Thermal transport with nanoparticles of fractional Oldroyd-B fluid under the effects of magnetic field, radiations, and viscous dissipation: Entropy generation; via finite difference method

Muhammad Imran Asjad; Muhammad Usman; Muhammad Madssar Kaleem; Dumitru Baleanu; Taseer Muhammad

doi:10.1515/phys-2022-0166

Article Open Access

Thermal transport with nanoparticles of fractional Oldroyd-B fluid under the effects of magnetic field, radiations, and viscous dissipation: Entropy generation; via finite difference method

Muhammad Imran Asjad , Muhammad Usman , Muhammad Madssar Kaleem , Dumitru Baleanu and Taseer Muhammad

Published/Copyright: December 12, 2022

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From the journal Open Physics Volume 20 Issue 1

Abstract

It is a well-known fact that functional effects like relaxation and retardation of materials, and heat transfer phenomena occur in a wide range of industrial and engineering problems. In this context, a mathematical model is developed in the view of Caputo fractional derivative for Oldroyd-B nano-fluid. Nano-sized particles of copper (Cu) are used to prepare nano-fluid taking water as the base fluid. The coupled non-linear governing equations of the problem are transformed into dimensionless form. Finite difference scheme is developed and applied successfully to get the numerical solutions of deliberated problem. Influence of different physical parameters on fluid velocity profile and temperature profile are analyzed briefly. It is observed that for increasing values of fractional parameter (α), fluid velocity increased, but opposite behavior was noticed for temperature profile. Nusselt number (Nu) decayed for advancement in values of heat source/sink parameter (Q₀), radiation parameter (Nr), volume fraction parameter of nano-fluid (ϕ), and viscous dissipation parameter (Ec). Skin friction (C_f) boosts for the increase in the values of magnetic field parameter (Ha). It can also be noticed that the extended finite difference scheme is an efficient tool and gives the accurate results of discussed problem. It can be extended for more numerous type heat transfer problems arising in physical nature with complex geometry.

Keywords: Oldroyd-B nano-fluid; Caputo time fractional model; viscous dissipation; thermal radiations; magnetic field, finite difference method

Nomenclature

u ( m / s ): velocity
θ (K): temperature
ρ nf ( kg / m 3 ): density
μ nf ( kg / ms ): dynamic viscosity
k nf ( W / m K ): thermal conductivity of nano-fluid
β θ ( K − 1 ): volumetric thermal expansion coefficient
g ( m / s 2 ): gravitational acceleration
( C p ) nf: heat capacity of nanoparticles
σ nf ( S / m ): electrical conductivity of nanoparticles
ν nf ( m 2 / s ): kinematic viscosity of nanoparticles
ϕ: volume fraction of nanoparticles

1 Introduction

In the modern era of science and technologies, a lot of researchers are engrossing by the non-Newtonian fluids because non-Newtonian fluids have much more applications then Newtonian fluids. Practically, precise forecasting of all connected features for such fluids is not credible through a single model because in non-Newtonian fluids, shear stress and shear rate are nonlinearly correlated to each other. Classical Navier–Stokes equation are effectual in characterizing numerous rheological properties and dynamics of these fluids, for e.g., stress differences, retardation, relaxation, elongation, yield stress, memory effects, and many others. A number of different models are suggested to incorporate the additional non-linear terms and to speculate the rheological properties of non-Newtonian fluids. Certain models which explained the non-Newtonian behavior of fluids are Maxwell model [1], Sisko’s model [2], Second grade fluid model [3], Burgers model [4], Jeffrey model [5], and Oldroyd-B model [6]. Many non-Newtonian fluids are present in foods like jams, mayonnaise, toothpaste, polymers, clay, and melts which enumerate the elastic feature. The Oldroyd-B model is a more appropriate and amenable model to manifest the elastic and viscous nature of the fluids because it has the ability to demonstrate the flow history and it discloses the retardation and relaxation mechanism. When the value of solvent viscosity is zero, then the Maxwell model is a limiting case of Oldroyd-B model. A considerable work in mathematical modeling of blood flow through restenosis arteries has been done in bio-science by Ali et al. [7]. A lot of contribution in this perspective can be seen in refs [7,8,9,10]. Fetecau et al. [11] elaborated apposite stress field for Oldroyd-B and fluid velocity distribution over a constantly moving plate. Fetecau et al. [12] extended this work further more for an Oldroyd-B flow model within a channel and found the analytical solution of the proposed model. Gul [13] solved the magnetohydrodynamic (MHD) flow of Oldroyd-B fluid model past an oscillating belt analytically by Homotopy perturbation method (HPM). Wang [14] scrutinized the dissipative non-linear flow of Oldroyd-B fluid generated by stretchable sheet via HPM. Song [15] elaborated the fractional Oldroyd-B model analytically via fluctuating duct. Hayat [16] investigated the nano-material stagnation point flow of Oldroyd-B model via optimal homotopy asymptotic method.

Applications of nano-fluids have become a topic of global interest for the last few decades because of their effective mode heat transfer ability. Choi and Eastman [17] in 1995, first time introduced the nano-fluid which is a revolutionary change in thermal transport and solved entropy problem in heat management systems. Nano-fluids are made by mixing nano-sized particles, ranging from 1 to 100 nm, of metals, oxides, or carbides in base fluids like water, carosene oil, etc. Nano-particles when added in fluid enhance thermal conductivity and surface area of thermal conduction, consequently enhancing the heat transfer rate. Nano-fluids are best coolants in heat transfer systems like heat exchangers, electronic cooling, and radiators. Polymerase chain reactions are efficiently enhanced by using graphene-based nano-fluids. In cold region, nano-fluids are used to heat the rooms more efficiently. By using nano-fluids, one can minimize the cost, loss of energy, and also the pollution. Kulkarni et al. [18] explored the application of nano-fluids in heating buildings and reducing pollution. Ramezanizadeh et al. [19] discussed in detail the application of nano-fluids in thermosiphons. Ebrahimnia-Bajestan et al. [20] worked on solar system heat exchanger with nano-fluids application. Li et al. [21] disclosed that the nano-fluids are best lubricants. Hamid [22] investigated the shape effects of nanoparticles on rotating flow of nano-fluids. A lot of contribution in this context can be found in refs [23,24,25,26].

The role of mathematical models in describing the different physical phenomena of daily life is more important. These models are composed of differential equations and some initial and boundary conditions. These differential equations are of different orders, may be integer order or non-integer fractional derivatives. But non-integer fractional derivatives have a vibrant scope in modern industrial era, as fractional models are more close to the geometry of physical systems. Also, fractional parameter is part of a solution which elaborated the memory effect more briefly helping to control the flow regime, consequently fractional models are considered superior than ordinary models. It was used to establish the models of the physical systems and many engineering processes by fractional differential equations. These models were found to be more accurately describing these fractional differential equations. Many physical phenomena of thermal transport, engineering, circuits, biotechnology, and signal process can be modified with the help of fractional derivative. Viscoelastic damping in polymers can be well modeled by using fractional derivative. Shape memory effect during deformation can be accurately described by using fractional calculus. Imran [27] explored the second grade fluid by using Caputo time fractional derivative analytically. Caputo and Fabrizio [28] presented a new fractional derivative having fruitful results of many physical problems. Ikram [29] investigated the flow of hybrid nano-fluid by using Caputo–Fabrizio (CF) derivative analytically. CF derivative involves non-singular kernel and it cannot produce original function when order of the derivative is zero. To overcome this problem, in 2016, Atangana and Baleanu [30] introduced new fractional derivative based on Mittag-Leffler function. Saqib et al. [31] investigated MHD channel flow of carbon nanotube-based nano-fluids by using ABC derivative. Wang [32] discussed heat and mass transfer of Oldroyd-B nano-fluid by using ABC derivative via Laplace transform approach.

