Unsteady Boundary Layer Equations for a Newtonian Fluid
9.3.3. Unsteady Boundary Layer Equations for a Newtonian Fluid
∂t∂y ∂y ∂x∂y
∂x ∂y 2 ∂y 3
This equation describes an unsteady hydrodynamic boundary layer on a flat plate; w is the stream function, x and y are the coordinates along and normal to the plate, respectively, and ν is the kinematic viscosity of the fluid. A similar equation describes an unsteady flow of a plane laminar jet out of a thin slit. Preliminary remarks. The system of unsteady hydrodynamic boundary layer equations
where u 1 and u 2 are the longitudinal and normal fluid velocity components, can be reduced to the equation in question by
the introduction of a stream function w such that u 1 = ∂w ∂y and u 2 =− ∂w ∂x .
1 ◦ . Suppose w(x, y, t) is a solution of the equation in question. Then the functions
w 1 = w(x, y + ϕ(x, t), t) +
ϕ(x, t) dx + χ(t),
∂t
2 C 1 w(C 2 x+C 2 C 3 t+C 4 , C 1 C 2 y+C 1 C 2 C 5 t+C 6 , C 1 C 2 t+C 7 )+ C 5 x−C 3 y+C 8 , where ϕ(x, t) and χ(t) are arbitrary functions and the C n are arbitrary constants, are also solutions
■✂❏ of the equation.
References : L. I. Vereshchagina (1973), L. V. Ovsiannikov (1982).
2 ◦ . Degenerate solutions linear and quadratic in y:
w=C 1 y + ϕ(x, t),
w=C 1 y 2
+ ϕ(x, t)y +
ϕ 2 ( x, t) +
ϕ(x, t) dx,
4 C 1 ∂t
where ϕ(x, t) is an arbitrary function of two variables and C 1 is an arbitrary constant. Here and henceforth, the additive arbitrary function of time, χ = χ(t), in exact solutions for the stream function is omitted. These solutions are independent of ν and correspond to inviscid fluid flows.
3 ◦ . Solutions involving arbitrary functions:
6 νx + C Z 1 C 2 ∂
w=
ϕ(x, t) dx,
y + ϕ(x, t) [ y + ϕ(x, t)] 2
∂t
w=C 1 exp − C 2 y−C 2 ϕ(x, t) + C 3 y+C 3 ϕ(x, t) + νC 2 x+
ϕ(x, t) dx,
∂t
1 y + ϕ(x, t)
ϕ(x, t) dx,
w = −6νC 1 x 1 /3 tan ξ+
y + ϕ(x, t)
ϕ(x, t) dx, ξ=C 1
∂t
x 2 /3
where ϕ(x, t) is an arbitrary function of two variables, and C 1 , C 2 , and C 3 are arbitrary constants. The construction of these solutions was based on the simpler, stationary solutions specified in 9.3.1.1. Note also the solution
w = f (x) exp − λy − λg(t) + νλ + g t ′ ( t) x,
where f (x) and g(t) are arbitrary functions and λ is an arbitrary constant. It can be obtained from ■✂❏ the second of the solutions specified above with ϕ(x, t) = − 1 λ ln f (x) + g(t), C 2 = λ, and C 3 = 0.
References : G. I. Burde (1995), A. D. Polyanin (2001 b, 2002), A. D. Polyanin and V. F. Zaitsev (2001).
Exact solutions of equation (2) in 9.3.3.1 Function
Remarks No.
F = F (y, t)
(or general form of solution) (or determining equations)
1 F = ψ(t)
ψ(t) is an arbitrary function
2 y F=
t+C 1 + ψ(t)
ψ(t) is an arbitrary function, C 1 is any
3 F= 6 y+ψ(t) ν + ψ ′ t ( t)
ψ(t) is an arbitrary function
4 F=C 1 exp − λy + λψ(t) − ψ t ′ ( t) + νλ
ψ(t) is an arbitrary function, C 1 , λ are any
5 F = F (ξ), ξ = y + λt
4 ◦ . Generalized separable solution linear in x:
(1) where the functions
w(x, y, t) = xF (y, t) + G(y, t),
F = F (y, t) and G = G(y, t) are determined from the simpler equations in two variables
F = F (y, t) is a solution of equation (2), then the functions
Equation (2) is solved independently of (3). If
F (y + ψ(t), t) + ψ t ′ ( t),
1 F (C 1 y+C 1 C 2 t+C 3 , C 1 t+C 4 )+ C 2 ,
where ψ(t) is an arbitrary function and C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation. Given a particular solution
F = F (y, t) of equation (2), the corresponding equation (3) can be reduced, with the substitution U= ∂G ∂y , to the second-order linear equation
Table 7 lists exact solutions of equation (2). The ordinary differential equations in the last two rows, determining a traveling-wave solution and a self-similar one, are both autonomous and, hence, their order can be reduced.
