Partial Fractional Equations and their ApplicationsView this Special Issue
Newtonian and Non-Newtonian Fluids through Permeable Boundaries
We considered the situation where a container with a permeable boundary is immersed in a larger body of fluid of the same kind. In this paper, we found mathematical expressions at the permeable interface of a domain , where . is defined as a smooth two-dimensional (at least class ) manifold in . The Sennet-Frenet formulas for curves without torsion were employed to find the expressions on the interface . We modelled the flow of Newtonian as well as non-Newtonian fluids through permeable boundaries which results in nonhomogeneous dynamic and kinematic boundary conditions. The flow is assumed to flow through the boundary only in the direction of the outer normal n, where the tangential components are assumed to be zero. These conditions take into account certain assumptions made on the curvature of the boundary regarding the surface density and the shape of ; namely, that the curvature is constrained in a certain way. Stability of the rest state and uniqueness are proved for a special case where a “shear flow” is assumed.
The flow of incompressible Navier-Stokes fluids and fluids of second grade through permeable boundaries and past porous walls has been studied under various conditions. The equation of motion for incompressible flows in Newtonian fluids (Navier-Stokes equations) under no-slip boundary conditions has been studied extensively from many perspectives. Since the pioneering papers of Leray [1–3] and Hopf  questions of the existence, stability [5, 6], and uniqueness of both classical and weak solutions have received more than their fair share of attention.
Recently the same issues have been studied for non-Newtonian fluids of second grade. The studies cover both weak solutions [7–12] and classical solutions for homogeneous Dirichlet boundary data  and nonhomogeneous boundary data [6, 14, 15].
Unlike Newtonian fluids, fluids of second grade (and other non-Newtonian species) have the property of developing “normal stresses differences” at boundaries. It was shown, for example, by Berker  that if an incompressible flow of a fluid of grade two satisfies the homogeneous Dirichlet boundary condition. The stress at the boundary is given by , where is the unit exterior normal to the boundary and is the vorticity. The wedge denotes a vector product. Thus there is a normal component of stress at the boundary in addition to the pressure. The question becomes what governs the flow across the boundary? Possible ways of circumventing this question may be to “prescribe” the normal component of the velocity field at the boundary or to prescribe mass or momentum flux. The prescription of shear stress has also been suggested. ([16, 17]). Nonlinear or non-Newtonian fluids are fluids like molten metals, multigrade oils, printing inks, paints, suspensions, polymer solutions, molten plastics, blood, protein solutions, and ice . These fluids cannot be described by the above model. The study of these interesting substances has proved to be very important with the growth of the polymer and plastics industry over the last four decades. Consequently, an interest has arisen to study the flow of these nonlinear fluids and, in the case of this model, second-grade fluids, through permeable boundaries. The boundary conditions alone in such circumstances are an interesting topic for study. Works by Berker  and Rajagopal and Gupta  can be mentioned in this regard.
In this study we shall provide an alternative approach through the formulation of “dynamics at the boundary,” the idea being that the normal component of velocity at the boundary is viewed as an unknown function which satisfies a differential equation intricately coupled to the flow in the region “enclosed” by the boundary.
A glimpse of the history of the research on non-Newtonian and Newtonian fluids around porous boundaries is given in Section 2. Notation and definitions precede Sections 4 and 5 which deal with the constitutive equations and the modelling of permeability. In Section 6 the expressions on the interface are given. The alternative model is studied and the stability and uniqueness are proved in Section 7. Section 8 concludes the study and further explorations are discussed.
Berker  studied the two-dimensional creeping flow of a second-order fluid with nonparallel porous walls. An additional velocity boundary condition was needed. The other conditions they used were due to the usual no-slip conditions. This additional velocity boundary condition was to prescribe the rate of shear at the wall. The problem was then solved numerically by a standard routine.