Viscous stresses destructing the fluctuating velocity gradient is known as viscous dissipation. Viscous dissipation effect is an irreversible process, which heats up the fluid because the viscosity of fluid takes energy from the motion of fluid and transforms it into internal energy of the fluid. Fruitful results with application of viscous dissipation, radiation, and magnetic field on nano-fluid are studied in refs [33,34]. Also, this effect was considered by Brinkman. Hussain et al. [35] explored the viscous dissipation effect in detail. Farooq [36] investigated the effect of viscous dissipation on entropy. Entropy is actually disordering or randomness of system. A loss of useful energy is known as entropy. Habib [37] discussed the entropy generation and heat transfer in power law fluid flow. Khan et al. [38] explored the entropy generation in nano-fluid flow between two rotating discs.

By following through literature discussed earlier, it is worth remarking that MHD Oldroyd-B nano-fluid model with thermal radiation, viscous dissipation, and heat source/sink in view of Caputo time fractional derivative is not considered and solved numerically via finite difference method (FDM). Furthermore, entropy generation, Bejan number, Nusselt number, and skin friction are calculated for the proposed model and compared with the existing literature.

2 Mathematical formulations

Let us consider the unsteady natural convection flow of incompressible fractional Oldroyd-B nano-fluid over infinite oscillating vertical plate as V = u 0 H ( t ) Cos ( ω t ) i , where H ( t ) is the unit step function, i is the unit vector in vertical flow direction, and ω is the frequency of oscillating plate. The plate is situated at y = 0 , parallel to x-axis and moving in x-direction. Initially, both the plate and fluid are at rest with constant temperature θ ∞ . For t > 0 , the plate started oscillating motion with velocity u 0 H ( t ) cos ( wt ) and the temperature of plate raised by θ w . Magnetic field is applied parallel to y-axis. Taking viscous dissipation as significant, velocity profile is of the form V ( u ( y , t ) , 0,0 ) as shown in Figure 1. The proposed model contains the following assumptions.

Flow is unsteady, incompressible, and unidirection
Pressure gradient is neglected
Applying magnetic field (neglecting induced magnetic field)
Considering thermal radiation along y-direction
Viscous dissipation is significant.

Figure 1

Geometry of flow dynamics.

Under the above described assumptions, momentum and energy equation are given as,

The basic governing equations of the problem are given [40] as,

(1) ∇ . V = 0 , ρ ∂ V ∂ t + ( V . ∇ ) V = ρ g + J × M + div ( T ) ,

where V is the velocity vector, M is the magnetic field parameter, ρ is the fluid density, the current density is represented by J, T is the Cauchy stress tensor, and t is the time and ρ g is the other body forces. In case of Oldroyd-B nano-fluid, the relation between the extra stress tensor and Cauchy stress tensor is given [40] as,

(2) T = − pI + S ,

where

(3) S + λ 1 D S D t = μ A 1 + λ 2 D A 1 D t ,

and

(4) D S D t = d S d t − LS − S L t , D A 1 D t = d A 1 d t − LA 1 − A 1 L t , A 1 = L + L t , L = ∇ V ,

where ρ is the density, T is the Cauchy stress tensor, p is the pressure, I is the identity matrix, b is the body force, S is the extra stress tensor, A 1 is the first Rivilin-Erikson tensor, λ 1 and λ 2 are the time relaxation parameters, λ r is the time retardation parameter, and L is the velocity gradient.

By using Eqs. (2)–(4) in Eq. (1),

(5) ρ hnf ∂ u ∂ t = ∂ S xy ∂ y + g ( ρ β θ ) nf ( θ − θ ∞ ) − ( σ ) nf B 0 2 u ,

where S xy is the extra stress tensor, β θ is the coefficient of thermal expansion, σ is the electrical conductivity, B o is the applied magnetic field, and θ is the temperature. Multiplying Eq. (5) by ( 1 + λ 1 α D t α )

(6) ρ nf ( 1 + λ 1 α D t α ) ∂ u ∂ t = ∂ ∂ y ( 1 + λ 1 α D t α ) S xy + g ( ρ β θ ) nf ( 1 + λ 1 α D t α ) ( θ − θ ∞ ) − ( σ ) nf B 0 2 ( 1 + λ 1 α D t α ) u ,

The constitutive relation for Oldroyd-B fluid is given [40] as,

(7) ( 1 + λ 1 α D t α ) S xy = μ nf ( 1 + λ r α D t α ) ∂ u ∂ y .

Applying Eq. (7) in Eq. (6)

(8) ( 1 + λ 1 α D t α ) ρ nf ∂ u ∂ t = ( 1 + λ r α D t α ) μ nf ∂ 2 u ∂ y 2 + ( 1 + λ 1 α D t α ) g ( ρ β θ ) nf ( θ − θ ∞ ) − ( 1 + λ 1 α D t α ) σ nf B 0 2 u .

Energy equation in the presence of thermal radiations and heat source/sink with viscous dissipation effect is given [41] as,

(9) ( ρ C p ) nf ∂ θ ∂ t = K nf ∂ 2 θ ∂ y 2 − ∂ q r ∂ y + μ nf ∂ u ∂ y 2 + Q ( θ − θ ∞ ) ,

where ( C p ) nf is the specific heat capacity of nanofluid at constant pressure, ( ρ ) nf is the density of nano-fluid, K nf is the thermal conductivity of nano-fluid, μ nf is the dynamic viscosity of nano-fluid, q r is the radiative heat flux, and Q is the heat source/sink parameter.

Under the Rosseland’s approximation in ref. [32], q r takes the following form

(10) q r = − 4 σ * 3 k * ∂ θ 4 ∂ y ,

where σ * is the Stefan–Boltzmann constant and k * is the absorption constant. Expanding θ 4 about θ _∞ by using Taylor series

(11) θ 4 = 4 θ ∞ 3 θ ' − θ ∞ 4 .

Neglecting higher power of θ and using Eqs. (10) and (11) in Eq. (9), also multiplying with ( 1 + λ 2 β D t β ) , Eq. (9) takes the following form

(12) ( ρ C p ) nf ( 1 + λ 2 β D t β ) ∂ θ ∂ t = K nf ( 1 + λ 2 β D t β ) ∂ 2 θ ∂ y 2 + 16 σ * θ ∞ 3 3 k * ( 1 + λ 2 β D t β ) ∂ 2 θ ∂ y 2 + ( 1 + λ 2 β D t β ) μ nf ∂ u ∂ y 2 + ( 1 + λ 2 β D t β ) Q ( θ − θ ∞ ) .

Associated initial and boundary conditions [42] are,

(13) u ( y , 0 ) = 0 , u ( 0 , t ) = u o H ( t ) cos ( wt ) , u ( ∞ , t ) = 0 ,

(14) θ ( y , 0 ) = θ ∞ , θ ( 0 , t ) = θ w , θ ( ∞ , t ) = θ ∞ .

