Exact solutions to vorticity of the fractional nonuniform Poiseuille flows

Nehad Ali Shah; Dumitru Vieru; Constantin Fetecau; Shalan Alkarni

doi:10.1515/phys-2024-0006

Article Open Access

Exact solutions to vorticity of the fractional nonuniform Poiseuille flows

Nehad Ali Shah , Dumitru Vieru , Constantin Fetecau and Shalan Alkarni

Published/Copyright: April 17, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

Closed-form expressions for the dimensionless velocity, shear stresses, and the flow vorticity fields corresponding to the isothermal unsteady Poiseuille flows of a fractional incompressible viscous fluid over an infinite flat plate are established. The fluid motion induced by a pressure gradient in the flow direction is also influenced by the flat plate that oscillates in its plane. The vorticity field is dependent on two spatial coordinate and time, and it is an arbitrary trigonometric polynomial in the horizontal coordinate. The exact solutions, obtained by generalized separation of variables and Laplace transform technique, are presented in terms of the Wright function and complementary error function of Gauss. Their advantage consists in the fact that the values of the fractional parameter can be chosen so that the predicted material properties by them to be in agreement with the corresponding experimental results. In addition, they describe motions for which the nontrivial shear stresses are influenced by history of the shear rates. It is found that the flow vorticity is stronger near the plate, but it could be attenuated in the case of fractional model.

Keywords: fractional viscous fluid; unsteady Poiseuille flow; flow vorticity; exact solutions

1 Introduction

Generally, the exact solutions of the initial-boundary value problems describe the behavior of a fluid in motion or a solid in deformation. They can also be used as tests to verify numerical schemes that are developed to study more complex flow or deformation problems. During the time such solutions for unsteady motions of the incompressible viscous fluids have been established by Berker [1], Schlichting [2], Wang [3], Polyanin [4], Baranovskii et al. [5], and Burmasheva and Prosviryakov [6]. In the last paper, the authors provide exact expressions for velocity, vorticity, and tangential stresses corresponding to the nonuniform Poiseuille flow of an incompressible viscous fluid. A characteristic feature of these expressions is polynomial dependence on one of the horizontal coordinates, which varies from zero to infinity. Their polynomial coefficients depend on the vertical coordinate and time. Consequently, the presented solutions can become endless for large values of horizontal coordinate.

The fractional models have been used in many domains including physics, chemistry, quantum mechanics, viscoelasticity, etc. Bagley and Torvik [7] were the first to apply fractional derivatives in viscoelasticity. A very good agreement with the experimental data has been obtained by Mainardi and Spada [8] using fractional derivatives to describe the viscoelastic behavior of different materials.

Fractional dynamics studies complex processes described by differential equations with derivatives of noninteger order. The equations that describe the fractional mathematical models are not a simple extension of differential equations with derivatives of integer order to noninteger order. These models can provide new insights into a large number of fundamental results and into the description of some new types of physical processes and systems such as the discrete maps with memory that are equivalent to the fractional differential equations of kicked motions, media for which the equations of motion with long-range interaction are mapped into equations of continuous medium with the fractional derivatives, the fractional dynamics of open quantum systems that interacts with its environment, and fractional models of fractal media dynamics. In the fractional dynamics, integral and derivatives of fractional orders are used to describe processes with a power-law nonlocality or a power-law long-term memory [9]. Another advantage of fractional models consists in the fact that the value of the fractional parameter can be chosen so that the predicted material properties to be in accord with the experimental results.

In this article, the boundary layer flow of a viscous fluid over a plate situated in the plane z = 0 is studied. The flow is generated by a time-dependent pressure gradient in the flow direction. The condition on the lower boundary of the fluid is the no-slip condition on the flow velocity. A generalized mathematical model is proposed by considering the fluid characterized by the fractional constitutive equations. In this model, the shear stresses are influenced not only by the velocity gradients but also by their history. If the fractional order of the fractional derivative tending to zero, the mathematical model of the ordinary fluid is recovered.

2 Formulation of the problem

Let us consider the isothermal unsteady nonuniform motion of an incompressible viscous fluid in an infinite horizontal layer. In the absence of body forces, this motion is governed by the following differential equations [10]:

(1) ( ∇ , v ) = 0 , ρ ∂ v ∂ t + ( v , ∇ ) v = − ∇ p + div τ .

