Analytical solutions of fractional couple stress fluid flow for an engineering problem

Rabia Naz; Muhammad Danish Ikram; Muhammad Imran Asjad

doi:10.1515/nleng-2022-0281

Article Open Access

Analytical solutions of fractional couple stress fluid flow for an engineering problem

Rabia Naz , Muhammad Danish Ikram and Muhammad Imran Asjad

Published/Copyright: May 3, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

In this article, analytical solutions of couple stress fluid flow modeled with a power law fractional differential operator are discussed. Stokes’ second problem for an incompressible couple stress fluid is studied for an horizontal plate of infinite length. The governing equations of the flow problem are expressed in terms of a partial differential operator and then converted into a non-dimensional model by using dimensional analysis. Then the integer order problem was formulated in terms of the non-integer order of three types of fractional derivatives and then solved with the help of the Laplace transform method. The obtained solutions are complex and expressed in terms of series. In order to check the memory index of the solutions obtained with three different fractional operators, we have plotted some graphs. It is found that the constant proportional operator provides us a better choice about the memory and maximum enhancement achieved in the comparison of Caputo and Caputo–Fabrizio. Furthermore, in order to check the accuracy of the present results, we have compared the obtained solutions with the existing literature and found a good agreement between them.

Keywords: constant proportional Caputo; couple stress fluids; Stokes’ second problem; analytical solutions

1 Introduction

A large portion of the readings underway elaborate on the viscous resources of the classical Naiver–Stokes associations. There are many multifaceted goods, e.g., paints and polymer solutions that cannot be considered over the classical Naiver–Stokes equations. These are famous as non-Newtonian liquids. Numerous models of non-Newtonian liquids are established in the mechanism. The couple stress liquid model is one of them. It has significant configurations because of the existence of body couples, couple stresses, and non-symmetric stress tensor. Inspiring illustrations of the couple stress liquids are blood, oils, and suspension liquids [1–7].

Umavathi et al. [8] investigated the steady-state heat transfer performance of a couple stress nano-fluid sandwiched between viscous fluids numerically. Gopal et al. [9] discussed the movement of a couple stress liquid over a porous layer bounded by parallel plates. The effects of Newtonian heating on magnetohydrodynamics (MHD) 3-dimensional movement past a stretching region have been studied by refs [10,11]. Turkyilmazoglu [12] analyzed the heat and movement of couple stress MHD liquids over permeable stretching/shrinking surfaces. Hayat et al. [13,14] presented an analysis of MHD extracting movement of couple stress nano-material between two parallel surfaces. Devakar and Iyengar [15] evaluated Stokes’ first and second problems for an incompressible couple stress liquid under isothermal constraints.

Fractional calculus has enlarged significant status and importance due to its valid applications in several branches of science and engineering [16–21]. Tahir et al. [22] discussed the natural convection of unsteady movement of Maxwell liquid over a waving upright flat plate with constant temperature at the boundary by Caputo–Fabrizio (CF) fractional operator. Saqib et al. [23] used the CF fractional model to analyze the free convection movement of Jeffrey fluid. Natural convection flow of differential type fluid with Caputo fractional derivatives is determined by Imran et al. [24]. Jarad et al. [25] explained the uniqueness conditions of solutions to a certain class of ODEs with ABC fractional derivative. Abro et al. [26] explored the impacts of carbon nanotubes on the MHD flow of methanol based nanofluids by ABC and CF fractional techniques. Danish Ikram et al. [27] analyzed a fractional model of extraction nanofluids using clay nanoparticles for different based fluids.

Recently, Baleanu et al. [28] introduced a new fractional operator called constant proportional Caputo (CPC) fractional derivative by combining Caputo and conformable fractional derivative. This operator is a power law and is a linear combination of two fractional differential operators. Imran et al. [29] used CPC fractional operator to study the astrophysical properties of clay nanofluids with Brinkman and Maxwell–Garnet terminologies. Ikram et al. [30] presented a comparison of CPC and CF fractional operators using hybrid nanofluids. They concluded that CPC gave better memory effects than CF fractional operator. Chu et al. [31] calculated the MHD free convection movement of hybrid nanofluid in a microchannel through CPC fractional operator. Asadullah et al. [32] explained the heat and mass transfer flow for adhesive fluid between two vertical parallel plates. They found an analytical solution by applying CPC fractional approach.

Hamid et al. [34], introduced an insubstantial method to investigate the nonlinear oscillatory fractional-order differential equations. Hamid et al. [35] proposed a linearized steady-state insubstantial technique to study two-dimensional nonlinear evolutionary and reaction–diffusion models. In the study of Hamid et al. [35], research is related to the growth of a hybrid semi-spectral computational scheme built on multidimensional Chelyshkov polynomials (CPs). In the study of Hamid et al. [36], finite difference method (FDM) and Chelyshkov polynomial method schemes are used. They are used to obtain nonlinear oscillatory difficulties of random orders that do not have particular solutions in the literature. In the study of Hamid et al. [37], an operational matrix-based spectral computational strategy was combined with the Picard system and successfully utilized to find the solution for a family of nonlinear development differential equations. The numerical solutions of non-dimensional system of partial differential equations finite difference method tied with Crank Nicolson technique is established and effectively applied [38]. Crank Nicolson finite difference method is unconditionally stable. Prabhakar fractional model is an alternative, newly proposed analytical approach without the appearance of singularity. This model grants modifications of both Atangana Baleanu (AB) and Caputo Fabrizio models in the literature. In the study of Li and Peng [39], a critical function of utilizing a set of laws or equations is to transform one signal into another. One may wonder why the Laplace transform technique is necessary when other methods are available. The straight forward answer is that this method simplifies ordinary differential equations (ODEs) into algebraic equations, making it easier to solve the ODE problem. There are several methods, each with their own significance. Rubbab et al. [40] used CF time-fractional derivative for the fractional model of the advection–diffusion process. In the study of Guo et al. [41], fractional-order simulations in mass and heat transfer analysis are limited by an elliptic inclined plate through slip effects. MHD fluid flow through a porous medium flowing on a vertical plate for this problem AB time-fractional derivative is used to find the numerical outcomes [42]. In the study of Raza et al. [43], a thermal study of the mixed free convection Casson nanofluid along with heat transfer effects over a vertical plate is discussed. To develop the fractional model, the partial derivative with respect to time is swapped with the recent interpretations of fractional derivatives AB and CF time-fractional derivatives, and the Laplace method is applied to determine the solution of the governing equations. AB fractional derivative is used to find the significances of exponential heating and Darcy’s law for second-grade fluid flow over a wavering plate [44]. Raza et al. [45] discussed the current classification of fractional derivative AB time-fractional derivative on second-grade fluid flowing on an infinite plate under the MHD effect.

