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Abstract

The radial flow between two flat disks plays a crucial role in engineering and industrial applications due to its relevance in lubrication systems, cooling technologies, and fluid transport mechanisms. This study investigates the laminar radial flow of an incompressible Jeffrey fluid between permeable flat disks under the influence of a magnetic field. The exact solutions for both accelerating and decelerating flows are derived using the Jacobi elliptic function of the first kind. The effects of key physical parameters on velocity profiles are analyzed graphically, while torque variations are presented in tabular form. Additionally, the streamline patterns and 2D velocity graphs exhibit consistent trends across parameter variations, reinforcing the accuracy of the findings. Results indicate that accelerating velocity increases with a stronger magnetic field but decreases for all other parameters. In contrast, decelerating velocity rises with increasing porosity and declines for all remaining factors. Torque analysis reveals that an increase in the Jeffrey fluid parameter enhances torque on both disks, whereas a higher Darcy number increases the torque on the upper disk but reduces it on the lower disk. Streamline analysis is conducted to illustrate flow behavior under varying parameter conditions. The results confirm a parabolic velocity profile, with maximum velocity at the central region and minimum velocity at the disk surfaces.

Keywords

accelerating and decelerating flow MHD Jeffrey fluid porous medium Jacobi elliptic sine-squared function exact solution

Introduction

Disk flow encompasses two fundamental types of non-swirling fluid motion: accelerating flow and decelerating flow. Accelerating flow occurs when fluid is induced from the boundaries and drained through a central outlet, whereas the reverse process defines decelerating flow. This type of flow is a core concept in fluid mechanics, with significant applications in engineering and industrial systems. It plays a crucial role in the design and optimization of various technologies, including air bearings, flow control valves, disk-type heat exchangers, atmospheric micrometers, viscosity meters, HVAC systems, artificial heart valves, centrifugal pumps, and turbines. Furthermore, the U.S. Federal Highway Administration Hydraulics Lab is investigating disk-type flow for studying bed-load inception, as it involves varying shear stress at disk surfaces. A comprehensive review of disk-type flow reveals that numerous studies have been conducted due to its widespread industrial significance. However, most research has primarily focused on numerical solutions, given the nonlinear nature of the governing equations. Over the years, various numerical techniques have been employed to tackle this problem. Willis¹ pioneered early research in this field, followed by Osterle et al.,² who used the Kármán integral technique to derive an approximate solution for high Reynolds number disk flows, highlighting the significance of inertial effects in the laminar flow regime. Livesey³ later applied the integral method to analyze viscous flow between parallel plates while considering inertia effects. Several researchers extended this work by introducing different solution techniques under varying physical conditions. Morgan and Saunders⁴ numerically solved the inertial effect in disk flow, while Savage⁵ used the perturbation technique to analyze creeping and laminar flows between flat parallel plates. Moller⁶ explored inflow and outflow in disk flows through simplifications of the Navier-Stokes equations. Boyack⁷ investigated inflow between two rotating disks via the integral method, later extending his study to non-rotating disks.⁸ Other key contributions include the work of Lee and Lin,⁹ who derived a numerical solution for radial pressure distribution, and Vatistas,^10,11 who provided a closed-form solution by simplifying inertial terms and analyzing Reynolds number variations in velocity profiles. Vatistas and Zitouni^12,13 further utilized series expansion, asymptotic methods, and numerical techniques to explore disk flow behavior, observing that velocity decreases in the central region while increasing at disk surfaces. Hossain et al.¹⁴ incorporated temperature-dependent viscosity and magnetohydrodynamics (MHD), employing perturbation, asymptotic, and finite difference methods. Arikoglu et al.¹⁵ examined the influence of temperature and velocity jumps on entropy generation, flow characteristics, and heat transfer, solving the nonlinear equations via the von Kármán substitution technique. Despite extensive research, most previous studies have relied on numerical or closed-form solutions for tackling nonlinear governing equations in disk flow problems. Given its practical significance, an exact analytical solution is necessary for simplifying computations and enhancing theoretical insights. Guo et al.¹⁶ derived an exact solution for fully developed flow between flat radial disks, using Jacobi elliptic sine-squared functions to simplify the nonlinear governing equations for both inflow and outflow. Their analysis detailed laminar flow with a parabolic velocity profile at low Reynolds numbers. Mahmud et al.¹⁷ extended disk flow studies by incorporating magnetic effects, while Naeem et al.^18–20 recently advanced the field by deriving exact solutions for accelerating and decelerating magnetized flows through porous disks under various conditions.

