## Abstract

A general simple continua can be enhanced by constitutive equations which depend on the acceleration and velocity gradients to model the effects of a material characteristic length. This paper shows that for irrotational flows of a class of incompressible materials this model yields a Bernoulli equation. Consequently, for this class of materials and flows, it is possible to satisfy the balance of linear momentum exactly, including the effect of a material characteristic length which introduces size dependence of solutions. An example of a rigid circular cylinder moving through an inviscid fluid is considered to demonstrate dependence of the motion on the size of the cylinder.

## 1. Introduction

The constitutive equations for most simple continua are formulated in terms of local quantities, such as deformation and rate of deformation which lead to solutions that are insensitive to scaling of all length dimensions in a special problem. However, a number of physical phenomena are observed to be size dependent. In particular, materials that exhibit softening behaviour are prone to localization where severe deformations and deformation rates exist in relatively small regions of the material. For example, it is of interest to understand the thickness of an adiabatic shear band that can occur during high-rate loading of metals. This localization process also causes severe numerical problems with important details of a solution being unphysically dependent on the size of the element mesh.

Consequently, from both theoretical and practical points of view, there is a need to propose phenomenological models that capture the main effects of a material characteristic length. Within the context of solid mechanics, it is clear that grains, grain boundaries, inclusions and other material microstructure can influence material response when the wavelengths of relevant phenomena are close to the characteristic dimensions of the microstructure. Examples of phenomenological models of a material characteristic length include enhanced Cosserat continuum models [1] and strain gradient plasticity theories [2].

Within the context of fluids, materials can exhibit size effects owing to a material characteristic length that is related to the size of particles, polymers, surfactants or the distributions of electric charges in the fluids. Examples of phenomenological models for a material characteristic length in fluids include: fluids of complexity 2 and fluids of second grade [3]; generalized Navier–Stokes equations for turbulence [4–6]; finite-scale equations for compressible fluids [7]; Cosserat models for granular flows [8,9] and micropolar fluids for modelling journal bearings [10].

Most of these enhanced theories which deal with a material characteristic length require additional boundary conditions that are not easy to interpret or propose. In an attempt to model the first-order effects of a material characteristic length, Rubin *et al*. [11] proposed a phenomenological model which enhances an arbitrary simple continuum by adding a term to the Helmholtz free energy (per unit mass) and a term to the Cauchy stress which introduce the dispersive effects of a material characteristic length with no dissipation. It was shown in Rubin *et al*. [11] that these terms have no influence on the thermodynamics of the continuum being enhanced. Moreover, this model does not require additional boundary conditions. More recently, Saccomandi and co-workers [12–15] have used this model to study a number of interesting problems in fluid mechanics.

The objective of this work is to show that it is possible to develop a Bernoulli equation for irrotational flows of a class of incompressible materials that are enhanced by the model of a material characteristic length proposed in the study of Rubin *et al*. [11], which depends on the acceleration and velocity gradients. This means that for this class of materials and flows, it is possible to satisfy the balance of linear momentum exactly, including the effect of a material characteristic length which introduces size dependence of the solutions.

An outline of the paper is as follows. Section 2 summarizes the model proposed in Rubin *et al*. [11], and §3 develops the Bernoulli equation. Finally, §4 presents an example of flow of a rigid circular cylinder through an inviscid fluid which demonstrates dependence on the size of the cylinder owing to the material characteristic length.