By the motivation of the above study, our aim is to report the couple stress fluid by applying CPC fractional operator and comparing its results with other fractional derivatives as well as the classical results obtained by Devakar and Iyengar [15]. There is no such result exists in the literature.

2 Preliminaries

In the literature, there are many basic definitions to deal with fractional derivatives; some of them are provided below.

2.1 Definition 1

The Caputo fractional derivative of order 0 < α < 1 of a function h : ( 0 , ∞ ) → R n is defined as follows [46]:

(1) D t α C g ( y , t ) = 1 Γ ( 1 − α ) ∫ 0 t ǵ ( y , s ) ( t − s ) − α d s = ( h α ∗ ǵ ) ( t ) ,

whereas h α ( t ) = t − α Γ ( 1 − α ) is the singular power-law kernel and ǵ ( y , s ) = ∂ g ( y , t ) ∂ t ∣ t = s = ∂ t g ( y , t ) ∣ t = s

The Laplace transform of above equation is given as follows:

(2) ℒ { D t α C g ( y , t ) } = s α ℒ { g ( y , s ) } − s α − 1 g ( y , o ) ,

2.2 Definition 2

The CPC hybrid operator may be defined as follows [46]:

(3) D t α 0 CPC g ( y , t ) = 1 Γ ( 1 − α ) ∫ 0 t ( K 1 ( α ) g ( y , s ) + K O ( α ) g ̀ ( τ ) ) ( t − τ ) − α d τ = K 1 ( α ) 0 R L I t 1 − α g ( t ) + K o ( α ) D t α 0 C ,

Here, CPC stands for constant proportional Caputo.

3 Problem formulation

The constitution equation between the force stress tensor t i j and rate of deformation tensor d i j is as follows:

t i j = ( − p + λ 1 ∇ ⋅ q ¯ ) δ i j + 2 μ d i j − 1 2 ε i j k [ m , k + 4 η ω k , r r + ρ c k ] .

Couple stress tensor m i j that present in the theory have linear constitutive relation

m i j = 1 3 m δ i j + 4 η ́ ω j , i + 4 η ω i , j ,

where ω = 1 2 ∇ × q ¯ is spin vector, ω i , j is the spin tensor, d i j is the rate of deformation tensor, m is the trace of couple stress tensor m i j , ρ c k is the body couple vector, and quantities λ 1 , μ and η , η ́ are viscosity and couple stress viscosity coefficients [15].

4 Mathematical formulation

Suppose that the incompressible and unsteady movement of a couple stress liquid that fill up the partial space y > 0 over a smooth (solid) plate covering the x z -plane.

The assumptions are the following:

At the start, both the plate and fluid are at rest.
On time t = 0 + , the plate is starting to oscillate in its own plane.
Beside x -axis, the flow happen in x -direction as presented in the following figure.
Consequently, the velocity is V = ( u ( y , t ) , 0 , 0 ) and it generally satisfies the equation of continuity.

The governing equation for couple stress fluid are as follows [15]:

(4) ρ ∂ u ( y , t ) ∂ t = μ ∂ 2 u ( y , t ) ∂ y 2 − η ∂ 4 u ( y , t ) ∂ y 4 .

The boundary and initial conditions are

(5) u ( y , t ) → 0 if y → ∞ ,

(6) u ( 0 , t ) = UH ( t ) cos ( ω t ) , when t > 0 .

(7) ∂ 2 u ( y , t ) ∂ y 2 = 0 , when y = 0 for t > 0 .

(8) u ( y , 0 ) ≡ 0 ∀ y .

Now the non-dimensional variables are introduced as follows:

(9) Φ = u U , ξ = y L , τ = U L t .

where

L 2 = η μ , Re = ρ U L μ .

As Eq. (4) reduces, we obtain

(10) Re ∂ Φ ( ξ , τ ) ∂ τ = ∂ 2 Φ ( ξ , τ ) ∂ ξ 2 − ∂ 4 Φ ( ξ , τ ) ∂ ξ 4 .

The dimensionless conditions are

(11) Φ ( ξ , 0 ) = 0 ∀ ξ ,

(12) Φ ( 0 , τ ) = H ( τ ) cos ( ω τ ) τ > 0 ,

(13) Φ ( ξ , τ ) → 0 as ξ → ∞ ,

(14) ∂ 2 Φ ( ξ , s ) ∂ ξ 2 = 0 , at ξ = 0 , τ > 0 .

4.1 Constant proportional model of couple stress fluid

This section will cover the fractional model of a physical problem provided in Eq. (10).

(15) Re CPC D τ α Φ ( ξ , τ ) = ∂ 2 Φ ( ξ , τ ) ∂ ξ 2 − ∂ 4 Φ ( ξ , τ ) ∂ ξ 4 .

With the help of the Laplace transform on Eqs. (10)–(14), we obtain the following:

(16) d 4 Φ ¯ ( ξ , s ) d ξ 4 − d 2 Φ ¯ ( ξ , s ) d ξ 2 + Re k 1 ( α ) s + k o ( α ) s α Φ ¯ ( ξ , s ) = 0 ,

satisfying the boundary conditions

(17) Φ ¯ ( 0 , s ) = s s 2 + w 2 ,

(18) Φ ¯ ( ξ , s ) → 0 if ξ → ∞ ,

(19) ∂ 2 Φ ¯ ( 0 , s ) ∂ ξ 2 = 0 .

After applying the conditions in (11)–(14), the solution of Eq. (17) is

(20) Φ ¯ ( ξ , s ) = s 1 − L 1 + k 2 ( α ) s s α − 1 2 ( s 2 + w 2 ) 1 − L 1 + k 2 ( α ) s s α e − ξ 1 + 1 − L 1 + k 2 ( α ) s s α 2 + s 1 − L 1 + k 2 ( α ) s s α + 1 2 ( s 2 + w 2 ) 1 − L 1 + k 2 ( α ) s s α × e − ξ 1 − 1 − L 1 + k 2 ( α ) s s α 2 ,

where

(21) L = 4 Re k 0 ( α ) , k 2 ( α ) = k 1 ( α ) k 0 ( α ) .

