Analytical Solutions of Some Fully Developed Flows of Couple Stress Fluid between Concentric Cylinders with Slip Boundary Conditions

Abstract

We establish, in this paper, the closed form analytical solutions of steady fully developed flows of couple stress fluid between two concentric cylinders, generated due to the constant pressure gradient or the translatory motion of the outer cylinder or both, using the slip boundary conditions. The classical solutions for Newtonian fluid in the hydrodynamic case appear as a limiting case of our solutions. The velocity profiles of the flows are presented and the effect of various parameters on velocity is discussed. The results indicate that the presence of couple stresses decreases the velocity of the fluid.

1. Introduction

The inadequacy of the classical Navier-Stokes theory for describing rheological complex fluids has led to the development of several theories of non-Newtonian fluids. Non-Newtonian fluid flows play important role in several industrial manufacturing processes. Few such examples include drilling mud, polymer solutions or melts, and certain oils and greases as well as many other emulsions. Due to the prominent applications in the modern technology and industries, many researchers made attempt to study different non-Newtonian fluid flow problems. One of the popular non-Newtonian fluid models which attracted the interest of several researchers in fluid mechanics is the couple stress fluid model proposed by Stokes [1]. Couple stress fluid model is a simple generalization of the classical Newtonian fluid model that allows the sustenance of couple stresses and body couples in the fluid medium. This theory completely describes the possible effects of couple stresses assuming that the fluid has no microstructure at the kinematical level, so that the kinematics of motion is fully determined by the velocity field [2]. The concept of couple stresses arises due to the way in which the mechanical interactions in the fluid medium are modeled and the stress tensor for this fluid is not symmetric. The couple stress fluid theory has many industrial and scientific applications and can be used to model the flow of synthetic fluids, polymer thickened oils, liquid crystals, animal blood, and synovial fluid present in synovial joints [2, 3]. It has interesting applications in the theory of lubrication. Numerous researchers have studied the hydrodynamic lubrication of squeeze film flows considering the lubricant to be couple stress fluid and found that the couple stress fluid increases the load carrying capacity of the journal bearing (see Naduvinamani et al. [4], Naduvinamani et al. [5, 6], and Lin and Hung [7] and the references therein).

The study of flows of non-Newtonian fluids with slip boundary condition has gained considerable attention in the recent past. Though many flow problems concerning Newtonian and several non-Newtonian fluids have been solved under the no-slip condition, the fluid slippage might occur at the solid boundary [8–11]. Several investigations indicate the existence of slip at the solid boundary [12, 13]. As a matter of fact, long back Navier [14] proposed a general boundary condition that permits the possibility of fluid slip at a solid boundary. This boundary condition assumes that the tangential velocity of the fluid relative to the solid at a point on its surface is proportional to the tangential stress acting at that point. An excellent review of experimental studies regarding the phenomenon of slip of Newtonian fluids at solid interface is given by Neto et al. [15].

Due to its vast applications in engineering and industry, pressure driven flow or the Poiseuille flow has attracted attention of various researchers. Although pressure driven flows are unidirectional and have been studied earlier for Newtonian and some non-Newtonian fluids, they still attract special attention in a number of emerging problems [16–19]. Ellahi [11] has examined the effect of the slip condition on flow of an Oldroyd 8-constant fluid in a channel. Chen and Zhu [20] obtained the analytical solution of Couette-Poiseuille flow of Bingham fluids between two porous parallel plates with slip conditions. Wang and Ng [21] have studied slip flow due to a stretching cylinder. Song and Chen [22] investigated the Poiseuille flow of simple fluids in cylindrical nanochannels. Yang and Zhu [23] have analyzed the squeeze flow of Bingham fluid in the small gap between parallel disks with the Navier slip condition. Ferrás et al. [24] presented analytical solutions of both Newtonian and inelastic non-Newtonian fluids with slip boundary conditions for Couette and Poiseuille flows. Hron et al. [25] established closed form analytical solution for the flows of incompressible non-Newtonian fluids with Navier’s slip conditions at the boundary. Ellahi et al. [26] have obtained the exact solutions of nonlinear problems of steady flow of third grade fluid between the concentric cylinders. To the best knowledge of the authors, nobody has attempted to find the analytical solutions of the Poiseuille, Couette, and generalized Couette flows of couple stress fluid between concentric cylinders subject to slip boundary conditions.

The aim of this paper is to establish the closed form analytical solutions for Poiseuille flow, Couette flow, and generalized Couette flows of an incompressible couple stress fluid between two concentric circular cylinders with slip boundary condition. The slip boundary conditions are applied at the boundaries of the inner and outer cylinders. Three cases have been discussed: in the first case it is assumed that both inner and outer cylinders are at rest and the flow is due to the constant pressure gradient which is also known as Poiseuille flow, in the second case the inner cylinder is at rest and the outer cylinder is moving with translatory velocity assuming the pressure is constant (the resultant flow is known as Couette flow), and in the last case it is assumed that the flow is generated due to the translatory motion of the outer cylinder when the inner cylinder is at rest and simultaneously a constant pressure gradient is applied. The flow in the last case is known to be generalized Couette flow or Couette-Poiseuille flow. The paper is organized in terms of seven sections. The next section is devoted to formulating the basic equations governing the flow of couple stress fluid in cylindrical polar coordinates for the presently considered flows. Poiseuille, Couette, and generalized Couette flows are investigated in Sections 3, 4, and 5, respectively, Section 6 is devoted for discussion of results, and the last section contains the concluding remarks.

2. Basic Equations Governing the Flow

The equations governing the flow of a couple stress fluid are given by [1] 𝑑𝜌𝑑𝑡+𝜌∇•𝑞=0𝜌𝑑𝑞𝑑𝑡=𝜌1𝑓+2∇×𝜌𝑐−∇𝑝−𝜇∇×∇×𝑞−𝜂∇×∇×∇×∇×𝑞+𝜆₁+2𝜇∇∇•𝑞, (1)(2) where 𝑞 and𝜌 are the velocity and the density of the fluid, respectively, 𝑝 is the fluid pressure at any point, and 𝑓 and 𝑐 are the body force per unit mass and body couple per unit mass, respectively.