Non-Newtonian fluids have attracted significant attention from researchers and scientists due to their wide-ranging applications in engineering, industrial processes, and biomedical sciences. Examples include paints, oils, clay coatings, blood, and drilling mud, where the shear stress and deformation rate do not exhibit a linear relationship. Due to this complexity, solving flow problems involving non-Newtonian fluids requires advanced mathematical models and computational techniques. To capture their unique rheological behavior, several fluid models have been developed, including Oldroyd-A, Oldroyd-B, Williamson, Maxwell, Jeffrey, and Casson fluid models. Among these, the Jeffrey fluid model is particularly important due to its ability to exhibit both viscous and elastic characteristics under deformation. This dual behavior makes it suitable for studying viscoelastic flows in various scientific and industrial applications. Unlike Newtonian fluids, Jeffrey fluids deviate from the linear shear stress-strain relationship, with deformation influenced by multiple factors such as strain rate, material properties, and external forces. Several researchers have investigated the hydrodynamics and heat transfer characteristics of Jeffrey fluid under different conditions.^21–23 Further, Hayat et al.²⁴ analyzed the effect of Newtonian heating and magnetohydrodynamics (MHD) on Jeffrey fluid flow along an impermeable stretching surface, concluding that fluid velocity decreases under magnetic effects, while temperature rises with Newtonian heating. Qasim²⁵ extended this study to include heat and mass transfer effects in Jeffrey fluid flow along a stretching sheet, incorporating the influence of a heat source and solving the equations using the power series method. Ellahi et al.²⁶ explored peristaltic flow of Jeffrey fluid in eccentric cylinders under MHD effects, deriving series solutions for the problem. Prasad et al.²⁷ examined the boundary layer flow of Jeffrey fluid along permeable circular cylinders, incorporating heat transfer analysis and solving the problem numerically. Similarly, Nallapu and Radhakrishnamacharya²⁸ analyzed Jeffrey fluid flow through narrow porous tubes under a magnetic field, deriving analytical solutions and illustrating the effects of key parameters graphically. Ramesh²⁹ explored Poiseuille, Couette, and generalized Couette flows of Jeffrey fluid between parallel plates, incorporating effects of Joule heating, thermal radiation, viscous dissipation, and slip boundary conditions, and obtaining analytical solutions for all governing equations. Later, Ramesh and Joshi³⁰ extended this study to unsteady flows, deriving solutions numerically. The effects of magnetic fields and thermal radiation on different non-Newtonian fluid models (Jeffrey, Maxwell, and Oldroyd-B) in a porous medium along a stretching sheet were examined by Khan et al.³¹ Naganthran et al.³² analyzed Jeffrey fluid flow over a porous stretching disk, focusing on heat generation and absorption effects. Rana et al.³³ investigated the transverse swimming of motile gyrotactic microorganisms in a Jeffrey fluid near a stretching wall, employing the shooting method and Runge-Kutta-Fehlberg technique for numerical solutions. Their study revealed that the first solution was more stable than the second. In lubrication studies, Ali et al.³⁴ examined the flow of incompressible Jeffrey fluid between two co-rotating rolls under MHD effects, utilizing lubrication approximation theory to simplify governing equations and obtain analytical solutions for velocity and pressure. Similarly, Sadiq et al.³⁵ analyzed Jeffrey’s fluid lubrication effects in spiraling disk flows, demonstrating that increased lubrication enhances flow speed while reducing skin friction at the disk boundary. More recently, Khan et al.³⁶ studied Jeffrey fluid flow with nanofluidics, MHD, and thermal radiation effects in a conical disk system, solving the problem numerically. Their findings highlight the complex interactions between magnetic fields, thermal properties, and fluid dynamics in confined geometries.

The literature review indicates that previous research primarily focused on solution methodologies for disk flow, with limited emphasis on non-Newtonian fluids and magnetic effects. Motivated by the growing significance of Jeffrey fluid in industrial and engineering applications, this study explores the electrically conducting Jeffrey fluid flow between permeable radially flat disks under the influence of a magnetic field. Disk-type flows are widely encountered in turbomachinery, centrifugal pumps, heat exchangers, cooling systems, and bioreactors, making this research highly relevant to both mechanical and biomedical engineering. Furthermore, it is observed that most existing studies have relied exclusively on numerical techniques for solving disk flow problems. In contrast, this work presents a novel analytical approach by deriving exact solutions for the governing equations using complete and incomplete Jacobi elliptic functions of the first kind. The novelty of the problem is given in Table 1. The effects of key physical parameters on the velocity profile are examined graphically, while the influence on torque distribution is systematically presented in tabular form. This analytical framework provides deeper physical insights into the behavior of Jeffrey fluid in disk flows, offering a valuable reference for future studies in fluid dynamics and magnetohydrodynamics (MHD) applications.

Table 1.

Novelty of the present problem.

Ref. no.	[16]	[18]	[19]	[20]	Present
Disk flow	Yes	Yes	Yes	Yes	Yes
Hybrid nanofluid	No	No	No	No	No
Jeffrey fluid	No	No	No	No	Yes
MHD	No	Yes	Yes	Yes	Yes
Porosity	No	Yes	No	Yes	Yes
Thermal radiation	No	Yes	No	Yes	No
Source parameter	No	Yes	No	Yes	No

Mathematical formulation

A fully developed accelerating and decelerating flow of an incompressible Jeffrey fluid is examined between two parallel disks having an inner radius $R_{1}$ and outer radius $R_{2}$ with porous medium. The distance between the disks is considered as 2h as illustrated in Figure 1. The velocity, pressure, and shear stress are calculated for the region where flow is steady and fully developed, the inlet and outlet effects are ignored, and the pressure $p_{1} - p_{0}$ decreases in the considered region and also by considering $R_{2} - R_{1} >> 2 h$ . The magnetic field $B = (0, 0, B_{0})$ is applied normally to the disk. From Ohm’s law, the induced current density can be expressed as $J = σ (E + V \times B)$ where $E$ shows the electric field and $σ$ shows the conductivity of the fluid. When the effect of an electric field is neglected, Ohm’s law gives $J ~ σ uB$ . So, the Lorentz force $F = J \times B = (- σ {uB}_{0}^{2}, 0, 0)$ is acting on the fluid in the direction opposite to the direction of fluid motion. The flow is considered purely radial, steady, and axisymmetric.

Figure 1.

Schematic geometry for flow between two narrowly flat radial disks.

Justification for the assumptions used in the study

In developing the mathematical model for the radial flow of an incompressible Jeffrey fluid between permeable flat disks under a magnetic field, certain assumptions were made to simplify the governing equations while preserving the essential physics of the problem. Below is a detailed explanation of the rationale behind these assumptions:

Incompressible fluid assumption

The fluid is assumed to be incompressible, meaning its density remains constant throughout the flow.

This is a reasonable assumption for liquids (such as lubricants, biofluids, and industrial coolants), where density variations due to pressure changes are negligible.

Laminar flow assumption

The study considers laminar flow, which is justified based on the low Reynolds number typically observed in applications like lubrication systems, cooling devices, and biomedical flows.

In laminar regimes, the flow is smooth and predictable, allowing for accurate analytical solutions.

Jeffrey fluid as a non-Newtonian model

The Jeffrey fluid model is chosen because it effectively captures both viscous and elastic effects, making it more suitable for describing complex fluids like biofluids, polymeric solutions, and industrial suspensions.

This assumption is well-supported by previous studies in non-Newtonian fluid mechanics.