Here, v = ( u , v , w ) is the velocity vector, ρ is the fluid density, p the hydrostatic pressure, ∇ is the gradient operator, and the parenthesis defines the scalar product. The extra-stress tensor τ corresponding to such fluids is given by the relation:

(2) τ = μ ( L + L T ) ,

where L = ∇ v and μ is the dynamic viscosity of the fluid.

In the following, we shall study unidirectional fluid motions whose velocity field, reported to a suitable Cartesian coordinate system x, y, and z, is

(3) v = u ( x , y , z , t ) e x ,

where e x is the unit vector along the x-direction (Figure 1).

Figure 1

Flow geometry.

Substituting v from the equality (3) in the constitutive Eq. (2) one finds that the components τ y y , τ y z , and τ z z of τ are zero, while τ x x , τ x y , and τ x z are given by the relations:

(4) τ x x = 2 μ ∂ u ∂ x , τ x y = μ ∂ u ∂ y , τ x z = μ ∂ u ∂ z .

The substitution of v from Eq. (3) into Eq. (1) leads to the following relations:

(5) ∂ u ∂ x = 0 , ρ ∂ u ∂ t = − ∂ p ∂ x + ∂ τ x x ∂ x + ∂ τ x y ∂ y + ∂ τ x z ∂ z ,

(6) ∂ p ∂ y = ∂ τ x y ∂ x , ∂ p ∂ z = ∂ τ x z ∂ x .

The equalities (4)₁ and (5)₁ clearly imply that τ x x = 0 .

These last relations clearly imply that the unidirectional fluid motion is nonuniform and the velocity and pressure fields have the next forms

(7) u = u ( y , z , t ) , p = x a ( t ) + b ( t ) .

Consequently, the fluid motion is characterized by the partial differential equation:

(8) ρ ∂ u ( y , z , t ) ∂ t = − a ( t ) + ∂ τ x y ( y , z , t ) ∂ y + ∂ τ x z ( y , z , t ) ∂ z ,

where

(9) τ x y ( y , z , t ) = μ ∂ u ( y , z , t ) ∂ y and τ x z ( y , z , t ) = μ ∂ u ( y , z , t ) ∂ z .

Introducing the following nondimensional variables and functions

(10) ( x ∗ , y ∗ , z ∗ ) = 1 L ( x , y , z ) , t ∗ = t t 0 , u ∗ = t 0 L u , ( a ∗ , τ x y ∗ , τ x z ∗ ) = t 0 μ ( a L , τ x y , τ x z ) ,

and dropping out the star notation, one obtains the dimensionless forms

(11) Re ∂ u ( y , z , t ) ∂ t = − a ( t ) + ∂ τ x y ( y , z , t ) ∂ y + ∂ τ x z ( y , z , t ) ∂ z ,

(12) τ x y ( y , z , t ) = ∂ u ( y , z , t ) ∂ y , τ x z ( y , z , t ) = ∂ u ( y , z , t ) ∂ z ,

of the governing Eqs. (8) and (9). In Eq. (10), L and t 0 are characteristic length and time, respectively, while

(13) Re = L V / ν ( V = L / t 0 being a characteristic velocity)

is the Reynolds number and ν = μ / ρ denotes the kinematic viscosity of the fluid. Along with the equalities (11) and (12), the initial condition

(14) u ( y , z , 0 ) = 0

has to be satisfied.

3 Fractional model and its solution

In the following, we consider the mathematical model described by the next nondimensional fractional constitutive equations:

(15) τ x y ( y , z , t ) = D t α c ∂ u ( y , z , t ) ∂ y , τ x z ( y , z , t ) = D t α c ∂ u ( y , z , t ) ∂ z ; 0 ≤ α < 1 ,

where

(16) D t α c f ( y , z , t ) = h c ( t , α ) ∗ ∂ f ( y , z , t ) ∂ t = ∫ 0 t h c ( t − τ , α ) ∂ f ( y , z , τ ) ∂ τ d τ ; 0 ≤ α < 1

is the time-fractional Caputo derivative [11], ∗ designates the convolution product and

(17) h c ( t , α ) = t − α Γ ( 1 − α ) ; t > 0 , 0 ≤ α < 1 ,

is its kernel.