The constitutive equation connecting the force stress tensor 𝑡_𝑖𝑗 and rate of deformation tensor 𝑑_𝑖𝑗 is given by 𝑡_𝑖𝑗=−𝑝+𝜆₁∇•𝑞𝛿_𝑖𝑗+2𝜇𝑑_𝑖𝑗−12𝜀_𝑖𝑗𝑘𝑚_,𝑘+4𝜂𝜔_{𝑘,𝑟𝑟}+𝜌𝑐_𝑟. (3) The couple stress tensor 𝑚_𝑖𝑗 that arises in the theory has the linear constitutive relation 𝑚_𝑖𝑗=13𝑚𝛿_𝑖𝑗+4𝜂^′𝜔_𝑗,𝑖+4𝜂𝜔_𝑖,𝑗. (4) In the above 𝜔=(1/2)∇×𝑞 is the spin vector, 𝜔_𝑖,𝑗 is the spin tensor, 𝑑_𝑖𝑗 is the rate of deformation tensor, 𝑚 is the trace of couple stress tensor 𝑚_𝑖𝑗, and 𝜌𝑐_𝑘 is the body couple vector. The quantities 𝜆₁ and 𝜇 are the viscosity coefficients and 𝜂 and 𝜂^′ are the couple stress viscosity coefficients. These material constants are constrained by the inequalities 𝜇≥0;2𝜇+3𝜆₁≥0;𝜂≥0;𝜂^′≤𝜂. (5) In the absence of body forces and body couples, the field equations governing the flow of an incompressible couples stress fluid are ∇•𝑞=0,𝜌𝑑𝑞𝑑𝑡=−∇𝑝−𝜇∇×∇×𝑞−𝜂∇×∇×∇×∇×𝑞. (6)(7) Here, note that, in the absence of couple stresses (i.e., as 𝜂→0), momentum equation (7) reduces to the Navier-Stokes equation of motion for classical viscous Newtonian fluid.

For unidirectional fully developed steady flow between two concentric horizontal circular cylinders, the velocity field is in the form 𝑞=(0,0,𝑤(𝑟)) and it automatically satisfies continuity equation (6). The momentum equation (7) governing the flow reduces to 𝜂∇⁴𝑤−𝜇∇²𝑤+𝜕𝑝𝜕𝑧=0, (8) where ∇²=𝑑²/𝑑𝑟²+(1/𝑟)(𝑑/𝑑𝑟).

In view of the higher order nature of governing equation (8), additional boundary conditions are required to find the solution. In addition to the slip boundary conditions, we use Stokes [2] boundary conditions to solve the governing equation of the flow under consideration. The Stokes boundary condition assumes that the couple stresses vanish on the boundary of the solid.

3. Fully Developed Flow Generated by the Applied Constant Pressure Gradient (Poiseuille Flow)

Consider the fully developed steady flow of an incompressible couple stress fluid between two concentric circular cylinders due to the application of constant pressure gradient in the positive 𝑧-direction. The radiuses of the inner and outer cylinders are, respectively, 𝑅₁ and 𝑅, and the cylinders are assumed to be at rest. With these, governing equation (8) is to be solved subject to the following slip boundary conditions: 𝑤(𝑅)𝑑−𝛾𝜂𝑑𝑟𝑤−𝜇∇²𝑤_𝑟=𝑅=0,𝑤𝑅₁𝑑+𝛾𝜂𝑑𝑟𝑤−𝜇∇²𝑤_𝑟=𝑅1=0,𝑑²𝑤𝑑𝑟²=0at𝑟=𝑅,𝑅₁Vanishingofcouplestresses, (9)(10)(11) where 𝑅>𝑅₁.

Introducing the following nondimensional parameters 𝜇𝑤=𝜌𝑅𝑤;𝑟=𝑅𝜇𝑟;𝑝=²𝜌𝑅²𝑝;𝑧=𝑅𝑧, (12) momentum equation (8) and boundary conditions (9)–(11), after dropping “”s, become ∇²∇²−1𝑎²𝐺𝑤=𝑎², (13) with the boundary conditions 𝑤(1)𝑑−𝛼𝑑𝑟𝑤−𝑎²∇²𝑤_𝑟=1=0,𝑤(𝑘)𝑑+𝛼𝑑𝑟𝑤−𝑎²∇²𝑤_𝑟=𝑘=0,𝑑²𝑤𝑑𝑟²=0at𝑟=1,𝑘Vanishingofcouplestresses, (14)(15)(16) where 𝑎²=𝜂/𝜇𝑅², 𝑘=𝑅₁/𝑅<1, 𝐺=−𝜕𝑝/𝜕𝑧, and 𝛼=𝛾/𝑅.

3.1. The Methodology to Find the Solution of the Governing Partial Differential Equation (13)

Consider the linear homogeneous differential equation 𝐿𝑤=0, where 𝐿 is the differential operator.

Suppose 𝐿 can be written as 𝐿=𝐿₁𝐿₂, where 𝐿₁ and 𝐿₂ are lower order linear operators than 𝐿.

If 𝑤₁ is a solution of the differential equation 𝐿₁𝑤=0 and 𝑤₂ is a solution of the differential equation 𝐿₂𝑤=0, then, in general, 𝑤₁+𝑤₂ need not be a solution of the differential equation 𝐿𝑤=0.

However, for the governing equation (13) the associated homogeneous differential equation is ∇²(∇²−1/𝑎²)𝑤=0.

Thus the operator 𝐿 in this case is 𝐿=∇²(∇²−1/𝑎²) in which 𝐿₁=∇² and 𝐿₂=∇²−1/𝑎².

Therefore, 𝐿₂=𝐿₁−1/𝑎².

With this, one can easily show that 𝑤₁+𝑤₂ is a solution of the differential equation 𝐿𝑤=0 or ∇²(∇²−1/𝑎²)𝑤=0.

