The fundamental governing equations for fluid mechanics are the NavierStokes equations. This inherently nonlinear set of partial differential equations has no general solution, and only a small number of exact solutions have been found.
Exaet solutions arc important for the following reasons:
1. The solutions represent fundamental fluid-dynamic flows. Also, owing to the uniform validity of exact solutions, the basic phenomena described by the Navier-Stokes equations can be more closely studied.
2. The exact solutions serve as standards for checking the accuracies of the many approximate methods, whether they are numerical, asymptotic, or empirical. Current advances in computer technology make the complete numerical integration of the Navier-Stokes equations more feasible. However, the accuracy of the results can only be ascertained by a comparison with an exact solution.
Now, let us define what we mean by an exact solution of the NavierStokes equations. Let q(x, t) be the velocity vector, a function of space x and time t. Let p(x, t) be the pressure. The constant-property NavierStokes equations are
(1)
159 0066--4189/91/0115-0159$02.00
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V'q = 0, (2)
where the density p and the kinematic viscosity v are constants. Conservative body forces may be absorbed into the pressure term. Although p can be eliminated by taking the curl of Equation (1), the constant v (or, in nondimensional form, the Reynolds number) is a basic parameter. An exact solution is defined as a solution of Equations ( 1) and (2) that is valid for all x, t and for all values of v. Obviously, all closed-form solutions of Equations (1) and (2) are exact solutions. Direct numerical solutions of the partial differential equations, no matter how accurate, are not exact solutions because the value of v has to be assigned for each solution. On the other hand, similarity solutions, where v is implicit in the similarity transforms, and where universal curves can be obtained once and for all, are exact solutions. We opt to exclude from our definition infinite-series solutions obtained from expansions or separation of variables. The reason is that the series could not be exact unless summed to infinity. The reader may disagree from this viewpoint. The degenerate potential-flow solutions, although satisfying the Navier-Stokes equations, are also excluded from our discussion.
The existing exact solutions have been published in a wide variety of journals, spanning a century or more. Most of the exact solutions are obtained by a variety of methods and address specific fluid-dynamic problems, resulting in minimal cross referencing. It would be difficult for a researcher in fluid mechanics to know, for a certain problem, whether an exact solution exists or not. A notable example is the oblique stagnation flow on a plate, which was solved independently three times within a span of 27 years!
The only comprehensive review of exact solutions of the Navier-Stokes equations is that due to Berker ( 1963), which expanded on the earlier works of Berker ( 1936) and Dryden et al ( 1932). Other sources include those of Whitham ( 1963) and Schlichting ( 1968, pp. 76-103). Since then, many new solutions have appeared, and a new review is necessary. The unsteady exact solutions were recently reviewed by Wang (1989a). The present work is the complement of that source, being a review of the steady exact solutions. Because of the difficult nature of the task, in spite of a careful search, there may be works that escaped our notice. To these authors, we offer our sincere apologies.
PARALLEL FLOWS
For parallel flows, the nonlinear convection terms in the equations are identically zero. Superposition of solutions in the same domain is pos-
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EXACT NAVIER-STOKES SOLUTIONS 161
sible if the governing equations and boundary conditions are linear and homogeneous.
Let (x, y, z) be Cartesian coordinates and velocity w be in the z direction only. The steady Navier-Stokes equations reduce to
(3)
The pressure gradient is necessarily a constant, say pz = cpv. lf the pressure gradient is zero, we have the Laplace equation in two dimensions:
(4)
There are infinitely many solutions to Equation (4). The physical problem involves flows due to longitudinally moving boundaries. A well-known example is the Couette flow between two plates. Other examples can be found in Berker ( 1963) and in potential-theory literature.
One important class of parallel flows is the flow in long cylinders driven by a pressure gradient. The governing equation is the Poisson equation,
(5)
with w = ° on a closed boundary. Exact solutions include the Poiseuille flow between plates and in a circular tube and other cross sectional geometries, such as annuli, eccentric circles, ellipses, confocal ellipses, equilateral triangles, circles with circular notches, lima�ons, lemniscates, epitrochoids, etc. An excellent review has been given by Shah & London (1978). Since the flow in tubes is analogous to the torsion of elastic cylinders, elasticity literature (such as Poschl 1921, Higgins 1942, Muskhelishvili 1963, pp. 571-667; Timoshenko & Goodier 1970, pp. 291-353) should be consulted. Note that we have excluded the solutions using infinite series, such as that of the flow through a rectangular duct.
