From Eq.\ (\ref{Qnsph}) it follows that the moduli $Q_{n}$ are the usual capillary multipoles defined at a locally flat interface (see Refs.~\cite{Dominguez2008a,Dominguez_book}). Accordingly, due to the conditions of force and torque balance on a flat interface~\cite{Dominguez2008a}, $Q_{0}$ represents a ``capillary monopole'', i.e., the total external force acting on the particle, $Q_{1}$ a ``capillary dipole'', i.e., the total external torque. In the case of a free particle, $Q_{2}$ represents the lowest non-vanishing multipole. Furthermore, we note that for each particle $i$ one can always choose the orientation $\phi_i$ of the coordinate frame $O_i$ such that the phase of the order $n$ vanishes, i.e., $\tilde{\phi}_{i,n}=0$. We can write the interaction energy $\Delta F_{nn'}$ for a pair of capillary multipoles $Q_{n}$ and $Q'_{n'}$ of arbitrary orders $n\geq0$ and $n'\geq0$ as a sum of terms such that in Eq.\ (\ref{multipole_expansion}) the sum over $l$ is taken under the constraint $l\geq\max \{2,n,n'\}$ with $m$ and $m'$ fixed to $m=\pm n$ and $m'=\pm n'$; this implies $\Delta F = \sum_{n,n'=0}^{\infty}\Delta F_{nn'}$. Accordingly, the interaction free energy $\Delta F_{nn'}$ for a pair of capillary multipoles $Q_n$ and $Q_{n'}$ scales with the droplet radius $R_0$ as $\Delta F_{nn'}\sim \gamma a^{n+1}a'^{\,n'+1}/R_0^{n+n'}$. Note, that in the case of two monopoles ($n=0$ and $n'=0$) the interaction does not depend on $R_0$ but only on the moduli $Q_{0}$ and $Q'_{0}$.

The Wigner d-matrix in Eq.\ (\ref{multipole_expansion}) can be expressed in terms of the Jacobi polynomials $P_n^{(\sigma,\rho)}(\cos\bar{\theta})$~\cite{Edmonds_book}:
\begin{multline}
d^{\, l}_{m,m'}(\bar{\theta})=\left[\dfrac{(l+m')!(l-m')!}{(l+m)!(l-m)!}\right]^{1/2}
\left(\sin\dfrac{\bar{\theta}}{2}\right)^{m'-m}\times\\
\times
\left(\cos\dfrac{\bar{\theta}}{2}\right)^{m'+m}P^{(m'-m,m'+m)}_{l-m'}(\cos\bar{\theta}).
\end{multline}
\begin{widetext}
By using certain properties of the Jacobi polynomials (see, e.g., Ref.~\cite{Edmonds_book}) we finally obtain
\begin{multline}
 	\dfrac{\Delta F_{nn'}}{\gamma a a'}\xrightarrow[a/R_0,a'/R_0\rightarrow0]{}
	\dfrac{Q_{n}Q'_{n'}}{(-2)^{n+n'+1}n!n'!\pi}\dfrac{a^{n}a'^{\,n'}}{R_0^{n+n'}}
	\sum_{l\geq \max\{2,n,n'\}}\dfrac{(2l+1)}{(l+2)(l-1)}\times\\
	\vspace*{-0.5cm}
	\times\left\{
\begin{array}{l}
		\dfrac{(l+n')!}{(l-n)!}\Bigg[(-1)^{n}\cos(n\phi_{1}+n'\phi_{2})\left(\cos\dfrac{\bar{\theta}}{2}\right)^{n'-n}\left(\sin\dfrac{\bar{\theta}}{2}\right)^{n'+n} P_{l-n'}^{(n'+n,n'-n)}(\cos\bar{\theta}) \\
		\hspace*{-0.2cm}
		\begin{array}{lll}
		      \ + \cos(n\phi_{1}-n'\phi_{2})\left(\cos\dfrac{\bar{\theta}}{2}\right)^{n'+n}\left(\sin\dfrac{\bar{\theta}}{2}\right)^{n'-n}P_{l-n'}^{(n'-n,n'+n)}(\cos\bar{\theta})\Bigg],                                     & n>0, & n'>0,\\
		      (-1)^n\cos(n\phi_{1})P_l^n(\cos\bar{\theta}),          & n>0, & n'=0,\\
		      2^{-1}P_l(\cos\bar{\theta}),                                                                  & n=0, & n'=0,
		\end{array}
\end{array}\right.
	\label{Fn1n2}
\end{multline}
\noindent where $P_l$ and $P_l^n$ are the Legendre and the associated Legendre polynomials, respectively.
\end{widetext}