For the proposed model, following are the non-dimensional quantities [42]

u * = u u 0 , t * = u 0 2 ν t , θ * = θ − θ ∞ θ w − θ ∞ , λ 1 * = u 0 2 ν λ 1 , λ 2 * = u 0 2 ν λ 2 , y * = u 0 y ν ,

Using these transformations in Eqs. (9) and (12–14)

(15) Gr = ν g ( β θ ) f ( θ w − θ ∞ ) u 0 3 , H a 2 = M = σ f B 0 2 ν ρ f u 0 2 , Pr = μ ( C p ) f K f , Nr = 16 σ * θ ∞ 3 a 5 3 k * K f , Ec = μ f k f u 0 2 ν ( θ w − θ ∞ ) , Q 0 = Q ν a 5 K f u 0 2 .

where relation for nano-fluids are given [43] as

(16) ρ nf ρ f = a 1 = ( 1 − ϕ ) + ϕ ρ s ρ f , ( ρ β θ ) nf ( ρ β θ ) f = a 2 = ( 1 − ϕ ) + ϕ ( ρ β θ ) s ( ρ β θ ) f , μ nf μ f = a 3 = 1 ( 1 − ϕ ) 2 . 5 , ( ρ C p ) nf ( ρ C p ) f = a 4 = ( 1 − ϕ ) + ϕ ( ρ C p ) s ( ρ C p ) f , k nf k f = a 5 = ( k s + 2 k f ) − 2 ϕ ( k f − k s ) ( k s + 2 k f ) + ϕ ( k f − k s ) , ( σ ) nf ( σ ) f = a 6 = 1 + 3 σ s σ f − 1 ϕ σ s σ f − 2 − σ s σ f − 1 ϕ ,

(17) b 1 = a 3 a 1 , b 2 a 2 a 1 , b 3 = a 6 a 1 , b 4 = a 4 a 5 , b 5 = a 3 a 5 , b 6 = a 6 a 5 .

The dimensionless forms of the momentum and heat equations are

(18) ( 1 + λ 1 α D t α ) ∂ u ∂ t = ( 1 + λ r α D t α ) b 1 ∂ 2 u ∂ y 2 + b 2 Gr ( 1 + λ 1 α D t α ) ( θ ) − b 3 H a 2 ( 1 + λ 1 α D t α ) ( u ) ,

(19) b 4 Pr ( 1 + λ 2 β D t β ) ∂ θ ∂ t = ( 1 + Nr ) ( 1 + λ 2 β D t β ) ∂ 2 θ ∂ y 2 + b 5 Ec ( 1 + λ 2 β D t β ) ∂ u ∂ y 2 + ( 1 + λ 2 β D t β ) Q 0 θ .

Also, the dimensionless initial and boundary conditions

(20) u ( y , 0 ) = 0 , u ( 0 , t ) = H ( t ) cos ( wt ) , u ( ∞ , t ) = 0 ,

(21) θ ( y , 0 ) = 0 , θ ( 0 , t ) = 1 , θ ( ∞ , t ) = 0 .

3 Entropy generation

For the Oldroyd-B fluid flow in magnetic field, the volumetric entropy generation of nano-fluid is given [44] as

(22) E G = E θ + E f + E m ,

where E θ , E f , and E m are entropy generated due to heat transfer, friction, and magnetic field effect, respectively. These terms are defined as,

(23) E θ = K nf θ ∞ 2 1 + 16 σ * θ ∞ 3 3 k * K nf ∂ θ ∂ y 2 , E f = μ nf θ ∞ ∂ u ∂ y 2 , E m = σ nf B 0 2 θ ∞ u 2 .

By using Eq. (23) in Eq. (22)

(24) Ns = E G E 0 = a 5 ( 1 + Nr ) ∂ θ ∂ y 2 + a 3 Br Ω ∂ u * ∂ y 2 + a 6 Br Ω H a ( u ) 2 .

Here E 0 is the characteristic entropy generation rate, Br is the Brinkman number, H a 2 is the Hartmann number, and Ω is dimensionless temperature difference given as

(25) E 0 = K f ν f 2 u 0 2 ( θ w − θ ∞ ) 2 θ ∞ 2 , Ω = ( θ w − θ ∞ ) θ ∞ , Nr = 16 σ * θ ∞ 3 3 k * K nf , Br = μ f u 0 2 K f ( θ w − θ ∞ ) , and H a 2 = σ f B 0 2 ν ρ f u 0 2 .

3.1 Bejan number

The Bejan number for the proposed model is defined [44] as

(26) Be = K nf θ ∞ 2 ∂ θ ∂ y 2 K nf θ ∞ 2 1 + 16 σ * θ ∞ 3 3 k * K nf ∂ θ ∂ y 2 + μ nf θ ∞ ∂ u ∂ y 2 + σ nf B 0 2 T ∞ u 2 ,

(27) Be = ∂ θ ∂ y 2 ( 1 + Nr ) ∂ θ ∂ y 2 + b 5 Br Ω ∂ u ∂ y 2 + b 6 Br Ω H a ( u ) 2 .

3.2 Skin friction coefficient

Skin friction for the considered model is defined [45] as

(28) C f = − u y ( 0 , t ) .

3.3 Local Nusselt number

Nusselt number for the proposed model is calculated by the given definition [45] as

(29) Nu = − θ y ( 0 , t ) .

4 Methodology

In this section, we proposed a finite difference scheme to handle the governing set of fractional-order fluid problem and heat transfer. For this, the discretization of the derivative of fractional-order of u , u t , and u yy are specified [46] as,

(30) D t j + 1 α 0 C u ( y i , t j + 1 ) = Δ t − α Γ ( 2 − α ) [ u i j + 1 − u i j ] + Δ t − α Γ ( 2 − α ) ∑ l = 1 j ( u i j − l + 1 − u i j − l ) d l α ,

(31) D t j + 1 1 + α 0 C u ( y i , t j + 1 ) = Δ t − ( 1 + α ) Γ ( 2 − α ) [ u i j + 1 − 2 u i j + u i j − 1 ] + Δ t − ( 1 + α ) Γ ( 2 − α ) × ∑ l = 1 j ( u i j − l + 1 − 2 u i j − l + u i j − l − 1 ) d l α ,

(32) ∂ ∂ t u ( y i , t j + 1 ) t = t j + 1 = 1 Δ t [ u i j + 1 − u i j ] ,

(33) ∂ 2 ∂ y 2 u ( y i + 1 , t j ) y = y i + 1 = 1 Δ y 2 [ u i + 1 j + 1 − 2 u i j + 1 + u i − 1 j + 1 ] .

In the above expressions, d l α = − l 1 − α + ( 1 + l ) 1 − α for l = 1,2,3 , … , j . Now, the rectilinear grid is assumed to examine the solution of the governing set of fractional-order fluid problem and heat transfer having grid spacing Δ y > 0 , Δ t > 0 in the direction of space and time separately, where Δ t = T N , Δ x = L M for Δ y , Δ t from Z + . The inner points ( y i , t j ) in the discussed domain Ω = [ 0 , T ] × [ 0 , L ] are given as i Δ y = y i and j Δ t = t j . The discretization of the governing set of fractional-order fluid problem and heat transfer at ( y i , t j ) is given as

