Stationary Hydrodynamic Equations (Navier–Stokes

10.3.2. Stationary Hydrodynamic Equations (Navier–Stokes

Equations)

( ∆w) = ν∆∆w,

Preliminary remarks. The two-dimensional stationary equations of a viscous incompressible fluid

∂y and u 2 =− ∂x followed by the elimination of the pressure

are reduced to the equation in question by the introduction of a stream function w such that u 1 = ∂w

∂w

p (with cross differentiation) from the first two equations.

Reference : L. G. Loitsyanskiy (1996).

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

w 1 =− w(y, x), w 2 = w(C 1 x+C 2 , C 1 y+C 3 )+ C 4 , w 3 = w(x cos α + y sin α, −x sin α + y cos α),

where 1 ,

C ...,C α are arbitrary constants, are also solutions of the equation.

4 and

Reference : V. V. Pukhnachov (1960).

2 ◦ . Any solution of the Poisson equation ∆ w = C is also a solution of the original equation (these are “inviscid” solutions). On the utilization of these solutions in the hydrodynamics of ideal fluids, see Lamb (1945), Batchelor (1970), Lavrent’ev and Shabat (1973), Sedov (1980), and Loitsyanskiy (1996).

3 ◦ . Solutions in the form of a one-variable function or the sum of functions with different arguments:

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

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

w(x, y) = C 2

1 exp(− λy) + C 2 y + C 3 y+C 4 + νλx,

w(x, y) = C 1 exp( λx) − νλx + C 2 exp( λy) + νλy + C 3 , w(x, y) = C 1 exp( λx) + νλx + C 2 exp(− λy) + νλy + C 3 ,

where C 1 , ...,C 5 and λ are arbitrary constants.

References : V. V. Pukhnachov (1960), L. G. Loitsyanskiy (1996), A. D. Polyanin (2001 c).

3 w(x, y) = A(kx + λy) 2 + B(kx + λy) + C(kx + λy) + D,

2 w(x, y) = Ae 2 + B(y + kx) + C(y + kx) + νλ(k + 1) x + D, where ✧✂★

− λ(y+kx)

A, B, C, D, k, β, and λ are arbitrary constants.

Reference : V. V. Pukhnachov (1960).

5 ◦ . Generalized separable solutions:

w(x, y) = 6νx(y + λ) −2 + A(y + λ) + B(y + λ) + C(y + λ) + D ( ν ≠ 0), w(x, y) = (Ax + B)e − λy + νλx + C,

w(x, y) = − A sinh(βx) + B cosh(βx) e λy ν 2 2 + ( β + λ ) x + C, λ

w(x, y) = − A sin(βx) + B cos(βx) e λy ν 2 2 + ( λ − β ) x + C, λ

p w(x, y) = Ae λy+βx

A, B, C, D, k, β, and λ are arbitrary constants.

Reference : A. D. Polyanin (2001 c). Special case. Setting

√ A = −νλ, B = C = 0, λ = p k/ν in the second solution, we obtain w = kν x

This solution describes the steady-state motion of a fluid due to the motion of the surface particles at y = 0 with a velocity u 1 | y=0 = kx.

6 ◦ . Generalized separable solution linear in x:

(1) where the functions

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

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

F y ′ F yy ′′ − FF yyy ′′′ = νF yyyy ′′′′ ,

(3) On integrating the equations once, we obtain the system of third-order equations

G ′ y F yy ′′ − FG ′′′ yyy = νG ′′′′ yyyy .

A and B are arbitrary constants. The order of the autonomous equation (4) can be reduced by one. Equation (2) has the following particular solutions:

F (y) = ay + b,

F (y) = 6ν(y + a) −1 ,

a, b, and λ are arbitrary constants. In the general case, equation (5) is reduced, with the substitution U=G ′ y , to the second-order linear nonhomogeneous equation

where U=G ′ y . (9) The corresponding homogeneous equation (with

νU yy ′′ + FU y ′ − F y ′ U + B = 0,

B = 0) has two linearly independent particular solutions:

where Φ = exp −

F dy ; (10)

F if F yy ≡ 0,

see Polyanin and Zaitsev (2003). The general solution of equation (3) corresponding to the particular solution (7) is expressed as

G(y) = e C 1 3 −1

( −2 y + a) +e C 2 +e C 3 ( y + a) +e C 4 ( y + a) , where e C 1 ,e C 2 ,e C 3 , and e C 4 are arbitrary constants (these are expressed in terms of C 1 , ...,C 4 ).