\subsection{Effective interaction potentials}
For a first discussion, we evaluate the interaction energy in Eq.\ (\ref{Fn1n2}) for $a=a'$ and $n'=n$.
In the case of two identical pointlike \textit{monopoles}, $n=n'=0$,
one obtains~\cite{Prudnikov_book2}
\begin{multline}
\frac{\Delta F_{00}(\bar{\theta})} {\gamma a^2}\xrightarrow[a/R_0\rightarrow0]{}\\
\frac{Q_0^2}{4\pi}\left[\frac{1}{2} + \frac{4}{3}\cos\bar{\theta} + 2\cos\bar{\theta}\ln\left(\sin\dfrac{\bar{\theta}}{2}\right) \right],
\label{greens_function}
\end{multline}
which gives back Green's function $G(\bar{\theta})\equiv-\Delta F_{00}(\bar{\theta})/(\gamma a^2 Q_0^2)$ as given in Ref.~\cite{Morse1993}. We note that according to Eq.\ (\ref{greens_function}) one has $G(\bar{\theta}\rightarrow0)\rightarrow -(1/(2\pi))\ln\bar{\theta}$, which renders the deformation $v(r)=-(1/(2\pi))\ln(r/R_0)$ due to a pointlike force, known for a flat interface, where $r=\bar{\theta}R_0\ll R_0$ is the arc length (see the dashed line in Fig.~\ref{sphere}) and $R_0$ plays the role of the capillary length.

In the case of pointlike \textit{dipoles}, $n=n'=1$, one can evaluate the series involving the Jacobi polynomials by using their generating function (see the Appendix):
\begin{multline}
	\dfrac{1}{\gamma a^2}\Delta F_{11}(\bar{\theta},\phi_1,\phi_2) \xrightarrow[a/R_0\rightarrow0]{}\dfrac{Q_1^2}{8\pi}\left(\dfrac{a}{R_0}\right)^2\times \\
\times
\Big[ \cos(\phi_1+\phi_2)f_+(\bar{\theta}) + \cos(\phi_1-\phi_2)f_-(\bar{\theta}) \Big]
\label{DF11-final}
\end{multline}
where $\phi_1$ and $\phi_2$ are the orientations of the particles as indicated in Fig.~\ref{sphere} and chosen such that the phases $\tilde{\phi}_{i,n}$ defined in Eq.\ (\ref{phin}) vanish for $n=1$, and where
\begin{align}
 f_+(\theta):=&\dfrac{1}{\sin^{2}(\theta/2)} - 4\sin^2\dfrac{\theta}{2}\ln\left(\sin\dfrac{\theta}{2}\right) - \dfrac{20}{3}\sin^2\dfrac{\theta}{2} + 2,\\
 f_-(\theta):=&4\left(\cos\dfrac{\theta}{2}\right)^2\ln\left(\sin\dfrac{\theta}{2}\right) + \dfrac{20}{3}\cos^2\dfrac{\theta}{2}.\label{f-}
\end{align}
We note that the dependences on the angular separation $\bar{\theta}$ and on the orientations $\phi_1,\phi_2$ do not factorize. Therefore, in order to minimize the free energy, the particles adopt orientations which depend on $\bar{\theta}$. According to Eq.\ (\ref{DF11-potential}) below one can distinguish three branches of the free energy (see the dotted green lines in Fig.~\ref{fig:potentials}) corresponding to three different minimal orientations, out of which the one with the lowest free energy is the equilibrium one. Assuming that the rotational relaxation of the particles is much faster than the translational one, an effective interaction potential can be obtained by minimizing the free energy with respect to $\phi_1$ and $\phi_2$:
\begin{multline}
	\dfrac{1}{\gamma a^2}\min_{\{\phi_1,\phi_2\}} \left\{\Delta F_{11}(\bar{\theta},\phi_1,\phi_2)\right\} \xrightarrow[a/R_0\rightarrow0]{}\\
\dfrac{Q_1^2}{8\pi}
\left\{\begin{array}{llll}\vspace*{0.1cm}
              -f_+(\bar{\theta})+f_-(\bar{\theta}), &\text{for}       & \bar{\theta}<\bar{\theta}_0,                   & \,\,\uparrow\,\,\,\uparrow \\ \vspace*{0.1cm}
              -f_+(\bar{\theta})-f_-(\bar{\theta}), &\text{for}       & \bar{\theta}_0<\bar{\theta}<\bar{\theta}_1, & \rightarrow\leftarrow \\ \vspace*{0.1cm}
              \quad f_+(\bar{\theta})-f_-(\bar{\theta}), &\text{for}  & \bar{\theta}>\bar{\theta}_1,                   & \,\,\uparrow\,\,\,\downarrow
             \end{array}\right.
\label{DF11-potential}
\end{multline}
where $\bar{\theta}_0=\arccos[1-2\exp(-10/3)]\simeq0.38$ and $\bar{\theta}_1\simeq1.83$ are the zeros of $f_-(\bar{\theta}_0)=0$ and $f_+(\bar{\theta}_1)=0$, respectively; the arrows indicate the minimal orientational configuration (e.g., $\uparrow\,\,\,\uparrow$ corresponds to $(\phi_1=\pi/2,\phi_2=\pi/2)$ etc).

In the case of pointlike \textit{quadrupoles}, $n=n'=2$, one obtains
\begin{multline}
 	\dfrac{1}{\gamma a^2}\Delta F_{22}(\bar{\theta},\phi_1,\phi_2)  \xrightarrow[a/R_0\rightarrow0]{}\\ -\dfrac{3Q_2^2}{64\pi} \left(\dfrac{a}{R_0}\right)^4 \cos(2\phi_1+2\phi_2) \dfrac{1}{\sin^4(\bar{\theta}/2)}.
\label{DF22-final}
\end{multline}
Distinct from the case of dipoles, the free energy has a minimum for $\phi_1+\phi_2=k\pi,\, k=0,1,\ldots$ (so that $\cos(2\phi_1+2\phi_2)=1$), independent of $\bar{\theta}$. This corresponds to a monotonic attraction (see the solid blue line in Fig.~\ref{fig:potentials}).