If the flow in a cylinder is symmetrical about a bisecting line, half of the solution represents the flow in an inclined trough. Gravity acceleration now replaces the pressure gradient.
The following subsections desribe some flows closely related to parallel flows.
Concentric Flows
Let (r, e, z) be cylindrical coordinates. Concentric flows are those in which the velocity v is in the e direction only. The Navier-Stokes equations give
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1 v Vrr+ - Vr- 2" = O. r r (6)
Examples are the flow between concentric rotating circular cylinders and the flow due to a single rotating cylinder in an infinite fluid. Concentric flows and parallel flows can be superposed, provided that the boundaries coincide.
Flows That are Essentially Parallel or Concentric
These flows are governed by linear equations. There are two main categories: asymptotic suction flows and spiral flows.
If a constant normal velocity -V is added to a parallel flow w(y), the constant-pressure Navier-Stokes equations become
(7)
A solution is the asymptotic suction profile
w = W[l-exp ( - Vy!v)], (8)
where W is a constant. Equation (7) is also applicable to the flow between two porous plates with injection on one plate and suction of the same magnitude on the other plate (Berman 1958). The axisymmetric analogue of the asymptotic suction profile on a porous plate is the longitudinal flow over a porous circular cylinder with constant suction. The governing equation is
- �a Wr = V ( wrr+ � W,) .
The solution,
[ (a)VaIV] w=Vl- - , r
(9)
(10)
was found independently by Wuest (1955), Lew (1956), and Yasuhara (1957). The flow in an annulus with radial velocity -Va!r was studied by Berman (1958), and that along a corner with suction was examined by Stuart (1966).
The fluid motion due to a rotating cylinder with uniform suction is governed by
- �a (Vr+ �) = v (Vrr+ � Vr- ;2). (11)
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EXACT NAVIER-STOKES SOLUTIONS 163
The spiral-flow solution was found by Hamel (1916). A potential stagnation flow superposed on a concentric flow gives the equation
- ar (Vr+ �) = v (Vrr+ � Vr- ;2). (12)
Burgers (1948) gave a solution that can be applied to a viscous vortex. It was later extended by Sullivan (1959). Many other spiral flows were given by Berker (1963).
Other Linear Flows
These flows are neither parallel nor essentially parallel. They are governed by nondegenerate linear equations, i.e. the order of the governing equations is the same as in the original Navier-Stokes equations. The solutions reviewed here are pseudo-plane flows, where all path lines lie in their own plane, or essentially pseudo-plane flows. Other pseudo-plane flows can be found in Berker's (1936, 1963) works.
The most important of these flows is Ekman flow. In a coordinate system rotating with angular velocity 0, let the Cartesian velocities be [u(z), v(z), 0]. The governing equations become
20u = vvzz• (13)
Ekman (1905) found the solution for a moving plate in a rotating system to be
(14)
Ekman flow was extended by Gupta (1972) to include suction or weak injection on the plate. The flow due to a pressure gradient between two plates in a rotating system was studied by Vidyanidhi & Nigam (1967) and Vidyanidhi et al (1975).
A related pseudo-plane flow is that between two noncoaxial rotating plates, studied by Abbott & Walters (1970) but first considered by Berker (1963). Nonunique solutions were found by Berker (1979, 1982). Addition of a constant suction was considered by Erdogan (1976) and Rajagopal (1984).
GENERALIZED BELTRAMI FLOWS
Let q be the velocity and, == V x q be the vorticity. The vorticity equation is obtained by taking the curl on the steady Navier-Stokes equation
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v X ({ X q) = -vV X V X {. (15)
For parallel or concentric flows, the nonlinear terms on the left-hand side of Equation (15) are identically zero. For essentially parallel and related flows, some convection effects are retained, resulting in a linearization of Equation (15). In order to seek a more general simplification of the nonlinear terms, one must place restrictions on the properties of the vector fields. This process in general reduces the order of the governing equations, such that boundary conditions cannot be assigned a priori. For example, the no-slip condition is almost never satisfied.