Eq. (20) can be written as follows:

(22) Φ ¯ ( ξ , s ) = s 2 ( s 2 + w 2 ) e − ξ 1 + 1 − L 1 + k 2 ( α ) s s α 2 − s 2 ( s 2 + w 2 ) 1 − L 1 + k 2 ( α ) s s α × e − ξ 1 + 1 − L 1 + k 2 ( α ) s s α 2 + s 2 ( s 2 + w 2 ) e − ξ 1 − 1 − L 1 + k 2 ( α ) s s α 2 + s 2 ( s 2 + w 2 ) 1 − L 1 + k 2 ( α ) s s α × e − ξ 1 − 1 − L 1 + k 2 ( α ) s s α 2 .

In equivalent form, it can be stated as follows:

(23) Φ ¯ ( ξ , s ) = s s 2 + w 2 + ∑ p 1 = 1 ∞ ∑ p 2 = 0 ∞ ∑ p 3 = 0 ∞ ∑ p 4 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) p 1 ( − L ) p 3 [ k 2 ( α ) ] p 4 ( − 1 ) r w 2 r 2 ( 2 ) p 1 p 1 ! p 2 ! p 3 ! p 4 ! s 1 − α p 3 + p 4 + 2 r × Γ p 1 2 + 1 Γ p 2 2 + 1 Γ ( p 3 + 1 ) Γ p 1 2 + 1 − p 2 Γ p 2 2 + 1 − p 3 Γ ( p 3 + 1 − p 4 ) − ∑ p 1 = 1 ∞ ∑ p 2 = 0 ∞ ∑ p 3 = 0 ∞ ∑ p 4 = 0 ∞ ∑ m = 0 ∞ ∑ k = 0 ∞ ∑ r = 0 ∞ ( − ξ ) p 1 ( − 1 ) p 3 + r w 2 r L p 3 + m [ k 2 ( α ) ] p 4 + k 2 ( 2 ) p 1 p 1 ! p 2 ! p 3 ! p 4 ! m ! k ! s 1 + 2 r − α p 3 + p 4 − α m + k × Γ p 1 2 + 1 Γ p 2 2 + 1 Γ ( p 3 + 1 ) Γ ( m + 1 2 ) Γ ( m + 1 ) Γ p 1 2 + 1 − p 2 Γ p 2 2 + 1 − p 3 Γ ( p 3 + 1 − p 4 ) Γ 1 2 Γ ( m + 1 − k ) + ∑ q 1 = 1 ∞ ∑ q 2 = 0 ∞ ∑ q 3 = 0 ∞ ∑ q 4 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) q 1 ( − 1 ) q 2 + q 3 + r w 2 r L q 3 [ k 2 ( α ) ] q 4 2 ( 2 ) q 1 q 1 ! q 2 ! q 3 ! q 4 ! s 1 + 2 r − α q 3 + q 4 × Γ q 1 2 + 1 Γ q 2 2 + 1 Γ ( q 3 + 1 ) Γ q 1 2 + 1 − q 2 Γ q 2 2 + 1 − q 3 Γ ( q 3 + 1 − q 4 ) + ∑ q 1 = 1 ∞ ∑ q 2 = 0 ∞ ∑ q 3 = 0 ∞ ∑ q 4 = 0 ∞ ∑ m = 0 ∞ ∑ k = 0 ∞ ∑ r = 0 ∞ ( − ξ ) q 1 ( − 1 ) q 2 + q 3 + r w 2 r L q 3 + m [ k 2 ( α ) ] q 4 + k 2 ( 2 ) q 1 q 1 ! q 2 ! q 3 ! q 4 ! m ! k ! s 1 + 2 r − α q 3 + q 4 − α m + k × Γ q 1 2 + 1 Γ q 2 2 + 1 Γ ( q 3 + 1 ) Γ ( m + 1 2 ) Γ ( m + 1 ) Γ q 1 2 + 1 − q 2 Γ q 2 2 + 1 − q 3 Γ ( q 3 + 1 − q 4 ) Γ 1 2 Γ ( m + 1 − k ) .

Taking the Laplace inverse transform to Eq. (23), we obtain

(24) Φ ( ξ , τ ) = cosw τ + ∑ p 1 = 1 ∞ ∑ p 2 = 0 ∞ ∑ p 3 = 0 ∞ ∑ p 4 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) p 1 ( − L ) p 3 [ k 2 ( α ) ] p 4 ( − 1 ) r w 2 r τ − α p 3 + p 4 + 2 r 2 ( 2 ) p 1 p 1 ! p 2 ! p 3 ! p 4 ! Γ ( 1 − α p 3 + p 4 + 2 r ) × Γ p 1 2 + 1 Γ p 2 2 + 1 Γ ( p 3 + 1 ) Γ p 1 2 + 1 − p 2 Γ p 2 2 + 1 − p 3 Γ ( p 3 + 1 − p 4 ) − ∑ p 1 = 1 ∞ ∑ p 2 = 0 ∞ ∑ p 3 = 0 ∞ ∑ p 4 = 0 ∞ ∑ m = 0 ∞ ∑ k = 0 ∞ ∑ r = 0 ∞ ( − ξ ) p 1 ( − 1 ) p 3 + r w 2 r L p 3 + m [ k 2 ( α ) ] p 4 + k τ 2 r − α p 3 + p 4 − α m + k 2 ( 2 ) p 1 p 1 ! p 2 ! p 3 ! p 4 ! m ! k ! Γ ( 1 + 2 r − α p 3 + p 4 − α m + k ) × Γ p 1 2 + 1 Γ p 2 2 + 1 Γ ( p 3 + 1 ) Γ ( m + 1 2 ) Γ ( m + 1 ) Γ p 1 2 + 1 − p 2 Γ p 2 2 + 1 − p 3 Γ ( p 3 + 1 − p 4 ) Γ 1 2 Γ ( m + 1 − k ) + ∑ q 1 = 1 ∞ ∑ q 2 = 0 ∞ ∑ q 3 = 0 ∞ ∑ q 4 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) q 1 ( − 1 ) q 2 + q 3 + r w 2 r L q 3 [ k 2 ( α ) ] q 4 τ 2 r − α q 3 + q 4 2 ( 2 ) q 1 q 1 ! q 2 ! q 3 ! q 4 ! Γ ( 1 + 2 r − α q 3 + q 4 ) × Γ q 1 2 + 1 Γ q 2 2 + 1 Γ ( q 3 + 1 ) Γ q 1 2 + 1 − q 2 Γ q 2 2 + 1 − q 3 Γ ( q 3 + 1 − q 4 ) + ∑ q 1 = 1 ∞ ∑ q 2 = 0 ∞ ∑ q 3 = 0 ∞ ∑ q 4 = 0 ∞ ∑ m = 0 ∞ ∑ k = 0 ∞ ∑ r = 0 ∞ ( − x i ) q 1 ( − 1 ) q 2 + q 3 + r w 2 r L q 3 + m [ k 2 ( α ) ] q 4 + k τ 2 r − α q 3 + q 4 − α m + k 2 ( 2 ) q 1 q 1 ! q 2 ! q 3 ! q 4 ! m ! k ! Γ ( 1 + 2 r − α q 3 + q 4 − α m + k ) × Γ q 1 2 + 1 Γ q 2 2 + 1 Γ ( q 3 + 1 ) Γ ( m + 1 2 ) Γ ( m + 1 ) Γ q 1 2 + 1 − q 2 Γ q 2 2 + 1 − q 3 Γ ( q 3 + 1 − q 4 ) Γ 1 2 Γ ( m + 1 − k ) .