\section{Discussion}
We discuss the influence of the curvature of the interface on the interaction potentials by comparing the expressions for the curved interfaces (Eqs.\ (\ref{DF11-final})-(\ref{DF22-final}), solid lines in Fig.\ \ref{fig:potentials}) with the well established results for the case of a flat interface (see, e.g., Ref.~\cite{Dominguez_book}, dashed lines in Fig.\ \ref{fig:potentials}), with the corresponding spatial separation $d$ between the particles given by $d=R_0\bar{\theta}$. A striking difference is the non-monotonicity for curved interfaces of the free energy for monopole-monopole and dipole-dipole interactions, which in both cases leads to short-ranged attraction and long-ranged repulsion. These peculiarities can be traced back to the presence of an external body force necessary for fixing the center of mass. For $n=0,1$ the expected reduction of the results for curved interfaces to those for a flat interface occurs only for very small angular separations. This is not the case for quadrupoles ($n=2$), for which at all separations there is no qualitative discrepancy and only a minor quantitative difference from the flat limit.
\begin{figure
	\centering
	\psfragscanon
		\psfrag{x}[l][l][1]{$\bar{\theta}$}
		\psfrag{x1}[l][l][1]{$\bar{\theta}_1$}
		\psfrag{x0}[l][l][1]{$\bar{\theta}_0$}
		\psfrag{as}[l][l][1]{$\bigg\}$ {\small flat}}
		\psfrag{y}[c][c][1]{$(R_0/a)^{2n}\times\min_{\phi_1,\phi_2}\{\Delta F_{nn}\}/(\gamma a^2Q_n^2)$}
		\psfrag{n0}[l][l][1]{$n=0$}
		\psfrag{n1}[l][l][1]{$n=1$}
		\psfrag{n2}[l][l][1]{$n=2$}

		\begin{overpic}[width=0.45\textwidth]{fig02}
		 \put(65,8.5){\includegraphics[width=0.15\textwidth]{fig03}}
		\end{overpic}
	\psfragscanoff
		\caption{Rescaled effective interaction potentials (Eqs.\ (\ref{greens_function})-(\ref{DF22-final}), solid lines) for two capillary multipoles of the same order $n=0,1,2$, each shifted as a whole such that $\Delta F_{nn}(\bar{\theta}=\pi)=0$. The dashed lines correspond to the following expressions: $[\ln(\bar{\theta})-\ln(\pi)]/(2\pi)$ for $n=0$, $[1/\pi^2-1/\bar{\theta}^2]/(2\pi)$ for $n=1$, and $3[1/\pi^4-1/\bar{\theta}^4]/(4\pi)$ for $n=2$, representing for comparison the case of a flat interface with the equivalent spatial separation $d=R_0\bar{\theta}$ of the particles. As expected, for $\bar{\theta}\rightarrow0$ each full  curve approaches its corresponding dashed curve. The green dotted lines correspond to three metastable branches of the free energy for $n=1$ (see Eq.\ (\ref{DF11-potential})) with two relevant intersection points (at $\bar{\theta}_1\simeq1.83$, where $f_+(\bar{\theta}_1)=0$, and $\bar{\theta}_0\simeq0.38$, where $f_-(\bar{\theta}_0)=0$) indicated by filled squares. In each interval $[0,\bar{\theta}_0]$, $[\bar{\theta}_0,\bar{\theta}_1]$, and $[\bar{\theta}_1,\pi]$ the full green curve corresponds to the lowest of the three branches and as such describes thermal equilibrium.}
	\label{fig:potentials}
\end{figure}

\begin{figure
	\centering
	\psfragscanon
		\psfrag{x}[l][l][1]{$\bar{\theta}$}
		\psfrag{y}[c][c][1]{$(R_0/a)^{4}\times\Delta F^{(2)}_{ell}/(\gamma a^2Q_2^2)$}
		\psfrag{r5}[l][l][1]{$a/R_0=0.599$}
		\psfrag{r6}[l][l][1]{$a/R_0=0.5$}
		\psfrag{r8}[l][l][1]{$a/R_0=0.375$}
		\psfrag{2a}[l][l][0.9]{$2a$}
		\psfrag{2b}[l][l][0.9]{$2b$}
		\psfrag{ps}[l][l][0.9]{$\psi_i$}
		\psfrag{om}[l][l][0.9]{$\Omega_i$}
		\psfrag{h}[l][l][0.9]{$h_i$}
		\psfrag{r0}[c][c][0.9]{$R_0$}
		\psfrag{tp}[l][l][0.9]{$\theta_p$}
		\psfrag{2r}[c][c][1]{$2R_0$}
		\psfrag{blank}[l][l][1]{}
		\psfrag{q}[l][l][1]{point-quadrupoles}
		\psfrag{ss}[l][l][1]{$\Bigg\}$

