#### Abstract

This research investigates an analytical solution for the slow squeeze flow of the slightly viscoelastic fluid film between two circular disks in which the upper disk approaches the lower disk with a constant velocity, and the lower disk is kept stationary. The determination of the study is to identify the behavior of the differential type fluid on the steady squeezing flow using Langlois recursive approach. The governing equations for the axisymmetric flow are expressed in cylindrical coordinates and yield the nonlinear system of partial differential equations. The analytical solution of the resulting equations with nonhomogeneous boundary conditions is obtained by the Langlois recursive approach. Flow variables such as stream function, velocity profiles, pressure distribution, shear, normal stresses, and normal force acting on the disk are determined. These flow variables are nondimensionalized by using suitable dimensional quantities. The influence of slightly viscoelastic parameter , radial distance , and aspect ratio on velocity components, pressure distribution, and normal squeeze force is examined mathematically and portrayed graphically. The results illustrate that the axial and radial velocities increase at the higher values of the slightly viscoelastic parameter , which confirms the shear thickening behavior. The obtained solutions of the flow variables satisfied the existing solutions on squeeze flow of viscous fluid upon vanishing the slightly viscoelastic parameters. This solution could elucidate the classical lubrication problems, particularly in load and thrust bearing characteristics of the human body joints, the compression molding process of materials, etc.

#### 1. Introduction

The squeezed flow has many real-life applications, such as dampers, motor bearing, lubrication, chewing between teeth or gums, and the compression molding processes of metals and polymers (filled or unfilled). The valves and arthritic joints are interesting real-world applications in biology and biotechnology of essential squeezed flows [1]. The entire synovial fluid in the human knee joint is not pumped immediately from the two sides of the joint during the heel-strike processes and toe-off period. In the presence of liquid viscous resistance, the contact surfaces require a specific period. During this interval, the pressure is generated, and the squeezed fluid film supports the force [2]. Several flows are often found in traditional lubrication products, such as cams, engine connecting rod bearings, and gears. In such cases, viscoelastic additives are capable of increasing the load-bearing strength of lubricants [3].

The study of squeezing flow can be traced down back to the 19th century. The work by Stefan [4], Reynolds [5], and Scott [6] can be considered as pioneering work. Earlier experimental works on squeezing flow between two circular disks were reported in [7–9]. In the beginning, creeping flow, limiting to Newtonian fluids, was studied using nonclassical lubrication approximation. Later, inertia was included in the Newtonian creeping flow by Jackson [10], and Kuzma [11] identified the error in the inertia term of the work done by Jackson [10]. Jones and Wilson [12] found the squeezing force of liquids with inertia for large Reynolds values. Numerical solutions for squeezing flow between parallel disks have been conducted by Hamza and Macdonald [13].

Further experimental results for slow squeezing are highlighted in power-law fluids, while rheological models are applied for fast squeezing flow, which describes the overshoot phenomena [14]. The applied forces and spaces between the disks in the squeezing flow of the power-law fluid were found by Lieder [14, 15] in the solution of shear-thinning polymers. From these results, it has been observed that force and distance parameters deviate from the Scott equation. Later on, McClelland and Finlayson [16] extended Oliver’s work and proposed a model that agrees with experimental data at fast squeezing rates by combining the effects of normal stress. The experimental results on a high viscosity of low-density polyethylene in constant load tests were compared with Criminale–Ericksen–Filbey fluid and found a close relationship between the results [3]. However, it has been analyzed that results produced in the case of slow squeezing are better as compared to fast squeezing. Kramer [17] repeated Tanner’s work and used only the lodge rubber-like fluid Maxwell model. Phan–Thien and Tanner [18] extended the work of [9, 19] by including the inertia effect but neglecting the body forces and edge effects of the motion of the nonlinear Maxwell model. Phan–Thien et al. [20] found the solution of creeping squeeze film flow of inelastic fluids such as the Carreau model and modified Phan–Thien–Tanner (MPTT) model by numerical method considering stress overshoot phenomena. Yousefi et al. [2] assumed the constitutive equation of the MPTT model by considering a synovial fluid and showed a valuable contribution to knee joint lubrication problems. Lee et al. [9] obtained a finite element solution to squeeze a convicted Maxwell fluid under the constant force on the upper disk with the inertia of the fluid. Muravleva has studied the squeeze flow of the Bingham fluid [21–23] in the plane and axisymmetric geometries by the numerical technique, that is, augmented Lagrangian method and asymptotic solution. After that, Singeetham and Puttana [24] extended the work of Muravleva and found the analytical solution by matching the asymptotic expansion technique of plane squeeze flow of Herschel–Bulkley and Casson fluid between two disks. The solution is divided into three regions, that is, shear stress dominant, pseudo-plug, and central pseudo-plug plastic regions due to the yielded stress. The behavior of squeezing flow of MHD Casson fluid with slip and no-slip conditions is investigated in [25–27]. The behavior of Newtonian [28–30] and non-Newtonian fluid [31–34] on different geometries is examined numerically and analytically. The theoretical and experimental researches of squeezing flows have been studied by many researchers [35, 36].