Table 8 presents transformations that simplify equation (4) corresponding to respective solutions of equation (2) in Table 7. It is apparent that in the first three cases, solutions of equation (4) are expressed via solutions of a linear constant-coefficient heat equation. In the other three cases, equation (4) is reduced to linear equations, which can be solved by the method of separation of variables.
The fourth equation in Table 8 has the following particular solutions (
A and B are any):
Z(η) = A + B
Z(η, t) = Aνλ t+A
dη.
Transformations of equation (4) for the respective exact solutions of equation (2); the number in the first column corresponds to the number of the exact solution
F = F (y, t) in Table 7 No.
Transformation of equation (4) Resulting equation
1 U = u(ζ, t), ζ = y + ψ(t) dt
∂u
∂t = ν ∂ 2 ∂ζ u 2
U= 1 t+C t+C 3 1 u(z, τ ), τ = 1 3 ( 1 ) + C 2 ,
∂u
∂ = 2 ν u 2 z = (t + C 1 ) y+ ψ(t)(t + C 1 ) dt + C 3 ∂τ
∂z
∂ 2 U=ζ u −3 u(ζ, t), ζ = y + ψ(t) ∂t = ν ∂ζ 2
∂u
4 U=e η Z(η, t), η = −λy + λψ(t)
5 2 U = u(ξ, t), ξ = y + λt ∂u
u(ξ, τ ), ξ = yt ∂u −1 /2 , τ = ln t ∂τ = ν ∂ξ 2 + H(ξ) ∂ξ + 1− H ξ ′ ( ξ) u
∂u
For other exact solutions of this equation, see the book by Polyanin (2002), where a more general equation, ∂ t w = f (x)∂ xx w + g(x)∂ x w, was considered.
Equation 5 in Table 8 has a stationary particular solution u 0 = F ξ ′ ( ξ) (cf. equation 5 in Table 7). Consequently, other particular solutions of this equation are given by
Ψ( ξ) dξ
u(ξ) = C 1 F ξ ′ ( ξ) + C 2 F ξ ′
u(ξ, t) = C 1 νtF ξ ′ ( ξ) + C 1 F ξ ′ ( ξ)
dξ, Φ( ξ) =
dξ; [ F ξ ′ ( ξ)] 2 Ψ( ξ)
see Polyanin (2002). ❑✂▲
References : D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (2000), A. D. Polyanin (2001 b, 2002), A. D. Polyanin and V. F. Zaitsev (2001, 2002).
Example 1. Solution exponentially dependent on time:
w(x, y, t) = f (y)x + e Z − λt g(y) dy,
where the functions f = f (y) and g = g(y) are determined by the system of ordinary differential equations
( ′ ) 2 f y − ff yy ′′ = νf yyy ′′′ , − λg + gf y ′ − fg ′ y = νg ′′ yy .
Example 2. Periodic solution:
w(x, y, t) = f (y)x + sin(λt) Z g(y) dy + cos(λt) h(y) dy,
where the functions f = f (y), g = g(y), and h = h(y) are determined by the system of ordinary differential equations
( f y ′ ) 2 − ff yy ′′ = νf yyy ′′′ , − λh + f y ′ g−fg ′ y = νg yy ′′ , λg + f y ′ h−fh ′ y = νh ′′ yy .
w(x, y, t) = A(t)e k 1 x + B(t)e k 2 x e λy + ϕ(t)x + ay,
A(t) = C 1 exp ( νλ 2 − ak 1 ) t+λ
ϕ(t) dt ,
B(t) = C 2 exp ( 2 νλ − ak 2 ) t+λ
ϕ(t) dt ,
where ϕ(t) is an arbitrary function and C 1 , C 2 ,
a, k 1 , k 2 , and λ are arbitrary constants.