		\psfrag{tt}[l][l][1]{$\Big\}$

		\begin{overpic}[width=0.45\textwidth]{fig04}
		  \put(53,48){\includegraphics[width=0.142\textwidth]{fig05}}
		  \put(70,33){\includegraphics[width=0.0675\textwidth]{fig06}}
		  \put(70,15){\includegraphics[width=0.0675\textwidth]{fig07}}
		  \put(86,33){\includegraphics[width=0.0336\textwidth]{fig08}
		\end{overpic}

	\psfragscanoff
		\caption{Rescaled effective interaction energy $\Delta F^{(2)}_{ell}$ for two ellipsoidal particles of aspect ratio $a/b=3$, contact angle $\theta_p=2\pi/3$, various droplet radii $R_0$, and two different orientational configurations: tip-to-tip and side-to-side (as visualized by the snapshots from the finite-element free energy minimization). In view of large relative numerical errors for small values of $\Delta F^{(2)}_{ell}$ the zero of the free energy is taken as the mean value of the data points with $\bar{\theta}>1.5$ (the error bars are comparable or smaller than the size of the symbols). The thick solid line corresponds to Eq.\ (\ref{DF22-final}) with $\phi_1=\phi_2=0$ or $\phi_1=\phi_2=\pi/2$ (in either case Eq.\ (\ref{DF22-final}) yields the same free energy). The capillary quadrupole of each of the particles is approximated by the expression $Q_2=2\pi\Delta u/a$; this relation is valid for a flat interface~\cite{Dominguez2008a} at large interparticle separations, with the undulation $\Delta u$ of the interface taken to be the one at the surface of a single ellipsoidal particle. As an approximation for a droplet we adopt the same relation, $Q_2(R_0) = 2\pi \Delta u(R_0)/a$, with the undulation $\Delta u\equiv\max_{\phi}r(\theta=a/R_0,\phi)-\min_{\phi}r(\theta=a/R_0,\phi)$ of the interface at the bottom part of a particle centered at the north pole of the droplet (see the green part of the upper inset). This undulation has been calculated numerically for three droplet radii: $\Delta u(a/R_0=0.599)/a=0.028$, $\Delta u(a/R_0=0.5)/a=0.031$, $\Delta u(a/R_0=0.375)/a=0.034$. $R_0$ is the radius of the droplet \textit{without} the particles. The immersion $h_i$ and the tilt angle $\psi_i$ of the long axis of each particle $i$ with respect to the plane perpendicular to the direction $\Omega_i$ are subjected to minimization. The colored lines end at small values of $\bar{\theta}$ because the particle surfaces touch each other for nonzero $\bar{\theta}$.}
	\label{fig:numerical}
\end{figure}

Next, we compare our approximate analytic theory with the numerical minimization of the free energy for floating ellipsoidal particles with semi-axes $a$ and $b$, which we performed by using a finite element method~\cite{Brakke}. The numerical procedure consists of minimizing the free energy functional
\begin{multline}
	\mathcal{F}[\{\boldsymbol{r}(\Omega)\},h_1,h_2,\psi_1,\psi_2;\bar{\theta},\phi_1,\phi_2,\theta_p,a,b,V_l,\lambda]=\\
	= \gamma S_{lg} -\gamma\sum_{i=1,2}\cos\theta_{p,i}S_{pl,i} - \lambda(V-V_l),
	\label{functional}
\end{multline}
with respect to the shape of the interface $\{\boldsymbol{r}(\Omega)\}$, immersions of the particles $h_1,h_2$, and their tilts $\psi_1,\psi_2$ (relative to planes perpendicular to the directions $\Omega_1$ and $\Omega_2$, respectively, see the inset in Fig.\ \ref{fig:numerical}), while the angular separation $\bar{\theta}$ and the tangential orientations $\phi_1,\phi_2$ are kept fixed; $S_{lg}$ is the area of the liquid-gas interface, and $S_{pl,i}$ and $\theta_{p,i}$ are the area of the particle-liquid interface and the contact angle, respectively, of particle $i$. The last term in Eq.\ (\ref{functional}) ensures conservation of the liquid volume $V_l$ with $\lambda$ as the corresponding Lagrange multiplier.