This research study explores the squeeze flow of the differential type fluids due to their significant features of rod climbing, shear thinning, shear thickening effects, and normal stress. The constitutive equation of a special class of third-order fluid named the slightly viscoelastic fluid has applications in journal bearing and slide bearing [37, 38]. Governing equations of the squeeze flow of slightly viscoelastic fluid film are the nonlinear system of partial differential equations in the axisymmetric form with nonhomogeneous boundary conditions. Such types of differential equations are complex to solve numerically. Therefore, in this research investigation, the recursive approach of Langlois [39] has been used to linearise these equations, and the analytical solution has been obtained on specified boundary conditions.

The aim of this research is to deliver the analytical solution of squeeze flow of steady incompressible viscoelastic fluid between two circular disks using the Langlois recursive approach. This approach is successfully applied by Ullah et al. [40–42] for the creeping flow of slightly viscoelastic fluid and Maxwell fluid through a porous plane slit with uniform reabsorption, no-slip wall, and slip wall. Bhatti et al. [43] also used this technique for the second-order fluid flowing through a porous circular tube. Thus, we have employed the recursive approach to study the slow axisymmetric squeezed flow of slightly viscoelastic fluid film between two disks. The analytical expressions for the velocity field, pressure distribution, shear, and normal stress for squeezing flow have been obtained. The effects of the different physical parameters on the motion of fluid have been illustrated graphically. This study was considered to deliver answers to the following related research questions:(1)What is the variation in velocity components, and pressure distribution of the squeeze flow due to the radial and axial direction?(2)At various radial points, what is the impact on velocity profile, pressure distribution, and normal squeeze force due to the rise of slightly viscoelastic parameter ?

This paper is structured as follows. In Section 2, the governing equations of motions with associated boundary conditions have been presented. In Section 3, the geometry of the problem and the model of the governing equations in cylindrical coordinates have been described. The construction of the velocity field for first, second, and third-order approximation problems has been explained in Section 4. In Section 5, the computations of the velocity field are given. The pressure distribution for squeezed flow has been obtained in section 6. The shear, normal stresses, and normal force are presented in Sections 7 and 8, respectively. The effects of different physical parameters have been investigated in section 9. Finally, in section 10, the concluding remarks are enlightened.

#### 2. The Governing Equations of Motion

The basic equations governing the motion of an isotropic incompressible fluid [42] are given aswhere denotes the velocity vector, represents the constant fluid density, represents the body force per unit mass, and the material time derivative is denoted by . The constitutive equation of third-order fluid for Cauchy stress by Truesdell and Noll [44] is given aswhere represents the indeterminate part of the stress due to constraint of incompressibility, is identity tensor, represents the coefficient of viscosity, the material constants are represented by and , and and are the Rivlin-Ericksen tensors and defined as follows:

In addition, thermodynamics analysis imposes conditions [45] if all the motions of the fluid meet the Clausius‐Duhem inequality, and assuming the Helmholtz free energy is minimum while the fluid is at rest, then the material coefficients met the following restrictions:

By making use of (6), the reduced form *of* (3) yields as follows:

The authors [45] presented (7) and studied the thermal effects in the variable viscosity of journal bearings. By considering the material constants equal to zero in (7), which becomes as follows:

Authors [37, 45] claimed that (8) represents the constitutive relation for the slightly viscoelastic fluid and can be considered the special class of differential type fluids or a special type of power-law model. Furthermore, in (8), the slightly viscoelastic parameter and represent the shear thickening and shear thinning behavior of the fluid, respectively. However, for , (8) reduces to the Newtonian fluid.

Substituting (8) in (2), yielding the result in vector form aswhere is the Laplacian operator and .

#### 3. Problem Statement

The slow squeeze flow of an incompressible slightly viscoelastic fluid in axisymmetric form has been considered in Figure 1. The lower disk is kept fixed while the above disk moves with constant velocity (*t*) under the force *F*, which approaches the lower disk. *H*(*t*) represents the fluid film thickness, which decreases with time and fluid flows outside the disks radially. The velocity components for the axisymmetric flows are assumed as follows:where and are the radial and axial velocity components.