The general solutions of equation (3) corresponding to the particular solutions (6) and (8) are given by (10) and (11), respectively. ✫✂✬

References : R. Berker (1963), A. D. Polyanin (2001 c). Special case. A solution of the form (1) with G(y) = kF (y) describes a laminar fluid flow in a plane channel with

porous walls. In this case, equation (3) is satisfied by virtue of (2). ✫✂✬

Reference : A. S. Berman (1953).

7 ◦ . Solution (generalizes the solution of Item 6 ◦ ):

w(x, y) = F (z)x + G(z),

z = y + kx,

where the functions

F = F (z) and G = G(z) are determined by the autonomous system of fourth-order ordinary differential equations

z F zz − FF zzz = ν(k + 1) F zzzz ′′′′ ,

zz . (13) On integrating the equations once, we obtain the system of third-order equations

G ′ z F zz ′′

− FG ′′′ zzz = ν(k + 1) G ′′′′ zzzz +4 kνF zzz ′′′ + 2 FF ′′

ν(k 2 + 1) G ′′′ zzz + ψ(z) + B,

A and B are arbitrary constants, and the function ψ(z) is defined by

2 k Z zz + 2 FF k ′′ +1 zz dz.

ψ(z) = 4kνF ′′

The order of the autonomous equation (14) can be reduced by one. Equation (12) has the following particular solutions:

F (z) = az + b, z = y + kx,

2 F (z) = 6ν(k −1 + 1)( z + a) ,

a, b, and λ are arbitrary constants. In the general case, equation (15) is reduced, with the substitution U=G ′ z , to a second-order linear nonhomogeneous equation, a particular solution of which, in the homogeneous case ψ = B = 0, is given by

F zz ′′ if F zz ′′ ✭ 0, U=

F if F zz ′′ ≡ 0.

Consequently, the general solution to (15) can be expressed by quadrature; see Polyanin and Zaitsev (2003). ✫✂✬

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

w=

F (z) dz + C 1 , z = arctan

where the function

F is determined by the first-order autonomous ordinary differential equation

(16) and C 1 , C 2 , and C 3 are arbitrary constants. The general solution of equation (16) can be written out ✮✂✯ in implicit form and also can be expressed in terms of the Weierstrass elliptic function.

3 ν(F ′ 2 3 z 2 ) −2 F + 12 νF + C 2 F+C 3 = 0,

Reference : L. G. Loitsyanskiy (1996).

9 ◦ . There is an exact solution of the form

1 w = a ln |x| + x V (z) dz + C , z = arctan .

To a = 0 there corresponds a self-similar solution of (16). ◮ For other exact solutions, see equation 10.3.2.4.

2 ∂w 2 ∂ ∂w ∂ ∂ w ∂ w

2. ( ∆w) –

( ∆w) = ν∆∆w + f (y),

Preliminary remarks. The system

u 1 ∂u 1 + u 2 ∂u 1 =− 1 ∂p + ν∆u 1 +

F (y),

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 . The above system of equations describes the plane flow of a viscous incompressible fluid under the action of a transverse

force. Here, f (y) = F y ′ ( y). The case

F (y) = a sin(λy) corresponds to A. N. Kolmogorov’s model, which is used for describing subcritical and transitional (laminar-to-turbulent) flow modes. ✮✂✯

Reference : O. M. Belotserkovskii and A. M. Oparin (2000).

1 ◦ . Solution in the form of a one-argument function:

3 3 w(y) = − 2 ( y − z) f (z) dz + C

1 y + C 2 y + C 3 y+C 4 ,

6 ν 0 where C 1 , ...,C 4 are arbitrary constants.

2 ◦ . Additive separable solution for arbitrary f (y):

w(x, y) = − 2 ( y − z) Φ( z) dz + C 1 e λy + C 2 y + C 3 y+C 4 + νλx,

Φ( − z) = e λz

e λz f (z) dz,

where C 1 , ...,C 4 and λ are arbitrary constants.