Magnetic field influence (MHD)

The inclusion of a magnetic field is based on its importance in MHD lubrication, electromagnetic pumps, and biomedical applications (such as targeted drug delivery using magnetic nanoparticles).

The assumption that the magnetic field is uniform and externally applied simplifies the governing equations while retaining the core physical effects of MHD forces.

Permeability of the disks

The assumption that the disks are permeable is justified by its relevance in filtration, cooling systems, and biological membranes where fluid injection or suction occurs.

This allows for a more realistic analysis of flows in porous media and engineering applications.

Use of the longitudinal symmetry in the radial direction

The flow is assumed to be radially symmetric, meaning variations in the azimuthal direction are neglected.

This simplification is valid for centrally symmetric systems, such as disk-shaped lubrication films and rotating machinery.

For the above-mentioned assumptions, the velocity field becomes¹⁶

u_{r} = u_{r} (r, z) .

(1)

The continuity equation becomes

\frac{1}{r} \frac{\partial}{\partial r} (r u_{r}) = 0,

(2)

which provides

r u_{r} = ν F (z),

(3)

where $υ$ expresses the kinematic viscosity and $F (z)$ is a non-dimensional function depending on $z .$ Moreover, $F (z) = \frac{r u_{r}}{υ}$ is a local Reynolds number which is invariant along a streamline (z = constant), particularly the midplane Reynolds number $R = r u_{max} (r) / ν$ remains constant with the subscript max indicates the local maximum velocity. So, the radial flow becomes self-similar which reduces the $u_{r} (r, z)$ from different $r$ should collapse into one in terms of appropriate similarity variables. This conjecture is proved subsequently.

The extra stress tensor for the Jeffrey fluid is given by²³:

S = \frac{μ}{1 + λ_{1}} (γ + λ_{2} \frac{D γ}{Dt}),

(4)

where $λ_{1}$ indicates the Jeffrey fluid parameter which is the ratio of relaxation time to the retardation time, $λ_{2}$ is the retardation time, $μ$ indicates the dynamic viscosity, and $γ$ is first Rivlin Erickson tensor having expression

γ = \nabla V + {(\nabla V)}^{t} .

(5)

Equation (2) gives

\frac{\partial u_{r}}{\partial r} = - \frac{u_{r}}{r} .

(6)

Equation (6) reduced the momentum equation in the radial direction $r$ as^18–20:

\begin{matrix} ρ u_{r} \frac{\partial u_{r}}{\partial r} = \frac{μ}{(1 + λ_{1})} [\frac{\partial^{2} u_{r}}{\partial z^{2}} + \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u_{r}}{\partial r}) - \frac{u_{r}}{r^{2}}] \\ - \frac{\partial p}{\partial r} - \frac{μ}{k_{0}} u_{r} - σ B_{0}^{2} u_{r}, \end{matrix}

(7)

where $p$ indicates the pressure, $ρ$ indicates the density of the fluid, and $κ_{0}$ is the permeability of the porous medium. Solving equations (6) and (7) gives

\frac{ρ u_{r}^{2}}{r} + \frac{μ}{(1 + λ_{1})} \frac{\partial^{2} u_{r}}{\partial z^{2}} = \frac{\partial p}{\partial r} + \frac{μ}{k_{0}} u_{r} + σ B_{0}^{2} u_{r},

(8)

which indicates that the pressure gradient speeds up or slows down the flow by controlling the viscous effect. The assumption of axisymmetric flow results as $\frac{\partial p}{\partial θ} = 0$ , which reduced the pressure as $p = p (r, z)$ . The impact of gravitational acceleration in the z-direction gives the momentum equation as

\frac{\partial p}{\partial z} = - ρ g .

(9)

Differentiating equation (8) w.r.t $z$ and equation (9) w.r.t $r$ and removing ( $\partial^{2} p / \partial r \partial z$ ), the equations become

\frac{\partial}{\partial z} (\frac{ρ u_{r}^{2}}{r} + \frac{μ}{(1 + λ_{1})} \frac{\partial^{2} u_{r}}{\partial z^{2}} - \frac{μ}{k_{0}} u_{r} - σ B_{0}^{2} u_{r}) = 0 .

(10)

By applying integration, the above equation becomes

\frac{ρ u_{r}^{2}}{r} + \frac{μ}{(1 + λ_{1})} \frac{\partial^{2} u_{r}}{\partial z^{2}} - \frac{μ}{k_{0}} u_{r} - σ B_{0}^{2} u_{r} = ψ (r),

(11)

where $ψ (r)$ indicates an unknown function which by comparison of equations (8) and (11) becomes

\frac{\partial p}{\partial r} = ψ (r),

(12)

which shows $\frac{\partial p}{\partial r}$ in equation (8) is invariant w.r.t $z$ . Equation (8) for $u_{r} (r, z)$ can be solved by using no-slip condition as

u_{r} (r, \pm h) = 0 .

(13)

Due to flow symmetry along $z = 0,$ the above boundary conditions become

u_{r} (r, 0) = \pm u_{max} (r), \frac{\partial u_{r}}{\partial z} (r, 0) = 0,

(14)

where the minus sign indicates the accelerating flow while the plus sign indicates the decelerating flow. The $u_{max} (r)$ can be obtained by using the integral form of the continuity equation. So, $p (r, z)$ can be evaluated by equations (9) and (12) while $u (r, z)$ can be evaluated by using boundary conditions from equation (14) into equation (8).

Non-dimensionalization of governing equation

The non-dimensional solution of equation (8) can be obtained by using equation (3) as

u_{r} (r, z) = \frac{ν}{r} F (z),

(15)

which gives

\frac{\partial^{2} u_{r}}{\partial z^{2}} = \frac{ν}{r} \frac{d^{2} F}{d z^{2}} .

(16)

Substitution of equations (15) and (16) into equation (8) gives

\frac{ρ ν^{2}}{r^{3}} F^{2} + \frac{μ}{(1 + λ_{1})} \frac{ν}{r} \frac{d^{2} F}{d z^{2}} = \frac{\partial p}{\partial r} + σ B_{0}^{2} \frac{ν}{r} F + \frac{μ}{k_{0}} \frac{ν}{r} F .