Consider the velocity field q. The flow is solenoidal due to the continuity equation (2). Here we have excluded all lamellar (irrotational) fields where , = O. The complex lamellar fields with C· q = 0 include all two-dimensional and axisymmetric flows but do not simplify Equation (15). The Beltrami flows or screw fields are those with the property { x q = 0 (i.e. vorticity is parallel to velocity and the flow is necessarily three dimensional). In the case of a Trkalian field with C = cq, one can show that no steady flow exists.
In what follows, we study the generalized Beltrami flows where
V X (C x q) = O. (16)
From Equation (15) we find the additional condition
V X (V X C) = O. (17)
Note that the order of Equation (16) is lowered. Generalized Beltrami flows are two-way flows, since the sign of the velocity can be changed without altering the streamlines. Two-way flows have been discussed by Irmay & Zuzovsky (1970). On the other hand, since viscosity is absent, steady generalized Beltrami flows are also called circulation-preserving flows or universal flows (e.g. Marris 1981). Although the solutions are inviscid, the flow is rotational and is generated by nonzero viscous stresses at some distant boundary. Marris (1969) has discussed the three-dimensional motion that satisfies Equations (2), (16), and (17), but so far no examples have been found.
Planar Case
Planar generalized Beltrami flows were first investigated by Kampe de Periet (1930, 1932). For planar flow, a streamfunction t{I(x,y) can be defined from Equation (2):
q = V x (t{lk). (18)
Here k is the unit vector normal to the plane. Let
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EXACT NAVIER-STOKES SOLUTIONS 165
Then the vorticity vector is given by
,,;,, V x q = -K(x, y)k.
Equations (16) and (17) give
(20)
(21)
(22)
Equation (21) shows that the vorticity K is a function of streamfunction only, i.e.
K = f(t/I). (23)
An exact solution results when Equations (19), (22), and (23) are made
compatible. For the special case of K = constant, the governing equations reduce to
(24)
Some useful examples, excluding parallel or concentric flows, are as follows:
Source or vortex in shear flow (Tsien 1943):
Shear flow over convection cells (Wang 1990a):
t/I = ay 2 +be-AY cos AX.
Elliptic vortex (Kirchhoff 1883):
t/I = ax2+by2.
Oblique impingement of two jets:
t/I = y(ay+bx).
Axisymmetric Case
(25)
(26)
(27)
(28)
(29)
Let (r, 0, z) be cylindrical coordinates and (u, 0, w) be the corresponding velocities. Define a streamfunction t/I such that
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IjJ = frudz = -frwdr.
Then the vorticity is given by
Equation (16) gives
or
K = rf(IjJ).
Equation (17) gives
(rKr) -r- r
+Kzz = O.
(30)
(31)
(32)
(33)
(34)
An exact solution is obtained when Equations (31), (33), and (34) are made compatible. Strakhovitch (1934) noted that for constant/, the solution to this set of equations is
(35)
where ljJirro is any axisymmetric potential flow. In a lengthy proof, Marris & Aswani (1977) showed that f = constant is the only admissible case for axisymmetric generalized Beltrami flows. The effect of swirl was discussed by Weinbaum & O'Brien (1967). Some useful axisymmetric solutions are given below:
Impingement of two rotating streams (Berker 1963):
(36)
Rotational stagnation flow over a plate (Agrawa1 1957):
(37)
This solution is very special for Beltrami-type flows, since a solid boundary is present at z = O.
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EXACT NAVIER-STOKES SOLUTIONS 167
Ellipsoidal and spherical vortices:
(38)
This fonn of the solution is much simpler than that of O'Brien (1961), who used ellipsoidal coordinates. The case b = c is Hill's spherical vortex (Hill 1894).
Long recirculating regions (Wang 1990b):
I/l = ar2[cr2-I+b(r4-12r2z2+8z4)].
Flow in a porous pipe (Terrill 1982):
I/l = a(r4 - 2r2) + L c.r J I (A.r)eA.z. a
(39)
(40)
Here A. is a zero of the Bessel function J Q. The constants c. can be adjusted for arbitrary normal injections on the cylinder at r = 1. A more general form of Equation (40) was given by Berker (1963).
