On Thermal Boundary Layer in a Viscous Non-Newtonian Medium

⎯ An existence and uniqueness theorem for a classical solution to the system of equations describing thermal boundary layers in viscous media with the Ladyzhenskaya rheological law is generalized.


INTRODUCTION
If the surface of a plate and the incoming flow have different temperatures, then, near the plate surface, a thermal boundary layer is formed in which the temperature varies from its value at the wall to the freestream temperature at the edge of the thermal boundary layer. Moreover, heat transfer occurs between the plate surface and the fluid flow (see [1]).
Thus, near the surface of a body placed in fluid flow, dynamic and thermal boundary layers are formed, which represent the boundaries of the corresponding perturbation fronts separating the perturbed and unperturbed flows.
A system of equations governing a thermal boundary layer developing in a plane-parallel steady forced convective flow of a Newtonian fluid was considered in [2].
In this paper, we study a system of equations describing thermal boundary layers in plane-parallel fluid flows with Ladyzhenskaya rheology (see Fig. 1 1 ). A thermal boundary layer is assumed to develop at the interface between two fluids or a fluid and a gas (Marangoni boundary layer).

FORMULATION OF THE PROBLEM
Consider the system of equations governing the thermal boundary layer developing in a plane-parallel fluid flow with Ladyzhenskaya rheological law: (1) (2) Assume that system (1), (2) is defined in the domain System (1), (2) is supplemented with boundary conditions of the form The unknowns in problem (1)-(4) are the streamwise and spanwise velocities , of the flow at the point (x, y) and the temperature of the medium at this point. The constants , a, and c are physical parameters of the considered fluid, which are assumed to be prescribed. The constant is the freestream temperature. The functions , , , and are also assumed to be prescribed; they denote the free-stream velocity, surface tension at the boundary , the blowing (suction) rate at the point x on the lower wall of the domain, and the wall temperature at the point x, respectively.
Since the functions and in problem (1), (3) do not depend on the temperature , problem (1), (3) can be solved separately.
A uniqueness theorem for the solution of problem (1), (3) was proved in [3]. Assume that the functions and satisfy Eqs.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A THERMAL BOUNDARY LAYER
In this section, taking into account the conditions imposed on the functions and , we prove the existence and uniqueness of a classical solution to problem (2), (4). Without loss of generality, the coefficient a is assumed to be equal to 1. For convenience, the right-hand side of Eq. (2) is denoted by . Theorem 1.
Suppose that functions satisfy system (1), conditions (3), and the solution uniqueness conditions. Assume that is a locally Hölder continuous function in with a Hölder exponent , while is a dif-  (2), (4) in the new variables (see (5)), we consider the partial derivatives involved in Eq. (2): In what follows, we assume that , , and . Taking into account the above expressions for the derivatives and the change of variables (5), Eq. (2) becomes Grouping terms in this equation, we finally represent the problem in terms of variables (5): Here, (1 ) = .  (1  ) ). c bx Let us prove the uniqueness of a solution to problem (6), (7). Suppose that and are two solutions of problem (6), (7  where is a continuous function on the interval having the properties The solution of problem (6), (8) exists and, according to the maximum principle for nondegenerate parabolic equations (see, e.g., [4]), where, for an arbitrary Holder continuous function the norms and in (10) are defined as (see, e.g., [4]). The function in (10) denotes boundary conditions (8).
Relying on estimates (10), we extract a subsequence , , that converges uniformly as , together with the derivatives involved in Eq. (6), in each closed domain lying strictly inside D. Passing to the limit as in the equation for , we conclude that the limit function satisfies Eq. (6) in the rectangle D.
To prove the fulfillment of the condition , we estimate the difference in the case of small . In the domain the function satisfies the equation

Suppose that
in . Next, we introduce the auxiliary function Y(η) = . Here, the constant N > 0 is chosen using the condition . This is possible, since the coefficient is bounded for bounded y or, according to (5), for . The constant K > 0 is chosen so that (11) where M is the same as in inequality (9) Combining boundary conditions (8) with inequalities (9) and (13), we conclude that on the boundary of lying on the straight lines , , and . According to the maximum principle, it follows that everywhere in . Therefore, which is uniform with respect to and x. Passing to the limit as and yields the condition .
To prove the fulfillment of the second boundary condition in (7)