Steady Hydrodynamic Boundary Layer Equations for a Newtonian Fluid

9.3.1. Steady Hydrodynamic Boundary Layer Equations for a Newtonian Fluid

∂y ∂x∂y

∂x ∂y 2 ∂y 3

This is an equation of a steady laminar hydrodynamic boundary layer on a flat plate; w is the stream function, x and y are the longitudinal and normal coordinates, respectively, and ν is the kinematic This is an equation of a steady laminar hydrodynamic boundary layer on a flat plate; w is the stream function, x and y are the longitudinal and normal coordinates, respectively, and ν is the kinematic

Preliminary remarks. The system of hydrodynamic boundary layer equations

where u 1 and u 2 are the longitudinal and normal components of the fluid velocity, respectively, is reduced to the equation in

question by the introduction of a stream function w such that u 1 = ∂w and u 2 =− ∂y ∂w ∂x .

References : H. Schlichting (1981), L. G. Loitsyanskiy (1996).

1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the functions

w 1 = w(x, y + ϕ(x)), w 2 = C 1 w(C 2 x+C 3 , C 1 C 2 y+C 4 )+ C 5 ,

where ϕ(x) is an arbitrary function and C 1 , ...,C 5 are arbitrary constants, are also solutions of the equation. ✳✂✴

References : Yu. N. Pavlovskii (1961), L. V. Ovsiannikov (1982).

2 ◦ . Degenerate solutions (linear and quadratic in y):

w(x, y) = C 1 y + ϕ(x),

w(x, y) = C 1 y + ϕ(x)y +

ϕ ( x) + C 2 ,

where C 1 and C 2 are arbitrary constants and ϕ(x) is an arbitrary function. These solutions are independent of ✳✂✴ ν and correspond to inviscid fluid flows.

Reference : D. Zwillinger (1989, pp. 396–397).

3 ◦ . Solutions involving arbitrary functions:

6 νx + C 1 C 2

w(x, y) =

y + ϕ(x)

[ y + ϕ(x)] 2

w(x, y) = ϕ(x) exp(−C 1 y) + νC 1 x+C 2 ,

w(x, y) = C 1 exp − C 2 y−C 2 ϕ(x) + C 3 y+C 3 ϕ(x) + νC 2 x+C 4 ,

w(x, y) = 6νC 1 x tanh ξ+C 2 ,

w(x, y) = −6νC 1 x /3 tan ξ+C 2 ,

where C 1 , ...,C 4 are arbitrary constants and ϕ(x) is an arbitrary function.

Special case 1. For C 1 = p k/ν and ϕ(x) = − √ kν x, the second solution becomes √

w= kν x

It describes a fluid flow induced by the motion of surface particles at y = 0 with a velocity of u 1 | y=0 = kx. The fluid velocity components in this case meet the boundary conditions

References : N. V. Ignatovich (1993), A. D. Polyanin (2001 a).

4 ◦ . Table 5 lists invariant solutions to the hydrodynamic boundary layer equation that are obtained with the classical group-theoretic methods. Solution 1 is expressed in additive separable form, solution 2 is in multiplicative separable form, solution 3 is self-similar, and solution 4 is generalized self-similar. Solution 5 degenerates at a = 0 into a self-similar solution (see solution 3 with λ = −1).

Invariant solutions to the hydrodynamic boundary layer equation (the additive constant is omitted) No.

Solution structure

Function

F or equation for F Remarks

C 1 exp(− λy) + C 2 y if λ ≠ 0,

2 w = F (x)y −1

F (x) = 6νx + C 1 —

3 w=x λ+1

F (z), z = x λ y

(2 λ + 1)(F z ′ ) 2 −( λ + 1)F F zz ′′ = νF zzz ′′′ λ is any

4 w=e λx

F (z), z = e λx y

2 λ(F z ′ ) 2 − λF F zz ′′ = νF zzz ′′′ λ is any

5 w = F (z) + a ln |x|, z = y/x −( F z ′ ) 2 − aF zz ′′ = νF zzz ′′′

a is any

Equations 3–5 for

F are autonomous and generalized homogeneous; hence, their order can be reduced by two. ✵✂✶

References : Yu. N. Pavlovskii (1961), H. Schlichting (1981), L. G. Loitsyanskiy (1996), G. I. Burde (1996).