Special case. If f (y) = aβ cos(βy), which corresponds to F (y) = a sin(βy), it follows from the preceding formula with C 1 = C 2 = C 4 =0 and

B = −νλ that w(x, y) = −

where B and C are arbitrary constants. This solution is specified in the book by Belotserkovskii and Oparin (2000); it describes a flow with a periodic structure.

3 ◦ . Additive separable solution for

f (y) = Ae − λy + Be λy :

w(x, y) = C 1 e λx + C 2 x−

where C 1 and C 2 are arbitrary constants.

w(x, y) = ϕ(y)x + ψ(y),

where the functions ϕ = ϕ(y) and ψ = ψ(y) are determined by the system of fourth-order ordinary differential equations

ϕ ′ y ϕ ′′ yy − ϕϕ ′′′ yyy = νϕ ′′′′ yyyy ,

(2) On integrating once, we obtain the system of third-order equations

ψ ′ y ϕ ′′ yy − ϕψ ′′′ yyy = νψ yyyy ′′′′ + f (y).

A and B are arbitrary constants. The order of the autonomous equation (3) can be reduced by one. Equation (1) has the following particular solutions:

ϕ(y) = ay + b, ϕ(y) = 6ν(y + a) −1 ,

ϕ(y) = ae − λy + λν,

where

a, b, and λ are arbitrary constants. In the general case, equation (4) is reduced, with the substitution U=ψ y ′ , to the second-order linear nonhomogeneous equation

f (y) dy + B. (5) The corresponding homogeneous equation (with

νU yy ′′ + ϕU y ′ − ϕ ′ y U + F = 0,

where U=ψ y ′ , F=

F = 0) has two linearly independent particular solutions:

Z U 1 = ϕ ′′ yy if ϕ ≠ ay + b,

2 = U 1 U dy 2 ,

where Φ = exp − ϕ dy ; ϕ

U 1 ν the first solution follows from the comparison of (1) and (5) with

if ϕ = ay + b,

F = 0. Consequently, the general solutions of equations (5) and (2) are given by

F Z U=C 1 U 1 + C 2 U 2 + U 1 U 2 Φ dy − U 2 U 1 Φ dy, ψ=

U dy + C 4 ;

see Polyanin and Zaitsev (2003). ∂w

∂w

( ∆w) + 2a∆w = ν∆∆w. ∂y

This equation is used for describing the motion of a viscous incompressible fluid induced by two parallel disks, moving towards each other; see Craik (1989) and equation 10.3.3.2 in the stationary case.

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

w 1 =− w(y, x), w 2 = w(x + C 1 , y+C 2 )− aC 2 x + aC 1 y+C 3 , w 3 = w(x cos β + y sin β, −x sin β + y cos β),

where C 1 , C 2 , C 3 , and β are arbitrary constants, are also solutions of the equation.

are “inviscid” solutions). For details about the Poisson, see, for example, the books by Tikhonov and Samarskii (1990) and Polyanin (2002).

3 ◦ . Solution dependent on a single coordinate x:

w(x) =

( x − ξ)U (ξ) dξ + C 1 x+C 2 ,

where C 1 and C 2 are arbitrary constants and the function U (x) is determined by the second-order linear ordinary differential equation

axU x ′ +2 aU = νU xx ′′ .

The general solution to this equation can be found in Polyanin and Zaitsev (2003). Likewise, we can obtain solutions of the form w = w(y).

4 ◦ . Generalized separable solution linear in x:

(1) where the functions

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

F = F (y) and G = G(y) are determined by the fourth-order ordinary differential equations

F y ′ F yy ′′ − FF yyy ′′′ + a(3F yy ′′ + yF yyy ′′′ )= νF yyyy ′′′′ ,

(3) Equation (2) is solved independently of equation (3). If

F yy ′′ G ′ y − FG ′′′ yyy + a(2G ′′ yy + yG yyy ′′′ )= νG ′′′′ yyyy .

F = F (y) is a solution to (2), then the function

F (y + C) − aC,

where

C is an arbitrary constant, is also a solution of the equation. Integrating (2) and (3) with respect to y yields

yy + a(2F y + yF yy )= νF yyy + C 1 ,

(5) where C 1 and C 2 are arbitrary constants.