. 1 Δ t ( u i j + 1 − u i j ) + λ 1 α Δ t − ( 1 + α ) Γ ( 2 − α ) ( u i j + 1 − 2 u i j + u i j − 1 ) + λ 1 α Δ t − ( 1 + α ) Γ ( 2 − α ) × ∑ l = 1 j ( u i j − l + 1 − 2 u i j − l + u i j − l − 1 ) b l α = b 1 Δ y 2 ( u i + 1 j + 1 − 2 u i j + 1 + u i − 1 j + 1 ) + b 1 λ 2 α Δ t − α Δ y 2 Γ ( 2 − α ) ∑ l = 0 j ( u i + 1 j − l + 1 − 2 u i + 1 j − l + u i + 1 j − l − 1 ) b l α − 2 b 1 λ 2 α Δ t − α Δ y 2 Γ ( 2 − α ) × ∑ l = 0 j ( u i j − l + 1 − 2 u i j − l + u i j − l − 1 ) b l α + λ 2 α Δ t − α Δ y 2 Γ ( 2 − α ) ∑ l = 0 j ( u i − 1 j − l + 1 − 2 u i − 1 j − l + u i − 1 j − l − 1 ) b l α + b 2 G r θ i j + 1 + c 2 G r Δ t − α Γ ( 2 − α ) ( θ i j + 1 − θ i j ) + b 2 G r Δ t − α Γ ( 2 − α ) ∑ l = 1 j ( θ i j − l + 1 − θ i j − l ) b l α − b 3 H a 2 u i j + 1 − b 3 H a 2 Δ t − α Γ ( 2 − α ) ( u i j + 1 − u i j ) + b 3 H a 2 Δ t − α Γ ( 2 − α ) ∑ l = 1 j ( u i j − l + 1 − u i j − l ) b l α , b 4 P r Δ t ( θ i j + 1 − θ i j ) + b 4 Pr λ 2 β Δ t − ( 1 + β ) Γ ( 2 − β ) ( θ i j + 1 − 2 θ i j + θ i j − 1 ) + b 4 Pr λ 2 β Δ t − ( 1 + β ) Γ ( 2 − β ) ∑ l = 1 j ( θ i j − l + 1 − 2 θ i j − l + θ i j − l − 1 ) b l β = 1 + Nr Δ y 2 ( θ i + 1 j + 1 − 2 θ i j + 1 + θ i − 1 j + 1 ) + ( 1 + Nr ) λ 2 β Δ t − β Δ y 2 Γ ( 2 − β ) ∑ l = 0 j ( θ i + 1 j − l + 1 − 2 θ i + 1 j − l + θ i + 1 j − l − 1 ) b l β − ( 1 + Nr ) 2 λ 2 β Δ t − β Δ y 2 Γ ( 2 − β ) × ∑ l = 0 j ( θ i j − l + 1 − 2 θ i j − l + θ i j − l − 1 ) b l β + ( 1 + Nr ) λ 2 β Δ t − β Δ y 2 Γ ( 2 − β ) ∑ l = 0 j ( θ i − 1 j − l + 1 − 2 θ i − 1 j − l + θ i − 1 j − l − 1 ) b l β + b 5 Δ y 2 E c ( u i + 1 j − u i j ) ( u i + 1 j + 1 − u i j + 1 ) + b 5 λ 2 β Δ t − β Δ y 2 Γ ( 2 − β ) E c ( u i + 1 j − u i j ) × u i + 1 j + 1 − u i + 1 j − ∑ l = 1 j ( u i + 1 j − l + 1 − u i + 1 j − l ) b l β − u i j + 1 + u i j + ∑ l = 1 j ( u i j − l + 1 − u i j − l ) b l β + Q 0 Δ t − β Γ ( 2 − β ) ( θ i j + 1 − θ i j ) + Q 0 Δ t − β Γ ( 2 − β ) ∑ l = 1 j ( θ i j − l + 1 − θ i j − l ) b l β + Q 0 θ i j + 1 .

for j = 1,2,3 , … , N − 1 , i = 1,2,3 , … , N − 1 , with the following initial and boundary conditions,

u i 0 = 0 , u i 1 = u i − 1 , θ i 0 = 0 , θ i 1 = θ i − 1 , for i = 0 , 1 , 2 , 3 , … , M , u 0 j = cos ( jw Δ t ) , u M j = 0 , θ 0 j = 1 , θ M j = 0 , for j = 1 , 2 , 3 , … , N − 1 .

Test Problem

In order to check the efficiency of the proposed algorithm, suppose the problem is

D t α 0 C U ( x , t ) = ∂ 2 ∂ x 2 U ( x , t ) − ∂ ∂ x U ( x , t ) + g ( x , t ) .

Here the initial and boundary conditions are specified below and inhomogeneous term can be designated against the choice of fractional-order derivative.

U ( x , 0 ) = U ( 0 , t ) = U ( L , t ) = 0 .

The exact solution of the considered problem is U ( x , t ) = x ( x − t ) t 2 . Several simulations have been accomplished to check the accuracy of the suggested method. Figure 2(a) and (b) are plotted between the various values of N and maximum absolute error (MAE), and the computational order of convergence (COC) given below,

MAE = max 1 ≤ i ≤ M 1 ≤ j ≤ N | U ( x i , t j ) − U i j | , COC = log MAE ( k ) MAE ( k + 1 ) / log N ( k + 1 ) N ( k ) .

Figure 2

Code validation of the proposed scheme and varying time mesh size against (a) COC, (b) MAE, and varying mesh size for (c) time and (d) space against L _∞-norm between consective solutions.

It is well known that the method is convergent beside each fractional-order derivative and its convergence order increases as α → 1 . Figure 2(c) and (d) comprises the L _∞-norm among the consecutive solutions i.e., | U j + 1 − U j | ∞ and | U i + 1 − U i | ∞ when 0 ≤ i, j ≤ N, M = 500. It is established that the suggested method is very effective, reliable, and accurate for the considered problem. Solutions are unchanged against the fractional order mesh which validated the given solutions.

5 Analysis and discussion of results

This section of the research work deals with the detailed overview of the key numerical findings and physical interpretation of different emerging parameters such as Gr , Pr , M , Ec , α , λ , Nr , Ns , Nu , C f , ϕ , Q 0 , and Be are the Grashof number, Prandtl number, magnetic field parameter, Eckert number, fractional parameter, time relaxation parameter, radiation parameter, entropy generation parameter, Nusselt number, skin friction, volume fraction parameter of nanoparticles, heat source/sink parameter, and Bejan number, respectively. The behavior of the velocity profile u ( y , t ) , temperature profile θ ( y , t ) , and the effects of the mentioned physical parameters are deliberated as well as graphical illustration is made via MAPLE. The velocity profile, temperature profile, skin friction, Nusselt number, Bejan number, and entropy generation are calculated by assuming the different ranges ( 0 ≤ Gr ≤ 5 ) , ( 0 ≤ Ha ≤ 5 ) , ( 3 . 94 ≤ Pr ≤ 15 ) , ( 0 ≤ Ec ≤ 5 ) , ( 0 . 2 ≤ α , β ≤ 1 ) , ( 0 . 1 ≤ λ 1 , λ 2 ≤ 0 . 5 ) , ( 0 ≤ Nr ≤ 5 ) , ( 0 . 1 ≤ ϕ ≤ 0 . 2 ) , and ( 0 ≤ Q 0 ≤ 5 ) . Discretization of time and spatial derivative is done using FDMs. The coupled, nonlinear, and fractional models have been solved numerically by using FDM which is a dominant tool to deal such kinds of problems.

Results are obtained by solving Eqs. (18) and (19) with initial and boundary conditions illustrated in Eqs. (20) and (21), and physical properties of nanoparticle in Eq. (16) and Table 1.

Table 1

Thermo-physical properties of different materials taken from ref. [39]

Materials	ρ ( kg m − 3 )	C p ( J K g − 1 k − 1 )	k ( W m − 1 k − 1 )	β × 10 − 5 ( k − 1 )	σ ( Ω m ) − 1
water	997	4,197	0.613	21	0.05
Cu ( ϕ )	8,933	385	401	1.67	5.96 × 10 7

Figure 3a depicts the effect of magnetic field on fluid velocity. Enhancement in the strength of magnetic field parameter (Ha), causes a decrease in the fluid velocity. It can be justified as, by enhancing the magnetic field strength, actually the Lorentz force increased which shows a high resistance to fluid flow. Enhancing the magnetic field parameter amounts to stronger Lorentz force and hence rate of heat transfer in the boundary layer is higher. Bejan number (Be) is the ratio of entropy generation due to heat transfer to total entropy generation. By increasing the amount of magnetic field parameter there is an increase in Be as shown in Figure 3c. It means increase in Be by increasing the values of H a 2 is due to the decrease in the total entropy generation as shown in Figure 3d. From Figure 3b, it can be seen that higher values of (Ha) maximizes the skin friction (C _f). Similar results are reported by Sarojamma et al. [47].

