Abstract
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.
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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 free-stream 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. 1Footnote 1). A thermal boundary layer is assumed to develop at the interface between two fluids or a fluid and a gas (Marangoni boundary layer).
1 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:
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 \(u(x,y)\), \(v(x,y)\) of the flow at the point (x, y) and the temperature \(T(x,y)\) of the medium at this point. The constants \(\nu \), a, and c are physical parameters of the considered fluid, which are assumed to be prescribed. The constant \({{T}_{\infty }}\) is the free-stream temperature. The functions \(U(x)\), \(\widehat A(x)\), \({{v}_{0}}(x)\), and \({{T}_{w}}(x)\) are also assumed to be prescribed; they denote the free-stream velocity, surface tension at the boundary \(\{ y = 0\} \), 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 \(u(x,y)\) and \(v(x,y)\) in problem (1), (3) do not depend on the temperature \(T(x,y)\), 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 \(u\) and \({v}\) satisfy Eqs. (1) in the domain
are continuous in \(\overline Q ,\) and obey conditions (3); additionally, suppose that
where \({{C}_{1}}\), \({{C}_{2}}\), \({{C}_{3}}\), \({{C}_{4}}\), and y0 are positive constants. Then \(u\), \({v}\) are a unique solution of problem (1), (3) with the indicated properties.
2 EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A THERMAL BOUNDARY LAYER
In this section, taking into account the conditions imposed on the functions \(u(x,y)\) and \(v(x,y)\), we prove the existence and uniqueness of a classical solution \(T(x,y) \in {{C}^{2}}(Q) \cup C(\overline Q )\) 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 \(f(x,y)\).
Theorem 1. Suppose that functions \(u(x,y),\,\,v(x,y) \in C(\overline Q )\) satisfy system (1), conditions (3), and the solution uniqueness conditions. Assume that \(f(x,y) \in C({{Q}_{X}})\) is a locally Hölder continuous function in \(Q\) with a Hölder exponent \(0 < \gamma \leqslant 1\), while \({{T}_{w}}\) is a differentiable function on \([0,X]\) and its derivative is bounded on \([0,X]\). Additionally, suppose that
where \(\beta (y) = \mathop {\max }\limits_{0 \leqslant x \leqslant X} v(x,y)\), and the function \(g(y) = \mathop {\max }\limits_{0 \leqslant x \leqslant X} {\text{|}}\,f(x,y){\text{|}}\) satisfies
Then problem (2), (4) has a unique bounded classical solution \(T(x,y)\) in Q.
Proof. By using the change of variables
the domain \(Q = \{ (x,y) \in {{\mathbb{R}}^{2}}{\text{:}}\,\,0 < x < X,\,\,0 < y < \infty \} \) is transformed into the open rectangle \(D = \{ (x,\eta ) \in {{\mathbb{R}}^{2}}{\text{:}}\,\,0 < x < X,\,\,0 < \eta < 1\} \). To rewrite problem (2), (4) in the new variables (see (5)), we consider the partial derivatives involved in Eq. (2):
In what follows, we assume that \(u = u(x,y(\eta ))\), \(v = v(x,y(\eta ))\), and \(f = f(x,y(\eta ))\). 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,
Let us prove the uniqueness of a solution to problem (6), (7). Suppose that \({{T}_{1}}(x,\eta )\) and \({{T}_{2}}(x,\eta )\) are two solutions of problem (6), (7). Their difference is denoted by \(T(x,\eta ) = {{T}_{1}}(x,\eta ) - {{T}_{2}}(x,\eta )\).
