ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2018, Vol. 59, No. 1, pp. 72–78.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
ANALYTIC APPROXIMATE SOLUTION FOR A FLOW
OF A SECOND-GRADE VISCOELASTIC FLUID
IN A CONVERGING POROUS CHANNEL
Abstract: The problem of a two-dimensional steady ﬂow of a second-grade ﬂuid in a converging
porous channel is considered. It is assumed that the ﬂuid is injected into the channel through one
wall and sucked from the channel through the other wall at the same velocity, which is inversely
proportional to the distance along the wall from the channel origin. The equations governing the
ﬂow are reduced to ordinary diﬀerential equations. The boundary-value problem described by the
latter equations is solved by the homotopy perturbation method. The eﬀects of the Reynolds and
crossﬂow Reynolds number on the ﬂow characteristics are examined.
Keywords: homotopy perturbation method, second-grade ﬂuid, converging channel, velocity equa-
Flows of Newtonian and non-Newtonian ﬂuids ﬁnd numerous applications in engineering practice, in partic-
ular, in boundary layer control, transpiration cooling, and gaseous diﬀusion.
There are many publications that describe the derivation of the governing equations, investigations of non-
Newtonian second-grade ﬂuids, and ﬁnding analytical solutions for such ﬂuid ﬂows [1–11].
The behavior of second-grade ﬂuids is strongly related to normal stress moduli and kinematical tensors.
Dunn and Fosdick  demonstrated that the ﬂuid becomes unstable if the normal stress modulus is lower than
zero. Fosdick and Rajagopal  showed that the ﬂuid exhibits an anomalous rheological behavior under these
conditions. A thorough discussion of the issues involved can be found in the critical review of Dunn and Rajagopal .
Theoretical research on steady ﬂows of non-Newtonian ﬂuids was initiated by Berman , who found a series solution
for a two-dimensional laminar ﬂow of a viscous incompressible ﬂuid in a parallel-walled channel in the case of a
very low crossﬂow Reynolds number. Results of later studies in this ﬁeld were summarized by Choi et al. . For
high Reynolds numbers, a solution for a laminar boundary layer in a ﬂuid ﬂow in a converging or diverging channel
with a permeable wall through which suction or injection is performed with a velocity inversely proportional to the
distance along the wall from the channel origin was obtained by Rosenhead . Terrill  derived an analytical
solution of a similar problem for a slow ﬂow through a channel where the velocity of ﬂuid injection at one wall
is taken to be equal to the velocity of ﬂuid suction at the other wall. However, Terrill’s solution contains several
errors, as was pointed out by Roy and Nayak . Converging and diverging ﬂows of power-law ﬂuids with wall
suction or injection were studied by Balmer and Kauzlarich . Roy and Nayak  re-examined Terrill’s problem
by replacing a Newtonian ﬂuid by a viscoelastic Walter’s B-ﬂuid. The same problem was analyzed in  for the
Reiner–Rivlin ﬂuid model, and an analytical solution was obtained in the case of a slow ﬂow.
Department of Mechanical Engineering, University of Qom, Qom, Iran; firstname.lastname@example.org. Translated
from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 59, No. 1, pp. 83–90, January–February, 2018. Original
article submitted September 22, 2016; revision submitted December 27, 2016.
2018 by Pleiades Publishing, Ltd.