Special case 2. The Blasius problem on a translational fluid flow with a velocity U i past a flat plate is characterized by the boundary conditions

∂ y w=U i at x = 0. The form of the solution to this problem (in the domain x ≥ 0, y ≥ 0) is given in the third row of Table 5 with λ = −1/2. The

∂ x w=∂ y w=0 at y = 0,

∂ y w→U i as y → ∞,

boundary conditions for

F (z) are as follows:

F=F z ′ =0 at z = 0,

F z ′ →U i as z → ∞.

For details, see Blasius (1908), Schlichting (1981), and Loitsyanskiy (1996). Special case 3. The Schlichting problem on the axisymmetric flow of a plane laminar jet out of a thin slit is characterized

by the boundary conditions

∂ y w → 0 as y → ∞, which are supplemented with the integral condition of conservation of momentum

∂ x w=∂ yy w=0 at y = 0,

2 0 ∂ y w) dy = A

A = const).

The form of the solution to this problem (in the domain x ≥ 0, y ≥ 0) is given in the third row of Table 5 with λ = −2/3. On integrating the ordinary differential equation for

F with appropriate boundary conditions,

F=F zz ′′ =0 at z = 0,

F z ′ → 0 as z → ∞,

and the integral condition

0 F z ′ ) 2 = A,

we finally obtain

k=3 2 /3 . For details, see the book by Schlichting (1981) and Loitsyanskiy (1996).

1 k(A/ν w(x, y) = k(Aνx) 2 ξ, ξ=

1 /3 tanh

6 ) 1 /3 yx −2 /3 ,

Special case 4. Note two cases where the equation specified in row 3 of Table 5 can be integrated. For λ = −1, the solution can be obtained in parametric form:

There is a solution 2 F = 6νz −1 . For λ=− 3 , the twofold integration yields the Riccati equation

′ + 1 νF 2 z 6 F + C 1 z+C 2 =0 .

Bessel functions or order 1 /3.

5 ◦ . Generalized separable solution linear in x:

(1) where the functions f = f (y) and g = g(y) are determined by the autonomous system of ordinary

w(x, y) = xf (y) + g(y),

differential equations

( f y ′ ) 2 − ff yy ′′ = νf yyy ′′′ ,

(3) The order of equation (2) can be reduced by two. Suppose a solution of equation (2) is known.

f y ′ g ′ y − fg ′′ yy = νg ′′′ yyy .

Equation (3) is linear in g and has two linearly independent solutions:

g 1 = 1,

g 2 = f (y).

The second particular solution follows from the comparison of (2) and (3). The general solution of equation (2) can be written out in the form

g(y) = C 1 + C 2 f+C 3 f ψ dy −

f ψ dy ,

f = f (y), ψ=

exp −

f dy ;

see Polyanin and Zaitsev (2003). It is not difficult to verify that equation (2) has the following particular solutions:

f (y) = 6ν(y + C) −1 ,

C and λ are arbitrary constants. The first solution in (5) leads, taking into account (1) and (4), to the first solution of Item 3 ◦ with ϕ(x) = const. Substituting the second expression of (5) into (1) and (4), one may obtain another solution. ✷✂✸

Reference : A. D. Polyanin (2001 a).

6 ◦ . Generalized separable solution (special case of solution 3 in Item 3 ◦ ):

w(x, y) = (a + be − λy ) z(x) + cy,

where

a, b, c, and λ are arbitrary constants, and the function z = z(x) is defined implicitly by

c ln |z| + aλz = νλ 2 x.

Reference : N. V. Ignatovich (1993), B. I. Burde (1996).

7 ◦ . Below are two transformations that reduce the order of the boundary layer equation.

7.1. The von Mises transformation

∂w

ξ = x, η = w, U (ξ, η) =

where w = w(x, y),

∂y

leads to a nonlinear heat equation of the form 1.10.1.1:

7.2. The Crocco transformation

∂w

where w = w(x, y), leads to the second-order nonlinear equation

Reference : L. G. Loitsyanskiy (1996).