For an arbitrary \(\varepsilon > 0\), we choose \(\delta = \delta (\varepsilon ) > 0\) such that \({\text{|}}T(x,\eta ){\text{|}} \leqslant \varepsilon \) for \(x \in [0,X]\) and \(\eta = 1 - \delta (\varepsilon )\). Then the function \(V(x,\eta ) = T(x,\eta ) - \varepsilon \) satisfies the equation \(L(V) = 0\) in D and the inequality \(V(x,\eta ) \leqslant 0\) for \(x \in [0,X]\), \(\eta = 0\), and \(\eta = 1 - \delta (\varepsilon )\). Let us show that \(V \leqslant 0\) in the rectangle
Let
where \(H(\eta )\) is assumed to be a positive function from \({{C}^{1}}([0,1 - \delta (\varepsilon )])\). Assume that \(H = H(\eta )\) and \(R = R(x,\eta )\). Consider the operator L(V). The partial derivatives involved in the equation \(L(v) = 0\) can be written as
Taking into account these expressions for the derivatives, the operator L(V) is given by
Transforming terms in this equation yields
Dividing both sides of the equation \(L(V) = 0\) by the function \(H(\eta )\), we derive the equation
where
The function \(H(\eta ) > 0\) is chosen so that, for some constant \({{F}_{0}} < 0\), the inequality \(F(x,\eta ) \leqslant {{F}_{0}}\) holds for all \(\eta \in [0,1 - \delta (\varepsilon )]\).
Next, we introduce the function
Then, taking into account the form of \(\Phi (x)\) and the conditions imposed on \(u(x,y(\eta ))\), we have in \({{D}_{\varepsilon }}\) the inequality
Since \({{F}_{0}} < 0\) and \(\Phi (x) \to + \infty \) as \(x \to 0 + 0\), it is possible to find a number \(0 < {{x}_{0}} < X\) such that \(L(\Phi ) \leqslant 0\) for all \(0 < x \leqslant {{x}_{0}}\) and \(0 < \eta < 1 - \delta (\varepsilon )\). Then, for any \(\lambda > 0,\) the linear operator L satisfies the inequality
Furthermore, for each \(\lambda > 0,\) there exists a number \(0 < {{x}_{1}}(\lambda ) < {{x}_{0}}\) such that \(\lambda \Phi (x) - R(x,\eta ) \geqslant 0\) for \(0 \leqslant \eta \leqslant 1 - \delta (\varepsilon )\), \(0 < x \leqslant {{x}_{1}}\). Additionally, \(\lambda \Phi (x) - R(x,\eta ) \geqslant 0\) for \(\eta = 0\), \(0 < x \leqslant X\) and for \(\eta = 1 - \delta (\varepsilon )\), \(0 < x \leqslant X\). By the maximum principle, \(\lambda \Phi (x) - R(x,\eta ) \geqslant 0\) for \(0 < x \leqslant {{x}_{0}}\), \(0 \leqslant \eta \leqslant 1 - \delta (\varepsilon )\). Since \(\lambda > 0\) is arbitrary, we obtain \(R(x,\eta ) \leqslant 0\) for \(0 \leqslant x \leqslant {{x}_{0}}\), \(0 \leqslant \eta \leqslant 1 - \delta (\varepsilon )\). Once again applying the maximum principle yields \(R(x,\eta ) \leqslant 0\) in \(\overline {{{D}_{\varepsilon }}} \). Therefore, \(T(x,\eta ) \leqslant \varepsilon \) in \(\overline {{{D}_{\varepsilon }}} \). In view of the symmetry between \({{T}_{1}}\) and \({{T}_{2}}\), we obtain \({\text{|}}T(x,\eta ){\text{|}} \leqslant \varepsilon \) in \(\overline {{{D}_{\varepsilon }}} \). Since \(\varepsilon > 0\) is arbitrary, it follows that \({{T}_{1}} = {{T}_{2}}\) in \(\overline D \). The uniqueness of a solution is established.
Now we prove the existence of a solution to problem (6), (7). For each \(0 < \delta < \frac{1}{4}\), Eq. (6) is considered in the rectangle
with boundary conditions
where \(T_{1}^{\delta }(\eta )\) is a continuous function on the interval \([\delta ,1 - \delta ]\) having the properties
The solution \({{T}^{\delta }}(x,\eta )\) of problem (6), (8) exists and, according to the maximum principle for nondegenerate parabolic equations (see, e.g., [4]),
where M does not depend on \(\delta \). By virtue of Schauder-type estimates [4], in any fixed rectangle \({{D}_{\delta }}\) for each \(0 < \delta < \frac{1}{4}\), the Hölder norms of the solution T δ and its derivatives \(\frac{{\partial {{T}^{\delta }}}}{{\partial \eta }}\), \(\frac{{{{\partial }^{2}}{{T}^{\delta }}}}{{\partial {{\eta }^{2}}}}\), and \(\frac{{\partial {{T}^{\delta }}}}{{\partial x}}\) satisfy the estimate
where, for an arbitrary Holder continuous function \(\varphi (x,\eta ),\) the norms \({\text{||}}\varphi {\text{|}}{{{\text{|}}}_{\alpha }}\) and \({\text{||}}\varphi {\text{|}}{{{\text{|}}}_{{2 + \alpha }}}\) in (10) are defi-ned as
(see, e.g., [4]). The function \(\psi \) in (10) denotes boundary conditions (8).