In order to write the components form of equations (9) and (10), substituting (11) in the first Rivlin tensor , we get

Using (12) in the definition of , we obtain

Putting (11)–(13) in terms of (10), we get the following expressions:where .

Neglecting the inertial term and body force and putting (11)–(17) in (9) and (10), we get the following nonlinear system of partial differential equations:

Similarly, by putting (12) and (13) into (8), four nonzero components form for the stress tensor are given aswhere .

We plan to solve the above set of nonlinear partial differential equations subject to the following nonhomogeneous boundary conditions:where .

The first two conditions in (23) are due to the adherence of the top disk, and the location of the top disk is unknown. In the other two conditions, there is no slip at the bottom disk. Velocity may be constant, or it may vary with time.

#### 4. Construction of an Analytical Approximation of Solution by Langlois Recursive Approach

The coupled system of nonlinear partial differential equations (18)–(22) subject to nonhomogeneous boundary conditions (23) is not easy to be solved exactly. This nonlinear system is solved by using the recursive approach suggested by Langlois [39], which linearises the governing equations of motion given in equations (18)–(22). To obtain the approximate analytical solution of equations (18)–(22) subject to boundary conditions (23), the following equations for the velocity field, pressure, and stresses are sought in the form of perturbation of the state of the rest.where is a small dimensionless number, these assumptions lead to the systems of linear partial differential equations corresponding to the boundary conditions for every set so that as given by equations (24)–(27) provide the solution to equations (18)–(22). The first-order equations corresponding to are describing the solution of the governing equation of the Newtonian fluid subject to nonhomogeneous boundary conditions. The second-order equations corresponding to are similar except the nonhomogeneous part containing the terms of with homogeneous boundary conditions. The third-order equations corresponding to are identical except the nonhomogeneous part containing the terms of with homogeneous boundary conditions; hence, third-order equations give the contribution of the slightly viscoelastic term.

Substituting equations (24)–(27) into equations (19)–(21) and collecting the coefficients of equal powers of , we get the following first, second, and third-order boundary value problems, and the aim is to solve these problems , and .

##### 4.1. First-Order Problem with Associated Nonhomogeneous Boundary Conditions

On equating the terms of from equations (18)–(22), the following linear system of partial differential equations subject to nonhomogeneous boundary conditions is obtained.subject to boundary conditions

##### 4.2. Second-Order Problem with Associated Homogeneous Boundary Conditions

On equating the terms of from equations (18)–(22), the following linear system of partial differential equations subject to nonhomogeneous boundary conditions is obtained.subject to homogeneous boundary conditions

##### 4.3. Third-Order Problem with Associated Homogeneous Boundary Conditions

On equating the terms of from equations (18)–(22), the following linear system of partial differential equations subject to nonhomogeneous boundary conditions is obtained.where subject to homogeneous boundary conditions

#### 5. Computation of Velocity Field

##### 5.1. Velocity Field for First-Order Problem

In this subsection, we compute the velocity field of first-order approximation by rewriting the system of partial differential equations (28a)–(28c) with associated boundary conditions (30) in terms of stream function. The radial and axial velocity components of the axisymmetric flow can be expressed in the form of scalar stream function as follows:

It should be noted that (28a) is identically satisfied by (37). By differentiating (28b) and (28c) partially with respect to and , respectively, and by eliminating the pressure, the compatibility equation of first-order approximation is obtained aswhere and Furthermore, by using (37) in (30), boundary conditions are obtained in terms of stream function as follows:

In order to obtain the solution for the boundary value problem given in (38) and (39) by using the inverse method [46], assuming the following solution for stream function, a priori:where is an unknown function, which needs to be determined. Thus, using Equation (40) into (38) and (39) yields

The unknown function can be obtained from (41) by integrating and using associated conditions (42).

The steam function, redial velocity, and the axial velocity can be obtained by mathematical simplification, using equation (43) into (40) and (37).

It is noticed that the first-order velocity components (45) and (46) are in coherence with those obtained in [4] for the creeping squeeze flow of viscous fluid between two disks.

##### 5.2. Velocity Field for Second-Order Problem

We determine the velocity field of second-order approximation from equations (31) and (32) subject to homogeneous boundary conditions (34a) by reducing the system of partial differential equations in terms of stream function. Defining the stream function for second-order approximation is as follows:

It is noted that (31) is identically satisfied by using (47). Therefore, using (47) into (31) and eliminating the pressure, it reduces into the following form:

Furthermore, associated boundary conditions (33) are reduced in terms of stream function as follows:

The inverse solution of (48) corresponding to conditions (49) for any assumed form is zero due to the homogeneous boundary conditions. Therefore, we get .