Eq. (24) can be stated in terms of M-function defined in [47],

(25) Φ ( ξ , τ ) = cosw τ + ∑ p 1 = 1 ∞ ∑ p 2 = 0 ∞ ∑ p 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) p 1 ( − L ) p 3 ( − 1 ) r w 2 r 2 ( 2 ) p 1 p 1 ! p 2 ! p 3 ! × M 4 3 ( k 2 ( α ) τ ) ∣ ( p 1 2 − p 2 + 1 , 0 ) , p 2 2 − p 3 + 1 , 0 , ( p 3 + 1 , − 1 ) , ( 1 − α p 3 + 2 r , 1 ) p 1 2 + 1 , 0 , p 2 2 + 1 , 0 , ( p 3 + 1 , 0 ) − ∑ p 1 = 1 ∞ ∑ p 2 = 0 ∞ ∑ p 3 = 0 ∞ ∑ p 4 = 0 ∞ ∑ m = 0 ∞ ∑ r = 0 ∞ ( − ξ ) p 1 ( − 1 ) p 3 + r w 2 r L p 3 + m [ k 2 ( α ) ] p 4 2 ( 2 ) p 1 p 1 ! p 2 ! p 3 ! p 4 ! m ! × M 6 5 ( [ k 2 ( α ) ] τ ) ∣ ( p 1 2 − p 2 + 1 , 0 ) , p 2 2 − p 3 + 1 , 0 , ( p 3 − p 4 + 1 , 0 ) , 1 2 , 0 , ( m + 1 , − 1 ) , ( 1 − α p 3 + p 4 − α m + 2 r , 1 ) p 1 2 + 1 , 0 , p 2 2 + 1 , 0 , ( p 3 + 1 , 0 ) , ( m + 1 2 , 0 ) , ( m + 1 , 0 ) + ∑ q 1 = 1 ∞ ∑ q 2 = 0 ∞ ∑ q 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) q 1 ( − 1 ) q 2 + q 3 + r w 2 r L q 3 2 ( 2 ) q 1 q 1 ! q 2 ! q 3 ! M 4 3 ( k 2 ( α ) τ ) ∣ q 1 2 − q 2 + 1 , 0 , q 2 2 − q 3 + 1 , 0 , ( q 3 + 1 , − 1 ) , ( 1 − α q 3 + 2 r , 1 ) q 1 2 + 1 , 0 , q 2 2 + 1 , 0 , ( q 3 + 1 , 0 ) + ∑ q 1 = 1 ∞ ∑ q 2 = 0 ∞ ∑ q 3 = 0 ∞ ∑ q 4 = 0 ∞ ∑ m = 0 ∞ ∑ r = 0 ∞ ( − ξ ) q 1 ( − 1 ) q 2 + q 3 + r w 2 r L q 3 + m [ k 2 ( α ) ] q 4 2 ( 2 ) q 1 q 1 ! q 2 ! q 3 ! q 4 ! m ! × M 6 5 ( [ k 2 ( α ) ] τ ) ∣ q 1 2 − q 2 + 1 , 0 , q 2 2 − q 3 + 1 , 0 , ( q 3 − q 4 + 1 , 0 ) , 1 2 , 0 , ( m + 1 , − 1 ) ( 1 − α q 3 + q 4 − α m + 2 r , 1 ) q 1 2 + 1 , 0 , q 2 2 + 1 , 0 , ( q 3 + 1 , 0 ) , ( m + 1 2 , 0 ) , ( m + 1 , 0 ) .

4.2 Analytical solutions with Caputo derivative

In this section, we will develop a fractional model of the physical problem given in Eq. (10), where

(26) Re C D τ α Φ ( ξ , τ ) = ∂ 2 Φ ( ξ , τ ) ∂ ξ 2 − ∂ 4 Φ ( ξ , τ ) ∂ ξ 4 .

Taking the Laplace transform on Eq. (26), we have

(27) d 4 Φ ¯ ( ξ , s ) d ξ 4 − d 2 Φ ¯ ( ξ , s ) d ξ 2 + Re s α Φ ¯ ( ξ , s ) = 0 .

The solution of Eq. (27) becomes, with the conditions given in (11)–(14),

(28) Φ ¯ ( ξ , s ) = s 2 ( s 2 + w 2 ) e − ξ 1 + 1 − 4 Re s α 2 − s 2 ( s 2 + w 2 ) 1 − 4 Re s α e − ξ 1 + 1 − 4 Re s α 2 + s 2 ( s 2 + w 2 ) e − ξ 1 − 1 − 4 Re s α 2 + s 2 ( s 2 + w 2 ) 1 − 4 Re s α e − ξ 1 − 1 − 4 Re s α 2 .