Invariant solutions to the hydrodynamic boundary layer equation with pressure gradient (

a, k, m, and β are arbitrary constants) No. Function f (x) Form of solution w = w(x, y)

Function u or equation for u

1 f (x) = 0

See equation 9.3.1.1

See equation 9.3.1.1

f (x) = ax m−1 m+1 ( u ′ ) 2 − 2 m+3

4 uu zz w=x ′′ u(z), z = x y = νu ′′′ zzz + a

m+3

3 f (x) = ae βx

w=e 1 4 βx u(z), z = e 4 1 βx

w = kx + u(y)

u(y) = − a

6 ν y 3 + C 2 y 2 + C 1 y if k=0

5 f (x) = ax −3

w = k ln |x| + u(z), z = y/x

−( u ′ z ) 2 − ku ′′ zz = νu ′′′ zzz + a

8 ◦ . Conservation law:

+ D y − w x w y − νw yy = 0,

where D x = ∂ and D y = ∂x ∂ ∂y . ∂w ∂ 2 w

∂w ∂ 2 w

+ f (x).

∂y ∂x∂y

∂x ∂y 2 ∂y 3

This is a hydrodynamic boundary layer equation with pressure gradient. The formula f (x) = U U x ′ holds true; U = U (x) is the fluid velocity in the stream core* at the interface between the core and the boundary layer.

1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the functions

w 1 = w(x, y + ϕ(x)) + C, w 2 =− w(x, −y + ϕ(x)) + C,

✹✂✺ where ϕ(x) is an arbitrary function and C is an arbitrary constant, are also solutions of the equation.

References : Yu. N. Pavlovskii (1961), L. V. Ovsiannikov (1982).

2 ◦ . Degenerate solutions (linear and quadratic in y) for arbitrary f (x):

w(x, y) = ✻ y 2 f (x) dx + C 1 + ϕ(x),

w(x, y) = C 1 y 2 + ϕ(x)y +

ϕ 2 ( x) − 2

f (x) dx + C 2 ,

where ϕ(x) is an arbitrary function, and C 1 and C 2 are arbitrary constants. These solutions are independent of ✹✂✺ ν and correspond to inviscid fluid flows.

Reference : A. D. Polyanin (2001 a).

3 ◦ . Table 6 lists invariant solutions to the hydrodynamic boundary layer equation with pressure gradient that are obtained with the classical group-theoretic methods. Note that the Falkner–Skan problem (see Falkner and Skan, 1931) on a symmetric fluid flow past a wedge is described by the equation specified in the second row of Table 6. The case m=1 corresponds to a fluid flow near a stagnation point, and the case m = 0 corresponds to a symmetric

✹✂✺ flow past a wedge with an angle of α= 2 3 π.

References : Yu. N. Pavlovskii (1961), H. Schlichting (1981), L. G. Loitsyanskiy (1996), G. I. Burde (1996). * The hydrodynamic problem on the flow of an ideal (inviscid) fluid about the body is solved in the stream core.

w(x, y) = xF (y) + G(y),

F = F (y) and G = G(y) are determined by the system of ordinary differential equations

where the functions

( F ′ y 2 ) − FF yy ′′ = νF ′′′ yyy + a,

F y ′ G ′ y − FG ′′ yy = νG ′′′ yyy + b. (2) The order of the autonomous equation (1) can be reduced by one. Given a particular solution of

equation (1), the corresponding equation (2) can be reduced with the substitution H(y) = G ′ y to a second-order equation. For

F (y) = ✼

a y + C, equation (2) is integrable by quadrature (since, for

b = 0, we know two of its particular solutions: G 1 = 1 and G 2 = ✼ 1 √ ✽✂✾ 2 2 ay + Cy).

Reference : A. D. Polyanin (2001 a).

5 ◦ . Solutions for f (x) = −ax −5 /3 :

6 νx

w(x, y) =

[ y + ϕ(x)],

y + ϕ(x)

x 1 /3

where ✽✂✾ ϕ(x) is an arbitrary function.

Reference : B. I. Burde (1996).

6 ◦ . Solutions for f (x) = ax −1 /3 − bx −5 /3 :

w(x, y) = ✼

3 bz+x 2 /3

θ(z), −1 z = yx /3 ,

where the function θ = θ(z) is determined by the ordinary differential equation

3 ( θ z ′ ) − 3 θθ zz ′′ = νθ zzz ′′′ + a.

Reference : B. I. Burde (1996).

7 ◦ . Generalized separable solution for f (x) = ae βx :

a νλ 2 2 νλ w(x, y) = ϕ(x)e λy −

e βx−λy

− νλx +

y+

ln | ϕ(x)|,

β where ✽✂✾ ϕ(x) is an arbitrary function and λ is an arbitrary constant.

2 βλ 2 ϕ(x)

References : A. D. Polyanin (2001 a, 2002).