The effective potentials (Fig.~\ref{fig:numerical}) obtained numerically for prolate spheroids (i.e., elongated spheres) with aspect ratio $a/b=3$ and droplet radii $a/R_0=0.599,0.5,0.375$ differ for particles oriented tip-to-tip ($\phi_1=\phi_2=\pi/2$) and side-to-side ($\phi_1=\phi_2=0$). This signals the importance of capillary multipoles higher than only a quadrupole, because the point-quadrupole approximation yields the same free energy for both orientations (see Eq.\ (\ref{DF22-final})). This discrepancy is not surprising because the longer particle axes (which set the length scale $a$) are actually not very small compared with the droplet radius $R_0$ and thus in this case the point-particle limit is a rather crude approximation. Our numerical analysis is restricted to such small droplets due to the limited effectiveness of the finite-element method for larger drops caused by too large numerical errors. For the same reason we consider only particles with a large aspect ratio ($a/b=3$) for which the deformation of the interface is sufficiently large so that the numerical noise does not smear out the angular dependence of the free energy. Nonetheless, even in this regime our analytic theory captures the main qualitative features of the effective interaction potential between floating ellipsoids.
Presumably the reason for this is that even for large particles the droplet shape deviates only slightly from a sphere. (Due to a fixed volume of the liquid  larger particles cause a larger displaced liquid volume $V_{displ}$; primarily this leads to an increased effective radius $\sim\sqrt[3]{V_l+V_{displ}}$ of the droplet and thus generates only a constant in the deformation field $v$.) Accordingly, away from the particles our perturbation theory remains applicable. Thus even for large particles the deformation of the droplet can be approximated by the superposition of contributions from pointlike quadrupoles substituting  the actual particles.
The additional dependence on the orientations revealed by our numerical calculations, which goes beyond the quadrupole approximation, resembles the well established case of a flat interface, which has been analyzed experimentally~\cite{Loudet2005} and theoretically~\cite{Lehle2008} in terms of a multipole expansion. Those studies showed that the point-quadrupole approximation, within which tip-to-tip and side-to-side configurations have the same free energy, is reliable only either for particles with a very small eccentricity ($b\simeq a$) or at large spatial separations $d\ll a$. Otherwise, one should rather consider an expansion in terms of elliptic coordinates, from which it follows that the tip-to-tip configuration is energetically more favourable. This prediction agrees with the experimental observations~\cite{Loudet2005}. Our calculations indicate that the same configuration is the preferred one also in the case of spherical interfaces. However, we have not analyzed all the possible orientations. It is noteworthy that optimal orientations other then tip-to-tip and side-to-side have been observed at a flat interface for a pair of ellipsoidal particles of \textit{different} sizes ~\cite{Loudet2009}.

We can also discuss the effective capillary potential $F^{(1)}_{ell}(\alpha,\phi_1;\theta_0)$ for a single capillary quadrupole on a \textit{sessile}, spherical caplike droplet forming a contact angle $\theta_0$ with the supporting substrate, where $\alpha$ is the angular position of the particle ($\alpha<\theta_0$) measured from the droplet symmetry axis and relative to the center of the sphere circumscribing the droplet, and $\phi_1$ the orientation of the particle. As before the tilt angle $\psi_1$ is taken to be the optimal one. In the case $\theta_0=\pi/2$ the method of images can be applied, analogous to the case of monopoles~\cite{Guzowski2010}. The substrate acts like a mirror and the free energy of a \textit{single} particle equals one half of the energy of \textit{two} particles on a \textit{full} droplet, i.e.,
\begin{multline}
\Delta F^{(1)}_{ell}(\alpha,\phi_1;\theta_0=\pi/2):=F^{(1)}_{ell}(\alpha,\phi_1;\theta_0=\pi/2)\\
-F^{(1)}_{ell}(0,\phi_1;\theta_0=\pi/2)=\\
=\pm\dfrac{1}{2}\Delta F_{22}(\pi-2\alpha,\phi_1,\pi-\phi_1),
\label{eq:DFell}
\end{multline}
where the $+$ sign corresponds to a free contact line at the substrate and the $-$ sign to a pinned one. (Since the particle is force-free, there are no additional images, distinct from the case of monopoles, see Ref.~\cite{Guzowski2010}.) For a free contact line Eq.\ (\ref{eq:DFell}) holds in general, i.e., for arbitrary particles (not necessarily ellipsoidal). In such a case $\Delta F_{22}$ is replaced by the actual interaction energy $\Delta F^{(2)}$ between the particles (possibly containing also higher multipoles).
On the other hand, for a pinned contact line Eq.\ (\ref{eq:DFell}) holds only for pointlike particles in conjunction with Eq.\ (\ref{DF22-final}).

For a pointlike quadrupole, according to the expression for $\Delta F_{22}$ in Eq.\ (\ref{DF22-final}), the dependence of $\Delta F^{(1)}_{ell}$ on the orientation $\phi_1$ drops out so that within this approximation the particle is always attracted to a free contact line and repelled from a pinned one. Thus the effective potential is given by the black line in Fig.\ \ref{fig:numerical} with $\bar{\theta}=\pi-2\alpha$ (with an additional minus sign in the case of a pinned contact line).

In the case of an actual ellipsoidal particle the free energy is no longer given by the rhs of Eq.\ (\ref{DF22-final}) so that the dependence on $\phi_1$ does no longer drop out but becomes important in the vicinity of the contact line at the substrate. As stated above, if this contact line is \textit{free} (still assuming $\theta_0=\pi/2$), due to symmetry arguments the free energy of a single particle on a sessile drop equals one half of the energy for two particles on a full droplet (as predicted by the method of images for the more restrictive case of pointlike quadrupoles), for which the full numerical results are plotted in Fig.\ \ref{fig:numerical}. Accordingly, for a fixed angular position $\alpha=\pi/2-\bar{\theta}/2$ of the particle its orientation perpendicular to the contact line (solid lines in Fig.\ \ref{fig:numerical}) is energetically more favorable. However, the global minimum of the free energy corresponds to the particle touching the substrate (i.e., $\alpha\rightarrow\pi/2$ so that $\bar{\theta}\rightarrow0$) and being oriented parallel to the contact line (dashed lines in Fig.\ \ref{fig:numerical} for $\bar{\theta}<0.5$). However, we cannot judge if the latter configuration remains the optimal one for arbitrary aspect ratios and contact angles at the particle. (This and analyzing the influence of the contact angle at the substrate actually call for a separate numerical study.)