Further, the solution of the differential equation ∇²𝑤=0 is 𝑤₁(𝑟)=𝐴ln(𝑟)+𝐵 and the solution of the differential equation (∇²−1/𝑎²)𝑤=0 is 𝑤₂(𝑟)=𝐶𝐼₀(𝑟/𝑎)+𝐷𝐾₀(𝑟/𝑎) (because the second operator gives a Bessel differential equation).

Hence, the solution of the homogeneous differential equation ∇²(∇²−1/𝑎²)𝑤=0 becomes 𝑤(𝑟)=𝐴ln(𝑟)+𝐵+𝐶𝐼₀(𝑟/𝑎)+𝐷𝐾₀(𝑟/𝑎), which is the complementary function for (13).

The above complementary function with the particular integral will give the solution of the nonhomogeneous partial differential equation (13).

Using the above methodology, the solution of the boundary value problem (13)–(16) is obtained as 𝑤(𝑟)=𝐴₁ln(𝑟)+𝐵₁+𝐶₁𝐼₀𝑟𝑎+𝐷₁𝐾₀𝑟𝑎−𝐺𝑟²4, (17) where 𝐴₁=1𝐷₅₅𝐺𝑘²𝛼2𝑎³(1+𝑘)−𝐺𝑘²4𝑎³1−𝑘²−𝐺𝑘²×𝐼2𝑎₀𝑘𝑎𝐾₁1𝑎+𝐼₁1𝑎𝐾₀𝑘𝑎+𝐺𝑘²4𝑎⁴𝑘²−1+𝐺𝑘²𝛼2𝑎⁴()×𝐼1+𝑘₀𝑘𝑎𝐾₀1𝑎−𝐼₀1𝑎𝐾₀𝑘𝑎+𝐺𝑘4𝑎²𝑘²−1+𝐺𝑘𝛼2𝑎²()×𝐼1+𝑘₁1𝑎𝐾₁𝑘𝑎−𝐼₁𝑘𝑎𝐾₁1𝑎−𝐺𝑘4𝑎³𝑘²−1+𝐺𝑘𝛼2𝑎³(1+𝑘)+𝐺𝑘×𝐼2𝑎₁𝑘𝑎𝐾₀1𝑎+𝐼₀1𝑎𝐾₁1𝑎+𝐺𝑘²2+𝐺2,𝐵₁=1𝐷₅₅𝐼₀𝑘𝑎𝐾₀1𝑎×−𝐺𝑘²2𝑎²ln𝑘−𝑘²−𝑎²×𝐺𝛼2𝑎²𝑘−𝐺𝛼4𝑎⁴𝑘+𝐺𝛼²2𝑎⁴𝑘+𝐺𝑘²4𝑎⁴−𝐺𝑘²𝛼2𝑎⁴ln𝑘−𝐺𝑘⁴𝛼4𝑎⁴𝑎²+−1𝐺𝛼²𝑘2𝑎⁴−𝐺𝛼𝑘2𝑎²𝑘²+𝑎²−𝛼1+𝑘×𝐺2+𝐺𝑘²𝛼2𝑎²−𝐺𝑘²2−𝐺4𝑎²−𝐺𝛼2𝑎²−𝐺𝑘⁴4𝑎²(1−𝛼)+𝐼₀𝑘𝑎𝐾₁1𝑎×−𝐺𝛼²2𝑎³𝑘+𝐺𝛼4𝑎³𝑘𝑘²−𝑎²+𝐺𝑘²4𝑎³−𝐺𝑘²𝛼2𝑎³ln𝑘−𝐺𝑘⁴𝛼4𝑎³𝑎²−−1𝐺𝛼²𝑘−2𝑎𝐺𝛼𝑘2𝑎²−𝐺𝛼²𝑘2𝑎³𝑘²+𝑎²+𝛼1+𝑘𝐺−4𝑎𝐺𝑘²𝛼−2𝑎𝐺𝛼+2𝑎𝐺𝑘⁴𝛼4𝑎+𝐼₀𝑘𝑎𝐾₁𝑘𝑎×−𝐺𝛼²𝑎+𝛼1+𝑘−𝐺𝑘+4𝑎𝐺𝑘𝛼𝑎+𝐺𝛼4𝑎+𝐼₁𝑘𝑎𝐾₀1𝑎×𝐺𝑘()−2𝑎1−𝛼𝐺𝑘4𝑎³+𝐺𝑘𝛼𝑎2𝑎²+𝑘+1ln𝑘₂−𝑎²𝐺𝛼2𝑎𝑘²−𝐺𝛼4𝑎³𝑘²+𝐺𝛼²2𝑎³𝑘²+𝐺𝑘³𝛼4𝑎³−𝐺𝛼²𝑎2𝑎²+−−1𝐺𝛼²2𝑎³+𝐺𝛼2𝑎²×𝑘²+𝑎²𝐺𝑎𝛼2𝑘²−𝐺𝛼𝑎2+𝐺𝑘³()−4𝑎1−𝛼𝐺𝛼4𝑎𝑘²+𝐺𝛼²2𝑎𝑘²1+𝑎²+𝐼₁𝑘𝑎𝐾₁1𝑎×−𝐺𝑘𝛼2−𝐺𝑘4𝑎²+𝐺𝛼𝑘2𝑎²1+𝑎²−ln𝑘𝐺𝛼4𝑎²+𝐺𝛼²2𝑎²+𝐺𝑘³𝛼4𝑎²𝑎²−−1𝐺𝑘²𝛼²2𝑎²−𝐺𝑘³𝛼4+𝐼₁𝑘𝑎𝐾₀𝑘𝑎𝐺𝑘𝛼𝑎𝛼1+𝑘−𝐺𝛼²𝑎−𝐺𝑘4𝑎+𝐼₀1𝑎𝐾₁1𝑎−𝐺𝑘²2𝑎ln𝑘−𝐺𝑘𝛼𝑎−𝐺𝑘²4𝑎+𝐼₀1𝑎𝐾₀𝑘𝑎×−𝐺4𝑎²+𝐺𝑘²𝛼2𝑎²+𝐺𝛼2𝑎²𝛼1+𝑘−𝐺𝛼2𝑎²𝑘−𝐺𝛼4𝑎⁴𝑘+𝐺𝛼²2𝑎⁴𝑘𝑘²−𝑎²+𝐺𝑘²2𝑎²+𝐺𝑘²𝛼2𝑎⁴−𝐺𝑘²4