In 1989 Rajagopal and Kaloni  wrote remarks on boundary conditions for flows of fluids of the differential type. Rajagopal  discusses a lot of related issues. Rajagopal and Gupta  studied the flow of an incompressible fluid of second grade past an infinite porous plate subjected to either suction or blowing at the plate. They studied fluids modelled by No assumptions were made about the material moduli and . For the boundary value problem they considered, it was found that the velocity distributions do not depend on the normal stress modulus , but the pressure does. They found that it was possible to produce an exact solution which is asymptotic in nature for both “suction” and “blowing” at the plate if the material modulus . For , they found that such solutions could not exist in the case of blowing, a result which was in keeping with the classical incompressible fluid. Fosdick and Rajagopal  have shown that the model (1) whose material modulus exhibits anomalous behaviour was not to be expected of any fluid of rheological interest (also see ). Proudman studied an example of steady laminar flow at a large Reynolds number .
Beavers and Joseph  studied the flow of a Newtonian fluid over a porous surface in 1967. They found that if the governing differential system was not to be underdetermined, it was necessary to specify some condition on the tangential component of the velocity of the free fluid at the porous interface. It is usual in these analyses to approximate the fluid motion near the true boundary with an adherence condition for the tangential component of velocity of the free fluid at some boundary. Because of a certain ambiguity which is implied by the notion of a “true” boundary for a permeable material, it was found useful to define a nominal boundary. They fixed a nominal boundary by first defining a smooth geometric surface and then assuming that the outermost perimeters of all the surface pores of the permeable material are in this surface. Thus, if the surface pores were filled with solid material to the level of their respective perimeters, a smooth impermeable boundary of the assumed shape would result. This definition is precise when the geometry is simple (planes, spheres, cylinders, etc.) but may not be fully adequate in more complex situations. Beavers and Joseph’s  experiment was designed to examine the tangential flow in the boundary region of a permeable interface. The results of this experiment indicate that the effects of viscous shear appear to penetrate into the permeable material in a boundary layer region, producing a velocity distribution similar to that depicted in the following figure. The tangential component of the velocity of the free fluid at the porous boundary can be considerably greater than the mean filter velocity within the body of the porous material.
In Figure 1 the plane defines a nominal surface for the permeable material. The flow through the body of the permeable material, which is homogeneous and isotropic, is assumed to be governed by Darcy’s Law. Read more of the status on Darcy’s Law in . In the absence of body forces Darcy’s Law may be written as , where is the “permeability” of the material and is the volume flow rate per unit of the cross-sectional area. As such, represents the filter velocity rather than the true velocity of the fluid in the pores. The measured pressure gradient is denoted by .
3. Basic Notation
We work in Euclidean 3 space. The following notation will be used throughout: If and are second order tensors we shall use the notations and . Let be a bounded domain with a smooth (at least ) boundary . Let denote the unit exterior normal to at . We shall be concerned with smooth vector fields defined in such that on it has the form , where is the trace operator denoting boundary values and is a smooth scalar field defined on . Associated with we define the symmetric and skew-symmetric tensors and as and , where denotes the transpose of the gradient of . The rate of deformation tensor is related to by . We note that if is solenoidal then trace and, for any vector , , where denotes the vorticity associated with .
4. The Constitutive Equations
The stress tensor for the linear viscous Newtonian model is , with as the pressure, as the coefficient of viscosity, and as the velocity of the fluid. This model describes the flow of fluids like water and other similar fluids. Lamb  and Ladyzhenskaya  wrote mathematical theories on viscous incompressible flow.
Fluids of a differential type [29–31], of which Rivlin-Ericksen fluids are a subclass, are depicted by a popular nonlinear model. Fluids of complexity n form an important subclass of the fluids of a differential type. For incompressible fluids of complexity the Cauchy stress tensor is of the form . The pressure is not a thermodynamic variable and the term reflects Pascal’s law, which is inherent to all fluids. are the first Rivlin-Ericksen tensors  defined recursively by
Fluids of grade n are examples of fluids of complexity . The stress tensors for fluids of grades and respectively, are assumed to be of the form where and are material coefficients (possibly temperature-dependent).