Extensions of Planar Generalized Beltrami Flows
From Equation (23) we find that if vorticity follows the streamlines, the nonlinear terms in the Navier-Stokes equations, although not individually zero, would sum to zero. Now if Equation (23) is linearized, i.e.
K = al/l+B(x,y), (41)
where a is a constant and B is a known function, the Navier-Stokes equations are linearized as well. This idea is implicit in some of the solutions in Berker (1963). Equations (18) and (19) become
K = I/lxx+l/lyy = mjJ+B,
v(Kxx+Kyy)-Kxl/ly+Kyl/lx = o.
Eliminating K in Equation (43), we find that
Bxl/ly-Byl/lx-va21/l = vaB+v(Bxx+Byy).
(42)
(43)
(44)
An exact solution results when Equations (42) and (44) arc made compatible. In particular, if B is linear, we set B = by without loss of generality and let \}I = at/! + by. Then thc equations become linear and homogeneous in \}I. The solutions are viscosity dependent and are no longer two-way flows. Some interesting examples are as follows:
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Flow downstream of a two-dimensional grid (Kovasznay 1948):
(45)
Impingement of a rotating flow and a uniform flow (Riabouchinsky 1924):
I/t = Vy[exp (- Vx/v) -1]. (46)
The surface of contact is the plane x = o. Symmetry exists only about y = O. Another impingement flow has a curved contact surface (Wang 1966):
I/t = - Vy+c sinh AY exp [( - R±J R2 -4) A;] . (47)
where R is a Reynolds number V/(VA), and A is a constant. This solution is similar to the flow near a plate with suction (Lin & Tobak 1986). When R < 2, the flow becomes oscillatory in x (Hui 1987).
SIMILARITY SOLUTIONS
So far we have discussed cases where the nonlinear terms in the NavierStokes equations are identically zero, or sum to zero, or are forced to linearize. Similarity solutions are the exact solutions that most fully take into account the nonlinearities of the convection terms. Phenomena such as nonexistence and nonuniqueness may occur, and analytic analyses of these and other phenomena, such as stability, are greatly facilitated by the reduction of Navier-Stokes equations to ordinary differential equations.
Similarity solutions exist only for problems showing certain physical symmetries. Similarity transforms that reduce the Navier-Stokes equations to nonlinear ordinary differential equations may be obtained by dimensional analysis or by the method of stretchings (e.g. Hansen 1964, Ames 1965, pp. 123-67). The general theory is that of infinitesimal Lie transformation groups applied to partial differential equations (Bluman & Cole 1974, Olver 1986). However, group-theoretic methods have not been successful in discovering new exact steady solutions of the Navier-Stokes equations, although some complicated unsteady solutions have been found (Cantwe111978, Boisvert et aI1983).
In what follows we describe some steady similarity solutions of the Navier-Stokes equations. Some of the solutions do have a length scale and thus are dependent on the Reynolds number. Strictly speaking, these solutions, in our definition, are not exact solutions, since for each Reynolds
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EXACT NAVIER-STOKES SOLUTIONS 169
number a new integration, even for ordinary differential equations, must be performed. However, since similarity solutions for the Navier-Stokes equations are so rare, we have included even the few with length scales for interested readers.
Radial Flows
Two-dimensional radial flow was first studied by Jeffrey (1915) and Hamel (1916). Since the flow is relatively well known, one may refer to Berker (1963) and Whitham (1963) for detailed discussions.
In planar cylindrical coordinates (r,8), let the radial velocity be given by
vF(8) u= -- .
r
The Navier-Stokes equations reduce to
FIII+2FF'+4F' = o.
(48)
(49)
The solution to this nonlinear differential equation can be expressed in terms of elliptic functions. For a nonuniform source in an unbounded fluid, F(8) is required to be 2n periodic. More interesting is the flow between two nonparallel converging or diverging plates. It was found, depending on the angle between the plates, that the flow may not be unique for a given total flow rate. In general, the vorticity of converging or accelerating flow tends to be confined in boundary layers near the walls. For diverging plates, reverse flow may be present.
There are no axisymmetric radial-flow solutions, although three-dimensional radial flows may exist.
Rotating Disks
The flow due to an infinite rotating disk with rotation rate Q is a classical example of a problem that leads to similarity solutions. In cylindrical coordinates (r, 8, z), let (u, D, w) be the corresponding velocities. Von Karffijin (1921) used the transformation
D = Qrg(�), (50)
where
(51)
to reduce the Navier-Stokes equations to
(52)
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g"-2f'g+2g'f = O.