(17)

As $F (z)$ is dimensionless, so multiplication of $r^{3} / ρ ν^{2}$ on both sides of equation (17) gives

F^{2} + r^{2} (\frac{1}{1 + λ_{1}}) \frac{d^{2} F}{d z^{2}} = \frac{r^{3}}{ρ ν^{2}} \frac{\partial p}{\partial r} + \frac{σ B_{0}^{2}}{ρ ν} r^{2} F + \frac{r^{2}}{k_{0}} F,

(18)

which indicates that $r$ is the suitable length scale for $z$ . So, by inserting $ξ = z / r$ , equation (18) becomes

F^{2} + (\frac{1}{1 + λ_{1}}) \frac{d^{2} F}{d ξ^{2}} = \frac{r^{3}}{ρ ν^{2}} \frac{\partial p}{\partial r} + \frac{σ B_{0}^{2}}{ρ ν} r^{2} F + \frac{r^{2}}{k_{0}} F,

(19)

And the boundary conditions from equation (14) changed as

F (0) = \pm \frac{r u_{max} (r)}{ν} = \pm R, \frac{dF}{d ξ} (0) = 0 .

(20)

The non-zero condition in equation (20) implies that the problem can be further simplified if rescaling $u_{r}$ with $u_{max} (r)$ namely $U = (u_{r} / u_{max} (r)) = (F (ξ) / R) .$ So, by substituting $F = RU,$ the equation (19) becomes

(\frac{1}{1 + λ_{1}}) \frac{d^{2} U}{d Z^{2}} + U^{2} = α + M^{2} U + \frac{1}{Da} U,

(21)

where the non-dimensional quantities can be expressed as

\begin{matrix} Z = ξ \sqrt{R} = \frac{z}{r} \sqrt{R}, α = \frac{1}{R^{2}} \frac{r^{3}}{ρ ν^{2}} \frac{\partial p}{\partial r} = \frac{r}{ρ u_{max}^{2}} \frac{\partial p}{\partial r}, \\ M = \sqrt{\frac{σ}{ρ ν}} r B_{0}, Da = \frac{k_{0}}{r^{2}} . \end{matrix}

(22)

The dimensionless boundary conditions become

U (0) = \pm 1, \frac{dU}{d ξ} (0) = 0 .

(23)

In equation (22), M is the magnetic parameter, $Da$ is the Darcy number, and $α = α (r)$ indicates the dimensionless pressure gradient defined as the ratio of a pressure gradient to the inertial forces, and the continuity equation in integral form is employed to derive its value. When $α (r)$ is derived, $\partial p / \partial r$ can also be calculated by using equations (8) and (9) as

\frac{\partial p}{\partial r} = \frac{R^{2} ρ ν^{2}}{r^{3}} α (r) .

(24)

Now, the solution for $U (Z)$ can be derived as

Velocity distribution

Accelerating flow

Equation (8) can be solved by multiplying with $dU / dZ$ and then integrating as

\frac{1}{2} (\frac{1}{1 + λ_{1}}) {(\frac{dU}{dZ})}^{2} + \frac{U^{3}}{3} - α U - \frac{M^{2} U^{2}}{2} - \frac{1}{Da} \frac{U^{2}}{2} = A_{1},

(25)

where $A_{1}$ is the constant of integration which can be calculated by using equation (23) with a minus sign as

\begin{matrix} \frac{1}{2} (\frac{1}{1 + λ_{1}}) {(\frac{dU}{dZ})}^{2} + \frac{1}{3} (U^{3} + 1) - α (U + 1) \\ - \frac{1}{2} (M^{2} + \frac{1}{Da}) (U^{2} - 1) = 0, \end{matrix}

(26)

which can be expressed as

\begin{matrix} \frac{3}{2} (\frac{1}{1 + λ_{1}}) {(\frac{dU}{dZ})}^{2} = - (U^{3} + 1) + 3 α (U + 1) \\ + \frac{3}{2} (M^{2} + \frac{1}{Da}) (U^{2} - 1) \\ = - (U + 1) [{(U - \frac{1}{2})}^{2} + \frac{3}{4} - 3 α - \frac{3}{2} (M^{2} + \frac{1}{Da}) (U - 1)] . \end{matrix}

(27)

For accelerating flow, the condition is $- 1 \leq U \leq 0$ which gives $U + 1 \geq 0 .$ So, equation (27) must satisfy

[{(U - \frac{1}{2})}^{2} + \frac{3}{4} - 3 α - \frac{3}{2} (M^{2} + \frac{1}{Da}) (U - 1)] \leq 0,

(28)

resulting as

α \geq \frac{1}{3} {[{(U - \frac{1}{2})}^{2} + \frac{3}{4} - \frac{3}{2} (M^{2} + \frac{1}{Da}) (U - 1)]}_{U = - 1} = 1 .

(29)

This is the accelerating flow condition, meaning that if $α < 1$ , accelerating flow does not occur. For example if $α = 0$ then equation (27) requires $U < - 1$ , which is physically impossible for this considered problem. Equation (27) can be rearranged as

\frac{3}{2} (\frac{1}{1 + λ_{1}}) {(\frac{dU}{dZ})}^{2} = (U + 1) (U_{1} - U) (U - U_{2}),

(30)

where the real roots for $α \geq 1$ are calculated as

U_{1, 2} = \frac{1}{2} [H_{1} \pm \sqrt{{(H_{1})}^{2} - 4 (H_{1} - 3 α)}],

(31)

H_{1} = 1 + \frac{3}{2} (M^{2} + \frac{1}{Da}) .

(32)

Simplification of equation (30) gives

\sqrt{\frac{3}{2 (1 + λ_{1})}} (\frac{dU}{dZ}) = \sqrt{(U + 1) (U_{1} - U) (U - U_{2})} .