6 ◦ . Generalized separable solution: w(x, y, t) = A(t) exp(kx + λy) + B(t) exp(βkx + βλy) + ϕ(t)x + ay,
A(t) = C 2
1 exp ( νλ − ak)t + λ ϕ(t) dt ,
Z B(t) = C 2 exp ( 2 νβ 2 λ − akβ)t + βλ ϕ(t) dt ,
where ϕ(t) is an arbitrary function and C 1 , C 2 ,
a, k, β, and λ are arbitrary constants.
References : A. D. Polyanin (2001 b), A. D. Polyanin and V. F. Zaitsev (2001).
7 ◦ . “Two-dimensional” solution:
z = kx + λy, where ϕ(t) and ψ(t) are arbitrary functions, k and λ are arbitrary constants, and the function u(z, t)
w(x, y, t) =
u(z, t) dz + ϕ(t)y + ψ(t)x,
is determined by the second-order linear differential equation
∂u
+ kϕ(t) − λψ(t)
The transformation
u = U (ξ, t) − ϕ(t),
ξ=z−
kϕ(t) − λψ(t) dt
brings it to the linear heat equation
References : A. D. Polyanin (2001 b), A. D. Polyanin and V. F. Zaitsev (2001).
ϕ(x, t) dx,
z = y + ϕ(x, t);
∂t
w=e νλ 2 t C 1 sin( λz) + C 2 cos( λz) +
ϕ(x, t) dx,
z = y + ϕ(x, t);
∂t
w=C 1 e νλ z sin( λz − 2νλ t+C 2 )+
ϕ(x, t) dx,
z = y + ϕ(x, t),
∂t
where ϕ(x, t) is an arbitrary function of two arguments; C 1 , C 2 , and λ are arbitrary constants. For periodic function p ϕ(x, t) = ϕ(x, t + T ), the last solution is also periodic, w(x, y, t) = w(x, y, t + T ), if λ= π/(νT ).
η=k 2 y+λ 2 t, where the function W is determined by the differential equation
∂ξ ∂η 2 2 ∂η 3 In the special case
we have the steady boundary layer equation 9.3.1.1:
10 ◦ . “Two-dimensional” solution:
where the function
V is determined by the differential equation
1 ∂ 2 V 1 ∂ 2 V ∂V ∂ 2 V ∂V ∂ 2 V ∂ 3 V −
1 ∂V
∂ξ ∂η 2 ∂η 3 ❖✂P For example, this equation has solutions of the form
V = F (η)ξ + G(η).
Reference : L. V. Ovsiannikov (1982).
∂t∂y ∂y ∂x∂y
∂x ∂y 2 ∂y 3
This equation describes an unsteady hydrodynamic boundary layer with pressure gradient.
1 ◦ . Suppose w(x, y, t) is a solution of the equation in question. Then the functions
w 1 = w(x, y + ϕ(x, t), t) +
ϕ(x, t) dx,
∂t w 2 =− w(x, −y, t) + ψ(t),
where ❖✂P ϕ(x, t) and ψ(t) are arbitrary functions, are also solutions of the equation.
References : L. I. Vereshchagina (1973), L. V. Ovsiannikov (1982).
2 ◦ . For f (x, t) = g(t), the transformation Z t w = u(ξ, y, t) − h ′ t ( t)y,
ξ = x + h(t),
where h(t) = −
( t − τ )g(τ ) dτ , (1)
leads to a simpler equation of the form 9.3.3.1:
∂y ∂ξ∂y
∂ξ ∂y 2 ∂y 3
Note that f = g(t) and h = h(t) are related by the simple constraint h ′′ tt =− g. In the general case, transformation (1) brings the equation in question to a similar equation with the function f (x, t) modified according to
f (x, t) transformation (1) −−−−−−−−−−−−→
f (x, t) − g(t).
❖✂P
Reference : L. V. Ovsiannikov (1982).
w(x, y, t) = Cy + ϕ(x, t)y +
where ϕ(x, t) is an arbitrary function of two arguments and C is an arbitrary constant. From now on, the arbitrary additive function of time ψ = ψ(t) is omitted in exact solutions for the stream function. These solutions are independent of ν and correspond to inviscid fluid flows.