The fractional constitutive Eq. (15) is a generalization of classical constitutive Eq. (12). The fractional mathematical model, which is based on these equations, describes fluid motions for which the nontrivial shear stresses τ x y and τ x z are influenced by the history of the shear rates. From Eq. (16), it results that for α = 0 ,

(18) D t 0 c f ( y , z , t ) = 1 ∗ ∂ f ( y , z , t ) ∂ t = f ( y , z , t ) − f ( y , z , 0 ) .

Consequently, if f ( y , z , 0 ) = 0 , we obtain

(19) D t 0 c f ( y , z , t ) = f ( y , z , t ) .

On the basis of definition (16) and the properties of the Laplace transform [12,13]:

(20) L { f ( y , z , t ) } = f ¯ ( y , z , s ) = ∫ 0 ∞ f ( y , z , t ) e − s t d t ,

where s is the transform parameter, and we obtian

(21) L { D t α c f ( y , z , t ) } = L { h c ( t , α } ⋅ L ∂ f ( y , z , t ) ∂ t = 1 s 1 − α [ s L { f ( y , z , t ) } − f ( y , z , 0 ) ] = s α f ¯ ( y , z , s ) − s α − 1 f ( y , z , 0 ) .

If f ( y , z , 0 ) = 0 , the last equality becomes

(22) L { D t α c f ( y , z , t ) } = s α f ¯ ( y , z , s ) .

3.1 Determination of the dimensionless velocity field

In this section, we will determine an analytical solution of the problem characterized by Eqs. (11), (14) and (15). Replacing τ x y ( y , z , t ) and τ x z ( y , z , t ) from Eq. (15) in (11), one obtains the following fractional partial differential equation:

(23) Re ∂ u ( y , z , t ) ∂ t = D t α c ∂ 2 u ( y , z , t ) ∂ y 2 + D t α c ∂ 2 u ( y , z , t ) ∂ z 2 − a ( t ) ,

for the dimensionless velocity field u ( y , z , t ) , which has to satisfy the initial condition:

(24) u ( y , z , 0 ) = 0 .

Regarding the proposed mathematical model, the following discussion is useful.

In continuum mechanics, Newton’s second law is a fundamental principle. The particular properties of a material are expressed by the constitutive relations. Therefore, the new model considered defines a material that is characterized by the new constitutive Eq. (15). Obviously, if the fractional parameter tending to zero, the shear stresses of the fractional material tend to the stress values of the ordinary case. Eq. (23) is not an “artificial” fractionalization of Newton’s law. Eq. (23) is a consequence of Newton’s law expressed by Eq. (11) and of the generalized constitutive Eq. (15).

For the equality (23), we are looking for a solution of the form

(25) u ( y , z , t ) = u 0 ( z , t ) + ∑ n = 1 m u n ( z , t ) sin ( n y ) .

Substituting u ( y , z , t ) from Eq. (25) into Eq. (23), one obtains the system

(26) Re ∂ u 0 ( z , t ) ∂ t = D t α c ∂ 2 u 0 ( z , t ) ∂ z 2 − a ( t ) , Re ∂ u 1 ( z , t ) ∂ t = − D t α c u 1 ( z , t ) + D t α c ∂ 2 u 1 ( z , t ) ∂ z 2 , Re ∂ u 2 ( z , t ) ∂ t = − 2 2 D t α c u 2 ( z , t ) + D t α c ∂ 2 u 2 ( z , t ) ∂ z 2 , ..................................................................... . Re ∂ u m ( z , t ) ∂ t = − m 2 D t α c u m ( z , t ) + D t α c ∂ 2 u m ( z , t ) ∂ z 2 ,

of m + 1 fractional differential equations with m + 1 unknown functions u 0 , u 1 , u 2 , .. . , u m . The first equation is a non-homogeneous fractional diffusion equation, while the next ones are homogeneous fractional diffusion equations. From Eqs. (24) and (25), it also result that the functions u k ( z , t ) with k = 0 , 1 , 2 , . . . , m have to satisfy the initial conditions:

(27) u k ( z , 0 ) = 0 for k = 0 , 1 , 2 , .. . , m .

To determine these functions, we also impose the next boundary conditions:

(28) u k ( 0 , t ) = sin ( ω t ) , ω > 0 , k = 0 , 1 , 2 , .. . , m ; lim z → ∞ u 0 ( z , t ) < ∞ , lim z → ∞ u n ( z , t ) = 0 , n = 1 , 2 , .. . , m .