(33)

The solution of above equation (33) can be obtained by integrating and using the condition $U (0) = - 1$ as

\begin{matrix} Z \sqrt{\frac{2 (1 + λ_{1})}{3}} = \int_{- 1}^{U} \frac{dU}{\sqrt{(U + 1) (U_{1} - U) (U - U_{2})}} \\ = \frac{- 2 [F (ϕ | m_{1}) - K]}{\sqrt{U_{1} - U_{2}}}, \end{matrix}

(34)

where $K = K (m_{1})$ indicates the complete elliptic integral of first kind with $m_{1}$ as parameter and $F (ϕ | m_{1})$ indicates the incomplete elliptic integral of first kind having $ϕ$ as amplitude and can be expressed as

m_{1} = \frac{1 + U_{1}}{U_{1} - U_{2}}, ϕ = \sin^{- 1} \sqrt{\frac{U_{1} - U}{1 + U_{1}}} .

(35)

Equation (34) can be rewritten as

F (ϕ | m_{1}) = K - Z \sqrt{\frac{(U_{1} - U_{2}) (1 + λ_{1})}{6}},

(36)

and the simplification of equation (36) by using the Jacobi elliptic sine function gives

\frac{U_{1} - U}{1 + U_{1}} = s n^{2} (K - Z \sqrt{\frac{(U_{1} - U_{2}) (1 + λ_{1})}{6}}) .

(37)

The above-calculated solution can be verified by putting equation (37) into equation (30).

Moreover, the solution in terms of Z with the no-slip condition can be re-written as

Z_{h} = \frac{- \sqrt{6} [F (ϕ | m_{1}) - K]}{\sqrt{(U_{1} - U_{2}) (1 + λ_{1})}}, ϕ_{1} = \sin^{- 1} \sqrt{\frac{U_{1}}{1 + U_{1}}} .

(38)

Equation (37) shows the exact solution of accelerating the flow of Jeffry fluid with the magnetic effect in the porous medium.

Decelerating flow

Equation (8) is also applicable for decelerating flow by using the boundary condition with a plus sign to obtain the constant of integration as

\begin{matrix} \frac{1}{2 (1 + λ_{1})} {(\frac{dU}{dZ})}^{2} + \frac{1}{3} (U^{3} - 1) - α (U - 1) \\ - \frac{1}{2} (U^{2} - 1) (M^{2} + \frac{1}{Da}) = 0, \end{matrix}

(39)

\begin{matrix} \frac{3}{2 (1 + λ_{1})} {(\frac{dU}{dZ})}^{2} = - (U^{3} - 1) + 3 α (U - 1) \\ + \frac{3}{2} (U^{2} - 1) (M^{2} + \frac{1}{Da}) \\ = (1 - U) [{(U + \frac{1}{2})}^{2} + \frac{3}{4} - 3 α - \frac{3}{2} (U + 1) (M^{2} + \frac{1}{Da})] . \end{matrix}

(40)

The condition for decelerating flow is $0 \leq U \leq 1$ which indicates $1 - U \geq 0 .$ So, equation (40) must require

{(U + \frac{1}{2})}^{2} + \frac{3}{4} - 3 α - \frac{3}{2} (U + 1) (M^{2} + \frac{1}{Da}) \geq 0,

(41)

resulting as

α \leq \frac{1}{3} {[{(U + \frac{1}{2})}^{2} + \frac{3}{4} - \frac{3}{2} (U + 1) (M^{2} + \frac{1}{Da})]}_{U = 0} = \frac{1}{3},

(42)

which indicates the condition for decelerating flow. Equation (40) can be rearranged as

\frac{3}{2 (1 + λ_{1})} {(\frac{dU}{dZ})}^{2} = (1 - U) (U - U_{3}) (U - U_{4}),

(43)

where

U_{3, 4} = \frac{1}{2} [- H_{2} \pm \sqrt{{(H_{2})}^{2} - 4 (H_{2} - 3 α)}],

(44)

with

H_{2} = 1 - \frac{3}{2} (M^{2} + \frac{1}{Da}) .

(45)

The roots $U_{3}$ and $U_{4}$ will be real for the condition $4 H_{2} - H_{2}^{2} \leq 12 α$ and complex for $4 H_{2} - H_{2}^{2} > 12 α .$

Equation (40) can be simplified by using $U (0) = 1$ as

Z \sqrt{\frac{(1 + λ_{1})}{6}} = - \frac{F (ϕ | m_{2})}{\sqrt{1 - U_{4}}},

(46)

with

m_{2} = \frac{U_{3} - 1}{U_{4} - 1}, ϕ = \sin^{- 1} \sqrt{\frac{U - 1}{U_{3} - 1}} .

(47)

Equation (46) can be expressed in terms of Jacobi elliptic sine function as

\frac{U - 1}{U_{3} - 1} = s n^{2} (Z \sqrt{\frac{(1 - U_{4}) (1 + λ_{1})}{6}}) .

(48)

Equation (48) is accurate for the whole region of decelerating flow due to the symmetry along the z-axis. By using $U = 0$ for upper boundary, the following expression is obtained

Z_{h} = \frac{\sqrt{6} F (ϕ_{2} | m_{2})}{\sqrt{(1 - U_{4}) (1 + λ_{1})}}, ϕ_{2} = \sin^{- 1} \sqrt{\frac{1}{1 - U_{3}}} .

(49)

Pressure distribution

The non-dimensional flow rate in terms of maximum velocity is given as

Q = 4 π h (r u_{max}) | U' | .

(50)

In equation (50), $(r u_{max})$ is constant which is obtained from equation (3). So, $U'$ must be constant which indicates that $α$ is also invariant. Moreover, equations (21) and (24) represent $α$ is not dependent on $r .$ So pressure distribution can be evaluated by using equations (9) and (24) as

p (r, z) = α ρ {(R ν)}^{2} \int \frac{dr}{r^{3}} - \int g ρ dz = - g ρ z - \frac{α ρ {(R ν)}^{2}}{2 r^{2}} + D,

(51)

where $D$ is a constant of integration and can be obtained by using the pressure $p_{0} = p (r_{0}, z_{0})$ which gives

p (r, z) = p_{0} + g ρ (z_{0} - z) + \frac{α ρ {(R ν)}^{2}}{2} (\frac{1}{r_{0}^{2}} - \frac{1}{r^{2}}) .