$Figure 3 Impact of Hartmann number on (a) dimensionless velocity profile, (b) coefficient of skin friction, (c) Bejan number, and (d) entropy generation, when, α = β = 0 . 6 , \alpha =\beta =0.6, Λ 1 = 0.1 , {\Lambda }_{1}=0.1, Λ 2 = 0.2 , {\Lambda }_{2}=0.2, Gr = 5 , {\rm{Gr}}=5, Pr = 6.2 , {\rm{\Pr }}=6.2, Q 0 = 2.5 , Nr = 5 , Ec = 2 , {Q}_{0}=2.5,{\rm{Nr}}=5,{\rm{Ec}}=2, Br = 2 , {\rm{Br}}=2, Ω = 10 , \Omega =10, and {\rm{and}} ϕ = 0.1 \phi =0.1 .$

Figure 3

Impact of Hartmann number on (a) dimensionless velocity profile, (b) coefficient of skin friction, (c) Bejan number, and (d) entropy generation, when, α = β = 0 . 6 , Λ 1 = 0.1 , Λ 2 = 0.2 , Gr = 5 , Pr = 6.2 , Q 0 = 2.5 , Nr = 5 , Ec = 2 , Br = 2 , Ω = 10 , and ϕ = 0.1 .

Figure 4a elaborates the effect of Grashof number Gr on the fluid velocity profile. Grashof number is the ratio of thermal buoyancy forces to the viscous forces. Fluid velocity increases by increasing the value of Gr as thermal buoyancy forces dominated the viscous force. There is an increase in C _f by enhancing the Gr as shown in Figure 4b. Higher values of Gr lowers the Be as shown in Figure 4c and higher entropy generation can be noticed in Figure 4d. These results are also reported in ref. [47].

$Figure 4 Influence of Grashof number on (a) dimensionless velocity profile, (b) coefficient of skin friction, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , \alpha =\beta =0.6, Λ 1 = 0.1 , {\Lambda }_{1}=0.1, Λ 2 = 0.2 , {\Lambda }_{2}=0.2, Ha = 2 , Pr = 6.2 , {\rm{Ha}}=2,{\rm{\Pr }}=6.2, Q 0 = 2.5 , Nr = 5 , Ec = 2 , {Q}_{0}=2.5,{\rm{Nr}}=5,{\rm{Ec}}=2, Br = 2 , {\rm{Br}}=2, Ω = 10 , \Omega =10, and {\rm{and}} ϕ = 0.1 \phi =0.1 .$

Figure 4

Influence of Grashof number on (a) dimensionless velocity profile, (b) coefficient of skin friction, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0.1 , Λ 2 = 0.2 , Ha = 2 , Pr = 6.2 , Q 0 = 2.5 , Nr = 5 , Ec = 2 , Br = 2 , Ω = 10 , and ϕ = 0.1 .

Effect of Λ 1 on the fluid velocity is elaborated in Figure 5a. It can be seen that the enhancement in Λ 1 causes an increase in the fluid velocity. Skin friction increases by increasing Λ 1 as shown in Figure 5b. Be slightly increases by an increase in Λ 1 as shown in Figure 5c, and Ns decreased by increasing Λ 1 , Figure 5d.

$Figure 5 Effect of time relaxation parameter on (a) dimensionless velocity profile, (b) coefficient of skin friction, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0.1 , Λ 2 = 0.2 , \alpha =\beta =0.6,{\Lambda }_{1}=0.1,{\Lambda }_{2}=0.2, Gr = 5 , {\rm{Gr}}=5, Ha = 2 , Pr = 6.2 , {\rm{Ha}}=2,{\rm{\Pr }}=6.2, Q 0 = 2.5 , Nr = 5 , Ec = 2 , {Q}_{0}=2.5,{\rm{Nr}}=5,{\rm{Ec}}=2, Br = 2 , {\rm{Br}}=2, Ω = 10 , \Omega =10, and ϕ = 0.1 {\rm{and}}\phi =0.1 .$

Figure 5

Effect of time relaxation parameter on (a) dimensionless velocity profile, (b) coefficient of skin friction, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0.1 , Λ 2 = 0.2 , Gr = 5 , Ha = 2 , Pr = 6.2 , Q 0 = 2.5 , Nr = 5 , Ec = 2 , Br = 2 , Ω = 10 , and ϕ = 0.1 .

Effect of ϕ of nanoparticles on the fluid velocity profile is depicted in Figure 6a. Advancement in control of ϕ reduced the fluid velocity due to the fact that by increasing ϕ fluid becomes more viscous. Since the addition of nanoparticles increase the viscosity, consequently the velocity profile decreased. By increasing ϕ , heat transfer from boundary layer becomes better due to the enhanced thermal conductivity of the base fluid caused by nanoparticles which results in increasing the internal fluid temperature as shown in Figure 6b. There is a decrease in Nu by increasing ϕ as seen in Figure 6f. Effects of ϕ on (C _f) is elaborated in Figure 6e, by increasing the values of ϕ causes decrease in C _f. Heat transfer rate becomes good by increasing the ϕ and hence the values of (Be) are enhanced as seen in Figure 6c. Figure 6d depicts the effect of ϕ on entropy generation. Entropy enhanced by advancement in the values of ϕ because viscosity of the fluid increased by increasing ϕ . These results are also reported in ref. [48].

$Figure 6 Effect of ϕ \phi on (a) dimensionless velocity profile, (b) dimensionless temperature profile, (c) Bejan number, (d) entropy generation, (e) skin friction, and (f) Nusselt number, when α = β = 0 . 6 , Λ 1 = 0.1 , {\alpha =\beta =0.6,\Lambda }_{1}=0.1, Λ 2 = 0.2 , {\Lambda }_{2}=0.2, Gr = 5 , {\rm{Gr}}=5, Ha = 2 , Pr = 6.2 , {\rm{Ha}}=2,{\rm{\Pr }}=6.2, Q 0 = 2.5 , Nr = 5 , Ec = 2 , {Q}_{0}=2.5,{\rm{Nr}}=5,{\rm{Ec}}=2, Br = 2 , {\rm{Br}}=2, and {\rm{and}} Ω = 10 \Omega =10 .$

Figure 6

Effect of ϕ on (a) dimensionless velocity profile, (b) dimensionless temperature profile, (c) Bejan number, (d) entropy generation, (e) skin friction, and (f) Nusselt number, when α = β = 0 . 6 , Λ 1 = 0.1 , Λ 2 = 0.2 , Gr = 5 , Ha = 2 , Pr = 6.2 , Q 0 = 2.5 , Nr = 5 , Ec = 2 , Br = 2 , and Ω = 10 .

Figure 7a illustrates the behavior of fluid velocity against different values of fractional parameter ( α = β ) . By increasing the values of α = β , thermal and momentum boundary layers are reduced and become thinnest for α = β = 1 , consequently increasing the fluid velocity, and decreasing the temperature as shown in Figure 7b. There is an increase in Be and Ns by enhancing the values of α = β as shown in Figure 7c and d, respectively.

$Figure 7 Impact of α = β \alpha =\beta on (a) dimensionless velocity profile, (b) dimensionless temperature profile, (c) Bejan number, and (d) entropy generation when Λ 1 = 0.1 , {\Lambda }_{1}=0.1, Λ 2 = 0.2 , {\Lambda }_{2}=0.2, Gr = 5 , {\rm{Gr}}=5, Ha = 2 , Pr = 6.2 , {\rm{Ha}}=2,{\rm{\Pr }}=6.2, Q 0 = 2.5 , Nr = 5 , Ec = 2 , {Q}_{0}=2.5,{\rm{Nr}}=5,{\rm{Ec}}=2, Br = 2 , {\rm{Br}}=2, Ω = 10 , and ϕ = 0.1 \Omega =10,{\ }{\rm{and}}\phi =0.1 .$

Figure 7

Impact of α = β on (a) dimensionless velocity profile, (b) dimensionless temperature profile, (c) Bejan number, and (d) entropy generation when Λ 1 = 0.1 , Λ 2 = 0.2 , Gr = 5 , Ha = 2 , Pr = 6.2 , Q 0 = 2.5 , Nr = 5 , Ec = 2 , Br = 2 , Ω = 10 , and ϕ = 0.1 .

