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 =