(52)

Specifically, for the upper boundary that is, $z = h$ , the pressure $p_{0}$ and $p_{1}$ at $(r_{0}, z_{0})$ and $(r_{1}, z_{0})$ gives the result as

α R^{2} = \frac{2 (p_{0} - p_{1})}{ρ ν^{2}} {(\frac{1}{r_{1}^{2}} - \frac{1}{r_{0}^{2}})}^{- 1} .

(53)

Equation (50) can also be written as

R = \frac{Q}{4 π h ν | U' |} .

(54)

Now, $U', R$ , and $α$ can be evaluated by using equations (50), (53), and (54) by using a graphical method. The different values of $α$ in the case of accelerating and decelerating flow can be used to find a convergent solution. This solution verifies the theoretical values of velocity and pressure for both accelerating and decelerating flows.

Disk torque

The torque $τ$ generated by a disk having a radius $r$ can be generated as

τ = 2 π \int_{0}^{r} S_{rz} rdr,

(55)

where $S_{rz}$ for the problem under consideration is defined as

S_{rz} = U_{max} \frac{μ}{(1 + λ_{1})} {\frac{dU}{dZ} |}_{Z = \pm 1} .

(56)

By using equation (56) in equation (55), the torque becomes

τ = π r^{2} U_{max} \frac{μ}{(1 + λ_{1})} \frac{dU}{dZ} .

(57)

Shear stress distribution

For the condition $u_{θ} = 0$ and $u_{z} = 0$ , the remaining four non-zero components of shear stresses are given as

\begin{matrix} S_{rr} = \frac{2 μ}{(1 + λ_{1})} \frac{\partial u_{r}}{\partial r} - p, S_{θ θ} = \frac{2 μ}{(1 + λ_{1})} \frac{u_{r}}{r} - p, \\ S_{zz} = - p, S_{rz} = S_{zr} = \frac{μ}{(1 + λ_{1})} \frac{\partial u_{r}}{\partial z} . \end{matrix}

(58)

Shear stresses at the surface of disks have much importance. So, shear stresses at the boundary can be evaluated by using

u_{max} = \frac{Q}{4 π rh | U' |}, u_{r} (r, z) = u_{max} (r) U (Z), Z = \frac{z}{r} \sqrt{R},

(59)

and

\frac{\partial u_{r}}{\partial z} = u_{max} \frac{dU}{dZ} \frac{\partial Z}{\partial z} = \frac{Q}{4 π h | U' |} \frac{\sqrt{R}}{r^{2}} \frac{dU}{dZ},

(60)

where using equation (25), $dU / dZ$ can be obtained. It is observed that when the above equation is substituted in equation (58), the shear stress $S_{rz}$ satisfies an inverse-square law.

Convergence criteria

To verify the convergence of Jacobi elliptic sine squared function, standard convergence criteria is applied as

Boundedness and periodicity

The Jacobi elliptic sine function is periodic with a real period $K (m)$ where $K (m)$ is the complete elliptic integral of the first kind is. Since the function is periodic and bounded within $0 \leq \sin^{2} (u, m) \leq 1$ the solution remains convergent for all real values of $Z$ .

Convergence of the elliptic integral representation

The function transformation uses incomplete elliptic integral relations as in equation (48) for the decelerating flow. For convergence, $m_{2}$ must remain within $0 \leq m_{2} \leq 1 .$ The real roots $U_{3}$ and $U_{4}$ satisfy the condition $4 H_{2} - H_{2}^{2} \leq 12 α$ ensuring that the modulus $m_{2}$ remains real and within the valid range.

Specific cases of $m$

Case $m = 0$ :

$\sin (u, 0) = \sin (u)$ leading to a simple trigonometric function that is clearly convergent.

Case $m = 1$ :

$\sin (u, 1) = \tanh u$ , implying that the function approaches 1 as $u \to \infty$ but remains bounded.

Case $0 < m < 1$ :

The function is periodic and remains within the range [0, 1], ensuring uniform convergence.

Boundary condition verification

The condition $U (0) = 1$ leads to:

Z \sqrt{\frac{(1 + λ_{1})}{6}} = - \frac{F (ϕ | m_{2})}{\sqrt{1 - U_{4}}},

which is well-defined and ensures a valid elliptic integral expression for all real $Z .$ Thus, the Jacobi elliptic sine squared function used in the study is fully convergent under the given conditions. The same behavior is followed for the accelerating flow.

Results and discussion

In this article, the radial flow of a Jeffrey fluid is considered between two flat disks. The effect of MHD and porous medium is studied in detail. The analytical solution for both accelerating and decelerating flow is derived by using the Jacobi elliptic function of the first kind to solve non-linear governing equations. The impact of Jeffrey fluid parameter, darcy number, magnetic and pressure gradient parameters on accelerating and decelerating velocity are expressed graphically. The streamlines for accelerating and decelerating flow are also shown graphically. Moreover, the torque on both disks is also derived and expressed in tabular form. Moreover, the accuracy and reliability of the present work have been validated through a comparison of the obtained velocity profiles with those of the previous study by Guo et al.¹⁶ For identical values of the dimensionless variables, the velocity profile from the current work, in the absence of a magnetic field, porosity, and fluid parameter $λ_{1}$ , shows a strong correlation with the results from Guo et al.¹⁶ This comparison underscores the authenticity and robustness of the current findings, providing confidence in the validity of the proposed analytical solution. The numerical comparison for a recent study with the Guo et al.¹⁶ is also portrayed in Table 2.

Table 2.

Comparison table for the validation of current work with Guo et al.¹⁶ with the variation in pressure gradient parameter $α$ .

