Slow flow in channels with porous walls
We consider the slow flow of a viscous incompressible liquid in a channel of constant but arbitrary cross section shape, driven by non-uniform suction or injection through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a set of coupled equations for the velocity potential in two dimensions. When the channel aspect ratio and Reynolds number are both small, the problem reduces to solving the biharmonic equation with constant forcing in two dimensions. With the relevant boundary conditions, determining the velocity field in a porous channels is thus equivalent to solving for the vertical displacement of a simply suspended thin plate under uniform load. This allows us to provide analytic solutions for flow in porous channels whose cross-section is e.g. a rectangle or an equilateral triangle, and provides a general framework for the extension of Berman flow (Journal of Applied Physics 24(9), p. 1232, 1953) to three dimensions.
Channel flows – liquid flows confined within a closed conduit with no free surfaces – are ubiquitous. In animals (LaBarbera, 1990) and plants (Holbrook & Zwieniecki, 2005) they serve as the building blocks of vascular systems, distributing energy to where it is needed and allowing distal parts of the organism to communicate. When constructed by humans, one of the major functions of channels is to transport liquids or gasses, e.g. water (irrigation and urban water systems) and energy (oil or natural gas) from sites of production to the consumer or industry.
In some cases, the channels have solid walls which are impermeable to the liquid flowing inside. In other cases, the channels have porous walls which allow the liquid to flow across the wall and thus modify the axial flow. Both are important. The first class of flow has been studied in great detail, and analytical solutions are known in a few, but important, cases (Batchelor, 1967). The latter class has received much less attention, although it is equally important. The effect of porous walls is especially important in the study of biological flows due to the presence of permeable cell walls (Holbrook & Zwieniecki, 2005) and in industrial filtration applications (Nielsen, 2012).
Some analytic solutions of the flow in porous walled channels are known, primarily due to a similarity technique first used in this context by Berman (1953). Berman’s method is closely related to those commonly used in boundary layer theory (Schlichting & Gersten, 2000) and allows for the solution of steady flows in geometries with symmetries which makes the problem two-dimensional. By demanding that the solution be of similarity form the Navier-Stokes equation is reduced to a single non-linear third-order differential equation for the velocity potential in one space dimension. The flow between parallel plates (Berman, 1953) and in a cylindrical (Yuan & Finkelstein, 1956) and annular tube (Berman, 1958) have been analyzed in this way. Time dependent flows, flows at high Reynolds numbers and questions of uniqueness and stability of these flows have since been address by a large number of workers, see e.g. Cox (1991).
In this paper, we extend Berman’s method to three dimensional similarity flows, and derive a set of equations for the velocity potential which are valid in channels of arbitrary cross section shape. At low Reynolds numbers, and when the channel is very long compared to its characteristic transverse dimension, the Navier-Stokes equation reduces to a single partial differential equation for a velocity potential in two space dimensions; the inhomogenous biharmonic equation with constant forcing. This equation, which is derived in Sec. II, has been widely studied in the literature as is describes the transverse displacement of a simply suspended thin elastic plate under uniform load. In Sec. II.3 we provide analytic solutions to four cases of porous channel flows in geometries where the solution of the corresponding elastic problem is known: Flow in a cylindrical tube, between parallel plates, in a triangular channel, and in a rectangular channel.
Ii Flow in channels with porous walls
We consider a long, straight channel parallel to the -direction, and assume that it is translational invariant along this axis with an arbitrary, but constant, cross section as shown in Fig. 1(a-b). The channel has length , perimeter , cross section area , and volume . The coordinates in the transverse -plane are denoted , so that the full coordinates are written as and likewise for the gradient operator , Laplace operator , and velocity field
where we have used the short hand notation .
We consider the case of incompressible Newtonian fluids of viscosity and density (in the laminar regime) which are governed by the Navier-Stokes equation
We assume that the flow is driven by a prescribed injection or suction of fluid through the porous wall which leads to a normal flow velocity component at the channel wall of typical magnitude . The boundary conditions thus require that the tangential velocity component vanishes on the channel wall and that the normal velocity component is
where is a outward pointing normal unit vector and is a tangent unit vector to the boundary in the plane.