The system (26) can be written in the equivalent form:

(29) Re ∂ u 0 ( z , t ) ∂ t = D t α c ∂ 2 u 0 ( z , t ) ∂ z 2 − a ( t ) , Re ∂ u n ( z , t ) ∂ t = − n 2 D t α c u n ( z , t ) + D t α c ∂ 2 u n ( z , t ) ∂ z 2 , n = 1 , 2 , 3 , .. . , m .

By applying the Laplace transform to the equalities (29) and using the initial conditions (27) and the relation (22), one obtains the transformed equations:

(30) s Re u ¯ 0 ( z , s ) = s α ∂ 2 u ¯ 0 ( z , s ) ∂ z 2 − a ¯ ( s ) ,

(31) ( s Re + n 2 s α ) u ¯ n ( z , s ) = s α ∂ 2 u ¯ n ( z , s ) ∂ z 2 ; n = 1 , 2 , .. . , m ,

(32) u ¯ k ( 0 , s ) = ω s 2 + ω 2 , k = 0 , 1 , 2 , .. . , m ; lim z → ∞ u ¯ 0 ( z , s ) < ∞ , lim z → ∞ u ¯ n ( z , s ) = 0 , n = 1 , 2 , .. . , m ,

where u ¯ k ( z , s ) and a ¯ ( s ) are the Laplace transforms of u k ( z , t ) and a ( t ) , respectively. The function u ¯ 0 ( z , s ) has to be bounded at infinity.

The general solutions of the ordinary differential Eqs. (30) and (31) are as follows:

(33) u ¯ 0 ( z , s ) = C 01 ( s ) exp ( − z Re s 1 − α ) + C 02 ( s ) exp ( z Re s 1 − α ) − a ¯ ( s ) s Re ,

(34) u ¯ n ( z , s ) = C n 1 ( s ) exp ( − z Re s 1 − α + n 2 ) + C n 2 ( s ) exp ( z Re s 1 − α + n 2 ) ; n = 1 , 2 , 3 , .. . , m ,

where C k 1 ( ⋅ ) and C k 2 ( ⋅ ) with k = 0 , 1 , 2 , .. . , m are unknown functions.

By using the boundary conditions (32), one obtains

(35) C 01 ( s ) = ω s 2 + ω 2 + a ¯ ( s ) s Re ; C n 1 ( s ) = ω s 2 + ω 2 for n = 1 , 2 , 3 , .. . , m

and C k 2 ( s ) = 0 for k = 0 , 1 , 2 , . . . , m . Consequently, the solutions of the system of differential Eqs. (30) and (31) with the boundary conditions (32) are given by the following relations:

(36) u ¯ 0 ( z , s ) = ω s 2 + ω 2 + a ¯ ( s ) s Re exp ( − z Re s 1 − α ) − a ¯ ( s ) s Re ,

(37) u ¯ n ( z , s ) = ω s 2 + ω 2 exp ( − z Re s 1 − α + n 2 ) ; n = 1 , 2 , 3 , .. . , m .

To determine the inverse Laplace transforms of the functions u ¯ k ( z , s ) from the previous relations, we shall use the auxiliary function:

(38) F ¯ ( z , s , α ) = exp ( − z Re s 1 − α ) = 1 for z = 0 , 0 ≤ α < 1 ; exp ( − z Re s ) for z > 0 , α = 0 ; exp ( − z Re s 1 − α ) for z > 0 , 0 < α < 1 ,

and the inversion formula (A1) and (A2) from Appendix. Consequently, by applying the inverse Laplace transform at the equality (38), it results that

(39) F ( z , t , α ) = L − 1 { F ¯ ( z , s , α ) } = δ ( t ) for z = 0 and 0 ≤ α < 1 ; z Re 2 t π t exp − Re z 2 4 t for z > 0 , α = 0 ; t − 1 Φ ( 0 , α − 1 2 , − z Re t α − 1 ) for z > 0 , 0 < α < 1 ,

where δ ( ⋅ ) is the Dirac’s distribution.