The boundary conditions are
f(O) = 1'(0) = 0, g(O) = 1,
(53)
1'(00) = g(oo) = o. (54)
Equations (52)-(54) cannot be solved analytically, and thus numerical integration was done by Cochran (1934) and others. The solution in terms of boundary values is /,,(0) = 0.510225, g'(O) = -0.615917, and f( 00) = 0.4417. Since fluid near the disk is thrown out by centrifugal forces, a uniform flow toward the disk is induced. This suction flow confines the vorticity to a boundary layer near the plate. Of a different nature is the rotating flow over a fixed disk, studied by Bodewadt (1940). The right-hand side of Equation (51) is replaced by 1, and the boundary conditions are
Accurate numerical values are f"(0) = - 0.93934, g'(O) = 0.77139, and f(oo) = -0.66939. Since the normal velocity is now away from the disk, vorticity is no longer confined, and oscillations and nonuniqueness occur.
There are many extensions of the flow due to von Karman (1921) and B6dewadt (1940), including combinations of the two and the addition of a second disk. A recent review (Zandbergen & Dijkstra 1987) included many references related to this topic. Researchers should consult the Science Citation Index for the numerous papers quoting von Karman and Bodewadt.
Stagnation Flows
Let (u, v, w) be the Cartesian coordinates in the (x, y, z) directions. Let
u = axf'(e), v = Aayg'(e), w = Fa(f+Ag), (56)
where
(57)
The Navier-Stokes equations then reduce to the ordinary differential equations
Hiemenz (1911) first solved the two-dimensional stagnation flow toward a plate (). = 9 = 0). The boundary conditions are
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f(O) = reO) = 0,
EXACT NAVIER-STOKES SOLUTIONS 171
r(oo) = 1. (60)
Numerical solution gives f" (0) = 1.232588. There is a confined boundary layer near the plate at z = O. Notice that a solution does not exist if the (potential) flow at infinity is reversed, i.e. r (00) = - 1. The axisymmetric case (A = 1,f = g) was solved by Homann (1936), and three-dimensional cases (A ¥= 0, 1) were considered by Howarth (1951) and Davey (1961). Axisymmetric stagnation flow on a cylinder was solved by Wang (1974a).
In the case of stagnation flow on a fluid surface of different property, the tangential velocity on the interface is not zero but is determined by balancing shear stress (see Wang 1985, 1987).
The two-dimensional oblique stagnation flow was solved by Stuart (1959) and later by Tamada (1979) and Dorrepaal (1986). The flow far from the plate is the inviscid rotational flow described by Equation (28).
Porous Boundaries
A porous boundary is approximated by a given normal velocity and zero tangential velocity. Such a boundary occurs in transpiration cooling and gaseous diffusion processes. The governing equations for the fluid are the same as for the nonporous cases, but the boundary conditions are changed slightly.
Suction or injection on one rotating disk has been studied by Stuart (1954), Rogers & Lance (1960), Evans (1969), Kuiken (1971), and Ackroyd (1978). The flow between two disks, one or both of which are porous, was studied by Dorfman (1966), Rasmussen (1970), Wang (1976), Wilson & Schryer (1978), and Wang & Watson (1979). Through use of a free surface, condensation and melting on a rotating disk can be modeled (Sparrow & Gregg 1959, Wang 1989b).
The flow between porous channels with a longitudinal pressure gradient was first investigated by Berman (1953). Numerical solutions were obtained by Terrill (1964), Terrill & Shrestha (1965), and Shrestha & Terrill (1968). The porous-tube problem was studied by Yuan & Finkelstein (1956), Terrill & Thomas (1969), and Skalak & Wang (1978). The addition of swirl was considered by Prager (1964) and Terrill & Thomas (1973). The above solutions depend strongly on an injection Reynolds number. In certain instances, nonexistence and nonuniqueness were found.
Addition of a Translating Boundary
An exact solution with a planar boundary may be extended by translating the boundary in its own plane. The effects of lateral translation are influenced by the original exact solution but not vice versa. This
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172 WANG
one-way coupling differs from the direct effects of a normal velocity on the boundary.