Introducing the Reynolds number based on the wall velocity and the aspect ratio , and dropping the primes for ease of reading we finally have that
ii.2 Similarity solutions
The form of the boundary conditions is such that the in-plane velocity could be irrotational and proportional to everywhere (see Eqns. (7)-(8)), while the axial velocity should be proportional to the total volume of liquid which has entered the channel at , i.e. . It is worthwhile to enquire if the differential equation permits solutions of this form, and we therefore write the velocity as a similarity solution
Since , and therefore , is irrotational by assumption, we may write where is a velocity potential. Eq. (17) then implies that . The velocity field we seek is thus of the form
These boundary condition may be considerably simplified by noting that Eq. (23) implies that is constant on the boundary . (If consists of several physically separate boundaries, may take on different values on each of these). With this, Eqns. (22)–(24) become
where we have assumed that , , and use the notations and for first and second order normal derivatives.
ii.3 The case and
When both the Reynolds number and aspect ratio are small, we find from Eqns. (20)–(LABEL:eq:221) that
Eq. (29) implies that the pressure is a function of only, and we may write . By introducing , we find the governing equation for the velocity potential
ii.4 Analogy with the theory of simply suspended plates
The equation of motion for a thin suspended plate under a uniform transverse load is the inhomogeneous biharmonic equation
where is the displacement at any point from the position of equilibrium and depends only on the tension and mass of the plate. This equation is of the same form as Eq. (30), and the amplitude of the displacement may be taken to represent if
It appears therefore that if a solution of the problem of a suspended plate has been obtained, a problem of viscous motion in a porous tube has also been solved. This analogy is exact, so long as the boundary conditions in Eq. (25)–(27) are also fulfilled in the plate problem. This occurs when the plate is simply suspended, i.e. when on the boundary and , in which case Eqns. (25) and (26) are fulfilled. The final boundary condition (Eq. (27)) which determines the angle of deflection or the normal flow velocity across the membrane is set by the choice of if is simply connected or by and on each of the boundaries if is multiply connected.
Iii Flow solutions
Solutions of the inhomogeneous biharmonic equation (30) must satisfy the governing differential equation and boundary conditions characterizing each geometry. The fulfillment of the boundary conditions often presents considerable mathematical difficulties and thus in general, analytic solutions are rare. Taking advantage of the solutions known from plate bending theory (see e.g. Timoshenko (1964); Ventsel & Krauthammer (2001)), we are able to provide analytic solutions to porous channel flows in geometries where the solution of the corresponding elastic problem is known, such as in channels of rectangular and triangular cross section shape. In these cases the solution relies on either a parametrization of the boundary or a series solution. Before we consider these three dimensional cases, however, we illustrate the solution technique on a few two dimensional flow problems which have been solved by other means in the literature.
iii.1 Flow in a cylindrical tube
which has the solution
Assuming further that the radial flow component vanishes at () such that we find for the for the velocity potential in Eq. (33)
iii.2 Flow between parallel plates
which has the solution
The solution for the special case is shown in Fig. 3. In that case the velocity components are
These were first by obtained Berman (1953) who considered the case .
iii.3 Flow in an equilateral triangle
Consider the flow of liquid in a porous channel whose cross-section is an equilateral triangle of height and side length , as shown in Fig. 1(e). Let the normal wall velocity be given by , and let the mean value of when averaged over the channel wall be . The governing equation (30) is
The solution of Eq. (47) with the boundary conditions and is
The magnitude of the wall normal flow velocity is the same on each of the three boundaries. At the wall we find
The mean velocity (when averaged over the boundary) is so by choosing , the average inflow velocity becomes . Using in Eq. (49), the velocity components can be found from Eqns. (18)-(19) recalling that , and
The velocity field is shown in Fig. 4.
iii.4 Flow in an rectangular channel
Consider the flow in a porous channel whose cross-section is a rectangle with sides of length and centered at as shown in Fig. 1(d). Let the normal wall velocity be given by , and let the mean value of when averaged over the channel walls be .
The equations of motion and boundary conditions are once again given by Eqns. (47)-(48). As we need to evaluate the first and second order and -derivatives to determine the velocity field, Levy’s solution of the corresponding plate bending problem is convenient
To determine such what the average inflow velocity is , we solve for in
In the following we determine for the special cases and .
iii.4.1 The case
From Eq. (58) we therefore find
which determine as a function of the average normal flow velocity
iii.4.2 The case
To determine the velocity field in the case we again use Eq. (59). Near the centerline parallel to the -axis, i.e. for , we find from Eq. (59) that the wall velocity so with the condition in Eq. (58) is fulfilled. For the transverse velocity along the centerline we obtain from Eq. (55)
This series can be summed
We have analyzed slow flow in channels with porous walls. A similarity transformation reduces the Navier-Stokes equations to a set of coupled equations for the velocity potential in two dimensions. We have shown that when the Reynolds number and channel aspect ratio is small, an analogy exists between flow in channels with porous walls and bending of simply suspended plates under uniform load. If a solution of the problem of a suspended plate has been obtained, a problem of viscous motion in a porous tube has thus also been solved. We have applied this result to flow in rectangular and triangular channels. Our results provide a general framework for the extension of Berman flow (Berman, 1953) to three dimensions.
The author acknowledge many fruitful discussions with Hassan Aref and Tomas Bohr. This work was supported by the Materials Research Science and Engineering Center (MRSEC) at Harvard University.
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