ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2018, Vol. 59, No. 1, pp. 61–71.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
A.I. Filippov, O.V. Akhmetova, A.A. Kovalskii.
COEFFICIENT-BY-COEFFICIENT AVERAGING IN A PROBLEM
OF LAMINAR GAS FLOW IN A WELL
A. I. Filippov, O. V. Akhmetova, and A. A. Kovalskii
Abstract: This paper describes the problem of determining the temperature of laminar gas ﬂow,
in which the equation of convective heat transfer contains two variable coeﬃcients, is reduced to
nonclassical problems for zeroth and ﬁrst asymptotic expansion coeﬃcient with respect to a formal
parameter. The Laplace–Carson transform are used to obtain analytical expressions for the tem-
perature ﬁeld of ascending laminar gas ﬂow in a well with account for the relationships of density
and velocity with spatial coordinates in zeroth and ﬁrst asymptotic approximations. Expressions
for the temperature asymptotically averaged along the cross section of the well and temperature
distributions over the cross-sectional radius are obtained.
Keywords: gas ﬂow, gas well, temperature ﬁeld, asymptotic method, laminar ﬂow regime.
Temperature calculations are widely used to solve various geophysical problems, such determination of
possible temperature ranges, at which gas hydrates are formed, and to interpret the results of thermal measurements
in a well bore.
An original approach to deriving analytical expressions for the temperature ﬁelds under study was developed
in  on the basis of a Newton formula for heat transfer on a surface in the assumption that a heat transfer coeﬃcient
of ﬂow in a well with surrounding rocks does not depend on time. An integral method for accounting for heat transfer
of ﬂow with surrounding rocks was proposed in , in which case the heat ﬂux was given in the form of convolution.
That approach was developed in [3–7], and the solutions for the temperature of incompressible liquid and gas
in a well bore, which were constructed in , are currently used to determine an average ﬂuid temperature in a
well. PC improvement contributed to the development of numerical methods of solving diﬀerential equations in
partial derivatives that describe nonstationary heat transfer in a well, including compressible gas [8, 9]. In turn, the
improvement of numerical methods accompanied the development of analytical  and numerical-analytical 
methods of solution of problems of heat and mass transfer for the reason that analytical solutions of corresponding
problems allow testing new high-performance algorithms, performing a parametric analysis of the temperature ﬁeld
of a system under study, and investigating the speciﬁc features of its formation process .
In order to obtain analytical solutions, one should both assume that the velocity proﬁle is aligned and
calculate the temperature that is average in the cross section.
It is shown in  that, in a stationary case, allowance for the compressibility of gas ﬂow with the help
of relationship between density and vertical coordinate ρ(z
/D) has an essential inﬂuence even on the
zeroth approximation in an asymptotic expansion.
Sterlitamak Branch of Bashkir State University, Sterlitamak, 453100 Russia; firstname.lastname@example.org; ahok-
email@example.com; firstname.lastname@example.org. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 59, No. 1,
pp. 71–82, January–February, 2018. Original article submitted September 5, 2016; revision submitted January 9,
2018 by Pleiades Publishing, Ltd.