Exact Relativistic Ideal Hydrodynamical Solutions in (1+3)D with Longitudinal and Transverse Flows Page: 4 of 9
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The two equations involving derivatives over y and p are
trivially solved by setting p(x) p(r, p) (and the same
for energy density e(T, p) due to the E.o.S) and accord-
ingly tP(x) 'P(r, p). We note, that the simple form
of Eqs.(18,20), are a direct consequence of the vanishing
components uv" 0 in the flow field ansatz, Eq.(12).
Finally we introduce a combined field variable K de-
T + p T2p
We then substitute the pressure p in the equations (17)
and (19) and obtain two equations for the fields K and
vP, which can be expressed as
D* - K,r + Dc ' K, P
The coefficients Da, Db, Dc, D1, D2 are given by:
Da (1 - v) + v ,
Di v+ (1 -v)D2
Dl -2v0,0P, ,7 1 - 0/T - 1P , P ,
From (22)(23) we obtain
(ln K),T =z2
(ln K) = a 2 b1
'P Da - D
first order derivatives a igf and dinf . For In K as a
single function of two variables r, p, the two equations can
be consistent only if the following constraint on second
order derivatives are satisfied a2Ln1K a2Ln1C i.e.
[OP (r, p)]= 0 .
Thus we only need to solve the above single equation
for the velocity field vP(r, p). Since B, already involve
the first derivatives of vP ,T and iP ,P, the reduced ve-
locity equation 28 is a second-order partial differential
equation for the velocity field. As a minor caveat, the
method applies to the case/region in which ln K is at
least second-order differentiable. This reduction method
can be demonstrated in the more explicit case of (1+1)D
hydrodynamics, see Appendices B and C.
Given the above constraints, we can then solve from
(25) the matter field S directly
K K0 64i'; dr'F[r',p] f9 dp'QLrp']]
with Ko being the value at arbitrary reference point
Finally let us summarize our approach: after including
into the flow field ansatz the physically desired longitu-
dinal and transverse flows, we have reduced the hydro-
dynamic equations into a single equation (28) involving
(24) ONLY the transverse velocity field 'P, and any solution
to this equation automatically leads to the pressure field
which together with the velocity field forms a solution to
the original hydrodynamics equations:
p constant x x e fT dr'Yir', f9 dp9'L' . (30)
with the functions , given by
_ ( 1
v(1 - v)(1 -v2)2
[(1- v)D - v]vP , P +
[(1 - 2v2) + 2v(v - 1)t ]ttP ,7
-v[1 - j] [v + (1- vj]/T - iivvJP
C. Examination of the Method
We now examine the correctness of the reduced equa-
tion (28) and the solution (30), using the two known sim-
ple analytic solutions (10) and (11) as both of them are
certain special cases of our embedding with longitudinal
and transverse radial flows.
For the 1-D Bjorken expansion, we have vP B-. 0
which leads to
Bj. - ,
v -1 7
v ]lpp ,(26)
_ (T 1
v(1 - v)(1 v2
[ + (v - i)]Dp,1 +
[(-1 + 4v - 2v2) + 2v(v - i)tp]t2 ,
+vvjp[ - t ]/T + v[1 -2 j][1 - i) + vD]/p} .(27)
In (25), the function ln K depends (via tP) on two
variables T and p, and we have two equations for the two
QBj. _ -
One can easily verify that the above .j, Q8j. satisfy
the reduced equation (28). Furthermore by inserting
Bj., Q8j. into the solution (30) one finds exactly the
pressure in (10).
For the 3-D Hubble expansion, we have vP Hu. p/T
which leads to
3 v -5/2 2T
T 1 - v T2- p2
1 v -5/2 -2p
QHu. -+ 32
p 1-v T2-p2 (2
D2 = 2(1 -v)vPP , P + v(1 - 0)/p
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Liao, Jinfeng & Koch, Volker. Exact Relativistic Ideal Hydrodynamical Solutions in (1+3)D with Longitudinal and Transverse Flows, article, May 20, 2009; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc932705/m1/4/: accessed January 20, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.