𝑎⁴ln𝑘−𝐺𝑘⁴𝛼4𝑎⁴−𝐺𝛼²𝑘2𝑎⁴𝑘²+𝑎²−𝐺𝑘²2+𝐺𝑘⁴4𝑎²+𝐺𝑘³𝛼2𝑎²+𝐼₀1𝑎𝐾₁𝑘𝑎×−𝐺𝛼4𝑎³𝑘²+𝐺𝛼²2𝑎³𝑘²𝑘²−𝑎²+𝐺𝑘𝛼2𝑎³−𝐺𝑘4𝑎³+𝐺𝑘−2𝑎ln𝑘𝐺𝑘³𝛼4𝑎³−𝐺𝑘²𝛼²2𝑎³+𝐺𝛼−2𝑎𝐺𝛼²𝑎2𝑘²−𝐺𝑎𝛼2+𝐺𝑘³+4𝑎𝐺𝛼𝑘2𝑎²+𝑎²−𝐺𝛼4𝑎𝑘²+𝐺𝛼²2𝑎𝑘²1+𝑎²+𝐼₁1𝑎𝐾₀1𝑎−𝐺𝑘²2𝑎ln𝑘−𝐺𝑘𝛼𝑎−𝐺𝑘²2𝑎+𝐼₁1𝑎𝐾₀𝑘𝑎×𝛼1+𝑘𝐺−4𝑎𝐺𝑘²𝛼−2𝑎𝐺𝛼+2𝑎𝐺𝑘⁴𝛼+4𝑎𝐺𝛼4𝑎³𝑘−𝐺𝛼²2𝑎³𝑘𝑘²−𝑎²+𝐺𝑘²4𝑎³−𝐺𝑘²𝛼2𝑎³ln𝑘−𝐺𝑘⁴𝛼4𝑎³𝑎²+−1𝐺𝛼²𝑘2𝑎³𝑘²+𝑎²+𝐼₁1𝑎𝐾₁𝑘𝑎×𝐺𝛼4𝑎²+−𝐺𝑘𝛼2𝑎²+𝐺𝑘4𝑎²−ln𝑘𝐺𝛼²2𝑎²+𝐺𝑘³𝛼4𝑎²+𝐺𝑘²𝛼²2𝑎²,𝐶₁=1𝐷₅₅𝐾₀1𝑎𝐺21−𝑘²+𝐺4𝑎²𝑘²+−1𝐺𝛼𝑎²(0.5+𝑘)+𝐺𝑘²2𝑎²(𝛼+ln𝑘)+𝐾₁1𝑎𝐺𝑘4𝑎²−1+𝐺𝛼()+2𝑎1+2𝑘𝐺𝑘²(2𝑎𝛼+ln𝑘)+𝐾₀𝑘𝑎𝐺2𝑘²−1−𝐺𝑘²4𝑎²𝑘²−−1𝐺𝛼𝑘²𝑎²(0.5𝑘+1)−𝐺𝑘2𝑎²(𝛼+𝑘ln𝑘)+𝐾₁𝑘𝑎−𝐺𝛼𝑘²−2𝑎𝐺𝛼𝑘𝑎−𝐺()−2𝑎𝛼+𝑘ln𝑘𝐺𝑘𝑘4𝑎²,𝐷−1₁=1𝐷₅₅𝐼₀1𝑎𝐺2𝑘²−𝐺−14𝑎²𝑘²−−1𝐺𝛼𝑎²(0.5+𝑘)−𝐺𝑘²2𝑎²(𝛼+ln𝑘)+𝐼₁1𝑎𝐺𝑘4𝑎²−1+𝐺𝛼()+2𝑎1+2𝑘𝐺𝑘²(2𝑎𝛼+ln𝑘)+𝐼₀𝑘𝑎𝐺21−𝑘²+𝐺𝑘²4𝑎²𝑘²+−1𝐺𝛼𝑘²2𝑎²(𝑘+2)+𝐺𝑘2𝑎²(𝛼+𝑘ln𝑘)+𝐼₁𝑘𝑎−𝐺𝛼𝑎(0.5+𝑘)−𝐺𝑘()−2𝑎𝑘𝛼+ln𝑘𝐺𝑘𝑘4𝑎²,𝐷−1₅₅=𝐼₁1𝑎𝐾₁𝑘𝑎−𝐼₁𝑘𝑎𝐾₁1𝑎×𝑘𝛼𝑎²+𝑘𝑎²𝛼ln𝑘+𝑎²+𝐼₁1𝑎𝐾₀𝑘𝑎+𝐼₀𝑘𝑎𝐾₁1𝑎×𝑘²𝛼𝑎³+𝑘²𝑎³ln𝑘+𝛼𝑘𝑎³+1𝑎−𝐼₀1𝑎𝐾₁𝑘𝑎+𝐼₁𝑘𝑎𝐾₀1𝑎×𝑘𝛼𝑎³+𝑘𝑎³𝛼ln𝑘+𝑎³−𝑘𝑎+𝐼₀𝑘𝑎𝐾₀1𝑎−𝐼₀1𝑎𝐾₀𝑘𝑎×𝑘²𝛼𝑎⁴+𝑘²𝑎⁴ln𝑘+𝛼𝑘𝑎⁴+1𝑎²−𝑘²𝑎²−2. (18)

4. Fully Developed Flow Generated by the Translatory Motion of the Outer Cylinder (Couette Flow)

Here, the flow of an incompressible couple stress fluid between two concentric circular cylinders is generated due to the translatory motion of the outer cylinder with constant velocity (0,0,𝑈) while the inner cylinder is assumed to be at rest. The flow geometry is as shown in Figure 1. Assuming that the pressure in the flow region is constant, the governing equation (8) takes the form 𝜂∇⁴𝑤−𝜇∇²𝑤=0, (19) with the boundary conditions 𝑤(𝑅)𝑑−𝛾𝜂𝑑𝑟𝑤−𝜇∇²𝑤_𝑟=𝑅=𝑈,𝑤𝑅₁𝑑+𝛾𝜂𝑑𝑟𝑤−𝜇∇²𝑤_𝑟=𝑅1=0,𝑑²𝑤𝑑𝑟²=0at𝑟=𝑅,𝑅₁Vanishingofcouplestresses, (20)(21)(22) where 𝑅>𝑅₁.