F y ′ G ′ y − FG ′′ yy + a(G ′ y + yG ′′ yy )= νG ′′′ yyy + C 2 ,

Equation (2) has a particular solution

(6) where

F (y) = Ay + B,

A and B are arbitrary constants. On substituting (6) into (5) and performing the change of variable Q=G ′′ yy , we obtain the second-order linear ordinary differential equation

A − a)y + B Q ′ y +2 aQ = νQ ′′ yy ,

whose general solution can be found in Polyanin and Zaitsev (2003).

Solutions of the form w(x, y) = f (x)y + g(x) can be obtained likewise.

Reference : S. N. Aristov and I. M. Gitman (2002).

5 ◦ . Note that equation (2) has the following particular solutions:

a < 0, where C 1 and C 2 are arbitrary constants.

F = ay + C 1 cos 2 − a/ν y + C 2 sin 2 − a/ν y

if

4. ( ∆w) –

( ∆w) = ν∆∆w,

Preliminary remarks. Equation 10.3.2.1 is reduced to the equation in question by passing to the polar coordinate system

with origin at ( x 0 , y 0 ), where x 0 and y 0 are any numbers, according to

x = r cos θ + x 0 ,

y = r sin θ + y 0 (direct transformation),

x−x (inverse transformation).

The radial and angular fluid velocity components are expressed via the stream function w as follows: u = 1 r ∂w

r ∂θ , u θ =− ∂r .

∂w

1 ◦ . Any solution of the Poisson equation ∆ w = C is also a solution of the original equation (these are “inviscid” solutions).

2 ◦ . Solutions in the form of a one-variable function and the sum of functions with different argu- ments:

2 w(r) = C 2 1 r ln r+C 2 r + C 3 ln r+C 4 ,

w(r, θ) = Aνθ + C 2 1 r A+2 + C 2 r + C 3 ln r+C 4 ,

where ✲✂✳

1 , A, C ...,C 4 are arbitrary constants.

References : G. B. Jeffery (1915), V. V. Pukhnachov (1960).

3 ◦ . Solution:

(1) where the function U (ξ) is determined by the autonomous ordinary differential equation

w = bθ + U (ξ),

ξ = θ + a ln r,

− a(b + 4ν)U ξξξ ′′′ + 2( b + 2ν)U ξξ ′′ +2 U ξ ′ U ξξ ′′ = 0. The onefold integration yields

ν(a 2 + 1) (4) U

ξξ ′′ + 2( b + 2ν)U ξ ′ +( U ξ ′ ) = C 1 , (2) where C 1 is an arbitrary constant. Equation (2) is autonomous and independent of U explicitly. The

ν(a 2 + 1) U

− 2 a(b + 4ν)U

transformation

z=U ′ ξ ,

u(z) = U ′′ ξξ

brings it to the Abel equation of the second kind

z − a(b + 4ν)u + 2(b + 2ν)z + z = C 1 , (3) which is integrable by quadrature in some cases; for example, in the cases a = 0 and b = −4ν, we

3 z + 2( b + 2ν)z =2 C 1 z+C 2 if

a = 0,

b = −4ν. Four other solvable cases for equation (3) are presented in the book by Polyanin and Zaitsev (2003);

2 2 2 3 ν(a 2 + 1) u +

3 z −4 νz =2 C 1 z+C 2 if

(3) is first reduced to a canonical form with the change of variable u = k ¯u, where k = const.

Note that to a = b = 0 in (1)–(3) there corresponds a solution dependent on the angle θ alone; this solution can be written out in implicit form, see equation 10.3.2.1, Item 8 ◦ ✲✂✳ .

Reference : L. G. Loitsyanskiy (1996).

4 ◦ . Generalized separable solution linear in θ:

w(r, θ) = f (r)θ + g(r). Here, f = f (r) and g = g(r) are determined by the system of ordinary differential equations

r ′ L( f ) + f [L(f )] ′ r = νrL ( f ),

(5) where L( −1 f)=r ( rf

− 2 g ′ r L( f ) + f [L(g)] r ′ = νrL ( g), − 2 g ′ r L( f ) + f [L(g)] r ′ = νrL ( g),

dr

f (r) = C 1 ln r+C 2 ,

g(r) = C 3 r + C 4 ln r+C 5 rQ(r) dr + C 6 , r

Q(r) =

r ( C /ν)−1 exp

ln r dr,

2 ν ✴✂✵ where C 1 , ...,C 6 are arbitrary constants.