Degenerate solution (linear in y) for any f (x, t):
w(x, y, t) = ψ(x, t)y + ϕ(x, t),
where ϕ(x, t) is an arbitrary function, and ψ = ψ(x, t) is determined by the first-order partial differential equation
For information about the methods of integration and exact solutions of such equations (for various f), see the books by Kamke (1965) and Polyanin, Zaitsev, and Moussiaux (2002).
Degenerate solutions for f (x, t) = f (x):
w(x, y, t) = ◗ y 2 f (x) dx + C 1 + ϕ(x, t),
where ϕ(x, t) is an arbitrary function.
4 ◦ . Generalized separable solution (linear in x) for f (x, t) = f 1 ( t)x + f 2 ( t):
(2) where the functions
w(x, y, t) = xF (y, t) + G(y, t),
F = F (y, t) and G = G(y, t) are determined by the simpler equations in two variables
Equation (3) is solved independently of equation (4). If F = F (y, t) is a solution to equation (3), then the function
F (y + ψ(t), t) + ψ ′ t ( t),
where ψ(t) is an arbitrary function, is also a solution of the equation. Table 9 lists exact solutions of equation (3) for various f 1 = f 1 ( t); two more complicated solutions of this equation are given at the end of Item 4 ◦ . Note that, for
G ≡ 0, solutions (2) specified in the first and the last rows of Table 9 were treated in the book by Ovsiannikov (1982). The substitution U= ∂G ∂y brings equation (4) to the second-order linear equation
Let us dwell on the first solution to (3) specified in Table 9:
a ′ t + a 2 = f 1 ( t). (6) The Riccati equation for a = a(t) is reduced by the substitution a = h ′ t /h to the second-order linear
F (y, t) = a(t)y + ψ(t),
where
equation h ′′ tt − f 1 ( t)h = 0. Exact solutions of this equation for various f 1 ( t) can be found in Kamke (1977) and Polyanin and Zaitsev (2003). In particular, for f 1 ( t) = const we have
C 1 cos( kt) − C 2 sin( kt)
a(t) = k
if f 1 =− k 2 < 0,
C 1 sin( kt) + C 2 cos( kt)
C 1 cosh( kt) + C 2 sinh( kt)
a(t) = k
if f 1 = k 2 > 0.
C 1 sinh( kt) + C 2 cosh( kt)
Exact solutions of equation (3) in 9.3.3.2 for various f 1 ( t); ψ(t) is an arbitrary function Function
Function
F = F (y, t)
Determining equation
(or determining coefficients) Any
f 1 = f 1 ( t)
(or general form of solution)
F = a(t)y + ψ(t)
a ′ t + a 2 = f 1 ( t)
f 1 ( t) = Ae − βt
, 1 F = Be − 2 βt sin[ λy + λψ(t)] + ψ ′
cos[ λy + λψ(t)] + ψ t ′ ( t)
f 1 ( t) = Ae βt ,
F = Be βt sinh[ λy + λψ(t)] + ψ ′ ( t)
F = Be 2 βt cosh[ λy + λψ(t)] + ψ ′
β λ= A is any, β > 0 ❘
ψ ′ ( F = ψ(t)e t) −
λy
2 ψ(t) + 4 t λ λψ(t) − νλ
Ae βt−λy
1 ( t) = At −2
F=t −1 /2 H(ξ) − 1 ξ , ξ = yt −1 /2
2 4 − A−2H ξ +( H ξ ) − HH ξξ ′′ = νH ′′′ ξξξ
1 ( t) = A
F = F (ξ), ξ = y + λt
A + λF ′′ ξξ +( F ξ ′ ) − FF ξξ ′′ = νF ′′′ ξξξ
On substituting solution (6), with arbitrary f 1 ( t), into equation (5), one obtains
∂U
2 + a(t)y + ψ(t)
− a(t)U + f 2 ( t).
The transformation (Polyanin, 2002)
U=
u(z, τ ) +
f 2 ( t)Φ(t) dt , τ=
Φ 2 ( t) dt + C 1 ,
Φ( t) Z
a(t) dt , leads to the linear heat equation
z = yΦ(t) +
ψ(t)Φ(t) dt + C 2 , Φ( t) = exp
References : D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (2000), A. D. Polyanin (2001 b, 2002), A. D. Polyanin and V. F. Zaitsev (2001, 2002).