Figure 1 

Flow geometry.

Using the nondimensional variables 𝑤=𝑈𝑤;𝑟=𝑅𝑟;𝑧=𝑅𝑧, (23) the boundary value problem (19)–(22), after dropping “”s, becomes ∇²∇²−1𝑎²𝑤=0, (24) with the boundary conditions 𝑤(1)𝑑−𝛼𝑑𝑟𝑤−𝑎²∇²𝑤_𝑟=1=1,𝑤(𝑘)𝑑+𝛼𝑑𝑟𝑤−𝑎²∇²𝑤_𝑟=𝑘=0,𝑑²𝑤𝑑𝑟²=0at𝑟=1,𝑘Vanishingofcouplestresses, (25)(26)(27) where 𝑎²=𝜂/𝜇𝑅², 𝑘=𝑅₁/𝑅<1, and 𝛼=𝛾/𝑅.

Now, using the methodology described earlier, the exact solution of boundary value problem (24)–(27) which is valid for the entire flow field is given by 𝑤(𝑟)=𝐴₂ln(𝑟)+𝐵₂+𝐶₂𝐼₀𝑟𝑎+𝐷₂𝐾₀𝑟𝑎, (28) where 𝐴₂=1𝐷₅₅×𝑘²𝑎⁴𝑎²𝑘𝐾₁1𝑎𝐼₁𝑘𝑎−𝐾₁𝑘𝑎𝐼₁1𝑎−𝑎𝐾₁1𝑎𝐼₀𝑘𝑎+𝐾₀𝑘𝑎𝐼₁1𝑎+𝑎𝑘𝐾₀1𝑎𝐼₁𝑘𝑎−𝐾₁𝑘𝑎𝐼₀1𝑎+𝐾₀𝑘𝑎𝐼₀1𝑎−𝐾₀1𝑎𝐼₀𝑘𝑎,𝐶₂=1𝐷₅₅1𝑎𝑘𝐾₁𝑘𝑎−𝐾₁1𝑎−1𝑎²𝑘²𝐾₀𝑘𝑎−𝐾₀1𝑎,𝐷₂=1𝐷₅₅1𝑎𝑘𝐼₁𝑘𝑎−𝐼₁1𝑎−1𝑎²𝑘²𝐼₀𝑘𝑎−𝐼₀1𝑎,𝐵₂=1𝐷₅₅𝑘²𝑎⁴ln𝑘+𝛼𝑘𝑎⁴×𝑎²𝑘𝐾₁𝑘𝑎𝐼₁1𝑎−𝐾₁1𝑎𝐼₁𝑘𝑎+𝑎𝐾₁1𝑎𝐼₀𝑘𝑎+𝐾₀𝑘𝑎𝐼₁1𝑎−𝑎𝑘𝐾₀1𝑎𝐼₁𝑘𝑎+𝐾₁𝑘𝑎𝐼₀1𝑎+𝐾₀1𝑎𝐼₀𝑘𝑎−𝐾₀𝑘𝑎𝐼₀1𝑎+1𝑎𝐾₁1𝑎𝐼₀𝑘𝑎+𝐾₀𝑘𝑎𝐼₁1𝑎+1𝑎²𝐾₀1𝑎𝐼₀𝑘𝑎−𝐾₀𝑘𝑎𝐼₀1𝑎,𝐷−1₅₅=𝐼₁1𝑎𝐾₁𝑘𝑎−𝐼₁𝑘𝑎𝐾₁1𝑎×𝑘𝛼𝑎²+𝑘𝑎²𝛼ln𝑘+𝑎²+𝐼₁1𝑎𝐾₀𝑘𝑎+𝐼₀𝑘𝑎𝐾₁1𝑎×𝑘²𝛼𝑎³+𝑘²𝑎³ln𝑘+𝛼𝑘𝑎³+1𝑎−𝐼₀1𝑎𝐾₁𝑘𝑎+𝐼₁𝑘𝑎𝐾₀1𝑎×𝑘𝛼𝑎³+𝑘𝑎³𝛼ln𝑘+𝑎³−𝑘𝑎+𝐼₀𝑘𝑎𝐾₀1𝑎−𝐼₀1𝑎𝐾₀𝑘𝑎×𝑘²𝛼𝑎⁴+𝑘²𝑎⁴ln𝑘+𝛼𝑘𝑎⁴+1𝑎²−𝑘²𝑎²−2. (29)

5. Generalized Couette Flow

Here again, the physical model is similar to that of the earlier two problems. The flow is generated due to the translatory motion of the outer cylinder with constant velocity (0,0,𝑈) while the inner cylinder is at rest and simultaneously a constant pressure gradient is applied in the positive 𝑧-direction. The momentum equation governing this flow is 𝜂∇⁴𝑤−𝜇∇²𝑤−𝐺=0, (30) subject to the conditions 𝑤(𝑅)𝑑−𝛾𝜂𝑑𝑟𝑤−𝜇∇²𝑤_𝑟=𝑅=𝑈,𝑤𝑅₁𝑑+𝛾𝜂𝑑𝑟𝑤−𝜇∇²𝑤_𝑟=𝑅1=0,𝑑²𝑤𝑑𝑟²=0at𝑟=𝑅,𝑅₁Vanishingofcouplestresses, (31)(32)(33) where 𝑅>𝑅₁.