8 ◦ . For

f (x) = a 2 ν 2 x −3 ( xgg ′ − g 2 ), g=− 1 4 2 a ✼ 1 2 /3 1 x /2 16 a + bx , there are exact solutions of the form

yg

w(x, y) = aνz + 6νg tanh z,

z=

Reference : B. I. Burde (1996).

9 ◦ . Below are two transformations that reduce the order of the boundary layer equation.

9.1. The von Mises transformation

∂w

ξ = x, η = w, U (ξ, η) =

where w = w(x, y),

∂y

leads to the nonlinear heat equation

∂U

∂U

= νU

+ f (ξ).

∂y

∂y 2

leads to the second-order nonlinear equation

Reference : L. G. Loitsyanskiy (1996).

10 ◦ . Conservation law:

F (x) + D y − w x w y − νw yy = 0,

where D x =

+ f (x).

∂z ∂x∂z

Preliminary remarks. The system of axisymmetric steady laminar hydrodynamic boundary layer equations

u ∂u

+ v ∂u = ν ∂ u

∂r 1 ∂r 2 + + f (x), ( )

where u and v are the axial and radial fluid velocity components, respectively, and x and r are cylindrical coordinates, can

be reduced to the equation in question by the introduction of a stream function w and a new variable z such that

System (1), (2) is used for describing an axisymmetric jet and a boundary layer on an extensive body of revolution. The function ✿✂❀

f (x) is expressed via the longitudinal fluid velocity U = U (x) in the inviscid flow core as f = U U x ′ . References : F. L. Crabtree, D. K ¨uchemann, and L. Sowerby (1963), H. Schlichting (1981), L. G. Loitsyanskiy (1996).

1 ◦ . Self-similar solution for f (x) = Ax k :

k−1

w(x, z) = xU (ζ),

ζ = zx 2 ,

where the function U = U (ζ) is determined by the ordinary differential equation

− 1 ( k + 1)(U ζ ′ ) 2 + UU ζζ ′′ +

A + ν(ζU ζζ 2 ′′ ) ′ ζ = 0.

Special case. An axisymmetric jet is characterized by the values A = 0 and k = −3. In this case, the solution of the equation just obtained with appropriate boundary conditions is given by

where the constant of integration ✿✂❀

C can be expressed via the jet momentum.

References : H. Schlichting (1981), L. G. Loitsyanskiy (1996).

2 ◦ . Generalized separable solutions (linear and quadratic in z) for arbitrary f (x):

w(x, z) = ❁ z 2 f (x) dx + C 1 + ϕ(x),

w(x, z) = C 1 z 2 + ϕ(x)z +

ϕ 2 ( x) −

f (x) dx − νx + C 2 ,

where ϕ(x) is an arbitrary function and C 1 and C 2 are arbitrary constants. The first solution is “inviscid” (independent of ✿✂❀ ν).

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

w(x, z) = 2νx + νF (x)

F (x) = ❂ νC 2 2 f (x) dx + C 3 , ❃✂❄ where C 1 , C 2 , and C 3 are arbitrary constants.

Reference : G. I. Burde (1994).

4 ◦ . Functional separable solution for f (x) = ax + b:

a w(x, z) = νλϕ(x) + ( ax + b) Ce λξ + λξ − 3 , ξ=z−ϕ ′ x ( x), λ= ❂ ,

ν where ❃✂❄

C and λ are arbitrary constants and ϕ(x) is an arbitrary function.

Reference : G. I. Burde (1994).

5 ◦ . Generalized separable solution (linear in x) for f (x) = ax + b:

w(x, z) = xϕ(z) + ψ(z),

where the functions ϕ = ϕ(z) and ψ = ψ(z) are determined by the system of ordinary differential equations

( ϕ ′ z ) 2 − ϕϕ ′′ zz = ν(zϕ ′′ zz ) ′ z + a,

z − ϕψ zz = ν(zψ zz ) z + b.

The first equation has particular solutions ϕ= ❂ ❃✂❄ √ a z + C.

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

6 ◦ . Additive separable solutions for f (x) = a:

w(x, z) = ν(1 − k)x + C 1 z +

z 2 + C 2 z+C 3 ,

2 ν(k − 2)

w(x, z) = −νx −

z ln z+C 1 z + C 2 z+C 3 ,

where C 1 , . . . , k are arbitrary constants.

7 ◦ . Conservation law:

F (x) + D z − w x w z − νzw zz = 0,

where D x =