If the contact line at the substrate is \textit{pinned}, the global free energy minimum corresponds to the particle being positioned at the drop apex. In such a case, due to a large distance from the contact line with the substrate, the pointlike quadrupole approximation remains reliable. One can estimate the depth of this effective confining potential as follows. According to Eqs.\ (\ref{eq:DFell}) and (\ref{DF22-final}) with $Q_2=2\pi\Delta u/a$ and assuming $\Delta u/a\simeq 0.05$ (which  applies to an ellipsoidal particle of aspect ratio $a/b=3$ and contact angle $\theta_p=2\pi/3$, see the caption of Fig.\ \ref{fig:numerical}), the displacement of the particle, e.g., from $\alpha=0$ to $\alpha=\pi/4$, leads to a free energy increase of approximately $k_BT\times(a/\mu m)^2$ for a droplet as large as $R_0/a\simeq 10$ (assuming a surface tension $\gamma\simeq0.07 N/m$ which corresponds to a water droplet in air). Hence, one could experimentally measure the capillary confining potential, e.g., by observing the Brownian motion of micron-sized particles on sessile drops. Finally, we emphasize that the effects of confinement should be also taken into account in determining the effective \textit{pair} potentials between particles on \textit{sessile} drops (for the case of monopoles see Ref.~\cite{Guzowski2011}). In particular, the confinement leads to closer packing of particles on a droplet, which should not be mistaken for an additional attractive pair-interaction.

\section{Acknowledgements}
On of the authors (J.G.) acknowledges financial support from the Polish Ministry of Science (2010/2011) through grant nr IP2010 012270.

\section{Appendix: Calculation of series containing Jacobi polynomials}

In Eq.\ (\ref{Fn1n2}) for $n=n'=1$ the following series appears:
\begin{equation}
 S_1^{(\sigma_1,\rho_1)}(x):
 \sum_{j\geq1}\dfrac{(j+1)(j+2)(2j+3)}{j(j+3)}P_j^{(\sigma_1,\rho_1)}(x)
\end{equation}
where $(\sigma_1,\rho_1)=(2,0)$ or $(0,2)$, and for $n=n'=2$:
\begin{equation}
  S_2^{(\sigma_2,\rho_2)}(x):
  \sum_{j\geq0}(j+2)(j+3)(2j+5)P_j^{(\sigma_2,\rho_2)}(x)
\end{equation}
where $(\sigma_2,\rho_2)=(4,0)$ or $(0,4)$. Those series can be evaluated using the generating
function~\cite{Abramowitz_book}:
\begin{multline}
  g^{(\sigma,\rho)}(x,z)=\dfrac{1}{R(1-z+R)^{\sigma}(1+z+R)^{\rho}}\\
 =\sum_{n=0}^{\infty}\dfrac{z^nP_n^{(\sigma,\rho)}(x)}{2^{\sigma+\rho}},
\end{multline}
where $R=\sqrt{1-2xz+z^2}$ and $|z|<1$. One finds the relationships
\begin{multline}
  S_1^{(\sigma_1,\rho_1)}(x)=2\int_0^1\!dy\, y^2 \int_0^y\!dz\,\times \\
\times z^{-2}\dfrac{\partial}{\partial z}z^{1/2}\dfrac{\partial}{\partial z}z^{2}\dfrac{\partial}{\partial z}z\left[2^{\sigma_1+\rho_1}g^{(\sigma_1,\rho_1)}(x,z)-1\right]
\end{multline}
and
\begin{multline}
 S_2^{(\sigma_2,\rho_2)}(x)=\\
 =2^{\sigma_2+\rho_2+1}\lim_{z\rightarrow1}\dfrac{\partial}{\partial z}z^{1/2}\dfrac{\partial}{\partial z}z^{2}
 \dfrac{\partial}{\partial z}z^2g^{(\sigma_2,\rho_2)}(x,z),
\end{multline}
which can be evaluated with the results
\begin{align}
 &S_1^{(2,0)}(x)=\dfrac{4}{(1-x)^2}+\dfrac{4}{1-x}-2\ln\left(\dfrac{1-x}{2}\right)-\dfrac{20}{3},\\
 &S_1^{(0,2)}(x)=-2\ln\left(\dfrac{1-x}{2}\right)-\dfrac{20}{3},\\
 &S_2^{(4,0)}(x)=\dfrac{96}{(1-x)^4} \label{sum0},
\end{align}
and
\begin{equation}
 S_2^{(0,4)}(x)=0.
\end{equation}
Taking $x=\cos\bar{\theta}$ and inserting these results into Eq.\ (\ref{Fn1n2}) finally yields Eqs.\ (\ref{DF11-final})-(\ref{DF22-final}).