Effects of Pr on fluid temperature are presented in Figure 8a. Increasing the values of Pr increased the viscous diffusion rate and decreased the thermal diffusion rate consequently, decreasing the fluid temperature. Pr reduced the values of Nu for enhancing the time parameter as can be seen in Figure 8b. It can be noticed from Figure 8c and d, respectively, that increasing the value of Pr decreased Be and Ns. Similar results are published in ref. [48].

$Figure 8 Effect of Pr {\rm{\Pr }} on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , \alpha =\beta =0.6, Λ 1 = 0.1 , {\Lambda }_{1}=0.1, Λ 2 = 0.2 , {\Lambda }_{2}=0.2, Gr = 5 , {\rm{Gr}}=5, Ha = 2 , {\rm{Ha}}=2, Q 0 = 2.5 , Nr = 5 , Ec = 2 , {Q}_{0}=2.5,{\rm{Nr}}=5,{\rm{Ec}}=2, Br = 2 , {\rm{Br}}=2, Ω = 10 , \Omega =10, and ϕ = 0.1 {\rm{and}}\phi =0.1 .$

Figure 8

Effect of Pr on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0.1 , Λ 2 = 0.2 , Gr = 5 , Ha = 2 , Q 0 = 2.5 , Nr = 5 , Ec = 2 , Br = 2 , Ω = 10 , and ϕ = 0.1 .

Effects of Nr on temperature profile is elaborated in Figure 9a. Enhancement in Nr enhanced the fluid temperature, and reduced Nu for increasing time parameter in Figure 9b. It is a well-known fact that heat transfer plays a prominent role in the thermodynamic irreversibility. As Nr increased, it is noticed that Ns also increased in as shown in Figure 9c. Relative effect of heat transfer entropy generation and total entropy generation is measured by Be. It is observed that Be decreased near the surface of plate, and a little away from the surface, Be increased for increasing values of Nr. It can be justified as fluid friction reversibility is prominent near the surface, and at a distance from the surface, heat transfer irreversibility becomes prominent. Similar results are reported in ref. [47].

$Figure 9 Effect of thermal radiations parameter (Nr) on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0 . 1 , \alpha =\beta =0.6,{\Lambda }_{1}=0.1, Λ 2 = 0 . 2 , Gr = 5 , Ha = 2 , Pr = 6 . 2 , {\Lambda }_{2}=0.2,{\rm{Gr}}=5,{\rm{Ha}}=2,{\rm{\Pr }}=6.2, Q 0 = 2 . 5 , Ec = 2 , {Q}_{0}=2.5,{\rm{Ec}}=2, Br = 2 , {\rm{Br}}=2, Ω = 10 , \Omega =10, and ϕ = 0 . 1 {\rm{and}}\phi =0.1 .$

Figure 9

Effect of thermal radiations parameter (Nr) on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0 . 1 , Λ 2 = 0 . 2 , Gr = 5 , Ha = 2 , Pr = 6 . 2 , Q 0 = 2 . 5 , Ec = 2 , Br = 2 , Ω = 10 , and ϕ = 0 . 1 .

Relative effect of kinetic energy of flow on the enthalpy difference across the thermal boundary layer is measured by Ec. Increase in the value of Ec means decreasing the viscous dissipation and consequently increasing the fluid temperature as shown in Figure 10a. It is also noticed in Figure 10b that Nu decreased on increasing the values of time parameter against Ec. A slight increase in Be against increasing values of Ec is noticed in Figure 10c. Figure 10d presents the effect of Ec on Ns, it can be noticed that for increasing values of Ec, the entropy generation also increased.

$Figure 10 Effect of Ec on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0 . 1 , \alpha =\beta =0.6,{\Lambda }_{1}=0.1, Λ 2 = 0 . 2 , Gr = 5 , Ha = 2 , Pr = 6 . 2 , {\Lambda }_{2}=0.2,{\rm{Gr}}=5,{\rm{Ha}}=2,{\rm{\Pr }}=6.2, Q 0 = 2 . 5 , Nr = 5 , Br = 2 , {Q}_{0}=2.5,{\rm{Nr}}=5,{\rm{Br}}=2, Ω = 10 , and ϕ = 0 . 1 \Omega =10,{\rm{and}}\phi =0.1 .$

Figure 10

Effect of Ec on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0 . 1 , Λ 2 = 0 . 2 , Gr = 5 , Ha = 2 , Pr = 6 . 2 , Q 0 = 2 . 5 , Nr = 5 , Br = 2 , Ω = 10 , and ϕ = 0 . 1 .

Figure 11a illustrates the effect of time relaxation parameter ( λ 2 = Λ 2 ) on fluid temperature. Advancement in Λ 2 causes decrease in fluid temperature. In Figure 11b, it is examined that for higher values of time parameter against Λ 2 , the values of Nu are reduced. Be and Ns have opposite behavior against Λ 2 , as can be noticed that for enhancing values of Λ 2 , the Be is also enhanced as shown in Figure 11c, but Ns decreased in as shown in Figure 11d.

$Figure 11 Effect of Λ 2 {\Lambda }_{2} on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0 . 1 , Gr = 5 , Ha = 2 , Pr = 6 . 2 , \alpha =\beta =0.6,{\Lambda }_{1}=0.1,{\rm{Gr}}=5,{\rm{Ha}}=2,{\rm{\Pr }}=6.2, Q 0 = 2 . 5 , Nr = 5 , Ec = 2 , Br = 2 , {Q}_{0}=2.5,{\rm{Nr}}=5,{\rm{Ec}}=2,{\rm{Br}}=2, Ω = 10 , and ϕ = 0 . 1 \Omega =10,{\rm{and}}\phi =0.1 .$

Figure 11

Effect of Λ 2 on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0 . 1 , Gr = 5 , Ha = 2 , Pr = 6 . 2 , Q 0 = 2 . 5 , Nr = 5 , Ec = 2 , Br = 2 , Ω = 10 , and ϕ = 0 . 1 .

Figure 12a depicts the effects of Q ₀ on fluid temperature. Fluid temperature enhanced for advancement in the values of Q ₀. Nu decreased for increasing values of Q ₀ as can be noticed from Figure 12b. Be and Ns also increased for increasing values of Q ₀. Similar results can be found in ref. [47], showing that present results have good agreement with the published results.

$Figure 12 Effect of Q 0 {Q}_{0} on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0.1 , {\alpha =\beta =0.6,\Lambda }_{1}=0.1, Λ 2 = 0.2 , {\Lambda }_{2}=0.2, Gr = 5 , {\rm{Gr}}=5, Ha = 2 , Pr = 6.2 , Nr = 5 , Ec = 2 , {\rm{Ha}}=2,{\rm{\Pr }}=6.2,{\rm{Nr}}=5,{\rm{Ec}}=2, Br = 2 , {\rm{Br}}=2, Ω = 10 , \Omega =10, and ϕ = 0.1 {\rm{and}}\phi =0.1 .$

Figure 12

Effect of Q 0 on (a) dimensionless temperature profile, (b) Nusslt number, (c) Bejan number, and (d) entropy generation, when α = β = 0 . 6 , Λ 1 = 0.1 , Λ 2 = 0.2 , Gr = 5 , Ha = 2 , Pr = 6.2 , Nr = 5 , Ec = 2 , Br = 2 , Ω = 10 , and ϕ = 0.1 .

Tables 2 and 3 show the behavior of physical quantities, i.e., skin friction coefficient and local Nusselt number on varying the physical parameters under discussion.