We stated it in its equivalent form

(29) Φ ¯ ( ξ , s ) = s s 2 + w 2 + ∑ k 1 = 1 ∞ ∑ k 2 = 0 ∞ ∑ k 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) k 1 ( − 4 Re ) k 3 ( − 1 ) r w 2 r Γ ( k 1 2 + 1 ) Γ ( k 2 2 + 1 ) 2 ( 2 ) k 1 k 1 ! k 2 ! k 3 ! s 1 + 2 r − α k 3 Γ ( k 1 2 + 1 − k 2 ) Γ ( k 2 2 + 1 − k 3 ) − ∑ k 1 = 1 ∞ ∑ k 2 = 0 ∞ ∑ k 3 = 0 ∞ ∑ k 4 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) k 1 ( − 1 ) k 3 + r ( 4 Re ) k 3 + k 4 w 2 r Γ ( k 1 2 + 1 ) Γ ( k 2 2 + 1 ) Γ ( k 4 + 1 2 ) 2 ( 2 ) k 1 k 1 ! k 2 ! k 3 ! k 4 ! s 1 + 2 r − α k 3 − α k 4 Γ ( k 1 2 + 1 − k 2 ) Γ ( k 2 2 + 1 − k 3 ) Γ 1 2 + ∑ l 1 = 1 ∞ ∑ l 2 = 0 ∞ ∑ l 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) l 1 ( − 1 ) l 2 + l 3 + r ( 4 Re ) l 3 w 2 r Γ l 1 2 + 1 Γ l 2 2 + 1 2 ( 2 ) l 1 l 1 ! l 2 ! l 3 ! s 1 + 2 r − α l 3 Γ l 1 2 + 1 − l 2 Γ l 2 2 + 1 − l 3 + ∑ l 1 = 1 ∞ ∑ l 2 = 0 ∞ ∑ l 3 = 0 ∞ ∑ l 4 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) l 1 ( − 1 ) l 2 + l 3 + r ( 4 Re ) l 3 + l 4 w 2 r Γ l 1 2 + 1 Γ l 2 2 + 1 Γ ( l 4 + 1 2 ) 2 ( 2 ) l 1 l 1 ! l 2 ! l 3 ! l 4 ! s 1 + 2 r − α l 3 − α l 4 Γ l 1 2 + 1 − l 2 Γ l 2 2 + 1 − l 3 Γ 1 2 .

Taking the Laplace inverse transform to Eq. (29), we have

(30) Φ ( ξ , τ ) = cosw τ + ∑ k 1 = 1 ∞ ∑ k 2 = 0 ∞ ∑ k 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) k 1 ( − 4 Re ) k 3 ( − 1 ) r w 2 r τ 2 r − α k 3 Γ ( k 1 2 + 1 ) Γ ( k 2 2 + 1 ) 2 ( 2 ) k 1 k 1 ! k 2 ! k 3 ! Γ ( 1 + 2 r − α k 3 ) Γ ( k 1 2 + 1 − k 2 ) Γ ( k 2 2 + 1 − k 3 ) − ∑ k 1 = 1 ∞ ∑ k 2 = 0 ∞ ∑ k 3 = 0 ∞ ∑ k 4 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) k 1 ( − 1 ) k 3 + r ( 4 Re ) k 3 + k 4 w 2 r τ 2 r − α k 3 − α k 4 Γ ( k 1 2 + 1 ) Γ ( k 2 2 + 1 ) Γ ( k 4 + 1 2 ) 2 ( 2 ) k 1 k 1 ! k 2 ! k 3 ! k 4 ! Γ ( 1 + 2 r − α k 3 − α k 4 ) Γ ( k 1 2 + 1 − k 2 ) Γ ( k 2 2 + 1 − k 3 ) Γ 1 2 + ∑ l 1 = 1 ∞ ∑ l 2 = 0 ∞ ∑ l 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) l 1 ( − 1 ) l 2 + l 3 + r ( 4 Re ) l 3 w 2 r τ 2 r − α l 3 Γ l 1 2 + 1 Γ l 2 2 + 1 2 ( 2 ) l 1 l 1 ! l 2 ! l 3 ! Γ ( 1 + 2 r − α l 3 ) Γ l 1 2 + 1 − l 2 Γ l 2 2 + 1 − l 3 + ∑ l 1 = 1 ∞ ∑ l 2 = 0 ∞ ∑ l 3 = 0 ∞ ∑ l 4 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) l 1 ( − 1 ) l 2 + l 3 + r ( 4 Re ) l 3 + l 4 w 2 r τ 2 r − α l 3 − α l 4 Γ l 1 2 + 1 Γ l 2 2 + 1 Γ ( l 4 + 1 2 ) 2 ( 2 ) l 1 l 1 ! l 2 ! l 3 ! l 4 ! Γ ( 1 + 2 r − α l 3 − α l 4 ) Γ l 1 2 + 1 − l 2 Γ l 2 2 + 1 − l 3 Γ 1 2 .

Eq. (30) can be stated in terms of M-function [47] as follows:

(31) Φ ( ξ , τ ) = cosw τ + ∑ k 1 = 1 ∞ ∑ k 2 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) k 1 ( − 1 ) r w 2 r 2 ( 2 ) k 1 k 1 ! k 2 ! M 3 2 ( − 4 Re τ − α ) ∣ k 1 2 − k 2 + 1 , 0 , k 2 2 + 1 , − 1 , ( 1 + 2 r , − α ) k 1 2 + 1 , 0 , k 2 2 + 1 , 0 − ∑ k 1 = 1 ∞ ∑ k 2 = 0 ∞ ∑ k 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) k 1 ( − 1 ) k 3 + r ( 4 Re ) k 3 w 2 r 2 ( 2 ) k 1 k 1 ! k 2 ! k 3 ! M 4 3 ( 4 Re τ − α ) ∣ k 1 2 − k 2 + 1 , 0 , ( k 2 2 − k 3 + 1 , 0 ) , 1 2 , 0 , ( 1 − α k 3 + 2 r , − α ) k 1 2 + 1 , 0 , k 2 2 + 1 , 0 , 1 2 , 1 + ∑ l 1 = 1 ∞ ∑ l 2 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) l 1 ( − 1 ) l 2 + r w 2 r 2 ( 2 ) l 1 l 1 ! l 2 ! M 3 2 ( − 4 Re τ − α ) ∣ l 1 2 − l 2 + 1 , 0 , l 2 2 + 1 , − 1 , ( 1 + 2 r , − α ) l 1 2 + 1 , 0 , l 2 2 + 1 , 0 + ∑ l 1 = 1 ∞ ∑ l 2 = 0 ∞ ∑ l 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) l 1 ( − 1 ) l 2 + l 3 + r ( 4 Re ) l 3 w 2 r 2 ( 2 ) l 1 l 1 ! l 2 ! l 3 ! M 4 3 ( 4 Re τ − α ) ∣ l 1 2 − l 2 + 1 , 0 , l 2 2 − l 3 + 1 , 0 , 1 2 , 0 , ( 1 − α l 3 + 2 r , − α ) l 1 2 + 1 , 0 , l 2 2 + 1 , 0 , 1 2 , 1 .