Relying on estimates (10), we extract a subsequence \({{T}^{{{{\delta }_{m}}}}}\), \(m = 1,2,...\), that converges uniformly as \(m \to \infty \), together with the derivatives involved in Eq. (6), in each closed domain lying strictly inside D. Passing to the limit as \(m \to \infty \) in the equation for \({{T}^{{{{\delta }_{m}}}}}\), we conclude that the limit function \(T(x,\eta )\) satisfies Eq. (6) in the rectangle D.
To prove the fulfillment of the condition \(T(x,0) = {{T}_{w}}(x)\), we estimate the difference
in the case of small \(\eta \). In the domain
the function \({{S}^{\delta }}(x,\eta )\) satisfies the equation
Suppose that
in \(D_{\delta }^{'}\).
Next, we introduce the auxiliary function Y(η) = \(K(1 - {{e}^{{ - N\eta }}})\). Here, the constant N > 0 is chosen using the condition \(c(\eta )N \geqslant {\text{|}}b(x,\eta ){\text{|}} + 1\). This is possible, since the coefficient \(v(x,y)\) is bounded for bounded y or, according to (5), for \(\eta \leqslant {{\eta }_{0}} < 1\). The constant K > 0 is chosen so that
where M is the same as in inequality (9). Given \(N > 0\), we calculate \(L(Y)\), namely,
Consider the function \(Y \pm {{S}^{\delta }}(x,\eta )\). Taking into account (11) and (12), \(L(Y \pm {{S}^{\delta }})\) in \(D_{\delta }^{'}\) is estimated as
Combining boundary conditions (8) with inequalities (9) and (13), we conclude that \(Y \pm {{S}^{\delta }} \geqslant 0\) on the boundary of \(D_{\delta }^{'}\) lying on the straight lines \(x = \delta \), \(\eta = \delta \), and \(\eta = \frac{1}{3}\). According to the maximum principle, it follows that \(Y \pm {{S}^{\delta }} \geqslant 0\) everywhere in \(D_{\delta }^{'}\). Therefore,
which is uniform with respect to \(\delta \) and x. Passing to the limit as \(\delta \to 0\) and \(\eta \to 0\) yields the condition \(T(x,0) = {{T}_{w}}(x)\).
To prove the fulfillment of the second boundary condition in (7), we estimate the difference
for small \(1 - \eta \). The equation \(L({{J}^{\delta }}) = f(x,\eta )\) satisfied by the function \({{J}^{\delta }}(x,\eta )\) is considered in the domain
Next, we introduce the new function
where
and the functions \(\beta (\tau )\), \(g(\tau )\) are the same as in Theorem 1. Integral (14) exists by virtue of the assumptions of the theorem and defines a positive function for \(\eta < 1\). The constant \({{K}_{1}} \geqslant 1\) is determined by the condition \(Z\left( {\frac{2}{3}} \right) \geqslant 2M\), where M is the constant from inequality (9). The function \(L(Z)\) is calculated as
Since
by the definition of the function \(\beta (\eta )\), it follows from (15) that \(L(Z) \leqslant - {{K}_{1}}g(\eta ).\) On the boundary of \(D_{\delta }^{{''}}\) lying on the straight lines \(x = \delta \), \(\eta = \frac{2}{3}\), and \(\eta = 1 - \delta \), it is true that \(Z(\eta ) \pm {{J}^{\delta }}(x,\eta ) \geqslant 0.\) Inside \(D_{\delta }^{{''}}\), by virtue of the choice of \({{K}_{1}} \geqslant 1\), we have
Therefore, the maximum principle implies that \(Z \pm {{J}^{\delta }} \geqslant 0\) everywhere in \(D_{\delta }^{{''}}\), or \({\text{|}}{{J}^{\delta }}(x,\eta ){\text{|}} \leqslant Z(\eta )\) in this domain. Passing to the limit as \(\delta \to 0\) in the last inequality, we conclude that \(T(x,\eta ) \to {{T}_{\infty }}\) uniformly as \(\eta \to 1\), as required.
