Boundary-layer theory

An asymptotic approximation of the solution of boundary value problems for differential equations containing a small parameter in front of the highest derivative (singular problems) in subregions where there is a substantial effect from the terms containing the highest derivatives on the solution. The boundary-layer phenomenon arises in narrow zones near the parts of the boundary on which there is a difference in the numbers of boundary conditions for the initial problem and the degenerate one (with the small parameter taking the value zero), as well as near the surfaces of discontinuity of the solution of the degenerate problem.

The solution of the singular problem may be represented as the sum of two expansions. The outer expansion is determined by the method of the small parameter with the part $ B _ {0} $ of the boundary conditions $ B = B _ {0} + B _ {1} $. The inner expansion decreases rapidly outside the boundary layer and is usually sought for as polynomials in powers of $ \epsilon $. To determine these, the differential equations are transformed by variables that depend on $ \epsilon $ and that stretch the subregions of the boundary layer. The equations of the boundary layer are derived by equating to zero the coefficients at various powers of $ \epsilon $ after substituting the polynomials into the transformed equations. To these one adds conditions $ B _ {1} $. The error in the outer expansion, after it has been found, indicates the necessary change of variables. In solving complicated applied problems, the necessary change of variables can be clarified on the basis of physical estimates for the terms in the initial equations and the simplifications corresponding to them. This change should eliminate the highest derivatives in front of $ \epsilon $. To solve the problem it is necessary to determine where the boundary layers lie and how the conditions $ B $ are to be separated into $ B _ {0} $ and $ B _ {1} $. A notable feature of this solution is that hyperbolic equations for the outer expansions and parabolic equations for the inner expansions may correspond to initially elliptic equations.

In the theory of systems of ordinary differential equations

$$ \tag{1 } x _ {t} = f(x, y, t),\ \epsilon y _ {t} = \ g(x, y, t),\ 0 \leq t \leq T, $$

where $ x $, $ f $ are $ m $- dimensional and $ y $, $ g $ are $ n $- dimensional vector functions, an existence and uniqueness theorem for the solution of the Cauchy problem under the conditions $ x(0, \epsilon ) = x ^ {0} $ and $ y(0, \epsilon ) = y ^ {0} $ and under certain properties of $ f $ and $ g $ has been proved. One has also derived properties of the solution as $ \epsilon \rightarrow 0 $( see [1]). In the case of a boundary value problem for (1) under the conditions

$$ x (0, \epsilon ) = x ^ {0} ,\ y _ {1} (0, \epsilon ) = y _ {1} ^ {0} , \ y _ {2} (T, \epsilon ) = y _ {2} ^ {0} , $$

where the sum of the numbers of components of the vectors $ y _ {1} $ and $ y _ {2} $ is $ n $, there exist, in general, boundary layers in neighbourhoods of the ends of the segment $ [0, T] $. An algorithm has been constructed for finding asymptotics of the solution for this problem. Under certain properties of $ f $ and $ g $ it has been shown that a solution exists and is unique, and it has been estimated (see [3]). If a solution to the limit equation $ g = 0 $ is not unique with respect to $ y $, one can construct an internal boundary layer (in a neighbourhood of $ \tau $, $ 0 < \tau < T $) that separates regions with different solutions of the limit equation. An algorithm has been constructed for a particular type of integro-differential equation giving an asymptotic expansion in $ \epsilon $ for the problem with initial conditions, and some features of the behaviour of the solution have been examined.

In the case of linear ordinary differential equations $ (L + \epsilon M)x = f(t) $, $ 0 \leq t \leq 1 $, where $ L $ and $ M $ are differential operators, with boundary conditions $ B _ {0} + B _ {1} $, one can distinguish the class of problems the solutions of which contain boundary layers, and the concept of regular degeneracy (the solution to the limit equation enables one to satisfy the conditions $ B _ {0} $, while the asymptotic solution for the boundary layer enables one to satisfy $ B _ {1} $) can been introduced (see ). An iterative process has been constructed for the asymptotic representation of the solution, and estimates have been given for the residual terms in the expansions.