4.3 Analytical solutions with CF derivative

The solution of the initial value problem Eqs. (11)–(14) with CF fractional derivatives can be obtained via the Laplace transform method. The obtained solution for the velocity field expressed in terms of M-function

(32) Φ ( ξ , τ ) = cosw τ + ∑ m 1 = 1 ∞ ∑ m 2 = 0 ∞ ∑ m 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) m 1 ( − 4 Re ) m 3 ( − 1 ) r w 2 r 2 ( 2 ) m 1 m 1 ! m 2 ! m 3 ! ( 1 − α ) m 3 × M 4 3 α 1 − α τ ∣ m 1 2 − m 2 + 1 , 0 , m 2 2 − m 3 + 1 , 0 , ( 0 , 1 ) , ( 1 + 2 r , 1 ) m 1 2 + 1 , 0 , m 2 2 + 1 , 0 , ( m 3 , 1 ) − ∑ m 1 = 1 ∞ ∑ m 2 = 0 ∞ ∑ m 3 = 0 ∞ ∑ m 4 = 0 ∞ ∑ m 5 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) m 1 ( − 4 Re ) m 3 + m 5 ( − 1 ) r w 2 r ( α ) m 4 2 ( 2 ) m 1 m 1 ! m 2 ! m 3 ! m 4 ! m 5 ! ( 1 − α ) m 3 + m 5 × M 6 5 ( α τ ) ∣ m 1 2 − m 2 + 1 , 0 , m 2 2 − m 3 + 1 , 0 , ( m 4 , 0 ) , ( m 5 , 0 ) , ( 1 + m 4 + 2 r , 1 ) , 1 2 , 0 m 1 2 + 1 , 0 , m 2 2 + 1 , 0 , ( m 3 + m 4 , 0 ) , m 5 + 1 2 , 0 , ( m 5 , 1 ) + ∑ n 1 = 1 ∞ ∑ n 2 = 0 ∞ ∑ n 3 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) n 1 ( − 1 ) r + n 2 + n 3 w 2 r ( 4 Re ) n 3 2 ( 2 ) n 1 n 1 ! n 2 ! n 3 ! ( 1 − α ) n 3 × M 4 3 α 1 − α τ ∣ n 1 2 − n 2 + 1 , 0 , n 2 2 − n 3 + 1 , 0 , ( 0 , 1 ) , ( 1 + 2 r , 1 ) n 1 2 + 1 , 0 , n 2 2 + 1 , 0 , ( n 3 , 1 ) + ∑ n 1 = 1 ∞ ∑ n 2 = 0 ∞ ∑ n 3 = 0 ∞ ∑ n 4 = 0 ∞ ∑ n 5 = 0 ∞ ∑ r = 0 ∞ ( − ξ ) n 1 ( − 1 ) r + n 2 + n 3 w 2 r ( 4 Re ) n 3 + n 5 ( α ) n 4 2 ( 2 ) n 1 n 1 ! n 2 ! n 3 ! n 4 ! n 5 ! ( 1 − α ) n 3 + n 5 × M 6 5 ( α τ ) ∣ n 1 2 − n 2 + 1 , 0 , n 2 2 − n 3 + 1 , 0 , ( n 4 , 0 ) , ( n 5 , 0 ) , ( 1 + n 4 + 2 r , 1 ) , 1 2 , 0 n 1 2 + 1 , 0 , n 2 2 + 1 , 0 , ( n 3 + n 4 , 0 ) , n 5 + 1 2 , 0 , ( n 5 , 1 ) .

5 Graphical outcomes and arguments

Figures 1, 2, 3, 4 are designed to show the comparison between CPC, Caputo (C) and CF fractional operators for different values of α . By increasing α , velocity is decreased due to loss in momentum boundary layer. It is concluded that CPC provides better results than all other fractional derivatives. Figures 5 and 6 are designed to show the comparison between CPC fractional operator and classical derivative [15] for different values of α . It is concluded that CPC provides better results than the classical derivative. Figure 7 is designed to show the effect of time on the velocity. With the passage of time, the velocity decreases. The impact of Reynolds number on the velocity is also understood from Figure 8. It is discovered that the velocity drops for larger values of Re.

$Figure 1 A comparison of CPC with C and CF when α = 0.2 \alpha =0.2 , t = 5 t=5 , w = 0.05 w=0.05 , and Re = 0.09 \hspace{0.1em}\text{Re}\hspace{0.1em}=0.09 .$

Figure 1

A comparison of CPC with C and CF when α = 0.2 , t = 5 , w = 0.05 , and Re = 0.09 .

$Figure 2 A comparison of CPC with C and CF when α = 0.4 \alpha =0.4 , t = 5 t=5 , w = 0.05 w=0.05 , and Re = 0.09 \hspace{0.1em}\text{Re}\hspace{0.1em}=0.09 .$

Figure 2

A comparison of CPC with C and CF when α = 0.4 , t = 5 , w = 0.05 , and Re = 0.09 .

$Figure 3 A comparison of CPC with C and CF when α = 0.6 \alpha =0.6 , t = 5 t=5 , w = 0.05 w=0.05 , and Re = 0.09 \hspace{0.1em}\text{Re}\hspace{0.1em}=0.09 .$

Figure 3

A comparison of CPC with C and CF when α = 0.6 , t = 5 , w = 0.05 , and Re = 0.09 .

$Figure 4 A comparison of CPC with C and CF when α = 0.8 \alpha =0.8 , t = 5 t=5 , w = 0.05 w=0.05 , and Re = 0.09 \hspace{0.1em}\text{Re}\hspace{0.1em}=0.09 .$

Figure 4

A comparison of CPC with C and CF when α = 0.8 , t = 5 , w = 0.05 , and Re = 0.09 .

$Figure 5 A comparison of CPC with classical derivative [15] when α = 0.2 \alpha =0.2 , w = 0.05 w=0.05 , Re = 0.09 \hspace{0.1em}\text{Re}\hspace{0.1em}=0.09 , and t = 5 , 7 t=5,7 .$

Figure 5

A comparison of CPC with classical derivative [15] when α = 0.2 , w = 0.05 , Re = 0.09 , and t = 5 , 7 .

$Figure 6 A comparison of CPC with classical derivative [15] when α = 0.9 \alpha =0.9 , w = 0.05 w=0.05 , Re = 0.09 \hspace{0.1em}\text{Re}\hspace{0.1em}=0.09 , and t = 5 , 10 t=5,10 .$

Figure 6

A comparison of CPC with classical derivative [15] when α = 0.9 , w = 0.05 , Re = 0.09 , and t = 5 , 10 .

$Figure 7 The effect of t t on velocity when α = 0.5 \alpha =0.5 , w = 2 w=2 , and Re = 0.5 \hspace{0.1em}\text{Re}\hspace{0.1em}=0.5 .$

Figure 7

The effect of t on velocity when α = 0.5 , w = 2 , and Re = 0.5 .

$Figure 8 The effect of Re on velocity when t = 0.05 t=0.05 , w = 2 w=2 , and α = 0.5 \alpha =0.5 .$

Figure 8

The effect of Re on velocity when t = 0.05 , w = 2 , and α = 0.5 .