For incompressible fluids of second grade, the stress-deformation relation then becomes where and are the pressure and the velocity fields. Here is the coefficient of viscosity and and are material coefficients or “normal stress moduli.” In this case .
To use the relation (5) for the modelling of a fluid, the fluid has to be compatible with thermodynamics in the sense that all flows of the fluid must satisfy the Clausius-Duhem inequality, and the assumption must be made that the specific Helmholtz free energy is at a minimum when the fluid is in equilibrium. Under these assumptions, and  must satisfy Considerations of stability of the rest state require the assumptions and to be nonnegative; that is, , . See . Under assumption (6), which we shall follow throughout, the form of the stress tensor given in (5) reduces to a more compact expression. To obtain this we note that and , so that Therefore, by (5) and (7) where we have set .
Remark 1. Please note that for the Navier-Stokes equations we take .
5. Modelling of Permeability
We study the motion of fluids around and through a fixed porous container filled with the same fluid. The interior of the porous container is an open bounded set and the porous boundary, , is smooth. The surrounding fluid domain, , is bounded and its outer boundary is denoted by . The exterior normal to on is denoted by . Figure 2 illustrates the situation where the curvature of the boundary of is nonnegative.
Permeability of the walls of the container is described by assuming that at the boundary the flow has the direction of the normal: The velocity component is treated as an unknown and an evolution equation has to be found for it. We model the surface as having an effective area measure which has a density function with respect to the area measure . Thus . The effective area through which fluid can permeate is not more than the surface area and therefore for any . If , the wall is impermeable and if , there is no wall.
In order to obtain expressions for mass and momentum in a boundary patch , we let the patch be heuristically represented by a volume built from copies of (Figure 3). This is in line with the Beavers-Joseph thinking which was discussed before. For this volume we set up a coordinate system consisting of a “radial part” , which has the direction of the normal vector , and a “surface part” made up by vectors tangential to . For the mass of we obtain where is some measure of thickness. With the aid of these concepts we introduce the surface density of the fluid at as where is the volume density of the fluid.
To obtain the equation of motion for fluid in the boundary, we assume that the rate of change of linear momentum in the boundary is explained by stress forces at both sides of the boundary.
Let and denote the stress tensors at the sides of the boundary facing and , respectively, and let and denote the transfer-of-momentum tensors on the two sides. On an arbitrary boundary patch the law of conservation of linear momentum is stated in the following way: with as defined in (11), and it follows that
From (13) we have . In the domain the momentum flux tensor is given by . In accordance with this, we shall take at the boundary. The tensor will be taken as zero.
We take to obtain from (13)
From the incompressibility of the flow in it follows that
6. Expressions at the Interface
In order to obtain expressions for the stress tensors and as well as the acceleration at the boundary through which only normal flow occurs, we obtain a formal expression for the symmetric tensor on a surface which is immersed in fluid. We shall eventually use these expressions in postulating the form of and and in formulating a boundary condition which expresses zero tangential acceleration at a wall.
We consider a smooth vector field defined on a domain and a smooth two-dimensional (at least class ) manifold so that and are defined on . Let be the unit normal to at the point .
At any point on we consider two orthogonal curves and in a neighbourhood of parametrised by arc lengths and , respectively. Let and be the unit tangents to the principal normal curves at a point on the surface. For local coordinates we use the orthogonal system formed by , , and . Under the convention that we have and . Let and represent the principal curvatures at a point on the surface and let denote the mean curvature.
Assumptions(1)We shall assume throughout that the surface density is bounded and bounded away from zero; that is, there exist constants and such that Also, we assume .(2)Apart from the smoothness of we make two additional assumptions regarding the shape of ; namely, that the curvatures , , and are constrained in the following way:(a)There exist constants and such that (b)There exists a constant such that Note that these assumptions allow cases where and can be of opposite signs.
The surface gradient of a scalar function may be written as where the trace operator denotes the normal derivative. Also consider If is a vector field defined on , the surface gradient is defined as the tensor Surface divergence and surface curl are defined as The relationship between the surface operators and the volume operators for a function defined in is given by We use (20)–(25) to prove more important results to make the calculations easier.
