Asymptotic Solutions of the Planar Squeeze Flow of a Herschel-Bulkley Fluid

In this study, we present the analysis of the squeeze flow of a Herschel-Bulkley fluid between parallel plates that are approaching each other with a constant squeeze motion. The classical lubrication analysis predicts the existence of a central unyielded zone bracketed between near-wall regions. This leads to the well-known squeeze flow paradox for viscoplastic fluids. Since the kinematic arguments show that there must be a finite velocity gradient even in the unyielded zone, thereby precluding the existence of such regions. This paradox may, however, be resolved within the framework of a matched asymptotic expansions approach where one postulates separate expansions within the yielded and apparently unyielded (plastic) zones. Based on the above technique, we circumvent the paradox, and develop complete asymptotic solutions for the squeeze flow of a Herschel-Bulkley fluid. We derive expressions for the velocity, pressure and squeeze force. The effects of the yield threshold on the pseudo-yield surface that separates the sheared and plastic zones, and squeeze force for different values of non-dimensional yield stress have been investigated.


Introduction
The squeeze film phenomenon occurs for the close approach of a pair of surfaces, and conforms to the classical lubrication paradigm. This approach leads to a sharp growth in the pressure within the narrow gap (between the surfaces), this growth being proportional to the fluid viscosity. While, squeeze flow problems have been analysed extensively for Newtonian fluids, we here consider the same for viscoplastic fluid between planar geometry. Bingham, Casson and Herschel-Bulkley are some of the most commonly used models to represent viscoplastic fluids. More details on these models can be found in Bird et.al [1] and Barnes [2] review articles. Here in this study, we use the Herschel-Bulkley model to represent the viscoplastic fluid, which is a generalized model of non-Newtonian fluids. For such fluids, flow occurs when the stress in the gap exceeds critical yield stress.
Practically, this type of flow behaviour occurs in many areas, such as slurries and suspensions, muds, clays, certain polymer solutions, lavas, etc. Therefore, such materials have applications in different fields, ranging from the oil, gas industries to the concrete used for construction. The non-Newtonian behaviour of fresh concrete, fly ash, mining slurries and cement flows has been verified by conducting field and laboratory experiments [3][4][5]. However, the Herschel-Bulkley model is more suitable to represent the true rheological behaviour of viscoplastic fluids over a sufficiently wide range of shear rates. The flow behaviour of viscoplastic materials is important to analyse, due to the existence of yield surface which separates yielded and unyielded regions.
The geometry with small aspect ratio, having viscoplastic material as a fluid medium have a long history. Some of the researchers have analysed the squeeze flow of viscoplastic fluid and the presence of yield surface. But most of the earlier works [6][7][8][9] debate on the existence of true unyielded plug regions in the problem of squeeze flow of viscoplastic fluids. The classical lubrication theory leads to the well-known "squeeze flow paradox", since the velocity component of the unyielded plug region varies in the flow direction. This implies that true unyielded plug regions cannot exist. However the paradox can be resolved by using numerical simulations or approximate solutions.
Some of the investigators have used discretized approximations such as finite difference or finite element methods to analyse the squeeze flow of viscoplastic materials and verified with the experimental results [10][11][12][13][14][15]. Several researchers have used the technique of asymptotic expansions to analyse the viscoplastic squeeze flow problems. Initially Walton and Bittleston [16] have used this technique to study the Bingham plastics in a narrow eccentric annulus. Balmforth and Craster [17], later Frigaard and Ryan [18] and Putz et al. [19] have used asymptotic expansions to resolve the squeeze flow paradox and developed the consistent solution for thin-layer problems. Recently, Muravleva [20] has analysed the squeeze flow of a Bingham fluid in a planar geometry using numerical simulations and asymptotic expansions [17].
In the present work, we study the squeeze flow problem of a viscoplastic fluid in a planar geometry using the Herschel-Bulkley model by applying the technique of asymptotic expansions and also suggest corrections to resolve the squeeze flow paradox.