With the nondimensional parameters 𝑤=𝑈𝑤;𝑟=𝑅𝑟;𝑝=𝜇𝑈𝑅𝑝;𝑧=𝑅𝑧, (34) (30)–(33), after dropping “”s, become∇²∇²−1𝑎²𝐺𝑤=𝑎², (35) with the boundary conditions 𝑤(1)𝑑−𝛼𝑑𝑟𝑤−𝑎²∇²𝑤_𝑟=1=1,𝑤(𝑘)𝑑+𝛼𝑑𝑟𝑤−𝑎²∇²𝑤_𝑟=𝑘=0,𝑑²𝑤𝑑𝑟²=0at𝑟=1,𝑘Vanishingofcouplestresses, (36)(37)(38) where 𝐺=−𝜕𝑝/𝜕𝑧, 𝑎²=𝜂/𝜇𝑅², 𝑘=𝑅₁/𝑅<1, and 𝛼=𝛾/𝑅.

Employing the procedure described in Section 3, the analytical solution of the boundary value problem (35)–(38) which is valid for the entire flow field is given by 𝑤(𝑟)=𝐴₃ln(𝑟)+𝐵₃+𝐶₃𝐼₀𝑟𝑎+𝐷₃𝐾₀𝑟𝑎−𝐺𝑟²4, (39) where 𝐴₃=1𝐷₅₅𝐾₀1𝑎𝐼₁𝑘𝑎×𝐺𝑘()−2𝑎𝛼−1𝐺𝑘4𝑎³𝑘²−−1𝐺𝛼𝑘2𝑎³1+𝑎²+𝑘+𝑘𝑎³+𝐾₀1𝑎𝐼₀𝑘𝑎×𝐺𝑘²4𝑎⁴𝑘²−1+𝐺𝛼𝑘²2𝑎⁴()−𝑘1+𝑘²𝑎⁴+𝐾₁1𝑎𝐼₁𝑘𝑎×𝐺𝑘4𝑎²1−𝑘²−𝐺𝑘𝛼2𝑎²()+𝑘1+𝑘𝑎²+𝐾₁1𝑎𝐼₀𝑘𝑎×−𝐺𝛼𝑘²+2𝑎𝐺𝑘²4𝑎³𝑘²+−1𝐺𝛼𝑘²2𝑎³1+𝑎²+𝑘−𝐺𝑘²−𝑘2𝑎𝑎²³+𝐾₀𝑘𝑎𝐼₁1𝑎×−𝐺𝑘²+2𝑎𝐺𝑘²4𝑎³𝑘²+−1𝐺𝛼𝑘²2𝑎³1+𝑎²+𝑘−𝐺𝛼𝑘²−𝑘2𝑎𝑎²³−𝐾₀𝑘𝑎𝐼₀1𝑎×𝐺𝑘²4𝑎⁴𝑘²−1+𝐺𝛼𝑘²2𝑎⁴()−𝑘1+𝑘²𝑎⁴−𝐾₁𝑘𝑎𝐼₁1𝑎×𝐺𝑘4𝑎²1−𝑘²−𝐺𝑘𝛼2𝑎²()+𝑘1+𝑘𝑎²+𝐾₁𝑘𝑎𝐼₀1𝑎×𝐺𝑘()−2𝑎𝛼−1𝐺𝑘4𝑎³𝑘²−−1𝐺𝛼𝑘2𝑎³1+𝑎²+𝑘+𝑘𝑎³+𝐺𝑘²2+𝐺2,𝐷₃=1𝐷₅₅𝐼₀1𝑎𝐺2𝑘²−𝐺−14𝑎²𝑘²−−1𝐺𝛼𝑎²()+10.5+𝑘𝑎²−𝐺𝑘²2𝑎²(𝛼+ln𝑘)+𝐼₁1𝑎𝐺𝑘4𝑎²−1+𝐺𝛼()−12𝑎1+2𝑘𝑎+𝐺𝑘²(2𝑎𝛼+ln𝑘)+𝐼₀𝑘𝑎𝐺21−𝑘²+𝐺𝑘²4𝑎²𝑘²+−1𝐺𝛼𝑘²2𝑎²()−𝑘𝑘+2²𝑎²+𝐺𝑘2𝑎²(𝛼+𝑘ln𝑘)+𝐼₁𝑘𝑎−𝐺𝛼𝑎()+𝑘0.5+𝑘𝑎−𝐺𝑘()−2𝑎𝑘𝛼+ln𝑘𝐺𝑘𝑘4𝑎²,𝐶−1₃=1𝐷₅₅𝐾₀1𝑎𝐺21−𝑘²+𝐺4𝑎²𝑘²+−1𝐺𝛼𝑎²()−10.5+𝑘𝑎²+𝐺𝑘²2𝑎²(𝛼+ln𝑘)+𝐾₁1𝑎𝐺𝑘4𝑎²−1+𝐺𝛼()−12𝑎1+2𝑘𝑎+𝐺𝑘²(2𝑎𝛼+ln𝑘)+𝐾₀𝑘𝑎𝐺2𝑘²−1−𝐺𝑘²4𝑎²𝑘²−−1𝐺𝛼𝑘²2𝑎²()+𝑘𝑘+1₂𝑎²−𝐺𝛼𝑘²2𝑎²−𝐺𝑘2𝑎²(𝛼+𝑘ln𝑘)+𝐾₁𝑘𝑎−𝐺𝛼𝑘²−2𝑎𝐺𝛼𝑘𝑎+𝑘𝑎−𝐺()−2𝑎𝛼+𝑘ln𝑘𝐺𝑘𝑘4𝑎²,𝐷−1₅₅=𝐼₁1𝑎𝐾₁𝑘𝑎−𝐼₁𝑘𝑎𝐾₁1𝑎×𝑘𝛼𝑎²+𝑘𝑎²𝛼ln𝑘+𝑎²+𝐼₁1𝑎𝐾₀𝑘𝑎+𝐼₀𝑘𝑎𝐾₁1𝑎×𝑘²𝛼𝑎³+𝑘²𝑎³ln𝑘+𝛼𝑘𝑎³+1𝑎−𝐼₀1𝑎𝐾₁𝑘𝑎+𝐼₁𝑘𝑎𝐾₀1𝑎×𝑘𝛼𝑎³+𝑘𝑎³𝛼ln𝑘+𝑎³−𝑘𝑎+𝐼₀𝑘𝑎𝐾₀1𝑎−𝐼₀1𝑎𝐾₀𝑘𝑎×𝑘²𝛼𝑎⁴+𝑘²𝑎⁴ln𝑘+𝛼𝑘𝑎⁴+1𝑎²−𝑘²𝑎²𝐵−2,₃=1𝐷₅₅𝐼₁𝑘𝑎𝐾₀𝑘𝑎×𝐺𝑘𝛼𝛼2𝑎1+𝑘−𝑎²𝛼𝑘−𝑘𝑎𝐺1+4−𝐺𝛼2+𝐺𝛼2𝑎+𝐼₁1𝑎𝐾₀𝑘𝑎×𝛼1+𝑘×1𝑎+𝐺−4𝑎𝐺𝛼−2𝑎𝐺𝑘²𝛼+𝑘2𝑎₂𝑎³𝐺1+4−𝐺𝛼2𝛼ln𝑘+𝑘³𝑘²−𝑎²+𝑘²𝛼𝑎³𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼2𝑘+𝐼₀1𝑎𝐾₀𝑘𝑎×𝐺𝑘²𝛼2𝑎²𝛼1+𝑘−𝑎²𝛼𝑘−𝑎²−𝛼1+𝑘×1𝑎²+𝐺4𝑎²−𝐺𝛼2−𝐺𝛼2𝑎²−𝐺2+𝐺𝑘²2−𝑘²𝑎⁴𝐺1+4−𝐺𝛼2𝛼ln𝑘+𝑘³𝑘²−𝑎²−𝑘²𝛼𝑎⁴𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼+𝑘2𝑘₂𝑎²𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼+2𝑘𝐺𝑘²2𝑎²𝛼ln𝑘+𝑘³𝑘²−𝑎²+𝛼𝛼+²𝑘−𝐺2+𝐺𝑘²2+𝐼₀𝑘𝑎𝐾₁𝑘𝑎×𝛼𝑎𝐺1+4−𝐺𝛼2−𝐺𝛼²+𝛼2𝑎1+𝑘−𝐺𝑘+4𝑎𝐺𝑘𝛼𝑎−𝑘𝑎+𝐼₁1𝑎𝐾₁𝑘𝑎×−𝐺𝛼²+𝛼2𝑎𝑘²𝐺1+4−𝐺𝛼2+𝑘𝛼𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼+𝑘2𝑘𝑎²𝐺1+4−𝐺𝛼2𝛼ln𝑘+𝑘³𝑘²−𝑎²−𝑘𝛼𝑎²𝑎²−1𝐺𝑘²2+𝐺𝑘𝛼2+𝐺𝑎²𝛼2𝑘+𝐼₀1𝑎𝐾₁𝑘𝑎×𝐺𝛼²2𝑎1−𝑎²−𝛼𝑎𝑘²𝐺1+4−𝐺𝛼2+𝐺𝑎²𝛼2+(1−𝛼)𝐺𝑘³+4𝑎𝐺𝑘²𝛼−𝑘2𝑎𝑎³𝐺1+4−𝐺𝛼2𝛼ln𝑘+𝑘³𝑘²−𝑎²+𝑘𝛼𝑎³𝑎²−1𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼+2𝑘𝐺𝑘𝛼2𝑎ln𝑘+𝑘³𝑘²−𝑎²+(1−𝛼)𝐺𝑎𝛼2𝑘²+𝐼₁𝑘𝑎𝐾₀1𝑎×(1−𝛼)𝐺𝑘𝑘2𝑎ln𝑘+𝑎𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼+𝛼2𝑘𝑎𝑘²−𝐺4+𝐺𝛼2+𝐺𝑎²𝛼2−𝑘𝑎³𝐺1+4−𝐺𝛼2−𝐺𝑎²𝛼2×𝛼ln𝑘+𝑘³𝑘²−𝑎²−𝑘𝛼𝑎³𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼−2𝑘𝐺𝛼²𝑎2𝑎²−1−𝑎𝛼𝑘²−𝑎𝛼²𝑘²×−𝐺2+𝐺𝑘²2+(1−𝛼)𝑘²−𝑎²𝐺𝛼2𝑎𝑘²−𝛼𝑎𝑘²+𝐼₀𝑘𝑎𝐾₀1𝑎×(𝛼−1)𝐺𝑘²2𝑎²−𝑘ln𝑘₂𝑎²𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼−𝛼2𝑘1+𝑘−𝐺4𝑎²+𝐺𝛼2+𝐺𝛼2𝑎²+𝐺𝑘²4𝑎⁴−𝐺𝑘²𝛼2𝑎⁴−𝐺𝑘²𝛼2𝑎²+𝑘²𝑎⁴×𝛼ln𝑘+𝑘³𝑘²−𝑎²+𝑘²𝛼𝑎⁴𝐺𝑘²4+𝐺𝑎²𝛼2−𝐺𝑘𝛼−2𝑘𝐺𝑘²𝛼2𝑎²𝛼1+𝑘−𝑎²−𝑎²𝛼𝑘+𝛼1+𝑘−𝐺2+𝐺𝑘²2−𝛼𝛼+²𝑘−𝐺2+𝐺𝑘²2−(1−𝛼)𝑘²−𝑎²𝐺𝛼2𝑎²𝑘+1𝑎²𝛼1+𝑘+𝐼₁1𝑎𝐾₀1𝑎×−𝐺𝑘²2𝑎²ln𝑘−(1−𝛼)𝐺𝑘²+4𝑎𝐺𝑘𝛼+2𝑎𝐺𝑎𝛼−2𝑘𝐺𝑘²𝛼+4𝑎𝐺𝛼²𝑘+2𝑎𝐺𝑎²𝛼²−𝑘2𝑎𝑘₂−𝑎²𝐺𝛼2𝑎𝑘+𝐼₁𝑘𝑎𝐾₁1𝑎×𝐺𝛼²21−𝑎²+𝛼𝑘²−𝐺4+𝐺𝛼2+𝐺𝑎²𝛼2−𝐺𝛼²2𝑘²𝑘²−𝑎²−𝐺𝑘𝛼2−ln𝑘𝐺𝑘4𝑎²−𝐺𝛼𝑘2𝑎²−𝐺𝑘𝛼2+𝑘𝑎²×𝛼ln𝑘+𝑘³𝑘²−𝑎²−𝑘𝛼𝑎²𝐺𝑘²4+𝐺𝑘𝛼+2𝑎𝐺𝑎²𝛼+𝑎2𝑘₂𝛼²𝑘²−𝐺2+𝐺𝑘²2−𝛼𝑘²+𝐼₀𝑘𝑎𝐾₁1𝑎×𝐺𝑘²𝛼𝛼2𝑎ln𝑘−1+𝑘×−𝐺+4𝑎𝐺𝛼+2𝑎𝐺𝑎𝛼2+𝐺𝛼²𝑘2𝑎𝑘²−𝑎²−𝛼ln𝑘+𝑘³𝑘²−𝑎²×𝐺𝑘²4𝑎³−𝐺𝛼𝑘²2𝑎³−𝐺𝑘²𝛼+𝑘2𝑎𝑎²³+𝑘²𝛼𝑎³𝐺𝑘²4+𝐺𝑘𝛼2+𝐺𝑎²𝛼+2𝑘𝐺𝑘²𝛼𝛼2𝑎1+𝑘𝑎²−𝛼−11+𝑘−𝐺𝑎𝛼2+𝐺𝑘²𝑎𝛼2+1𝑎𝛼1+𝑘+𝐼₀1𝑎𝐾₁1𝑎×−1𝑎𝐺𝑘²4+𝐺𝑘𝛼−2𝑎𝐺𝑘²2𝑎ln𝑘−𝐺𝑘𝛼.2𝑎 (40)