$α$	$U (Z)$ ₁₆	$- U (Z)$ ₁₆	$U (Z)$ _Present	$- U (Z)$ _Present
2	1.4153	0.4210	1.4152	0.4212
4	2.2074	−0.6776	2.2073	−0.6777
6	2.9506	−1.7022	2.9510	−1.7017

Accelerating flow

Figure 2(a) to (d) shows the velocity profile of accelerating flow $- U$ as a function of $Z .$ It is observed from Figure 2(a) that with the increase in Darcy number, the velocity profile decreases Physically, the increase in Darcy number causes an increment in the permeability of the medium which in return induces a resistance to the fluid’s flow causing a decline in the velocity while the upsurge in Jeffrey fluid’s parameter induces a decline in the velocity of fluid. This phenomenon can be explained since $λ_{1}$ characterizes the fluid’s viscoelastic properties. So when $λ_{1}$ is heightened, it signifies an increase in the fluid’s resistance to deformation and its tendency to exhibit elastic behavior. Consequently, within the flow, the fluid resists shearing forces and deformations more vigorously, resulting in a reduction in its radial velocity profile. In essence, higher values of $λ_{1}$ indicate a more elastic and less flow-prone behavior, leading to a gradual decline in the velocity distribution within the fluid. Figure 2(c) reveals that the velocity profile for inflow diminishes as the pressure gradient parameter increases. Physically, a higher pressure gradient enhances the resistance to fluid motion, causing a reduction in velocity. Additionally, at lower values of the pressure gradient parameter, the flow profile appears flatter, indicating a more uniform velocity distribution across the channel. However, as the pressure gradient increases, the velocity distribution becomes more parabolic. This behavior signifies that a stronger pressure gradient accelerates the fluid near the center while slowing it near the boundaries, creating a steeper velocity gradient typical of parabolic flow profiles. The flattening of the velocity profile at higher values of the magnetic parameter is observed in Figure 2(d). It can be attributed to the influence of the Lorentz force generated by the magnetic field, which acts as a damping mechanism on fluid motion. The increased magnetic parameter introduces a stabilizing effect on the flow, reducing fluctuations and creating a more uniform velocity distribution. Furthermore, the increase in velocity for inflow with an elevated magnetic parameter suggests that the magnetic influence counteracts viscous resistance to some extent, promoting fluid acceleration. The symmetry observed in the flow profiles is a result of the laminar characteristics and the geometric constraints imposed by the parallel disks. In the lower half, the velocity increases from the lower boundary wall to reach a maximum at the center due to the balance of driving forces and viscous effects. For the upper half flow, the velocity decreases symmetrically from the center toward the upper boundary wall, eventually reaching zero at the disk surface due to the no-slip condition imposed at the boundary. This symmetric behavior underscores the impact of both magnetic influence and boundary conditions on the flow dynamics.

Figure 2.

(a–d) Deviation in accelerating velocity $- U (Z)$ for distinct values of (a) $Da$ , (b) $λ_{1}$ , (c) $α$ , and (d) $M$ .

Decelerating flow

Figure 3(a) to (d) shows the velocity profile of decelerating flow $U$ as a function of $Z .$ It is observed from Figure 3(a) that with the increase in Darcy number, velocity profile increases and decelerating velocity increases due to permeable disks. Figure 3(b) to (d) shows that the increase in Jeffrey fluid, pressure gradient, and magnetic parameter causes a decrease in fluid velocity. In Figure 3(c), it is evident that increasing the values of the pressure gradient parameter leads to a decrease in velocity within the flow. This reduction occurs because a higher pressure gradient parameter imposes a stronger opposing force against the fluid’s motion, thereby slowing down the flow. Notably, the velocity becomes more concentrated in the center of the region between the disks, creating a more pronounced difference relative to the velocity at the disk surfaces. This behavior indicates that the central portion of the flow is more susceptible to pressure gradient variations, while the boundary regions maintain a lower and steadier velocity due to the no-slip condition at the disks’ surfaces. The symmetry observed for both the upper and lower halves of the flow demonstrates consistent behavior across the channel, confirming that the flow responds uniformly to changes in the pressure gradient across both regions. This symmetric flow pattern reflects the influence of the radial geometry and laminar nature of the fluid motion, resulting in mirrored velocity profiles between the two flat disks. Figure 3(d) illustrates that as the magnetic parameter increases, the velocity of the fluid also rises. This increase is attributed to the Lorentz force induced by the magnetic field, which can act as a driving mechanism for fluid motion in magnetohydrodynamic (MHD) flows. The enhanced magnetic interaction reduces the resistive drag effects on the fluid particles, thereby accelerating the fluid and increasing its velocity. Moreover, the velocity reaches its maximum value in the central region between the disks due to the reduced influence of wall friction compared to the boundaries. At the surfaces of the disks, the velocity approaches its minimum value (often zero) because of the no-slip boundary condition, which requires the fluid’s velocity to match that of the stationary walls. This distribution reflects the balance between viscous and magnetic forces throughout the flow field, leading to a parabolic velocity profile with a peak at the center. Moreover, the symmetry for both the upper and lower half-planes is observed. The decelerating velocity increases from the lower disk surface where the velocity is zero due to the no-slip condition to the middle region while it starts decreasing again from the middle region to the upper disk wall. Maximum velocity is observed at the center of the region between the disks.

Figure 3.

(a–d) Deviation in decelerating velocity $U (Z)$ for distinct values of (a) $Da$ , (b) $λ_{1}$ , (c) $α$ , and (d) $M$ .

Streamlines

Streamlines are a crucial tool for visualizing the flow patterns of fluids, providing insights into the direction and behavior of fluid motion under varying conditions and geometries. Figures 4(a) and (b) and 5(a) and (b) show the streamlines for the accelerating flow at different values of Darcy’s number and Jeffrey’s fluid parameter. Figure 4(a) and (b) shows the streamlines of accelerating flow for different values of Darcy number. It is examined that with the increase in $Da,$ the permeability of the medium increases which causes an incline in the resistance of fluid so, the number of streamlines decreases which shows the decrease in the velocity. Similarly, from Figure 5(a) and (b), it is observed that with the increase in Jeffry fluid parameter $λ_{1},$ the velocity decreases as the rise in $λ_{1}$ induces an increment in the resistance of fluid resulting in the decline of liquid’s motion. Moreover, the velocity profile flattened more at lower values of $Da$ and $λ_{1} .$

Figure 4.