## 2. Modelling a material characteristic length

Recall that for a simple continua a material point located by the position vector **X** in a fixed reference configuration is deformed to its location **x** in the present configuration at time *t*, such that
2.1This mapping is restricted so that the deformation gradient ** F** is non-singular with the dilatation

*J*being positive 2.2and the right Cauchy–Green deformation tensor

**C**is given by 2.3Also, the velocity

**v**and acceleration

**a**of a material point are defined by 2.4where a superposed (⋅) denotes material time differentiation holding

**X**fixed. Then, the velocity gradient

**L**and rate of deformation tensor

**D**are given by 2.5For a simple continua, the conservation of mass and balance of linear momentum can be expressed in the forms 2.6aand 2.6bwhere

*ρ*is the current mass density,

*ρ*

_{0}its reference value,

**b**the body force per unit mass,

**T**the Cauchy stress tensor and div denotes the divergence operator with respect to the current position

**x**. Also, the reduced form of the balance of angular momentum requires

**T**to be a symmetric tensor 2.7Within the context of the general thermomechanical theory proposed by Green & Naghdi [16,17], an arbitrary simple continua is characterized by a Helmholtz free energy , Cauchy stress and additional quantities that satisfy the balance laws for all thermomechanical processes. It was shown in the study of Rubin

*et al*. [11] that the effects of a characteristic length can be modelled by separating the total Helmholtz free energy

*ψ*and total Cauchy stress

**T**additively into two parts 2.8where characterize a general continua and characterize the influence of the characteristic length. More specifically, is taken to be a function of the scalar

*δ*, such that 2.9and is determined by requiring the effect of the characteristic length to be dispersive but non-dissipative so the mechanical power associated with is balanced by the rate of change of the energy 2.10for all processes. In this expression,

**A**⋅

**B**=tr(

**AB**

^{T}) is the inner product between two second-order tensors {

**A**,

**B**}. This leads to the constitutive equation 2.11In addition, it was shown in the appendix of Rubin

*et al*. [11] that the tensor

**A**can be rewritten in the form 2.12It should be mentioned that the theory of dipolar fluids discussed by Bleustein & Green [18] and Green & Naghdi [19] includes a special form for dipolar inertia with the kinetic energy depending linearly on

*δ*in (2.9). This theory of dipolar fluids is different from the simple theory presented in the study of Rubin

*et al*. [11] for which (2.10) and (2.11) hold for all motions.

Under superposed rigid body motions (SRBM), the material point **x** at time *t* is mapped to the point **x**^{+} at time *t*^{+} in the superposed configuration, such that
2.13where **c**(*t*) is a arbitrary vector function of time only associated with superposed translation, **Q**(*t*) an arbitrary proper orthogonal tensor function of time only associated with superposed rotation and *c* an arbitrary constant shift in time. Moreover, under SRBM, it is known that {**F**,**C**,**D**} transform to {**F**^{+},**C**^{+},**D**^{+}}, such that
2.14It then follows from (2.9), (2.11), (2.12) and (2.14) that under SRBM
2.15a
2.15b
and
2.15cso the constitutive equation (2.12) is properly invariant under SRBM.

It was shown in Destrade & Saccomandi [13] that characterizes a special Rivlin–Ericksen fluid as **A** in (2.12) can be expressed in terms of the first two Rivlin–Ericksen tensors {**A**_{1},**A**_{2}}[20,21]
2.16In general, the model (2.11) is valid for arbitrary compressible flows and the dependence on a characteristic length is nonlinear since is a general function of *δ*. However, here attention will be focused on a simple special case with and taken in the forms
2.17where *α* is the constant characteristic length.

## 3. Bernoulli equation for potential flow of an incompressible material

The model (2.8) which includes the effect of a characteristic length is valid for general materials characterized by and for general velocity fields **v**. However, sometimes it is of interest to consider irrotational potential flow of an incompressible material for which the velocity **v** is derivable from a potential function *ϕ*(**x**,*t*), such that
3.1a
3.1bwhere ∇^{2}*ϕ* denotes the Laplacian of *ϕ* relative to **x**. Since the flow is isochoric the dilatation *J* in (2.6a) equals unity and the density is independent of time
3.2For these flows, the constitutive equation for stress **T** is modified to take the form
3.3where the constraint response is an arbitrary function of {**x**,*t*} that is determined by the equations of motion and boundary conditions, is the deviatoric part of
3.4and the pressure *p* is influenced by the stress associated with the characteristic length. Also, without loss in generality, the pressure term due to has been absorbed into the arbitrary function .