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In this episode Elyse tells us about Ile de la Cité, the place where Paris was born. We explain why you should take a stroll and enjoy the beautiful and romantic Ile de la Cite in Paris. We suggest you listen to this episode while on your Paris walk around the Ile de la Cite. You could also visit the Crypt on the Parvis de Notre Dame, or the visit the Conciergerie where Marie Antoinette was imprisoned, or maybe walk up the tower of Notre Dame and look for a very famous hunchback? You should definitely enjoy a relaxing moment in one of the many Brasseries on Ile de la Cite. Even if you only have two hours in Paris, you must go to the Ile de la Cite!


text:
Merritt Skidmore-Hess
Banning Kosher Meats Shows European Weakness
Mar 15, 2016, 8:29 AM
One of the most disappointing things that has come out of Europe in the past few years are the bans and attempts to ban Kosher and Halal meat. These bans on shechita, such as the 2014 ban in Denmark, are done under the auspices of animal rights. The advocates for these laws state that non-Kosher pre-stunned animal slaughter is more humane than meat created by Kosher and Halal slaughter.
The arguments for these laws on the basis of animal welfare are weak when most meat in Europe is produced in factory farms which are by their very nature cruel and inhumane. When Denmark instituted a ban on religious slaughter the Danish pork industry was killing 25,000 piglets through maltreatment and not using the carcasses for food. Despite this obvious animal cruelty happening in his country Danish minister Dan Jørgensen defended the ban on Kosher and Halal meat saying "Animal rights come before religion"
Furthermore those advocating for these bans seem to ignore the writings of experts that contradict them such as Temple Grandin who wrote that Kosher and Halal meat can be made as humanely as non-Kosher pre-stunned meat stating on her website that "When the cut is done correctly, the animal appears not to feel it."
These bans and attempts at bans reflect not the strength of Europe's liberalism but its weakness. It reflects a belief that Jewish culture and Islamic culture are inferior and need to become more like the "civilized" and secularized Christian cultures that dominate Europe. I do not see an effective fight for political moderation in Europe in the face of extremist like Golden Dawn, the National Front, and Jobbik when more mainstream political forces are passing such slanted laws against their religious minorities in multiple counties throughout the continent. A continent whose governments lecture on animal rights, yet ignore the ghastly industrialized meat factories in their backyards because the meat from them is cheap.
Merritt Skidmore-Hess is a graduate at of Georgia Southern University where he studied the history of Islam and the Middle East. Merritt also studied Judaic Studies with the Conservative Yeshiva in Jerusalem.
European Jewry

redpajama_set_name: RedPajamaCommonCrawl


text:
Il doppio del torneo di tennis ABN AMRO World Tennis Tournament 1987, facente parte della categoria Grand Prix, ha avuto come vincitori Stefan Edberg e Anders Järryd che hanno battuto in finale Chip Hooper e Mike Leach 3-6, 6-3, 6-4.

Teste di serie

 Stefan Edberg /  Anders Järryd (campioni)
 Peter Fleming /  John McEnroe (semifinali)

 Guy Forget /  Jakob Hlasek (quarti di finale)
 Chip Hooper /  Mike Leach (finale)