Table 2

Skin friction analysis against different physical parameters and α = β = ν

Ha	Gr	λ 1 α	ν = 0 . 4	ν = 0 . 7	ν = 1
0.1	0.5	0.1	1.3535	0.6187	−0.1437
0.2			1.3854	0.6524	−0.1086
0.3			1.4163	0.6850	−0.7445
2.0	0.1	0.1	1.0966	1.2040	0.4722
	0.2		1.8841	1.1824	0.4495
	0.3		1.8636	1.1607	0.4268
2.0	0.5	0.1	1.8226	1.1174	0.3814
		0.2	3.0058	1.9186	0.5525
		0.3	4.0504	2.7811	1.1557

Table 3

Nusselt number analysis against different physical parameters and α = β = ν

Pr	Nr	Ec	ν = 0 . 4	ν = 0 . 7	ν = 1
1	0.5	0.5	1.7262	1.0182	0.2809
2			1.7758	1.0685	0.3308
3			1.7951	1.0885	0.3512
6.2	0.1	0.5	3.5795	2.3869	0.9993
	0.2		3.5451	2.5582	0.9652
	0.3		3.5141	2.3281	0.9351
	0.5	0.1	3.5021	2.3271	0.9348
		0.2	3.4917	2.3155	0.9220
		0.3	3.4812	2.3040	0.9093

6 Conclusion

The fractional Oldroyd-B nano-fluid model in view of Caputo time fractional derivative along with infinite oscillating vertical plate has been investigated. The prevailing system of nonlinear partial differential equations is converted into dimensionless form. For the purpose of physical observations of the proposed model, a numerical scheme (FDM) is applied. Numerical solutions with parametric effects are elaborated by the mean of graphical figures portrayed using MAPLE. Key results are as follows:

Reduction in the velocity profile of fluid flow is observed against change in Ha, λ 1 , and ϕ .
Temperature profile boosts for increasing values of Nr, ϕ , Q ₀, and viscous dissipation parameter (Ec).
Ns enhanced by enhancing the values of Gr, Ec, α, ϕ , Nr, and Q ₀, and declined with the variation in Pr, Ha, λ 1 , and λ 2 .
Be increased with the increase in the values of Ha, λ 1 , λ 2 , ϕ , α, and Q ₀, and decreased for Gr and Pr. Dual nature for Nr and Ec) is noticed initially, i.e., they decreased and then increased gradually.
Nu decayed for the advancement in the values of Q ₀, Nr, ϕ , λ 2 , Ec, and Pr. And C _f boosts for increase in the values of Ha, Gr, ϕ , and λ 1 .

The core contribution to literature is that enhancement occurs in temperature by enlarging the values of Ec , Nr , and Q 0 . Ns is controlled by the addition of nano-particles.

This problem can be extended for further effects like variable thermal conductivity and porous medium. Also, FDM is a very strong tool for solving such numerous type’s problem and complex geometry.

Acknowledgments

The authors are thankful to the University of Management and Technology Lahore, 54770, Lahore Pakistan.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Ali A, Asjad MI. Numerical solutions of a heat transfer for fractional maxwell fluid flow with water based clay nanoparticles; A finite difference approach. Fractal Fract. 2021;5(4):242.10.3390/fractalfract5040242Search in Google Scholar

[2] Khan AS, Nie Y, Shah Z. Impact of thermal radiation on magnetohydrodynamic unsteady thin film flow of Sisko fluid over a stretching surface. Processes. 2019;7(6):369.10.3390/pr7060369Search in Google Scholar

[3] Asjad MI. Heat transfer analysis of fractional second-grade fluid subject to Newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: A comparison. Eur Phys J Plus. 2017;132(8):1–19.10.1140/epjp/i2017-11606-6Search in Google Scholar

[4] Javaid M. Unsteady flow of fractional Burgers’ fluid in a rotating annulus region with power law kernel. Alex Eng J. 2022;61(1):17–27.10.1016/j.aej.2021.04.106Search in Google Scholar

[5] Aleem M, Asjad MI, Ahmadian A, Salimi M, Ferrara M. Heat transfer analysis of channel flow of MHD Jeffrey fluid subject to generalized boundary conditions. Eur Phys J Plus. 2020;135(1):1–15.10.1140/epjp/s13360-019-00071-6Search in Google Scholar

[6] Riaz MB, Imran M, Shabbir K. Analytic solutions of Oldroyd-B fluid with fractional derivatives in a circular duct that applies a constant couple. Alex Eng J. 2016;55(4):3267–75.10.1016/j.aej.2016.07.032Search in Google Scholar

[7] Ali A, Hussain M, Anwar MS, Inc M. Mathematical modeling and parametric investigation of blood flow through a stenosis artery. Appl Math Mech. 2021;42(11):1675–84.10.1007/s10483-021-2791-8Search in Google Scholar

[8] Hussain Z. Mixed convective flow of CNTs nanofluid subject to varying viscosity and reactions. Sci Rep. 2021;11(1):1–14.10.1038/s41598-021-02228-9Search in Google Scholar PubMed PubMed Central

[9] Irfan M. Evaluating the performance of new mass flux theory on Carreau nanofluid using the thermal aspects of convective heat transport. Pramana. 2021;95(4):1–9.10.1007/s12043-021-02217-7Search in Google Scholar

[10] Rafiq K. Arrhenius activation energy theory in radiative flow of Maxwell nanofluid. Phys Scr. 2021;96(4):045002.10.1088/1402-4896/abd903Search in Google Scholar

[11] Fetecau C, Prasad SC, Rajagopal KR. A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid. Appl Math Model. 2007;31(4):647–54.10.1016/j.apm.2005.11.032Search in Google Scholar

[12] Fetecau C, Hayat T, Khan M. Unsteady flow of an Oldroyd-B fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate. Acta mechanica. 2008;198(1):21–33.10.1007/s00707-007-0522-0Search in Google Scholar

[13] Gul T. Unsteady MHD thin film flow of an Oldroyd-B fluid over an oscillating inclined belt. PLoS one. 2015;10(7):e0126698.10.1371/journal.pone.0126698Search in Google Scholar PubMed PubMed Central

[14] Wang J. Transportation of heat generation/absorption and radiative heat flux in homogeneous–heterogeneous catalytic reactions of non-Newtonian fluid (Oldroyd-B model). Comput methods Prog biomedicine. 2020;189:105310.10.1016/j.cmpb.2019.105310Search in Google Scholar PubMed

[15] Song YQ, Farooq A, Kamran M, Rehman S, Tamoor M, Khan R, et al. Analytical solution of fractional Oldroyd-B fluid via fluctuating duct. Complexity. 2021;2021:16.10.1155/2021/9576873Search in Google Scholar

[16] Hayat T. Cattaneo-Christov (CC) heat flux model for nanomaterial stagnation point flow of Oldroyd-B fluid. Comput Methods Prog Biomedicine. 2020;187:105247.10.1016/j.cmpb.2019.105247Search in Google Scholar PubMed

[17] Choi SU, Eastman JA. Enhancing thermal conductivity of fluids with nanoparticles. IL (United States): Argonne National Lab; 1995.Search in Google Scholar

[18] Kulkarni DP, Das DK, Vajjha RS. Application of nanofluids in heating buildings and reducing pollution. Appl Energy. 2009;86(12):2566–73.10.1016/j.apenergy.2009.03.021Search in Google Scholar

[19] Ramezanizadeh M, Nazari MA, Ahmadi MH, Açıkkalp E. Application of nanofluids in thermosyphons: a review. J Mol Liq. 2018;272:395–402.10.1016/j.molliq.2018.09.101Search in Google Scholar