6 Conclusion

In this article, analytical solutions of couple stress non-Newtonian fluid have been explained. By using the Laplace transform method and fractional derivatives, velocity results are obtained and shown graphically. The key findings of this article are as follows:

The fractional approach provides us the choice to control the boundary layers by choosing different values of the fractional parameter.
In the limiting case when α → 1 , our results are identical to classical obtained by [15].
The results obtained with novel power law fractional derivative exhibits the flow property (velocity) decay better in comparison of comparatively (C) and (CF).
Velocity falls with the rising values of Re and t .

Acknowledgement

We extend a sense of gratitude to the University of Management and Technology, Lahore, Pakistan, for facilitating and supporting our research.

Funding information: This study did not receive any funding in any form.
Author contributions: All of the authors substantially contributed to this study.
Conflict of interest: The authors declare no conflict of interest in the submission of this article.
Data availability statement: The data used to support the findings of this study are included within the article.

References

[1] Stokes VK. Couple stresses in fluids. Phys Fluids. 1966;9:1709–15. 10.1063/1.1761925Search in Google Scholar

[2] Srinivasacharya D, Kaladhar K. Mixed convection flow of couple stress fluid in a non-Darcy porous medium with Soret and Dufour effects. J Appl Sci Eng. 2012;15:415–22. Search in Google Scholar

[3] Hayat T, Muhammad T, Alsaedi A, Alhuthali MS. Magnetohydrodynamic three-dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation. J Magn Magn Mater. 2015;385:222–9. 10.1016/j.jmmm.2015.02.046Search in Google Scholar

[4] Liu CM. Complete solutions to extended Stokes’ problems. Math Probl Eng. 2008;2008:18 pages. 10.1155/2008/754262Search in Google Scholar

[5] Adesanya SO, Falade JA, Ukaegbu JC, Lebelo RS. Thermo dynamics analysis for a reacting couple stress fluid flow through a vertical channel. Int J Pure Appl Math. 2017;114:419–34. 10.12732/ijpam.v114i3.1Search in Google Scholar

[6] Ahmed S, Bég OA, Ghosh SK. A couple stress fluid modeling on free convection oscillatory hydromagnetic flow in an inclined rotating channel. Ain Shams Eng. 2014;5(4):1249–65. 10.1016/j.asej.2014.04.006Search in Google Scholar

[7] Sha Z, Dawar A, Alzahrani EO, Kumam P, Khan AJ, Islam S. Hall effect on couple stress 3D nanofluid flow over an exponentially stretched surface with Cattaneo Christov heat flux model. IEEE Access. 2019;7:64844–55. 10.1109/ACCESS.2019.2916162Search in Google Scholar

[8] Umavathi JC, Chamkha AJ, Manjula MH, Al-Mudhaf A. Flow and heat transfer of a couple-stress fluid sandwiched between viscous fluid layers. Can J Phys. 2005;83:705–20. 10.1139/p05-032Search in Google Scholar

[9] Gopal KN, Sreenadh S, Vijaya kumar AG. Flow of a couple stess fluid through a porous layer bounded by parallel plates. Int J Eng Res Technol. 2012;1(10). Search in Google Scholar

[10] Ramzan M, Farooq M, Alsaedi A, Hayat T. MHD three-dimensional flow of couple stress fluid with Newtonian heating. Eur Phys J Plus. 2013;49:128. 10.1140/epjp/i2013-13049-5Search in Google Scholar

[11] Hayat T, Mustafa M, Iqbal Z, Alsaedi A. Stagnation-point flow of couple stress fluid with melting heat transfer. Appl Math Mech. 2013;34:167–76. 10.1007/s10483-013-1661-9Search in Google Scholar

[12] Turkyilmazoglu M. Exact solutions for two-dimensional laminar flow over a continuously stretching or shrinking sheet in an electrically conducting quiescent couple stress fluid. Int J Heat Mass Transf. 2014;72:1–8. 10.1016/j.ijheatmasstransfer.2014.01.009Search in Google Scholar

[13] Hayat T, Aziz A, Muhammad T, Ahmad B. Influence of magnetic field in three-dimensional flow of couple stress nanofluid over a nonlinearly stretching surface with convective condition. PLoS One. 2015;10(12):18. 10.1371/journal.pone.0145332Search in Google Scholar PubMed PubMed Central

[14] Hayat T, Sajjad R, Alsaedi A, Muhammad T, Ellahi R. On squeezed flow of couple stress nanofluid between two parallel plates. Results Phys. 2017;7(20):553–61. 10.1016/j.rinp.2016.12.038Search in Google Scholar

[15] Devakar M, Iyengar TKV. Stokes’ problems for an incompressible couple stress fluid. Nonlinear Anal Modell Control. 2008;1(2):181–90. 10.15388/NA.2008.13.2.14578Search in Google Scholar

[16] Baleanu D, Diethelm K, Scalas E, Trujillo JJ. Fractional calculus: models and numerical methods: 3 (Series on complexity, nonlinearity, and chaos). Singapore: World Scientific Publishing Company; 2012. 10.1142/8180Search in Google Scholar

[17] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: Elsevier Science; 2006. p. 469–520. Search in Google Scholar

[18] Atangana A, Botha JF. A generalized ground water flow equation using the concept of variable order derivative. Boundary Value Problem. 2013;1:53–60. 10.1186/1687-2770-2013-53Search in Google Scholar

[19] Atangana A and Baleanu D. New fractional derivatives with nonlocal and non-singular kernel, theory and application to heat transfer model. Therm Sci. 2016;20:763–9. 10.2298/TSCI160111018ASearch in Google Scholar

[20] Atangana A. Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals. 2017;102:396–406. 10.1016/j.chaos.2017.04.027Search in Google Scholar

[21] Fernandez A, Özarslan MA, Baleanu D. On fractional calculus with general analytic kernels. Appl Math Comput. 2019;354:248–65. 10.1016/j.amc.2019.02.045Search in Google Scholar

[22] Tahir M, Imran MA, Raza N, Abdullah M, Aleem 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

[23] Saqib M, Ali F, Khan I, Sheikh NA, Jan SA. Exact solutions for free convection flow of generalized Jeffrey fluid: a Caputo-Fabrizio fractional model. Alexandr Eng J. 2017;57:1849–58. 10.1016/j.aej.2017.03.017Search in Google Scholar

[24] Imran MA, Khan I, Ahmad M, Shah NA, Nazar 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