Remark 1. In problem (2), (4), the free-stream temperature \({{T}_{\infty }}\) is assumed to be constant. It turns out that, for a solution \(T(x,y)\) of Eq. (2) that is bounded in the strip Q, a value other than a constant cannot be specified at infinity.
Proof. Assume that the solution \(T(x,y)\) of problem (2), (4) is such that \(\mathop {\lim }\limits_{y \to \infty } T(x,y) = {{T}_{\infty }}(x)\). In view of substitution (5), this is equivalent to the condition \(T(x,\eta ){{{\text{|}}}_{{\eta = 1}}}\) = T∞(x). For any \(\varepsilon > 0\), there exists \(\delta (\varepsilon )\) such that
Let \({{J}^{\delta }}(x,\eta ) = {{T}^{\delta }}(x,\eta ) - {{T}_{\infty }}(\delta )\). Equation (6) for the function \({{J}^{\delta }}(x,\eta )\) is considered in the domain
Consider the new function \(Z(\eta ) + \varepsilon \pm {{J}^{\delta }}(x,\eta )\), where \(Z(\eta )\) is defined by equality (14). Repeating word for word the entire argument used to prove the fulfillment of the second condition in (4), we obtain an estimate in \(\overline D _{\delta }^{{''}}\) that is uniform with respect to \(\delta \) and \(x\), namely,
Sending \(\delta \) to zero and noting that \(Z(\eta ) \to 0\) as \(\eta \to 1\), we have
Since \(\varepsilon \) is arbitrary, it follows that \(T(x,1) \equiv {{T}_{\infty }}(0)\), i.e., \({{T}_{\infty }}(x) \equiv {{T}_{\infty }}(0) = {\text{const}}\).
Remark 2. If only the first condition is set in (4), while the second condition is dropped, then a bounded solution satisfying Eq. (2) and the boundary condition \(T{{{\text{|}}}_{{y = 0}}} = {{T}_{0}}(x)\), where \({{T}_{0}}(x)\) is a continuous function defined in \(0 \leqslant x \leqslant X\), may not be unique.
Proof. Equation (2) with \(a = 1\), \(u = {{x}^{k}}\), \(v = \lambda y\), and \(f \equiv 0\) \((k \geqslant 1,{\text{ }}\lambda = {\text{const}} < 0)\) and the boundary condition \(T{{{\text{|}}}_{{y = 0}}} = 0\) are satisfied by the function
which is bounded and nonzero in the strip Q. It follows that, under the considered conditions, the solution of problem (2), (4) is not unique.
Change history
17 March 2023
An Erratum to this paper has been published: https://doi.org/10.1134/S1064562422340026
REFERENCES
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Butterworth-Heinemann, Oxford, 1987).
T. D. Dzhuraev, “On single-valued solution of the fundamental boundary-value problem in the theory of the thermal boundary layer,” J. Appl. Math. Mech. 38 (1), 152–158 (1974).
M. A. Kisatov, “System of equations for the Marangoni boundary layer in media with Ladyzhenskaya rheological law,” Dokl. Math. 103 (3), 130–132 (2021).
A. Friedman, Partial Differential Equations of Parabolic Type (Prentice Hall, Englewood Cliffs, N.J., 1964).
Funding
This work was supported by the Russian Science Foundation, project no. 20-11-20272.
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Translated by I. Ruzanova
The original online version of this article was revised: Due to a retrospective Open Access order.
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Kisatov, M.A., Samokhin, V.N. & Chechkin, G.A. On Thermal Boundary Layer in a Viscous Non-Newtonian Medium. Dokl. Math. 105, 23–27 (2022). https://doi.org/10.1134/S1064562422010070
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DOI: https://doi.org/10.1134/S1064562422010070