In the remainder of this work, attention will be limited to a uniform homogeneous material for which the density *ρ* is constant. Next, referring all tensors to the rectangular Cartesian base vectors **e**_{i} (*i*=1,2,3), it can be shown that for potential flow
3.5where a comma denotes partial differentiation with respect to the components *x*_{i}=**x**⋅**e**_{i} of **x**, **b**⊗**c** denotes the tensor product of two vectors {**b**,**c**}, the usual summation convention is used for repeated indices and use has been made of (3.1b). Now, with the help (2.11), (2.17), (3.2) and (3.5) it follows that
3.6which shows that the divergence of is equal to the gradient of a scalar.

If the body force is derivable from another potential *V* _{b}
3.7then with the help of (3.3)–(3.6), it can be shown that the balance of linear momentum (2.6b) simplifies to the form
3.8For some materials and some motions, the constitutive equation for has the property that its divergence is derivable from yet another potential
3.9so that (3.8) can be integrated exactly pointwise to obtain the Bernoulli equation
3.10where *g*(*t*) is an arbitrary function of time determined by boundary conditions.

For an incompressible Newtonian viscous fluid with constant viscosity *μ*
3.11and (3.10) is again valid for all potential flows. Of course, the expression (3.11) includes the special case of an inviscid fluid by setting *μ* equal to zero. Also, for a rigid-plastic material with constant yield strength *Y* the stress in the plastic region can be expressed in the form
3.12Although (3.9) for this material is not valid for general flows, it was shown in Yarin *et al*. [22] that exists for penetration of a rigid projectile with the shape of an Ovoid of Rankine. This model was used to develop the drag force on the projectile and to study the influence of separation on the velocity dependence of the drag force [23].

Thus, the Bernoulli equation (3.10) can be used to generalize all of these problems to include the effect of a constant characteristic length *α*. However, it is also necessary to generalize the boundary conditions to include the term so the traction vector **t** on a surface with unit outward normal vector **n** is given by
3.13Since in general, has a non-zero deviatoric part the traction vector **t** will have shearing components on an arbitrary surface. Therefore, even if the simple case of an inviscid fluid is considered the shear stress on a boundary will not necessary vanish. Of course, non-slip boundary conditions for viscous fluids typically yield velocity fields which are not potential flows. Consequently, the usefulness of the Bernoulli equation developed here for viscous fluids depends on the extent to which the actual flow fields can be approximated by potential flows. Moreover, the use of potential flows as an approximation for penetration of projectiles into metal targets is partially justified by the statement by Hill [24]: ‘The frictional component can be disregarded because of surface melting’. Based on this notion, the influence of the shear stress on the drag force was neglected in Yarin *et al*. [22].

## 4. Example of a rigid circular cylinder moving through an inviscid fluid

As a specific example, consider the two-dimensional motion of a rigid circular cylinder of radius *b* moving through in inviscid incompressible fluid in the positive **e**_{1} direction (figure 1). From section 6.22 in Milne-Thomson [25], it is possible to write the velocity potential for this motion in the form
4.1where *x*_{1}=*z*(*t*) defines the position of the centre of the cylinder whose surface is characterized by the function
4.2and the sign convention for the velocity potential is consistent with (3.1). Moreover, the velocity field **v**, acceleration field **a**, the velocity gradient **L** and the rate of deformation tensor **D** associated with this potential are given by
4.3and it can be shown that the surface of the cylinder is a material surface since the material derivative of *f* in (4.2) vanishes. Also, for this material (3.11) is valid (*μ*=0) so with the help of (3.3), (3.6) and (3.10) the stress is given by
4.4