It has been shown (see [5]) in the theory of boundary layers of a general non-linear second-order ordinary differential equation, subject to certain assumptions, that the solution of the first boundary value problem is made up of an external solution, a boundary layer and a residual term that, together with its first-order derivative, is of order $ \epsilon $ on the segment.

Studies have been made on the behaviour of solutions of boundary value problems of basic types for a linear partial differential equation of the form

$$ \epsilon \Delta u + A(x, y) u _ {x} + B(x, y) u _ {y} + C(x, y) u = \ f(x, y), $$

where $ \Delta $ is the Laplace operator, in a region $ D $ with boundary $ S $. Conditions on the functions $ A $, $ B $, $ C $, and $ f $, the boundary $ S $, and the functions $ a $ and $ \phi $ of the points $ P $ on $ S $ appearing in the boundary condition $ u _ {n} + a(P) = \phi (P) $ have been given such that $ u $ in $ D \cup S $ tends uniformly to the solution of the limit equation with this boundary condition on a certain part of $ S $( absence of a boundary layer) [6].

For an elliptic second-order equation in a region $ D $ with boundary $ S $, using the example of two independent variables

$$ \epsilon [ a(x, y) u _ {xx} + 2b(x, y) u _ {xy} + c(x, y) u _ {yy} + $$

$$ + {} d(x, y) u _ {x} + e(x, y) u _ {y} + g(x, y) u] + u _ {x} - $$

$$ - h(x, y) u = f(x, y),\ h \geq \alpha ^ {2} > 0 , $$

iterative processes have been constructed solving the problem with the condition $ u = 0 $ on $ S $, theorems have been proved on the structure of the expansion of $ u $ with respect to $ \epsilon $ and estimates have been made of the residual term in this expansion . Similar results have been obtained for equations of higher orders.

A method has been devised [7] for combining asymptotic expansions for the equation

$$ \epsilon \Delta u - a(x, y) u _ {y} = f(x, y) $$

in a rectangle with $ u $ given on the boundary.

Research on boundary-layer theory for non-linear partial differential equations is related mainly to aerohydrodynamics and is based on the Navier–Stokes equations or generalizations of them. Practical requirements have led to the development of the mathematical theory and to methods of handling various problems. Below only laminar flows are considered (see [8]–[10]).

The Navier–Stokes equations in the case of the hydrodynamics of planar ( $ k = 0 $) and axi-symmetric ( $ k = 1 $) flows of an incompressible fluid with constant viscosity coefficient $ \nu $ are:

$$ \tag{2 } \left . \begin{array}{c} ( \eta ^ {k} u) _ \xi + ( \eta ^ {k} v) _ \eta = 0,\ \ u _ \tau + uu _ \xi + vu _ \eta = \epsilon ^ {2} \Delta u - p _ \xi , \\ v _ \tau + uv _ \xi + vv _ \eta = \ \epsilon ^ {2} \Delta v - p _ \eta , \end{array} \right \} $$

where $ \epsilon = \mathop{\rm Re} ^ {-1/2} $, $ \mathop{\rm Re} = wX/ \nu $ is the Reynolds number, which is represented in terms of the characteristic values of the velocity $ w $ and the linear dimension $ X $. The solutions are defined by the boundary conditions on the closed boundary $ S $ of a region $ D $, where on the solid contour $ \Gamma $ one has the conditions $ \overline{u}\; = 0 $, $ \overline{v}\; = v _ {0} ( \xi , H( \xi )) $ for $ \eta = H( \xi ) $, where $ \overline{u}\; $ and $ \overline{v}\; $ are the tangential and normal components to $ \Gamma $ of the vector $ ( u , v ) $. The initial values of $ u $, $ v $ and $ p $ are given on $ D \cup S $.