The two-dimensional stagnation flow toward a plate moving with constant velocity in its own plane was obtained by Rott (1956). Axisymmetric stagnation flow on a moving plate was considered by Wang (1973) and Libby (1974), and stagnation flow on a translating and rotating cylinder was studied by Gorla (1978). The flow between moving porous plates can be applied to air-cushioned sliders (Wang 1974b, Wang & Skalak 1975, Watson et aI1978). The cases of rotating and translating flow or a rotating and translating disk were solved by Rott & Lewellen (1967). Shear flow over a rotating disk was considered by Wang (1989c).
Stretching Flows
The flow due to a stretching surface may be applied to the extrusion of sheet materials. The transformation is similar to Equations (56) and (57). Due to the zero tangential pressure gradient, the constant terms in Equations (58) and (59) are absent. The boundary conditions also differ slightly. For two-dimensional flow (A = 0), these conditions are
f(O) = F( CIJ) = 0, F(O) = 1. (61)
Crane (1970) obtained the rare closed-form similarity solution
f = 1 - exp ( - �). (62)
Axisymmetric and three-dimensional cases were studied by Wang (1984). The flow inside a stretching channel or tube was solved by Brady & Acrivos (1981), and the flow outside a stretching tube was investigated by Wang (1988a). Other extensions include suction or injection from the surface (Gupta & Gupta 1977), stretching in a rotating system (Wang 1988b), and flow over a stretching surface (Danberg 1976).
Spherical Coordinates
Slezkin (1934) was the first to note that when velocities are inversely proportional to the distance from the origin, the Navier-Stokes equations admit axisymmetric similarity solutions. Let (u, v, w) be the velocities in the directions of spherical coordinates (R, I/J, 8), respectively. Further, let
F(x)
v =
RsinI/J'
O(x) w = RsinI/J'
where x == cos I/J. The Navier-Stokes equations then reduce to
v(1-x2)F"" -4vxF'" +FF'" +3F' F" = -200'j(l-xl),
(63)
(64)
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EXACT NAVIER-STOKES SOLUTIONS 173
v(1 - X2)n" + Fn' = O. (65)
When swirl is absent (n = 0), Landau (1944) first found the closedform exact solution that describes the flow due to a momentum jet in an infinite fluid:
l -x2 F = 2v --. x -a
(66)
The same solution was rediscovered by Squire (1951), while Yatseyev (1950) used an ingenious transform to linearize Equation (64) and thus obtained the general solution in terms of hyper geometric functions. Squire (1952) extended the momentum-jet solution to a jet emerging from a wall, but the no-slip condition cannot be satisfied. Other jet solutions, some with infinite velocities on the axis, were considered by Yatseyev (1950), Squire (1955), and Paull & Pillow (1985a). Wang (1971) found solutions that can be applied to the spreading of material on the surface of a fluid. Squire's (1952) solution should be interpreted as a special case of Wang's solution.
The equations with nonzero swirl n were first analyzed by Goldshtik (1960). Applications to vortices interacting with solid boundaries were given by Long (1958), Guilloud & Amault (1971), Serrin (1972), Guilloud et al (1973), Yih & Wu (1982), and Paull & Pillow (1985b). As in Squire's (1952) work, it was not possible to satisfy the no-slip condition (Morgan 1956, Potsch 1981).
DISCUSSION
An exact solution of the Navier-Stokes equations, although theoretically valid for all Reynolds numbers, may be limited by instability. The first kind of instability is well known: the transition to turbulence at higher Reynolds numbers. The second kind has to do with sensitivity to undefined or altered boundary conditions. For example, nonuniqueness often occurs in most exact solutions where the vorticity is transported into the region from infinity. In these cases, the exact solutions are unlikely to be realized in practice.
In this brief review, we have considered the steady exact solutions of the Navier-Stokes equations. These solutions may be classified into three major types: parallel and related flows, generalized Beltrami and related flows, and similarity solutions. Recently, the method of hodograph transformation has been used to study steady plane viscous flows (LeFur 1978, Chandra et aI1982). However, this method does not seem to lead to new exact solutions. In time, additional exact solutions will be discovered. It
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should be emphasized, however, that in order for an exact solution to be
significant, it must have some potential physical application.
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