Mathematical formulation
The objective is to develop a consistent solution for the squeeze flow of an incompressible viscoplastic fluid for a planar geometry using Herschel-Bulkley fluid model, which resolves the squeeze flow paradox. Figure 1 shows the schematic of the squeeze flow problem. The gap width * 2H between the plates of length * 2L is filled with a viscoplastic material, while the plates approach each other with a constant velocity * s v . The governing equations of the problem in two dimensional form is as follows: (3) The constitutive equation for Herschel-Bulkley fluid model in three dimensional form is given by is the Reynolds number, where *  and * K are the density and consistency coefficient of the viscoplastic material.
Assuming the effect of fluid inertia is negligible, equations (1)-(3) can be written in dimensionless form as: Here * * L H   is called the aspect ratio. The dimensionless form of constitutive equations are given by And the strain rate tensor ij  is given by The second invariants  and  are denoted by  and  respectively, i.e. , In the above equations dimensionless Herschel-Bulkley number N is defined by where 0  is the yield stress.
The above equations (4)-(6) are solved by applying appropriate boundary conditions given below.
, and on the plane symmetry

Asymptotic expansions
The equations (4)-(6) along with the boundary conditions are solved by introducing asymptotic expansions.
Substituting these expansions in equations (4)-(6) and comparing the leading order terms (i.e. 0  terms), we get y Now consider the first order approximations (i.e. 1  terms), we get  (14) and (15), we obtain Solving equation (17) with the boundary condition, we get here g(x) is the unknown function of integration.

Plastic region
To determine the pseudo-yield surface ) ( 0 x y y  , one can use integral form of equation (9), i.e., The algebraic equation (21) can be solved by using any of the numerical method to obtain the yield surface, . From the equation (19), it can be observed that is purely a function of x such that, From above relations, it is obvious that the leading order velocity x direction which implies that the plug region is not a true plug region. This is the basis of the lubrication paradox for yield stress fluids. This problem arises due to the absence of diagonal components of the stress.
This paradox may, however, be resolved within the framework of a matched asymptotic expansions approach where one postulates separate expansions within the yielded and apparently unyielded (plastic) zones. The yielded zones conform to the lubrication paradigm with the shear stress being much greater than all other stress components. On the other hand, the shear and extensional stresses are comparable in the "plastic region", with the overall stress magnitude being asymptotically close to but just above the yield threshold.
These shear and plastic regions are separated by an interface represented by a smooth pseudo-yield surface , the asymptotic expansion described above breaks down. To find the appropriate solution in this region, which incorporates changes in horizontal velocity component we modify u velocity as follows: Using these expansions, we can find stress components From these components, we obtain Solving for where ) ( * 1 x u is an unknown function of integration. For the first order approximation we have the following equations: y v x u        Solving above equations (23)-(24), we get is an unknown function of integration.

Matching
Using the technique of matching, one can calculate the unknown functions of integration. Since shear stress is continuous at Similarly from continuity of velocity at (28) Substituting above integral constants in equations (13), (18), (19) and (22), we get the velocity distribution in both shear an plastic regions. Further, using the equation of continuity equation (6)

Pressure Distribution and Squeeze force
One of the important aspects of squeeze flow problem is squeeze force. To get this, we need to know the pressure distribution at the plate along the flow direction. The pressure gradient in shear region is given by , integrating above expression, we get the pressure distribution in shear region as follows: The constant 1 C can be calculated using boundary condition Further the squeeze force can be calculated by using the relation: can be substituted by using equations (30) and (31).

Results and Discussion
The core thickness along the principal flow direction for different values of Herschel-Bulkley number (N) and power law index (n) have been computed and the results are shown in figures 2 and 3. From these figures, we can observe that core thickness decreases along the horizontal direction from center plane ( 0  x ) to the edge of the plate. The rate of change of core thickness increases with increase in Herschel-Bulkley number. Hence the core thickness in general for fluids with higher Herschel-Bulkley numbers is more than the fluids with lower Herschel-Bulkley numbers. Further, as power law index increases core thickness decreases and also the rate of change of core thickness increases with increase in power-law index.  . From these figures, we can observe that, squeeze force increases considerably with the increase in Herschel-Bulkley number for a particular value of aspect ratio. Also, squeeze force increases with increasing power-law index. Further, as aspect ratio increases squeeze force decreases for large Herschel-Bulkley numbers and the rate of decrease in squeeze force is marginal with increase in aspect ratio.

Conclusions
The squeeze flow behaviour between two parallel plates lubricated by a Herschel-Bulkley fluid with constant squeeze motion is theoretically analysed using the technique of a matched asymptotic expansions. We obtain consistent asymptotic solutions which are free from "squeeze flow paradox".