For $ \epsilon $ small, in a first approximation the asymptotic solution is composed of the solution of (2) for $ \epsilon = 0 $ with some of the conditions on $ S $( only the condition $ \overline{v}\; = v _ {0} $ is imposed on $ \Gamma $) and the solution of the boundary-layer equations. The equations for the dynamic boundary layer are derived on the assumption that the conditional thickness $ \delta $ of the boundary layer and the value of $ v $ have orders $ \delta \sim X \epsilon $, $ v \sim w \epsilon $, and that the terms on the left-hand sides of the latter equations in (2) are of the order of the terms containing $ \epsilon ^ {2} $. Introduction of the variables $ t = \tau , x = \xi , y = \eta / \epsilon $ and $ V = v/ \epsilon $ leads, as $ \epsilon \rightarrow 0 $, to the Prandtl equations (cf. Prandtl equation):

$$ \tag{3 } (r ^ {k} u) _ {x} + (r ^ {k} V) _ {y} = 0,\ \ u _ {t} + uu _ {x} + Vu _ {y} = u _ {yy} - p _ {x,} $$

$$ p _ {y} = 0,\ 0 \leq t \leq T,\ 0 \leq x \leq X _ {0} ,\ 0 \leq y < \infty , $$

with the conditions

$$ u \mid _ {t=0 } = u _ {0} (x, y), \ v \mid _ {t=0 } = v _ {0} (x, y), \ u \mid _ {y=0 } = 0, $$

$$ V \mid _ {y=0 } = v _ {0} (t, x), $$

$$ u \rightarrow W(t, x) ,\ p _ {x} = -WW _ {x} - W _ {t} ,\ \textrm{ as } y \rightarrow \infty ,\ u \mid _ {x=0 } = 0, $$

where $ r $ is the distance from the symmetry axis for $ k = 1 $ and $ W(x) $ is a known function. These equations and conditions apply for any curvilinear contour with radius of curvature much larger than $ \delta $. In the latter case, $ x $ and $ y $ are the coordinates along the contour and along the normal to it.

If $ W $ is constant, the problem reduces to a boundary value problem for an ordinary differential equation. There are also other classes of analogous solutions.

Conditions under which solutions of boundary-layer problems exist are known; one has investigated the problem of uniqueness and stability of the solutions as well as how they result from the solutions for stationary cases [11]. Solutions have been constructed by the method of straight lines, and it has been shown that they converge.

Boundary-layer equations for a compressible liquid can be derived from the equations for the flow of a viscous and heat-conducting gas; they are much more complicated than (3). Their number is also larger. There is an integral transformation that simplifies these equations in the general case and reduces them to (3) when the Prandtl number $ \mathop{\rm Pr} = c _ {p} / K = 1 $, where $ c _ {p} $ is the heat capacity of the gas at constant pressure and $ K $ is the coefficient of heat conductivity [12]. Several modifications of the transformation exist. In the general case, the boundary-layer equations describe so-called natural convective flows. If $ \nu $ is independent of the temperature and if the Archimedean force is negligible, then the energy equation splits off from the system of boundary layer equations and one speaks of forced convective flow. The energy equation determines the thermal boundary layer, whose thickness differs from $ \delta $.

Boundary layers also arise in zones separating flows with different characteristic velocities. Shock waves are also boundary layers.

A distinct class of two-dimensional boundary-layer problems is associated with flows in rotating axi-symmetric plates and bodies.

Not only have methods been developed for solving non-stationary problems, but also problems in which $ W $ is periodic, when there is stepwise motion from a state of rest, problems with accelerated motion, for the boundary layer behind a shock wave, and problems with a variable temperature over the surface of the solid around which the flow takes place have been solved.

In the boundary-layer theory for three-dimensional flows, methods for obtaining a solution have been developed and cases in which the equations simplify have been studied. The boundary-layer equations for a sliding cylindrical wing of infinite span are analogous to the equations for a two-dimensional boundary layer. Approximate solutions have been obtained for the problem of a boundary layer on a rotating cylindrical propeller blade and on a rotating cylinder in a skew flow, as well as for the boundary-layer problem near the line of intersection between two planes.

These researches on boundary layers in aerohydrodynamics relate to a first approximation in boundary-layer theory. Higher approximations enable one to examine the interactions of boundary layers with the external flow, and to make calculations for moderate values of $ R $[13].

The stability of boundary layers enables one to determine the limits of applicability of the theory. There are studies [14] based on methods of small perturbations with periodic and local initial perturbations. In the case of two-dimensional flows, the analysis of three-dimensional perturbations reduces in linear approximation, on the basis of Squire's theorem, to a two-dimensional analysis with an altered value of $ \nu $. Non-linear analysis applied to stability loss shows that longitudinal vortices occur.