where the influence of the body force has been neglected (*V* _{b}=0) and it has been assumed that far ahead of the cylinder () the fluid is at rest and the pressure *p* in (3.3) is equal to the atmospheric pressure *p*_{0} (so that *ρg*(*t*)=*p*_{0}). Moreover, using the cylindrical polar base vectors {**e**_{r},**e**_{θ}} defined by
4.5it follows that on the surface of the cylinder
4.6so the traction vector **t** in (3.13) applied by the fluid on the surface of the cylinder is given by
4.7The fluid will remain in contact with the cylinder as long as the contact pressure *P*
4.8is non-negative. In particular, cavitation first occurs at the angles *θ*=±*β* where the flow first separates from the cylindrical surface
4.9Also, the contact pressure at the leading tip of the cylinder (*θ*=0) is given by
4.10In the study of Milne-Thomson [25], it is stated that the pressure distribution obtained for an inviscid fluid agrees with experiments on the leading edge of the cylinder but not elsewhere on the cylinder's surface.

Assuming that the fluid only remains in contact with the surface of the cylinder on its leading edge (−*β*≤*θ*≤*β*) with *β*≤*π*/2, the total drag force *F*_{T} per unit length applied by the fluid on the cylindrical surface in the (−**e**_{1}) direction over the contact region is given by
4.11Alternatively, the effect of the shear stress *σ*_{θ} can be neglected in the calculation of the drag force (as was done in [22]) to obtain the drag force *F*_{n} due only to the normal component *σ*_{r} of the traction
4.12The balance of linear momentum of the rigid cylinder is given by
4.13where *ρ*_{c} is the constant mass density of the cylinder and the drag force *F* is given by either of the expressions {*F*_{T},*F*_{n}}. In particular, using the approximation (*F*=*F*_{n}) yields the equation of motion
4.14which must be solved subject to the initial conditions that are specified by
4.15Next, using (4.9) to determine the separation angle *β* the equation (4.14) can be solved numerically.

The main objective of this example is to demonstrate the influence of the characteristic length *α* on modelling-size dependence of the solution. To this end, an approximate analytical solution is obtained by noting that for large values of *U*_{0} the influence of the atmospheric pressure *p*_{0} can be neglected except when *U* is very small. Consequently, for the approximate solution *p*_{0} is set equal to zero
4.16Also, the influence of in the separation condition (4.9) is neglected to deduce the approximate solution
4.17Then, the equation of motion (4.14) reduces to
4.18where the constant *C* is given by
4.19

Equation (4.18) can be integrated with respect to time *t* or the depth of penetration *z* to obtain
4.20In the absence of a material characteristic length (*α*=0) the separation angle *β* in (4.17) and the constant *C* in (4.19) are both independent of the radius *b* of the cylinder. Consequently, the curves (4.20) for *U* as a function of the normalized variables (*U*_{0}*t*/*b*) and (*z*/*b*) are unique. However, if the material characteristic length *α* is finite then *β* and these curves depend on the size of the cylinder. Specifically, figures 2–4 show the size dependence of the separation angle *β* (figure 2*a*), the constant *C* (figure 2*b*), the curve *U* as a function of time (figure 3) and the curve *U* as a function of the depth of penetration (figure 4). The density of the cylinder for these curves has been specified by *ρ*_{c}/*ρ*=2.7, which is representative of an aluminium cylinder. From these figures, it can be seen that the solution is well behaved for the full range of *b*/*α*. Since the functional forms of these quantities depend on (*b*^{2}/*α*^{2}) the values for *b*/*α*=100 are very close to the limiting values that would be obtained for a material that has no material characteristic length (*α*=0). In particular, from these results, it can be seen that the separation angle and the drag force increase with decreasing size of the cylinder. In particular, the size effect becomes significant when the radius *b* of the cylinder is of the order of the characteristic length *α*.

## Acknowledgements

This research was partially supported by M.B.R.'s Gerard Swope Chair in Mechanics.

- Received October 28, 2012.
- Accepted December 4, 2012.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.