Tabellone

Collegamenti esterni

ABN AMRO World Tennis Tournament 1987

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text:
EXCLUSIVE EXCERPT Missed Connections by Tamara Mataya @feakysnucker @SourcebooksCasa
by Tamara Mataya
Published by Sourcebooks, Inc.
Missed Connections by Tamara Mataya
Series: Summer Love, Book 1
Missed Connection: I saw you standing there, and I was struck by your eyes. Gorgeous, but not as gorgeous as your smile.
Thanks to her job at a crazy New Age spa, what should have been a sizzling NYC summer is being hijacked by demanding hippie bosses. To unwind, Sarah spends her nights cruising Missed Connections, dreaming of finding an uber-romantic entry all about her. Of course, the moment she finds that Missed Connection, real life comes crashing down around her in a night of unbridled passion with someone completely different: totally off-limits Jack.
Best. Hookup. Ever.
Gorgeous and wealthy, hot as sin, Jack can give Sarah everything she needs—except an emotional connection. That she gets from her Missed Connection, the romantic stranger who never fails to make her swoon. But there's only so much of Sarah to go around. Torn between the bad boy she can't keep and the sensitive stranger who bares his soul online, her heart and body are soon in two very different relationships…or are they?
iBooks:
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Exclusive Excerpt from Missed Connections
"How did your interview go?"
I smooth my ponytail in what I hope is a casual manner, feeling self-conscious about my tiny shorts and tank top. "I think it went well, but they said they had a few more applicants to go through. And they're a little strange."
"Strange how?"
How much can I say without sounding judgmental? "They're hippies."
"As in cool stoners? That might be kind of sweet having them as your bosses."
"I don't know about that, but I suspect there's going to be a lot of talk of chi and auras."
"Ah, New Agers."
"Yes." Resting an elbow on the desk, I prop my chin in one hand. "And they haven't let the person whose job I'd be taking know they're fired yet."
He grimaces. "Harsh."
"But I'd work there in a heartbeat if they'd have me."
He crosses his arms, and I try not to ogle them. "They'd be stupid not to hire you."
"Thanks, Jack." His earnestness makes me smile.
"I know you said before you couldn't waitress, but they make awesome tips. I could—"
"I know you have all kinds of connections, but I couldn't work as a waitress at one of those clubs. I don't have the coordination. There's a reason you guys never let me carry the drinks back to the table. I'd end shifts owing more than I'd made."
"Fair enough. So—" he says, as my cell phone vibrates against the desktop.
"I've got to get this." I hold up my hand. "Sarah speaking."
"Hi, Sarah, this is Fern. From Inner Space?"
"Hi, Fern." It's them, I mouth at Jack. "How are you?"
"I'm fine. Listen, I just wanted to call to let you know that unfortunately"—damn it—"our old receptionist found out we were interviewing and came in for an ugly confrontation before she stormed out, so we're going to need you to come in tomorrow morning."
Wait. "You mean I got the job?"
"Oh yes, didn't I say?"
I punch the air. "No! Thank you, Fern. I will definitely be there. What time do you need me?"
"You'll be working Monday to Friday, leaving around six. Is ten too early?"
Too early? The firm had me start work at seven—and I'll get weekends off. "Ten is perfect."
"Great. See you tomorrow."
"Bye." I end the call and spring to my feet. "I have a job."
"Congratulations!" Jack holds his hand up, but our high five turns into an enthusiastic hug.
And here, pressed up against his warm, muscular length, with my face to his chest, I remember why he's off-limits. Because I want him so very much, and he's so very wrong for me.
But for once, I don't care.
I tighten my embrace and breathe deeply, holding his scent in my lungs because I want any part of him inside me right now. His hand splays across my lower back and presses me closer, but no lines are crossed except those in my mind…where we've already done everything. Twice. My skin's cooled from the air-conditioning, but he's still warm from the heat outside, making the difference even more interesting. How would those heated hands feel trailing up my thighs…
Pulling back, I slowly drag my gaze from his chest to his face. I've wanted Jack from the moment I saw him six years ago at a house party, spinning records in the basement. Ten minutes later, I'd learned his nickname. DJ Madhead. My gay best friend's identical twin.
He licks his lips.
Oh, Jack is sex personified and he knows it. The trouble is, a lot of other women know it too. A lot of women. Too many women.
And I refuse to be just another car on the train. Man whores are firmly off-limits; I've seen what cheating can do to people. My mom tore my dad's heart out again and again. The worst part is he always takes her back. His pride is the least of my worries. It's the stress that she puts on his heart that worries me and reminds me to never date a potential cheater—no matter how pretty they are, or how pretty their words are. I've heard variations on every justification in the book from my mother's lips.
But even if all that changed, Jack's rampant Peter Pan syndrome would still keep him from being an option. He's a DJ. His office is a dance floor covered with intoxicated people. Late nights, flashing lights. How could I live like that, never seeing him? Never getting to spend more than a few hours a week, or a stolen moment on a noisy dance floor? How could I compete with all the women who throw themselves at him? I want more for myself—I need more. As lame as it may make me, I need someone who's serious about the future, about me—not just a hot guy who refuses to grow up.
So, despite the quickening of my pulse every time Jack comes near, nothing will ever happen between us. With a sigh, I step back, breaking contact, and head to the living room, hyper-aware of him as he follows and sits on the opposite end of the couch, giving me the space I don't want but need.
He picks up the conversation as if he hadn't noticed the weirdness.
"They want you to start tomorrow? That's awesome."
"Definitely." Though it's weird that the old receptionist had to be the deciding factor in me getting hired. Maybe it just sped up the timeline and they had chosen me already.
The door swings open and bangs against the wall. "Honey, I'm homo!" Pete calls out.
"God, you're such a caricature," I call with a grin.
"I'm a campy delight." He and his shopping bags rustle into the kitchen.
"Your brother's here."
"Good. I could use a big, strong man to help me with these heavy bags while I freshen up. I'm sweating like a hooker in church."
Jack rolls his eyes at me but moves to help Pete. I follow, trying not to notice how great Jack's ass looks in those jeans. Pete's already deposited the grocery bags on the counters when we reach the kitchen, so I stay out of the way while Jack helps him put stuff away.
They move with a similar grace, but Pete's a little softer and flows more, while Jack's like a slinky jungle cat. There's something about the way he walks that has always hit me right in the nether regions. Other than style choices and Jack's adorable mole, they are shockingly identical. Jack's hair is still their natural light brown and lacks the dyed, lacquered finesse of Pete's. Pete's eyebrows are also more groomed, but they don't look overdone. He's a junior aesthetician and stylist at a trendy, upscale salon in Manhattan, and he does amazing work. I trust no one else with my hair.
"Guess who has news?" Jack asks Pete.
"What? Who? Spill!"
I laugh. "I have a job! Soon, you'll have your couch back."
"Thank God," he exclaims with relief.
I narrow my eyes. "You could sound a little less excited to spare my feelings."
"Honey." He smooths an eyebrow with the tip of his ring finger, managing to look long suffering with that small action. "I love you, but if I had to see one more thong hanging over my towel rack, I was going to lose it."
"Pete!" My cheeks flame, and I look at Jack.
Tamara Mataya is a librarian with a great love for recommending books to patrons. Certified to teach English as a second language, she is also a musician. She writes erotic romance and contemporary romance that appeals to the New Adult audience. She lives in Alberta, Canada.
Connect with Tamara
Goodreads:
#Giveaway Excerpt Come Home to Deep River by Jack
$15 #Giveaway FORBIDDEN DESIRE by Robin Lovett @L
$10 #Giveaway EXTREME by Joan Gelfand @JoanGelfan
$10 #Giveaway Excerpt DEVIL'S WAYS ANTHOLOGY
#Giveaway Excerpt The Rebel Wears Plaid by Eliza
Tamara Mataya says
Thanks for having me!
Sounds like a great book!


text:
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