[20] Ebrahimnia-Bajestan E, Moghadam MC, Niazmand H, Daungthongsuk W, Wongwises S. Experimental and numerical investigation of nanofluids heat transfer characteristics for application in solar heat exchangers. Int J Heat Mass Transf. 2016;92:1041–52.10.1016/j.ijheatmasstransfer.2015.08.107Search in Google Scholar

[21] Li B, Li C, Zhang Y, Wang Y, Jia D, Yang M, et al. Heat transfer performance of MQL grinding with different nanofluids for Ni-based alloys using vegetable oil. J Clean Prod. 2017;154:1–11.10.1016/j.jclepro.2017.03.213Search in Google Scholar

[22] Hamid M, Usman M, Zubair T, Haq RU, Wang W. Shape effects of MoS2 nanoparticles on rotating flow of nanofluid along a stretching surface with variable thermal conductivity: A Galerkin approach. Int J Heat Mass Transf. 2018;124:706–14.10.1016/j.ijheatmasstransfer.2018.03.108Search in Google Scholar

[23] Alkanhal TA, Sheikholeslami M, Usman M, Haq RU, Shafee A, Al-Ahmadi AS, et al. Thermal management of MHD nanofluid within the porous medium enclosed in a wavy shaped cavity with square obstacle in the presence of radiation heat source. Int J Heat Mass Transf. 2019;139:87–94.10.1016/j.ijheatmasstransfer.2019.05.006Search in Google Scholar

[24] Usman M. Differential transform method for unsteady nanofluid flow and heat transfer. Alex Eng J. 2018;57(3):1867–75.10.1016/j.aej.2017.03.052Search in Google Scholar

[25] Usman M. Thermal and velocity slip effects on Casson nanofluid flow over an inclined permeable stretching cylinder via collocation method. Int J Heat Mass Transf. 2018;122:1255–63.10.1016/j.ijheatmasstransfer.2018.02.045Search in Google Scholar

[26] Usman M. Novel modification in wavelets method to analyze unsteady flow of nanofluid between two infinitely parallel plates. Chin J Phys. 2020;66:222–36.10.1016/j.cjph.2020.03.031Search in Google Scholar

[27] Imran M. Heat and mass transport of differential type fluid with non-integer order time-fractional Caputo derivatives. J Mol Liq. 2017;229:67–75.10.1016/j.molliq.2016.11.095Search in Google Scholar

[28] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl. 2015;1(2):1–13.10.18576/pfda/020101Search in Google Scholar

[29] Ikram MD. Effects of hybrid nanofluid on novel fractional model of heat transfer flow between two parallel plates. Alex Eng J. 2021;60(4):3593–604.10.1016/j.aej.2021.01.054Search in Google Scholar

[30] Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408; 2016.10.2298/TSCI160111018ASearch in Google Scholar

[31] Saqib M, Khan I, Shafie S. Application of Atangana–Baleanu fractional derivative to MHD channel flow of CMC-based CNT’s nanofluid through a porous medium. Chaos Solitons Fractals. 2018;116:79–85.10.1016/j.chaos.2018.09.007Search in Google Scholar

[32] Wang F. Comparative study of heat and mass transfer of generalized MHD Oldroyd-B bio-nano fluid in a permeable medium with ramped conditions. Sci Rep. 2021;11(1):1–32.10.1038/s41598-021-02326-8Search in Google Scholar PubMed PubMed Central

[33] Hassan M. Impact of iron oxide particles concentration under a highly oscillating magnetic field on ferrofluid flow. Eur Phys J Plus. 2018;133(6):1–14.10.1140/epjp/i2018-12045-7Search in Google Scholar

[34] Hassan M. Effects of iron nanoparticles’ shape on convective flow of ferrofluid under highly oscillating magnetic field over stretchable rotating disk. J Magnetism Magnetic Mater. 2018;465:531–9.10.1016/j.jmmm.2018.06.019Search in Google Scholar

[35] Hussain A, Ghafoor S, Malik MY, Jamal S. An exploration of viscosity models in the realm of kinetic theory of liquids originated fluids. Results Phys. 2017;7:2352–60.10.1016/j.rinp.2017.06.036Search in Google Scholar

[36] Farooq U. Transpiration and viscous dissipation effects on entropy generation in hybrid nanofluid flow over a nonlinear radially stretching disk. Entropy. 2018;20(9):668.10.3390/e20090668Search in Google Scholar PubMed PubMed Central

[37] Ullah H, Hayat T, Ahmad S, Alhodaly MS. Entropy generation and heat transfer analysis in power-law fluid flow: Finite difference method. Int Commun Heat Mass Transf. 2021;122:105111.10.1016/j.icheatmasstransfer.2021.105111Search in Google Scholar

[38] Khan MI, Kadry S, Chu Y, Waqas M. Modeling and numerical analysis of nanoliquid (titanium oxide, graphene oxide) flow viscous fluid with second order velocity slip and entropy generation. Chin J Chem Eng. 2021;31:17–25.10.1016/j.cjche.2020.08.005Search in Google Scholar

[39] Khan I. Convective heat transfer in drilling nanofluid with clay nanoparticles: applications in water cleaning process. BioNanoScience. 2019;9(2):453–60.10.1007/s12668-019-00623-1Search in Google Scholar

[40] Anwar T. Generalized thermal investigation of unsteady MHD flow of Oldroyd-B fluid with slip effects and Newtonian heating; a Caputo-Fabrizio fractional model. Alex Eng J. 2022;61(3):2188–202.10.1016/j.aej.2021.06.090Search in Google Scholar

[41] Maripala S, Naikoti K. Joule heat parameter effects on unsteady MHD flow over a stretching sheet with viscous dissipation and heat source. Appl Appl Math An Int J (AAM). 2019;14(4):4.Search in Google Scholar

[42] Tahir M. Wall slip and non-integer order derivative effects on the heat transfer flow of Maxwell fluid over an oscillating vertical plate with new definition of fractional Caputo-Fabrizio derivatives. Results Phys. 2017;7:1887–98.10.1016/j.rinp.2017.06.001Search in Google Scholar

[43] Yusuf T. Irreversibility analysis of Cu-TiO2-H2O hybrid-nanofluid impinging on a 3-D stretching sheet in a porous medium with nonlinear radiation: Darcy-Forchheimer’s model. Alex Eng J. 2020;59(6):5247–61.10.1016/j.aej.2020.09.053Search in Google Scholar

[44] Qing J. Entropy generation on MHD Casson nanofluid flow over a porous stretching/shrinking surface. Entropy. 2016;18(4):123.10.3390/e18040123Search in Google Scholar

[45] Ali Akgül, Asjad MI. Power law memory of natural convection flow of hybrid nanofluids with constant proportional Caputo fractional derivative due to pressure gradient. Pramana. 2020;94(1):1–11.10.1007/s12043-020-01997-8Search in Google Scholar

[46] Zhuang P, Liu F. Implicit difference approximation for the time fractional diffusion equation. J Appl Math Comput. 2006;22(3):87–99.10.1007/BF02832039Search in Google Scholar

[47] Sarojamma G, Vajravelu K, Sreelakshmi K. A study on entropy generation on thin film flow over an unsteady stretching sheet under the influence of magnetic field, thermocapillarity, thermal radiation and internal heat generation/absorption. Commun Numer Anal. 2017;2:141–56.10.5899/2017/cna-00319Search in Google Scholar

[48] Esfahani J. Influences of wavy wall and nanoparticles on entropy generation over heat exchanger plat. Int J Heat Mass Transf. 2017;109:1162–71.10.1016/j.ijheatmasstransfer.2017.03.006Search in Google Scholar

Received: 2022-02-14

Revised: 2022-04-26

Accepted: 2022-05-30

Published Online: 2022-12-12

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2022-0166

Keywords for this article

Oldroyd-B nano-fluid; Caputo time fractional model; viscous dissipation; thermal radiations; magnetic field, finite difference method

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