Physical generalizations of problems in the theory of boundary layers (see [15]–[17]) are related to research on multi-phase flows, to the use of real equations of state and transfer coefficients (a complication of the equations), to the study of non-equilibrium flows with diffusion (an extension of the system of equations, which then becomes of parabolic-hyperbolic type), to taking into account ablation of the surface around which the flow takes place (a complication of the boundary conditions and it becomes necessary to consider the thermal conduction in the solid), and also to considering radiation transport (integro-differential equations).

A further development of boundary-layer theory in aerohydrodynamics has been obtained in the study of flows that do not satisfy Prandtl's assumptions, where solutions are obtained as multi-layer asymptotic expansions. The causes of the complexity in the structure of the solution are that there are additional small parameters in the boundary conditions (for example, because the radius of curvature of the contour around which the flow takes place is small), there may be singular points, lines or surfaces in the first approximation of the theory, and bifurcation of the solution is possible.

This class also contains flows around points of separation or connections of the boundary layers to the contour around which the flow takes place and around the points of incidence of shock waves on the boundary layers. The characteristic solutions have ([18]–[20]) a three-layer structure. The external solution defines the potential flow perturbed by the boundary layer and is described by equations in perturbations. The middle layer of order $ X \epsilon $ can be described by the equations for non-viscous vortex flows with $ p _ {y} = 0 $, and it receives gas from the main part of the preceding boundary layer.

The equations for the third (thinnest) layer nearest to the wall can be derived on the assumption that its length and thickness are, correspondingly, of order $ X \epsilon ^ {3/4} $ and $ X \epsilon ^ {5/4} $. The following variables are introduced in the stationary case:

$$ x = \xi \epsilon ^ {-3/4 } ,\ y = \eta \epsilon ^ {-5/4 } , $$

$$ U = u \epsilon ^ {-1/4 } ,\ V = v \epsilon ^ {-3/4 } ,\ P = p \epsilon ^ {-1/2 } . $$

This leads, as $ \epsilon \rightarrow 0 $, again to (3) in which $ u $ is replaced by $ U $ and $ p $ by $ P $. This structure for the solution of a problem is characteristic for a wide class of flows with small perturbations.

Many studies have been made on the dynamics of flows for small $ \epsilon $, where the pressure in the outer supersonic flow varies considerably over short distances ( $ M > 1 $, $ M = w/c $, where $ w $ is the velocity of the gas and $ c $ is the velocity of sound). This includes problems on calculating flows around contours of large local curvature and flow attachment to the surface of the body. In these cases $ p _ {y} \neq 0 $ in the middle layer of the three-layer scheme for the solution.

The class of problems in which the perturbed solution occupies a finite region has also been studied. These solutions occur when there is moderate or strong interaction of the boundary layer with an external hypersonic flow ( $ M \rightarrow \infty $) or in supersonic flow around a body of finite length in which there are intense gas injections through the surface ( $ v _ {0} > 0 $). In these cases, the pressure on the outer boundary of the boundary layer is determined from the solution to the complete problem. The perturbation from the trailing edge of the body propagates upstream, which is caused by the non-uniqueness of the solution in a neighbourhood of the leading edge.