Remark 1. The ordinary differential equations in the last two rows of Table 9 (see the last column), which determine a self-similar and a traveling-wave solution, are both autonomous and, hence, their order can be reduced.
Remark 2. Suppose w(x, y, t) is a solution of the unsteady hydrodynamic boundary layer
equation with f (x, t) = f 1 ( t)x + f 2 ( t). Then the function w 1 = w(x + h(t), y, t) − h ′ t ( t)y,
where h ′′ tt − f 1 ( t)h = 0, is also a solution of the equation. ❙✂❚
Reference : L. V. Ovsiannikov (1982).
Remark 3. In the special case f 2 ( t) = 0, equation (4) admits a particular solution G = G(t), where G(t) is an arbitrary function.
w(x, y, t) = xg(y) + e − λt Z h(y) dy,
where the functions g = g(y) and h = h(y) are determined by the system of ordinary differential equations
( g ′ ) 2 − gg ′′ = νg ′′′ + A, − λh + hg ′ − gh ′ = νh ′′ + B.
The prime denotes a derivative with respect to y. Example 2+. Periodic solution with
f (x, t) = Ax + B 1 sin( λt) + B 2 cos( λt):
h 2 ( y) dy, where the functions
w(x, y, t) = xg(y) + sin(λt) Z h
1 ( y) dy + cos(λt)
g = g(y), h 1 = h 1 ( y), and h 2 = h 2 ( y) are determined by the system of ordinary differential equations
( g ′ ) 2 − gg ′′ = νg ′′′ + A, − λh 2 + g ′ h 1 − gh 1 ′ = νh ′′ 1 + B 1 , λh 1 + g ′ h 2 − gh ′ 2 = νh ′′ 2 + B 2 .
Below are two more complex solutions of equation (3). The solution
F (y, t) = − t y+γ 3 exp ν
dt
A cosh +
B sinh
γ where
A and B are arbitrary constants and γ = γ(t) is an arbitrary function, corresponds to the right-hand side of equation (3) in the form
Z dt
. The solution
A and B are arbitrary constants and γ = γ(t) is an arbitrary function, corresponds to the right-hand side of equation (3) in the form
γ 2 This solution was obtained in Burde (1995) for the case
A = 0.
5 ◦ . Generalized separable solution for f (x, t) = g(x)e βt , β > 0:
w(x, y, t) = ϕ(x, t)e λy +
ψ(x, t)e − λy 1 + ∂
ln | ϕ(x, t)| dx − νλx,
λ ∂t
e βt
ψ(x, t) = −
g(x) dx,
2 λ 2 ϕ(x, t)
where ❱✂❲ ϕ(x, t) is an arbitrary function of two arguments.
References : A. D. Polyanin (2001 b), A. D. Polyanin and V. F. Zaitsev (2002).
6 ◦ . Generalized separable solutions for f (x, t) = g(x)e βt , β > 0: w(x, y, t) = ❯
exp 1
2 βt
ψ(x) sinh λy + ϕ(x, t) +
ϕ(x, t) dx,
∂t
exp 1 w(x, y, t) = ∂
2 βt
ψ(x) cosh λy + ϕ(x, t) +
ϕ(x, t) dx,
∂t
ψ(x) = 2
g(x) dx + C 1 ,
where ❱✂❲ ϕ(x, t) is an arbitrary function of two arguments.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
w(x, y, t) = ❳
exp − 1
βt
ψ(x) sin λy + ϕ(x, t) +
ϕ(x, t) dx,
2 ∂t
w(x, y, t) = ❳
exp − 1 2 βt
ψ(x) cos λy + ϕ(x, t) +
ϕ(x, t) dx,
∂t
ψ(x) = 2
g(x) dx + C 1 ,
where ❨✂❩ ϕ(x, t) is an arbitrary function of two arguments.
Reference : A. D. Polyanin and V. F. Zaitsev (2002).
8 ◦ . Solution for f (x, t) = xg(t):
w(x, y, t) = t xy +
2 − νψ x + ϕ(z) exp(ψy),
z=
, ψ = ψ(t),
where ϕ(z) is an arbitrary function and the function ψ = ψ(t) is determined by the second-order linear ordinary differential equation
ψ tt ′′ = g(t)ψ.
Reference : D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (2000).