Lemma 2. Let and be two orthogonal unit tangential vectors and let be the exterior unit normal vector to . Let , , and be scalar functions; then
Proof. Consider the following:
We shall apply the expressions above to . By the Frenet-Serret formulae (torsion is zero) , and, similarly, . Hence The transpose is given by To find an expression for at , we need an expression for on the boundary: Although we know that the divergence of will be zero, it is helpful to observe that , where Hence .
We proceed to find expressions for , , and .
Thus we have
Let us define the symmetrical tensors and by with a tangential vector. Then, for a vector field of the form on , from (38) we have In local coordinates we have the representations If , it follows that tr, which is in line with incompressibility.
We would further like to obtain expressions for the terms , , and on the boundary .
Lemma 3. Let be the exterior normal to the boundary , , and with and as defined in (39). We assume that and , which implies that . Then where denotes the Gauss-curvature.
Proof. (a) We have chosen , , and so that . In view of the incompressibility and the fact that there is zero tangential velocity
(b) Consider the tensor built from “row vectors” with a basis for . Then Hence, .
Furthermore, and now Here we used the fact that . Determine term by term to obtain denotes the Gauss-curvature and is bounded by assumptions (17) and (18). Hence The term we use in the proof of (57) is therefore (c).
Here we make use of the additional boundary conditions (52) and (55) and the fact that to obtain that
6.1. Explicit Form of the Dynamic Boundary Condition
It is shown that for a smooth two-dimensional manifold contained in a domain the following is true for a vector field which is of the form on : where and are defined in (39).
If is solenoidal, as in the case under consideration, . A straightforward application of Stokes’ theorem shows that is tangential to the boundary, which implies that is tangential to the boundary. Indeed, let be any patch of the surface ; then where is a vector tangential to the boundary. Now if , then , and that implies that for all , which in turn implies that .
In the problem under consideration we shall assume that the “rate of deformation” tensor has precisely the form (52) on the boundary with the unit exterior normal (the traditional rate of deformation is defined as ).
We shall consider a kinematic boundary condition, which has a physical meaning in that there are no tangential components of deformation at the interface boundary. This concerns the form of the tensor .
Guided by these expressions and (8), we assume that, at , For the stress tensor in the fluid exterior to we assume that . This amounts to the situation where the fluid in is at rest. As a result we have
Equation (61) is the explicit form of the dynamic boundary condition.
7. An Alternative Model: Problem
Although it was possible to prove stability and uniqueness for the original model (see [33, 37]), we could not find a way to a possible proof of existence for a classical solution. In this chapter we describe an alternative model which displays all the properties of the original problem with respect to stability and uniqueness.
In the alternative model the dynamics at the boundary are formulated by assuming a “shear flow” of the form with and as the surface parameters (like arc length). It is assumed that the “body force” acting on the shearing fluid at the boundary is proportional to the difference between the pressures and . Under these assumptions the equation governing the evolution of is where , and is the resulting pressure through the boundary with thickness . is the Laplace-Beltrami operator () and denotes the surface gradient. The parameter has the physical dimension of length and may be thought of as the “thickness” of the “shear layer” (see , Sect 123, p. 506). Equation (63) is derived by calculating the stress tensor for a shear flow and noticing that terms of the form vanish. The term may be left out since, as before, it disappears when projections are taken. The kinematic boundary condition is still imposed.