On the other hand,

(40) L − 1 ω s 2 + ω 2 + a ¯ ( s ) s Re = sin ( ω t ) + 1 Re ∫ 0 t a ( τ ) d τ = h ( t ) ,

and the inverse Laplace transform of u ¯ 0 ( z , s ) is given by the next relation

(41) u 0 ( z , t ) = sin ( ω t ) for z = 0 , 0 ≤ α < 1 ; h ( t ) ∗ z Re 2 t π t exp − Re z 2 4 t − 1 Re ∫ 0 t a ( τ ) d τ for z > 0 , α = 0 ; h ( t ) ∗ t − 1 Φ 0 , α − 1 2 , − z Re t α − 1 − 1 Re ∫ 0 t a ( τ ) d τ for z > 0 , 0 < α < 1 .

To determine the inverse Laplace transforms of u ¯ n ( z , s ) with n = 1 , 2 , 3 , .. . , m , we write these functions in equivalent forms, namely,

(42) u ¯ n ( z , s ) = ω ( s 1 − α + a n ) s 2 + ω 2 1 s 1 − α + a n exp [ − z Re s 1 − α + a n ] , n = 1 , 2 , 3 , .. . , m

and use the following functions:

(43) g ¯ 1 n ( s , α ) = ω ( s 1 − α + a n ) s 2 + ω 2 = a n ω s 2 + ω 2 + ω s s 2 + ω 2 for α = 0 ; a n ω s 2 + ω 2 + ω s α s s 2 + ω 2 for 0 < α < 1

and

(44) G ¯ n ( z , s , α ) = 1 s 1 − α + a n exp [ − z Re s 1 − α + a n ] ,

where a n = n 2 / Re with n = 1 , 2 , 3 , .. . , m . The Laplace transform of g ¯ 1 n ( s ) is

(45) g 1 n ( t ) = a n sin ( ω t ) + ω cos ( ω t ) if α = 0 ; a n sin ( ω t ) + ω Γ ( α ) ∫ 0 t ( t − τ ) α − 1 cos ( ω τ ) d τ if 0 < α < 1 .

The functions G ¯ n ( z , s , α ) can be presented as compose functions, i.e.,

(46) G ¯ n ( z , s , α ) = g ¯ ( w ¯ n ( s ) ) ,

where the functions g ¯ ( s ) and w ¯ n ( s ) are defined by the relations:

(47) g ¯ ( s ) = 1 s exp [ − z Re s ] and w ¯ n ( s ) = s 1 − α + a n .

By using the inverse Laplace transforms of composite functions, we obtain

(48) G n ( z , t , α ) = L − 1 { G ¯ n ( z , s , α ) } = L − 1 { g ¯ ( w ¯ n ( s ) ) } = ∫ 0 ∞ g ( ξ ) h n ( ξ , t , α ) d ξ ,

where, according to the equality (A3)

(49) g ( t ) = erfc z Re 2 t ,

and (see again Eq. (A2) from Appendix)

(50) h n ( ξ , t , α ) = L − 1 { exp [ − ξ w ¯ n ( s ) ] } = exp ( − ξ a n ) δ ( t − ξ ) if α = 0 ; exp ( − ξ a n ) t Φ ( 0 , α − 1 , − ξ t α − 1 ) if 0 < α < 1 .

Finally, using the property (A3) from Appendix, it results that

(51) G n ( z , t , α ) = ∫ 0 ∞ e − ξ a n erfc z Re 2 ξ δ ( t − ξ ) d ξ if α = 0 1 t ∫ 0 ∞ e − ξ a n erfc z Re 2 ξ Φ ( 0 , α − 1 , − ξ t α − 1 ) d ξ if 0 < α < 1 = e − a n t erfc z Re 2 t if α = 0 ; 1 t ∫ 0 ∞ e − ξ a n erfc z Re 2 ξ Φ ( 0 , α − 1 , − ξ t α − 1 ) d ξ if 0 < α < 1 .

Consequently, bearing in mind Eqs. (39), (41), (42), and (47), we find that

(52) u n ( z , t ) = [ a n sin ( ω t ) + ω cos ( ω t ) ] ∗ e − a n t erfc z Re 2 t if α = 0 ; a n sin ( ω t ) + ω Γ ( α ) ∫ 0 t ( t − τ ) α − 1 cos ( ω τ ) d τ ∗ 1 t ∫ 0 ∞ e − ξ a n erfc z Re 2 ξ Φ ( 0 , α − 1 , − ξ t α − 1 ) d ξ if 0 < α < 1 .