9 ◦ . Generalized separable solution for f (x, t) = ae βx−γt : w(x, y, t) = ϕ(x, t)e λy
e βx−λy−γt
2 βλ 2 ϕ(x, t)
ln | ϕ(x, t)| dx − νλx +
y+ ln | ϕ(x, t)| ,
λ ∂t
where ❨✂❩ ϕ(x, t) is an arbitrary function of two arguments and λ is an arbitrary constant.
References : A. D. Polyanin (2001 b), A. D. Polyanin and V. F. Zaitsev (2002).
10 ◦ . Generalized separable solution for f (x, t) = f (t):
z = kx + λy, where ϕ(t) and ψ(t) are arbitrary functions, k and λ are arbitrary constants, and the function u(z, t)
w(x, y, t) =
u(z, t) dz + ϕ(t)y + ψ(t)x,
is determined by the second-order linear equation
∂u
+ kϕ(t) − λψ(t)
2 − ϕ t ′ ( t) +
f (t).
The transformation
u = U (ξ, t) − ϕ(t) +
f (t) dt,
ξ=z−
kϕ(t) − λψ(t) dt
brings it to the linear heat equation
References : A. D. Polyanin (2001 b), A. D. Polyanin and V. F. Zaitsev (2002).
11 ◦ . Generalized separable solution for f (x, t) = f (t):
Z w(x, y, t) = Ce − λy+λϕ(x,t)
− a(t)ϕ(x, t) −
ϕ(x, t) dx + a(t)y + νλx, a(t) =
f (t) dt,
∂t
where ϕ(x, t) is an arbitrary function of two arguments; C and λ are arbitrary constants.
ψ(x, t)e − λy + χ(x, t) + a(t)y, where λ is any, ϕ(x, t) is an arbitrary function of two arguments, and the remaining functions are
w(x, y, t) = ϕ(x, t)e λy +
given by
Cνe 2 νλ 2 t
2 ψ(x, t) = 2 x− a(t) dt , a(t) = f (t) dt + Ce νλ t ,
ϕ(x, t)
χ(x, t) = a(t) ln |ϕ(x, t)| + ln | ϕ(x, t)| dx − νλx.
λ ∂t
13 ◦ . Solutions for f (x, t) = f (t):
ϕ(x, t) dx + z
f (t) dt,
z = y + ϕ(x, t);
∂t
w=e − νλ t C
1 sin( λz) + C 2 cos( λz) +
ϕ(x, t) dx + z
f (t) dt, z = y + ϕ(x, t);
∂t
− λz
w=C 1 e sin( λz − 2νλ t+C 2 )+
ϕ(x, t) dx + z
f (t) dt, z = y + ϕ(x, t),
∂t
where ϕ(x, t) is an arbitrary function of two arguments; C 1 , C 2 , and λ are arbitrary constants. For
periodic function f (t) = f (t + T ) satisfying the condition
f (t) dt = 0; the last solution is also
periodic, w(x, y, t) = w(x, y, t + T ), if ϕ(x, t) = ϕ(x) and λ = π/(νT ).
14 ◦ . Solutions for f (x, t) = A:
w=−
C 2 z + C 1 z+
ϕ(x, t) dx,
z = y + ϕ(x, t);
ϕ(x, t) dx,
z = y + ϕ(x, t),
∂t
where ϕ(x, t) is an arbitrary function of two arguments; C 1 , C 2 , and k are arbitrary constants.
15 ◦ . Table 10 presents solutions of the unsteady hydrodynamic boundary layer equation with pressure gradient that depends on two generalized variables (used results of group-theoretic analyses in Ovsiannikov, 1982).
For f (x, t) = f (k 1 x+λ 1 t), there is also a wide class of “two-dimensional” solutions with the form
η=k 2 y+λ 2 t, where the function z is determined by the differential equation
w = z(ξ, η) + a 1 x+a 2 y,
= νk 2 2 + f (ξ). ∂ξ∂η
X = x + b(t), where a(t) and b(t) are some functions, a solution is given by
f (x, t) = a ′ ( t)X −1 /3 − 1 3 a 2 ( t)X −5 /3 − b ′′ ( t),
w = [a(t)X −1 /3 − b ′ ( t)]y + 6νXy −1 .
Reference : Burde (1995).
Solutions of the unsteady hydrodynamic boundary layer equation that depends on two generalized variables. Notation: R[z] = νz ηηη + z ξ z ηη − z η z ξη and g = g(u) is an arbitrary function.