(a, b) Deviation in streamlines of accelerating flow for (a) $Da = 0.4$ and (b) $Da = 0.8 .$

Figure 5.

(a, b) Deviation in streamlines of accelerating flow for (a) $λ_{1} = 1.5$ and (b) $λ_{1} = 2.5 .$

Figures 6(a) and (b) and 7(a) and (b) show the streamlines for the decelerating flow at different values of $Da$ and $λ_{1} .$ Decelerating velocity increase for the increase in Darcy number while decreases for the increase in Jeffrey fluid parameter. This behavior is in good agreement with the graphical representations. A higher $Da$ means that the porous medium is more permeable, reducing resistance to flow. As permeability increases, fluid can move more freely, leading to a higher velocity. This happens because the porous structure no longer restricts fluid movement as much, allowing for greater flow rates. On the other hand, a higher $λ_{1}$ means that the fluid retains its previous deformation longer, resisting motion and slowing down the flow. This viscoelastic effect increases internal resistance, which reduces the overall velocity. Physically, this is because the Jeffrey fluid does not behave like a simple Newtonian fluid; it has an additional elastic resistance that dampens the fluid motion.

Figure 6.

(a, b) Deviation in streamlines of decelerating flow for (a) $Da = 0.15$ and (b) $Da = 0.17 .$

Figure 7.

(a, b) Deviation in streamlines of accelerating flow for (a) $λ_{1} = 2.5$ and (b) $λ_{1} = 3.5 .$

Table 3 reveals the effect of variations in $Da$ and $λ_{1}$ on the torque at both disks. It is noted that with the increase in Darcy number, torque at the lower disk decreased while at the upper disk increased. The lower disk is more affected by the permeability of the medium. As $Da$ increases, the porous medium allows more fluid to pass through it rather than accumulating near the lower disk. This reduces the shear stress on the lower disk because less fluid remains near it to exert resistance. As a result, torque at the lower disk decreases with increasing $Da$ . Since increasing $Da$ allows more fluid to move freely, the fluid near the upper disk retains more motion and experiences greater shear stress. This leads to increased momentum transfer between the fluid and the upper disk, thereby increasing torque at the upper disk. Furthermore, an increase in the Jeffrey fluid parameter causes an increase in torque in both disks. It is because an increased value of $λ_{1}$ makes the fluid more resistant to deformation, meaning it retains more of its motion and resists shear. This increased resistance results in higher shear stress on both the upper and lower disks, leading to an increase in torque at both disks.

Table 3.

Effect of $λ_{1}$ and $Da$ on torque on both disks.

Parameters		Torque
$λ_{1}$	$Da$	$Z = - 1$	$Z = 1$
2	0.2	422.573	−602.731
	0.4	−10.9719	−176.364
	0.6	−16.2162	−115.784
	0.8	−15.42	−93.2056
2	0.5	−15.5818	−137.248
4		−11.0096	−8544.6
6		1.59371	1444.1
8		8.82122	96.5668

Conclusions

The considered problem deals with the steady, laminar, radial flow of Jeffrey fluid between two flat disks with the magnetic effect and porous medium. The exact solution of accelerating and decelerating flow for the considered non-linear problem is obtained by using the Jacobi elliptic sine-squared function of the first kind with the effect of MHD and porosity. The inspiration for this work is the wide applications of disk flow in biomedical, aerospace, and chemical engineering, rotating machinery, viscometers and rheometers, microfluidic devices, centrifugal separators, etc. In the future, this work can be extended by employing different fluid models like the Couple stress fluid model, Williamson’s fluid model, hyperbolic fluid model, and many other non-Newtonian fluid models along with numerous effects like viscous dissipation, entropy generation, Joule heating, Hall effect, and Sorret-Dufour impacts, etc. Following are some important results obtained during the considered problem:

The velocity and pressure gradient remain unchanged in the radial direction.

Both accelerating and decelerating flow verify the Poiseuille parabolic law at a low Reynolds number while becoming uniform at a high Reynolds number.

For accelerating flow, velocity increases with the escalation in magnetic effect while declines for Jeffrey fluid parameter.

For decelerating flow, velocity increases with the upsurge in porosity and decreases for all the remaining parameters.

Due to no-slip condition and stationary disks, the velocity at the disks becomes zero, increases away from the disk surface, and becomes maximum at the center.

Torque on the upper disk increases with the Darcy number while decreasing at the lower disk while the increase in Jeffrey fluid parameter, torque on both disks increases.

A comparison of recent work with the previous study¹⁶ is shown graphically and also in Table 2.

Footnotes

Acknowledgements

We are thankful to the reviewers for their encouraging comments and constructive suggestions to improve the quality of the manuscript.

Handling Editor: Sharmili Pandian

ORCID iDs

Ayesha Naeem

Muhammad Yousuf Rafiq

Statements and declarations

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Dynamics of accelerating and decelerating flow of an electrically conducting Jeffrey fluid between two narrowly flat permeable disks

Abstract

Keywords

Introduction

Mathematical formulation

Justification for the assumptions used in the study

Incompressible fluid assumption

Laminar flow assumption

Jeffrey fluid as a non-Newtonian model

Magnetic field influence (MHD)

Permeability of the disks

Use of the longitudinal symmetry in the radial direction

Non-dimensionalization of governing equation

Velocity distribution

Accelerating flow

Decelerating flow

Pressure distribution

Disk torque

Shear stress distribution

Convergence criteria

Boundedness and periodicity

Convergence of the elliptic integral representation

Specific cases of m

Boundary condition verification

Results and discussion

Accelerating flow

Decelerating flow

Streamlines

Conclusions

Footnotes

Acknowledgements

ORCID iDs

Statements and declarations

References

Specific cases of $m$