6. Discussion of Results

In all of the three considered problems, the governing equations give rise to Bessel differential equations. The solutions are obtained in terms of Bessel function and are simplified using recurrence relations of Bessel functions [27]. The velocity profiles for different flow parameters are presented and the effect of slip parameter, couple stress parameter, and the pressure gradient on velocity for each of the problems is described. We have the couple stress parameter 𝑎²=𝜂/𝜇𝑅²; in the absence of couple stresses, that is, as 𝜂→0, the parameter 𝑎→0 and the corresponding limiting solutions to the problems considered in the paper are given in Table 1. It is noted that these limiting solutions are the same as that of the classical solutions for Newtonian fluid.

Table 1 

Limiting solutions of (i) Poiseuille flow, (ii) Couette flow, and (iii) generalized Couette flow.

From Figure 2, it can be observed that the increasing of slip parameter has an increasing effect on the velocity. That is, the more the fluid slips at the boundary, the less its velocity is affected by the motion of the boundary. The graphs also indicate that, as the couple stress parameter 𝑎 increases, there is a decrease in the velocity. Since 𝑎²=𝜂/𝜇𝑅², 𝑎 increases as the couple stress viscosity 𝜂 increases. Thus, increasing of couple stresses has a decreasing effect on the fluid velocity. It can also be seen from the above figures that the velocity of the fluid increases when we increase the pressure gradient.

(a) 𝐺=10; 𝑎=0.4

(b) 𝛼=0.02; 𝐺=10

(c) 𝑎=0.4; 𝛼=0.02

(d) 𝐺=10; 𝛼=0 (no-slip boundary condition)

(e) 𝑎=0.4; 𝛼=0 (no-slip boundary condition)

Figure 2 

Velocity profiles for Poiseuille flow.

It can be seen from Figure 3(a) that, for Couette flow, as the slip parameter increases, the velocity decreases near the stationary cylinder while the trend is reversed near the moving cylinder. It is observed from Figure 3(b) that increasing of couple stress parameter 𝑎 has a decreasing effect on the velocity. Moreover, under the slip boundary condition the velocity of the fluid increases from couple stress fluid to Newtonian fluid and a similar trend is seen under the no-slip boundary condition as well.

(a) 𝑎=0.4

(b) 𝛼=0.02

(c) 𝛼=0 (no-slip boundary condition)

Figure 3 

Velocity profiles for Couette flow.

Here again, it is found that the increasing of couple stress parameter has a decreasing effect on the velocity and as the slip parameter increases, the velocity decreases near the stationary cylinder while the trend is reversed near the moving cylinder (see Figure 4). As expected, the velocity shows increasing trend with the increasing of pressure gradient. It is also seen from Figures 4(a) and 4(d) that, under the slip boundary condition, the velocity of the fluid increases gradually from couple stress fluid to Newtonian fluid when compared to the no-slip boundary condition.

(a) 𝐺=10; 𝑎=0.4

(b) 𝛼=0.02; 𝐺=10

(c) 𝑎=0.4; 𝛼=0.02

(d) 𝛼=0; 𝐺=10, (no-slip boundary condition)

(e) 𝑎=0.4; 𝛼=0 (no-slip boundary condition)

Figure 4 

Velocity profiles for generalized Couette flow.

7. Concluding Remarks

In this paper, three types of unidirectional fully developed flows of couple stress fluid between two concentric circular cylinders are presented, namely, the fully developed flow due to the applied constant pressure gradient, the fully developed flow due to the translatory motion of the outer cylinder, and the flow generated due to the simultaneous application of constant pressure gradient and the translatory motion of the outer cylinder. The closed form analytical expressions of the velocity are obtained in terms of Bessel functions for each of the problems. As expected, increase in the pressure gradient has corresponding increase in the velocity of the fluid. The observations of the present study are the following.(i)The classical solutions for Newtonian fluid in the hydrodynamic case appear as limiting case of our solutions.(ii)The presence of couple stresses has a decreasing effect on the fluid velocity.(iii)The more the fluid slips at the boundary (i.e., as the slip parameter increases), the less its velocity is affected by the boundary.(iv)Under the slip boundary condition the fluid velocity changes gradually as compared to the no-slip boundary condition.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.