The solutions corresponding to classical incompressible viscous fluids performing the same motion are given by the relations:

(53) u 0 c ( z , t ) = z Re 2 π ∫ 0 t h ( t − τ ) τ τ exp − Re z 2 4 τ d τ − 1 Re ∫ 0 t a ( τ ) d τ ,

(54) u n c ( z , t ) = ∫ 0 t { a n sin [ ω ( t − τ ) ] + ω cos [ ω ( t − τ ) ] } e − a n τ erfc z Re 2 τ d τ .

Direct computations clearly show that the initial and boundary conditions (27) and (28), respectively, are satisfied.

3.2 Determination of the dimensionless shear stresses

By applying the Laplace transform to Eq. (15) and bearing in mind the initial condition (24), one obtains

(55) τ ¯ x y ( y , z , s ) = s α ∂ u ¯ ( y , z , s ) ∂ y , τ ¯ x z ( y , z , s ) = s α ∂ u ¯ ( y , z , s ) ∂ z ,

where (see Eqs. (25), (36) and (37))

(56) u ¯ ( y , z , s ) = u ¯ 0 ( z , s ) + ∑ n = 1 m u ¯ n ( z , s ) sin ( n y ) = ω s 2 + ω 2 + a ¯ ( s ) s Re exp [ − z Re s 1 − α ] − a ¯ ( s ) s Re + ∑ n = 1 m ω sin ( n y ) s 2 + ω 2 × exp [ − z Re s 1 − α + n 2 ] .

Consequently, the transformed shear stresses τ ¯ x y ( y , z , s ) and τ ¯ x z ( y , z , s ) are given by the relations

(57) τ ¯ x y ( y , z , s ) = ∑ n = 1 m n ω s cos ( n y ) s 2 + ω 2 exp ( − z Re s 1 − α + n 2 ) s 1 − α ,

(58) τ ¯ x z ( y , z , s ) = − Re ω s s 2 + ω 2 + a ¯ ( s ) Re exp ( − z Re s 1 − α ) s 1 − α − ω Re ∑ n = 1 m s s 2 + ω 2 + a n s 2 − ( 2 − α ) s 2 + ω 2 exp ( − z Re s 1 − α + a n ) s 1 − α + a n .

To determine the inverse Laplace transform of the equality (57), we write τ ¯ x y ( y , z , s ) in the suitable form

(59) τ ¯ x y ( y , z , s ) = ∑ n = 1 m n ω T ¯ 1 ( s ) G ¯ n ( z , s , α ) cos ( n y ) ,

where G ¯ n ( z , s , α ) is given by the equality (44) and

(60) T ¯ 1 ( s ) = s s 2 + ω 2 + a n s 2 − ( 2 − α ) s 2 + ω 2 .

By applying the inverse Laplace transform to Eq. (59) and using the identity (A4) from Appendix, it results that

(61) τ x y ( y , z , t ) = ω ∑ n = 1 m n cos ( n y ) T 1 ( t ) ∗ G n ( z , t , α ) ,

where

(62) T 1 ( t ) = cos ( ω t ) + a n t 1 − α E 2 , 2 − α ( − ω 2 t 2 )

is the inverse Laplace transform of T ¯ 1 ( s ) and E α , β ( ⋅ ) is the two parameters Mittag–Leffler function [14]. In the case of classical incompressible viscous fluids when α = 0

(63) T 1 ( t ) = cos ( ω t ) + a n t E 2 , 2 ( − ω 2 t 2 ) = cos ( ω t ) + a n sin ( ω t ) ω ,

because E 2 , 2 ( − z 2 ) = sin ( z ) z .

To determine the inverse Laplace transform of τ ¯ x z ( y , z , s ) given by Eq. (58), we consider the auxiliary functions

(64) T ¯ 2 ( z , s ) = 1 s + c exp [ − z Re s + c ] and T ¯ 3 ( z , s ) = T ¯ 2 ( z , w ¯ 1 ( s ) ) ,

where w ¯ 1 ( s ) = s 1 − α . Using Eq. (A5) from Appendix, it results that

(65) T 2 ( z , t ) = L − 1 { T ¯ 2 ( z , s ) } = 1 π t exp − Re z 2 4 t − c t .