All spaces of vector fields are denoted by boldface letters.(1)Let be a bounded domain in with a smooth boundary (of class ), , and .(2), for a nonnegative integer and , is the usual Sobolev space (of real-valued functions) embedded in with norm . , for a nonnegative integer, denotes the Sobolev space of order . By this agreement is the Sobolev space of three vector fields and the components are elements of . In particular, the norm and scalar products in are defined by and .(3)With the above notation denotes the Hilbert space of vector functions , with , for which is integrable on . The norm and scalar products for are defined as and .(4)There exists a linear continuous operator , called the trace operator, such that = the “restriction” of to for every function which is continuous in . The space is the kernel of . The image space is a dense subspace of denoted by . The trace operator is bounded. Indeed, there exists a constant such that Reference [39, Theorem 9.4, page 41]. We shall refer to this result (64) as the Trace theorem.(5)For the deformation we use the following notation for the norm and scalar products: and .(6)We define the domain by is a closed subspace of . Elements of satisfy the kinematical boundary conditions (55).(7) denotes the closure of in with respect to the Sobolev norm .(8)The norm of on the boundary is chosen as The associated scalar product is According to assumption (16) this is equivalent to the usual metric. It is assumed that the function .(9)For the purpose of stability and uniqueness we define the following norm: (10)We shall deal extensively with the energy associated with fluids of second grade defined for the purpose of this study by with and . is evidently a norm on . We shall refer to the quantity as the energy norm of .(11)The constant , which appears in inequalities, denotes a generic positive constant. Sometimes it is necessary to indicate the quantities on which a constant depends in brackets or by a subscript.
7.2. Important Identities
Identity 1. For any symmetric tensor and any arbitrary tensor , we have , with .
Proof. Consider the following:
Expressions are necessary for inner products of the form , where is either a vector or a second order tensor. is the material derivative. In order to obtain simple expressions for the scalar product, we notice that if denotes either the usual scalar product or the “colon” product, then provided . Hence the following identity.
Identity 2. For any smooth vector or tensor quantity and any , we have
Proof. By the divergence theorem
Later in this study we shall employ the energy method. It will become necessary to use the various boundary conditions in order to prove stability. The following is important to obtain the required results.
Identity 3. If is a scalar field and , then
Proof. Integrating by parts and using the fact that is solenoidal
We note that, in particular for , the imbedding of in the space of bounded continuous functions makes the choice possible, and it follows from Identity 3 that For we may also choose , and it follows that since on .
The following will be of immediate importance.
Identity 4. For any ,
Proof. From the definition of it is evident that . Now , and, since , , integration over and Identity 1 yield
Thus, if the curvature is positive everywhere on , it becomes apparent that if and only if .
Identity 5. For any bilinear form on a Hilbert space , we have, for any and with that .
Identity 6. Let and be tensor fields of the same order and let denote the “scalar product” in such tensor fields. For it is true that
Proof. Consider the following: thus
The following two lemmas are important in establishing a Poincaré inequality.
Lemma 5. The bilinear forms and are bounded in the space . and are positive constants.
Lemma 6. The bilinear form is coercive in the sense that there exist constants and such that
Proof. From (83) we have
We can now obtain a generalised Poincaré inequality.
Lemma 7. There exists a smallest possible constant such that, for every ,
Proof. From the smoothness of (which is always assumed), it follows that the embedding is compact . From the boundedness and coerciveness proved above it follows that there exists a smallest eigenvalue and associated eigenfunction for which (see ): , for if it is zero, it follows that , which cannot be. It follows from (90) that for any the inequality holds and that is the largest such constant. Finally, we set .
Remark 8. It is now easy to see that is a norm on .
In fact, we have the following lemma.
Lemma 9. For all we have
Proof. Add to both sides of the inequality (89): From the definition of the energy norm it is clear that and the result follows.
From Lemma 7 it is clear that these are the best estimates of this form.
Lemma 10. The norms and are equivalent to the norm in the Sobolev space .
Proof. From (83) and (89) it follows that
The addition of the two inequalities above yields
Let = min ; then Equation (83) yields and from the Trace theorem it follows that From (92) it is evident that the energy norm is equivalent to the norm .
Remark 11. From the above lemma we may claim from the embedding , , that there exists a constant such that
7.4. Stability and Uniqueness for the Model Problem
satisfies the system of evolution equations
We now derive an energy identity for the solutions of (101). Take the , scalar product with on both sides of (101)1. This produces According to the formulation of the original problem on the boundary where , we obtain From (101)2 we obtain Substitute (104) into (103) to obtain