ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 1, pp. 145–154. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © E. N. Cheremnykh, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 1, pp. 147–157.
A Priori Estimates of the Solution of the Problem
of the Unidirectional Thermogravitational Motion
of a Viscous Liquid in the Plane Channel
E. N. Cheremnykh
Institute of Numerical Simulation, Russian Academy of Sciences, Krasnoyarsk, Russia
Siberian Federal University, Krasnoyarsk, Russia
Received October 25, 2016; in ﬁnal form, April 28, 2017
Abstract—We consider an initial boundary-value problem describing the unidirectional motion of a
liquid in the Oberbeck–Boussinesq model in a plane channel with rigid immovable walls on which
the temperature distribution is given (or the upper wall is heat-insulated). For this problem, we
obtain a priori estimates, ﬁnd an exact stationary solution, and determine conditions under which
the solution converges to its stationary regime.
Keywords: initial boundary-value problem, inverse problem, a priori estimate.
1. INTRODUCTION AND STATEMENT OF THE PROBLEM
It is known that motion arises in a nonuniformly heated liquid. If the liquid has no internal and free
interfaces, then the main reason for the motion is that the colder liquid sinks in the gravitational ﬁeld.
The motion due to this is called thermal gravitational convection. For its description, the following
Oberbeck–Boussinesq equations are used:
+ v ·∇v +
∇p = νΔv − βθg, ∇·v =0, Θ
+ v ·∇θ = χΔθ. (1.1)
In system (1.1), v is the velocity vector, g is the acceleration of gravity, p is the deviation of pressure
from its hydrostatic value, θ is the deviation of temperature from its mean value, ρ is the mean liquid
density, ν is the liquid kinematic viscosity coeﬃcient, β is the volumetric expansion coeﬃcient, and χ is
the thermal diﬀusivity. The positive quantities ρ, ν, χ, β, and the vector g are assumed constant.
Suppose that the motion is plane and unidirectional; then v =(u(x, y, t), 0), g =(0,−g) and
Eqs. (1.1) can be signiﬁcantly simpliﬁed:
= ρgβθ, θ
For stationary plane ﬂows, system (1.2) was studied in . It was shown that the function u is a cubic
polynomial and the other unknown functions can be expressed as
θ = −Ax + T (y),p= −Aρgxy + q(y),
where T and q are polynomials of the ﬁfth and sixth degree, respectively. It was established in  that the
Birikh solution is an invariant solution of system (1.1) with respect to the group with the main operators
(A =const) admitted by this system. In what follows, the consistency of
the system of equations (1.2) will be established in general form.
The ﬁrst three equations in (1.2) imply that the functions p and θ depend linearly on the variable x.
u = u(y, t),θ= −A(y,t)x + T (y, t),p= −B(y, t)x + q(y, t). (1.3)