The radial and axial velocity components for second-order approximation are obtained by substituting stream function into (47). We get and .

##### 5.3. Velocity Field for Third-Order Problem

In this subsection, we compute the third-order approximations of the velocity field from equations (34a–34c) corresponding to homogeneous boundary conditions (36), and by using the first-order solution in (34a), we get

Defining the stream function for the third-order problem is as follows:

(50) is identically satisfied, and using (53) in equations (51) and (52) by eliminating the pressure, the following equation is obtained:

Associated boundary conditions in terms of stream function are written as follows:

Similarly, an inverse solution is sought for stream function , a priori as follows:where and are unknowns and to be determined. On substituting (56) into (54), it takes the following form:

The following equations (58) and (59) are obtained by comparing the coefficients of and in (57).

Using the assumption (56), the corresponding boundary conditions (55) reduce in (60).

The solutions of an ordinary differential (58) and (59) corresponding to boundary conditions (60) are obtained as

The expressions for stream function and velocity components are obtained by using equations (61) and (62) into equations (53) and (56) as follows:

It is observed that the third-order approximate solution of the velocity profile and stream function contains the terms of the slightly viscoelastic parameter , which is the key feature of the present study.

##### 5.4. Stream Function and Velocity Field Correct to Third-Order Approximations

The expressions for stream function correct to third-order approximation obtained by adding equations (44) and (63) as follows:

In order to obtain the velocity components, correct to third-order by substituting equations (45) and (64) into (24), we get

Similarly, by substituting equations (46) and (65) into (25), we get

Results achieved from this proposed approach agree with the results presented in the literature [47, 48] for viscous fluids when *β* = 0.

#### 6. Computation of Pressure Distribution

##### 6.1. First-Order Approximation of Pressure Distribution

Substituting the first-order approximation of velocity profile (45) and (46) into equations (28b) and (28c), the following equations of pressure distribution are obtained as

Integrating (69) with respect to , we obtainwhere is an arbitrary function of . Differentiating (71) with respect to *z* and comparing with (70) yield the followingwhere is the constant of integration. Using equation (72) into (71), the first-order pressure distribution gets as follows:

##### 6.2. Second-Order Approximation of Pressure Distribution

Subsequently, substituting the second-order approximation of velocity profile into (31), the following equations of pressure distribution are obtained as

After integrating (74) and simplification, the second-order pressure distribution is obtained where *m* is the constant of integration.

##### 6.3. Third-Order Approximation of Pressure Distribution

Similarly, substituting the third-order approximation of velocity profile (64) and (65) into equations (34b) and (34c), the following equations of pressure distribution are obtained as

The solution of the third-order approximation of pressure distribution is obtained from equations (75) and (76) by using the procedure of the first-order pressure distribution as follows:

##### 6.4. Pressure Distribution Correct to Third-Order Approximation

For obtaining the pressure distribution, (78) corrects to third-order approximation, adding equations (73) and (77).where . For the evaluation of the constant , there is required an extra boundary condition at the outer edge of the disk.

In this research investigation, it is observed that the edge of the disk of fluid that resides is subjected to atmospheric pressure; therefore, at the free surface, the balance of normal stress is equivalent to atmospheric pressure. There is no loss of generality in taking [9].

From (68), at , we get ; hence, (79) reduces to . Using this condition in Equation (78) to obtain the constant

Putting the value of in (78), we have pressure distribution correct to third-order approximation:

#### 7. Computation of Shear and Normal Stresses

##### 7.1. First-Order Shear and Normal Stresses

The expressions for first-order approximations of shear and normal stresses are obtained by substituting first-order velocity approximations (45) and (46) and pressure distribution (73) into (29) as follows:

##### 7.2. Second-Order Shear and Normal Stresses

Similarly, the expressions for second-order shear and normal stresses are obtained by substituting second-order velocity approximation and pressure distribution into (32) as follows:

##### 7.3. Third-Order Shear and Normal Stresses

Similarly, the expressions for third-order shear and normal stresses are obtained by substituting third-order velocity approximation (67) and (68) and pressure distribution (77) into Equation (35a) as follows:

#### 8. Normal Force on the Upper Disk

The total force applied on the upper disk calculated by integral of the negative of that is exerted by the slightly viscoelastic fluid film in the positive *z*-direction at is estimated as

Substituting in (80) to obtain and using in (84), then obtained the following

#### 9. Analysis of Results

The convergence of the solution using the Langlois recursive approach is validated by determining the absolute residual error for different fluid parameters as given in Table 1. It can be observed that the absolute residual error of the solution for third-order approximation is approaching to zero that confirmed the convergence of the solution.