References

[1]	A.N. Tikhonov, "Systems of differential equations containing small parameters in front of the derivatives" Mat. Sb. , 31 : 3 (1952) pp. 575–586 (In Russian)
[2]	W. Wasov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)
[3]	A.B. Vasil'eva, V.F. Butuzov, "Asymptotic expansions of solutions of singularly perturbed equations" , Moscow (1973) (In Russian)
[4a]	M.I. Vishik, L.A. Lyusternik, "Regular degeneracy and boundary layers for linear differential equations with a small parameter" Uspekhi Mat. Nauk , 12 : 5 (1957) pp. 3–122 (In Russian)
[4b]	M.I. Vishik, L.A. Lyusternik, "Solutions to some problems on perturbations in the case of matrices and non-self-adjoint differential equations" Uspekhi Mat. Nauk , 15 : 3 (1960) pp. 3–80 (In Russian)
[5]	J.D. Cole, "Perturbation methods in applied mathematics" , Blaisdell (1968)
[6]	O.A. Oleinik, "on equations of elliptic type with a small parameter in front of the highest derivatives" Mat. Sb. , 31 : 1 (1952) pp. 104–117 (In Russian)
[7]	A.M. Il'in, E.F. Lelikova, "The method of matched asymptotic expansions for the equation in a rectangle" Mat. Sb. , 96 : 4 (1975) pp. 568–583 (In Russian)
[8]	N.E. Kochin, I.A. Kibel', N.V. Roze, "Theoretical hydrodynamics" , 2 , Interscience (1964) (Translated from Russian)
[9]	L.G. Loitsyanskii, "Laminar boundary layers" , Moscow (1962) (In Russian)
[10]	H. von Schlichting, "Grenzschicht-Theorie" , Karlsruhe (1951)
[11]	O.A. Oleinik, "Mathematical problems of boundary-layer theory" Uspekhi Mat. Nauk , 23 : 3 (1968) pp. 3–65 (In Russian)
[12]	A.A. Dorodnitsyn, "Boundary layers in an incompressible gas" Prikl. Mat. i Mekh. , 6 : 6 (1942) pp. 449–486 (In Russian)
[13]	M. van Dyke, "Perturbation methods in fluid mechanics" , Parabolic Press (1975)
[14]	R. Betchov, V.O. Criminale, "Stability of parallel flows" , Acad. Press (1967)
[15]	S. Soo, "Fluid dynamics of multiphase systems" , Blaisdell (1971)
[16]	W.H. Dorrance, "Viscous hypersonic flow" , McGraw-Hill (1962)
[17]	W.M. Kays, "Convective heat and mass transfer" , McGraw-Hill (1971)
[18]	V.Ya. Neiland, "On the theory of separation of laminar boundary layers in supersonic flow" Izv. Akad. Nauk SSSR Mekh. Zhidk. i Gaza : 4 (1969) pp. 53–57 (In Russian)
[19]	V.Ya. Neiland, "Asymptotic problems in the theory of viscous flows" Trudy TsAGI : 1529 (1974) (In Russian)
[20]	K. Stewartson, "Multistructured boundary layers on flat plates and related bodies" Adv. App. Mech. , 14 (1974) pp. 145–239

Comments

The article above presents a one-sided view of the subject and is in many aspects not up-to-date. Furthermore, very substantial contributions in the Western literature are neglected. Mathematically boundary layers arise in problems of singular perturbations and are treated by methods of asymptotic analysis. In linear and non-linear, ordinary and partial, differential equations much more has been accomplished than described above in the first part. Overviews that are partially complementary can be found in [a1]–[a3].

From the point of view of applications boundary-layer theory is described above as a branch of fluid- and gasdynamics, where it originated. However, boundary layers and transition layers also occur in many other fields, such as for example combustion, geophysics, free boundary problems, or epidemology. Samples and further literature can be found in [a4], [a5].

Systems of (ordinary) differential equations of the type (1) above are often referred to as singular perturbations (cf. also the editorial comments to Perturbation theory).

For an idea of the role of singular perturbations, boundary layers and multiple time scales in control and system theory, cf. [a6].

References

[a1]	J.L. Lions, "Perturbation singulières dans les problèmes aux limits et en contrôle optimal" , Lect. notes in math. , 323 , Springer (1973)
[a2]	W. Eckhaus, "Asymptotic analysis of singular perturbations" , North-Holland (1979)
[a3]	K.W. Chang, F.A. Howes, "Nonlinear singular perturbation phenomena: theory and application" , Springer (1984)
[a4]	C.M. Brauner (ed.) B. Gay (ed.) J. Mathieu (ed.) , Singular perturbations and boundary layer theory , Lect. notes in math. , 594 , Springer (1977)
[a5]	F. Verhulst (ed.) , Asymptotic analysis II , Lect. notes in math. , 985 , Springer (1983)
[a6]	P. Kokotovic (ed.) , Singular perturbations and time scales in modelling and control of dynamical systems , Univ. Illinois at Urbana-Champaign (1980)