Function f = f (x, t)
General form of solution
Equation for z = z(ξ, η)
f = f (x + λt)
w = z(ξ, y) − λy, ξ = x + λt
νz yyy + z ξ z yy − z y z ξy + f (ξ) = 0
f = g(x)t −2 w = z(x, η)t −1 /2 , η = yt −1 /2 νz ηηη + z x z ηη − z η z xη + 1 2 ηz ηη + z η + g(x) = 0 f=e λt g(xe − λt )
νz yyy + z ξ z yy − z y z ξy + λξz ξy − λz y + g(ξ) = 0 w = z(ξ, η)t − −(2 n+1)/2 n−2 n ,
w=e λt z(ξ, y), ξ = xe − λt
f=t 1 g(xt )
R[z] + 2 ηz ηη − nξz ξη + (1 + n)z η + g(ξ) = 0
ξ = xt n , η = yt −1 /2
f = ax n
w = z(ξ, η)t
R[z] + 2 ηz ηη − n−1 z ξη + n−1 z η + aξ n ξ = xt =0 , η = yt
−( n+3)/(2n−2)
w = z(ξ, η)t −1 /2 ,
1 ηz − f = ae 2
λx
R[z] + 2 ηη λ z ξη + ξ=x+ z
∂z∂t ∂z ∂x∂z
∂x ∂z 2 ∂z
∂z 2
Preliminary remarks. The system of axisymmetric unsteady laminar boundary layer equations
where u and v are the axial and radial components of the fluid velocity, respectively, and x and r the axial and radial coordinates, is reduced to the equation in question by the introduction of a stream function w and a new variable z such that
System (1), (2) describes an axisymmetric jet ( f ≡ 0) and a boundary layer on an extensive body of revolution (f ❪ 0 ).
1 ◦ . The equation remains the same under the replacement of w by w+ϕ(t), where ϕ(t) is an arbitrary function.
2 ◦ . Generalized separable solution (quadratic in z) for arbitrary f (x, t):
w(x, z, t) = Cz
ϕ(x, t)z +
ϕ ( x, t) +
ϕ(x, t) dx −
f (x, t) dx − νx + ψ(t),
4 C 2 C ∂t
where ϕ(x, t) and ψ(t) are arbitrary functions and C is an arbitrary constant. The equation also has an “inviscid” solution of the form w = ϕ(x, t)z + ψ(x, t), where ψ(x, t) is an arbitrary function, and the function ϕ = ϕ(x, t) is described by the first-order partial differential equation ∂ t ϕ + ϕ∂ x ϕ = f (x, t).
3 ◦ . Generalized separable solution (linear in x) for f (x, t) = a(t)x + b(t):
w(x, z, t) = xϕ(z, t) + ψ(z, t),
where the functions ϕ = ϕ(z, t) and ψ = ψ(z, t) are described by the system of partial differential equations
+ a(t),
+ b(t).
∂z∂t
∂z ∂z
∂z 2 ∂z
∂z 2 ∂z 2
4 ◦ . “Two-dimensional” solution for f (x, t) = f (x + λt):
w(x, z, t) = U (ξ, z) − λz, ξ = x + λt,
where the function U = U (ξ, z) is determined by the differential equation
∂z ∂ξ∂z
∂ξ ∂z 2 ∂z
∂z 2
which coincides, up to renaming, with the stationary equation (see equation 9.3.1.3 and its solutions).
5 ◦ . Generalized separable solution (linear in x) for f (x, t) = f (t):
w(x, z, t) = A(t)x + B(t) + z
f (t) dt + u(z, t),
where A(t) and B(t) are arbitrary functions, and the function u = u(z, t) is determined by the second-order linear parabolic differential equation
− A(t)
= νz
6 ◦ . Suppose w(x, z, t) is a solution of the unsteady axisymmetric boundary layer equation with
f (x, t) = a(t)x + b(t). Then the function
w 1 = w(ξ, z, t) − ϕ ′ t ( t)z + ψ(t),
ξ = x + ϕ(t),
where ψ(t) is an arbitrary function and ϕ = ϕ(t) is a solution of the linear ordinary differential equation ϕ ′′ tt − a(t)ϕ = 0, is also a solution of the equation.