#### 10. Discussion of Results

In this research study, an approximate analytical solution of the slow squeeze flow of slightly viscoelastic fluid films between two disks is obtained successfully using the recursive approach of Langlois. The following dimensionless variables are introduced to examine the effect of the slightly viscoelastic parameter and radial distance on flow variables such as velocity profile, pressure distribution, and normal force and depicted in Figures 2–7, drawn using the mathematics-based Maple software.

The impact of the radial velocity component at various radial points keeping fixed values and is shown in Figure 2. It is observed that the radial velocity increases as the radius increases and velocity is maximum at the axial distance . The effect of the slightly viscoelastic parameter on the radial velocity with fixed values and illustrated in Figure 3 and shows the behavior of the radial velocity is parabolic in the case of viscous fluid [4, 48, 49]. The radial velocity increases and occurs backward flow at the edges with the rise in . Thus, physically slightly viscoelastic parameter is inversely proportional to the fluid viscosity. The rise of this parameter *β* demotes the viscosity, and as a result, the fluid velocity intensified. This profile proved the behavior of the shear thickening fluid. Furthermore, behavior of the radial velocity in the case of viscous fluid, when = 0, is taken as a special case which is also illustrated in Figure 3.

Figure 4 represents the axial velocity profile at different radial values keeping and . Axial velocity rises in contrast to an increase in the value of . Figure 5 depicts the impact of the slightly viscoelastic parameter on the axial velocity . It is noticed that the negative magnitude of axial velocity indicates that the direction of flow is downward; also, reverse flow is occurring in axial velocity due to the rise of , and it is gradually increased with the maximum magnitude at . In Figures 1–4, the shear-thickening behavior of the velocity profile appears at the parameter .

Figure 6 demonstrates the effect of on pressure distribution at and . It shows that the pressure distribution is directly proportional to the thickness of the fluid and reduces radially approaching to zero at the edge. Physically when the viscoelasticity of the material rises, then pressure distribution increases to squeeze the fluid film. In Figure 7, force swiftly increases at the significant values of the aspect ratio of the film thickness and high values of .

The results obtained from this proposed approach for *β* = 0 agree with the results presented in the literature [47,48]. Furthermore, it also satisfies the physical behavior of the shear thickening fluid.

#### 11. Conclusion and Future Recommendations

This research study is focused on the analytical solution of the sluggish squeeze flow of the slightly viscoelastic fluid film between two circular disks where the upper disk is moving with constant velocity and the lower disk is stationary. The recursive approach of Langlois has been applied successfully for the analytical solution of the governing equations of the flow in the axisymmetric form. In order to examine the physical behavior of the flow, analysis is taken up to third-order linearization by this approach. The physical parameters of the flow such as velocity, pressure distribution, normal squeeze force, and stresses are determined as a function of aspect ratio H/R, radial distance *r*, and viscoelastic parameter , which are key flow variables of the study. The obtained results by this approach are in good agreement with the results of the squeezing flow of viscous fluid by vanishing slightly viscoelastic parameter which also confirms the physical behavior of shear thickening fluid. The following key points are drawn from the theoretical investigations.(1)The axial and radial velocities increase with the increment in the radial distance and observed that the maximum velocity occurs at the *z* = 0.5(2)The axial and radial velocities increase at the higher values of the slightly viscoelastic parameter , and backward flow occurs at the boundaries of the channel due to the slightly viscoelastic term(3)The pressure distribution elevates with the rise of the slightly viscoelastic parameter and crushes at the edges of the channel(4)The presence of a slightly viscoelastic parameter boosted the normal squeeze force on the upper disk

This study attempted to get an analytical solution for the slow squeeze flow of the slightly viscoelastic fluid film between two circular disks at no-slip conditions by the Langlois recursive approach. As a future recommendation, it is proposed that this method can also be used to get an analytical solution of the nonlinear partial differential equations arising from the squeeze flow of differential type fluids with slip conditions and inertia effects.

#### Nomenclature

: | Velocity vector |

: | Radial velocity component |

: | Axial velocity component |

: | Constant fluid density |

: | Body force |

: | Cylindrical coordinates |

: | Material derivative |

: | Rivilin Ericksen tensors |

: | Slightly Viscoelastic parameter |

: | Cauchy stress tensor |

: | Pressure |

: | Aspect ratio |

F: | Squeeze force on the upper disk . |

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no potential conflicts of interest with respect to the research, authorship, and publication of this article.