[25] Jarad F, Abdeljawad T, Hammouch Z. On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative. Chaos Solitons Fractals. 2018;117:16–20. 10.1016/j.chaos.2018.10.006Search in Google Scholar

[26] Abro KA, Khan I. Effects of carbon nanotubes on magnetohydrodynamic flow of methanol based nanofluids via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. Thermal Sci. 2019;23(2):883–93. 10.2298/TSCI180116165ASearch in Google Scholar

[27] Danish Ikram M, Imran Asjad M, Ahmadian A, Ferrara M. A new fractional mathematical model of extraction nanofluids using clay nanoparticles for different based fluids. Math Meth Appl Sci. 2020:1–14. 10.1002/mma.6568Search in Google Scholar

[28] Baleanu D, Fernandez A, Akgül A. On a fractional operator combining proportional and classical differintegrals. Mathematics 2020;8(3):360. Search in Google Scholar

[29] Imran MA, Ikram MD, Rizwan A, Baleanu D, Alshomrani A. New analytical solutions of heat transfer flow of clay-water base nanoparticles with the application of novel hybrid fractional derivative. Thermal Sci. 2020;24(1):343–50. 10.2298/TSCI20S1343ASearch in Google Scholar

[30] Ikram MD, Asjad MI, Akgül A, Baleanu D. Effects of hybrid nanofluid on novel fractional model of heat transfer flow between two parallel plates. Alexandr Eng J. 2021;60:3593–604. 10.1016/j.aej.2021.01.054Search in Google Scholar

[31] Chu YM, Ikram MD, Asjad MI, Ahmadian A, Ghaemi F. Influence of hybrid nanofluids and heat generation on coupled heat and mass transfer flow of a viscous fluid with novel fractional derivative. J Thermal Anal Calorim. 2021;144(6):2057–77.10.1007/s10973-021-10692-8Search in Google Scholar

[32] Asadullah M, Raza A, IIIkram MD, Asjad MI, Naz R, Inc M. Mathematical fractional modeling of transpot phenomena of viscous fluid-flow between two plates. Thermal Sci. 2021;25(2):417–21. 10.2298/TSCI21S2417ASearch in Google Scholar

[33] Hamid M, Usman M, Haq RU, Tian Z. A spectral approach to analyze the nonlinear oscillatory fractional-order differential equations. Chaos Solitons Fractals. 2021;146:110921. 10.1016/j.chaos.2021.110921Search in Google Scholar

[34] Hamid M, Usman M, Haq RU, Tian Z, Wang W. Linearized stable spectral method to analyze two-dimensional nonlinear evolutionary and reaction-diffusion models. Numer Meth Partial Differ Equ. 2022;38(2):243–61. 10.1002/num.22659Search in Google Scholar

[35] Hamid M, Usman M, Wang W, Tian Z. A stable computational approach to analyze semi-relativistic behavior of fractional evolutionary problems. Numer Meth Partial Differ Equ. 2022;38(2):122–36. 10.1002/num.22617Search in Google Scholar

[36] Hamid M, Usman M, Zubair T, Haq RU, Wang W. Innovative operational matrices based computational scheme for fractional diffusion problems with the Riesz derivative. Eur Phys J Plus 2019;134:1–7. 10.1140/epjp/i2019-12871-ySearch in Google Scholar

[37] Hamid M, Usman M, Wang W, Tian Z. Hybrid fully spectral linearized scheme for time-fractional evolutionary equations. Math Meth Appl Sci. 2021;44(5):3890–912. 10.1002/mma.6996Search in Google Scholar

[38] Hamid M, Zubair T, Usman M, Haq RU. Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel. AIMS Math. 2019;4(5):1416–29. 10.3934/math.2019.5.1416Search in Google Scholar

[39] Li K, Peng J. Laplace transform and fractional differential equations. Appl Math Lett. 2011;24(12):2019–23. 10.1016/j.aml.2011.05.035Search in Google Scholar

[40] Rubbab Q, Nazeer M, Ahmad F, Chu YM, Khan MI, Kadry S. Numerical simulation of advection-diffusion equation with Caputo–Fabrizio time fractional derivative in cylindrical domains: applications of pseudo-spectral collocation method. AEJ - Alexandr Eng J. 2022;60(1):1731–8. 10.1016/j.aej.2020.11.022Search in Google Scholar

[41] Guo B, Raza A, Al-Khaled K, Khan SU, Farid S, Wang Y, et al. Fractional-order simulations for heat and mass transfer analysis confined by elliptic inclined plate with slip effects: a comparative fractional analysis. Case Stud Therm Eng. 2021;28:101359. 10.1016/j.csite.2021.101359Search in Google Scholar

[42] Raza A, Ghaffari A, Khan SU, Haq AU, Khan MI, Khan MR. Non-singular fractional computations for the radiative heat and mass transfer phenomenon subject to mixed convection and slip boundary effects. Chaos Solitons Fractals. 2022;155:111708. 10.1016/j.chaos.2021.111708Search in Google Scholar

[43] Raza A, Khan SU, Khan MI, Farid S, Muhammad T, Khan MI, et al. Fractional order simulations for the thermal determination of graphene oxide (GO) and molybdenum disulphide (MoS2) nanoparticles with slip effects. Case Stud Therm Eng. 2021;28:101453. 10.1016/j.csite.2021.101453Search in Google Scholar

[44] Song YQ, Raza A, Al-Khaled K, Farid S, Khan MI, Khan SU, et al. Significances of exponential heating and Darcy’s law for second grade fluid flow over oscillating plate by using Atangana-Baleanu fractional derivatives. Case Stud Therm Eng. 2021;27(6):101266. Search in Google Scholar

[45] Raza A, Song Y-Q, Kamel AK, Farid S, Khan M, Sami UKh et al. Prabhakar fractional model is alternative newly proposed analytical approach without the appearance of singularity. This model grants modifications of both the AB and CF models. Case Stud Therm Eng. 2021;27:101266.10.1016/j.csite.2021.101266Search in Google Scholar

[46] Baleanu D, Fernandez A, Akgul A. On a fractional operator combing proportional and classical differintegral. Mathematics. 2020;8:360.10.3390/math8030360Search in Google Scholar

[47] Haubold HJ, Mathai AM, Saxena RK. Mittag-Leffler functions and their applications. J Appl Math. 2011;2011. 10.1155/2011/298628Search in Google Scholar

Received: 2022-04-21

Revised: 2022-12-30

Accepted: 2023-02-08

Published Online: 2023-05-03

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0281

Keywords for this article

constant proportional Caputo; couple stress fluids; Stokes’ second problem; analytical solutions

Creative Commons

BY 4.0