References : R. Berker (1963), A. D. Polyanin (2001 c).

where E w=r

+ 2 , E w = E(Ew).

∂r r ∂r

∂z 2

Preliminary remarks. The stationary Navier–Stokes equations written in cylindrical coordinates for the axisymmetric case can be reduced to the equation in question by the introduction of a stream function

r ∂z and u z =− r ∂r , ✴✂✵ where r= px + y , and u r and u z are the radial and axial fluid velocity components.

1 w such that u 1 r = ∂w ∂w

Reference : J. Happel and H. Brenner (1965).

1 ◦ . Any function w = w(r, z) that solves the second-order linear equation Ew = 0 is also a solution of the original equation.

2 ◦ . Solutions in the form of a one-argument function and the sum of functions with different arguments:

w(r) = C 1 4 2 r 2 + C 2 r ln r+C 3 r + C 4 ,

w(r, z) = Aνz + C 1 r A+2

where

1 , A, C ...,C 4 are arbitrary constants.

3 ◦ . Multiplicative separable solution:

w(r, z) = r 2 f (z),

where the function f = f (z) is determined by the ordinary differential equation (C is an arbitrary constant):

zzz +2 ff zz −( f z ) = C. (1) This solution describes an axisymmetric fluid flow towards a plane (flow near a stagnation point). ✴✂✵

Reference : H. Schlichting (1981).

4 ◦ . Generalized separable solution quadratic in r (generalizes the solution of Item 3 ◦ ):

w(r, z) = r 2 f (z) + Az + B,

where

A and B are arbitrary constants, and the function f = f (z) is determined by the ordinary differential equation (1).

5 ◦ . Generalized separable solution linear in z:

w(r, z) = ϕ(r)z + ψ(r). Here, ϕ = ϕ(r) and ψ = ψ(r) are determined by the system of ordinary differential equations

ϕ[L(ϕ)] ′

r L( ϕ) − 2r ϕ L(ϕ) = νrL ( ϕ),

(3) where L( −1 ϕ) = ϕ ′′

ϕ[L(ψ)] 2

r − ψ r L( ϕ) − 2r ϕ L(ψ) = νrL ( ψ),

rr − r ϕ ′ r .

ϕ(r) = C 1 r 2 + C 2 ,

where C 1 and C 2 are arbitrary constants. In this case, the change of variable U = L(ψ) brings (3) to

a second-order linear equation.

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

1 ∂w where E 2 w=

sin θ ∂

E w = E(Ew).

∂r

r ∂θ sin θ ∂θ

Preliminary remarks. The stationary Navier–Stokes equations written in spherical coordinates for the axisymmetric case are reduced to the given equation through the introduction of a stream function w such that u r =

r sin θ ∂r , where r= px + y + z , and u r and u θ are the radial and angular fluid velocity components. References : A. Nayfeh (1973), M. D. Van Dyke (1975).

∂w

1 ◦ . Any function w = w(r, θ) that solves the second-order linear equation Ew = 0 is also a solution of the equation in question.

Example. Solution:

w(r, θ) = (C 1 2 r −1 + C 2 r ) sin 2 θ,

where C 1 and C 2 are arbitrary constants.

2 ◦ . Self-similar solution:

w(r, θ) = νrf (ξ), ξ = cos θ,

where the function f = f (ξ) is determined by the first-order ordinary differential equation

and C 1 , C 2 , and C 3 are arbitrary constants.

The Riccati equation (1) is reduced, with the change of variable 2 f = −2(1 − ξ ) g ′ ξ /g, to the hypergeometric equation

2 2 (1 − 2 ξ ) g

ξξ ′′ +( C 1 ξ + C 2 ξ+C 3 ) g = 0,

which, in the case

2 C 2 1 ξ + C 2 ξ+C 3 = A(1 − ξ ), has power-law solutions:

Special case. In the Landau problem on the outflow of an axisymmetric submerged jet source, the solution of equation (1) is given by