On the other hand, the inverse Laplace transform of the compound function T ¯ 3 ( z , s ) = T ¯ 2 ( z , w ¯ 1 ( s ) ) is

(66) T 3 ( z , t ) = L − 1 { T ¯ 2 ( z , w ¯ 1 ( s ) ) } = L − 1 1 s 1 − α + c exp [ − z Re s 1 − α + c ] = ∫ 0 ∞ 1 π u g 3 ( u , t ) exp − Re z 2 4 u − c u d u ,

where

(67) g 3 ( u , t ) = L − 1 { exp [ − u w ¯ 1 ( s ) ] } = δ ( t − u ) if α = 0 ; 1 t Φ ( 0 , α − 1 , − u t α − 1 ) if 0 < α < 1 .

Finally, using these last results and the equality (58), it results that

(68) τ x z ( y , z , t ) = − Re ω cos ( ω t ) + a ( t ) Re ∗ τ 1 ( z , t ) − ω Re ∑ n = 1 m [ cos ( ω t ) + a n t 1 − α E 2 , 2 − α ( − ω 2 t 2 ) ] ∗ τ 2 ( z , t ) ,

where τ 1 ( z , t ) and τ 2 ( z , t ) are given by the relations:

(69) τ 1 ( z , t ) = L − 1 1 s 1 − α exp ( − z Re s 1 − α ) = ∫ 0 ∞ 1 π u g 3 ( u , t ) exp − Re z 2 4 u d u = 1 π t exp − Re z 2 4 t if α = 0 ; 1 t ∫ 0 ∞ 1 π u Φ ( 0 , α − 1 , − u t α − 1 ) exp − Re z 2 4 u d u if 0 < α < 1 ,

(70) τ 2 ( z , t ) = L − 1 1 s 1 − α + a n exp [ − z Re s 1 − α + a n ] = ∫ 0 ∞ g 3 ( u , t ) π u exp − Re z 2 4 u + a n u d u = 1 π t exp − Re z 2 4 t + a n t if α = 0 ; 1 t ∫ 0 ∞ 1 π u Φ ( 0 , α − 1 , − u t α − 1 ) exp − Re z 2 4 t + a n u d u if 0 < α < 1 .

3.3 Determination of the flow vorticity

The flow vorticity for the motion problem that has been previously studied is given by the relation

(71) Ω = rot v = e x e y e z ∂ ∂ x ∂ ∂ y ∂ ∂ z u ( y , z , t ) 0 0 = ∂ u ( y , z , t ) ∂ z e y − ∂ u ( y , z , t ) ∂ y e z = ∂ u 0 ( z , t ) ∂ z + ∑ n = 1 m ∂ u n ( z , t ) ∂ z sin ( n y ) e y − ∑ n = 1 m n u n ( z , t ) cos ( n y ) e z .

From Eq. (36), it results that

(72) ∂ u ¯ 0 ( z , s ) ∂ z = − Re s 1 − α ω s 2 + ω 2 + a ¯ ( s ) s Re exp [ − z Re s 1 − α ] = − Re s α ω s s 2 + ω 2 + a ¯ ( s ) Re exp [ − z Re s 1 − α ] s 1 − α .

By applying the inverse Laplace transform to Eq. (72) and bearing in mind the relation (69), it results that

(73) ∂ u 0 ( z , t ) ∂ z = − Re ω cos ( ω t ) + a ( t ) Re ∗ 1 π t exp − z 2 Re 4 t if α = 0 ; ω cos ( ω t ) + a ( t ) Re ∗ t α − 1 Γ ( α ) ∗ τ 1 ( z , t ) if 0 < α < 1 .

On the other hand, from Eq. (37), it results that

(74) ∂ u ¯ n ( z , s ) ∂ z = − Re ω s 2 + ω 2 s 1 − α + a n exp [ − z Re s 1 − α + a n ] = − Re ω s 2 − ( 1 + α ) s 2 + ω 2 + a n ω s 2 + ω 2 exp [ − z Re s 1 − α + a n ] s 1 − α + a n .

Now, bearing in mind Eq. (70), it results that

(75) ∂ u n ( z , t ) ∂ z = − Re [ ω t 2 E 2 , 1 + α ( − ω 2 t 2 ) + a n sin ( ω t ) ] ∗ τ 2 ( z , t ) .

Substituting the expressions of u n ( z , t ) , ∂ u 0 ( z , t ) / ∂ z , and ∂ u n ( z , t ) / ∂ z from Eqs. (37), (73), and (75), respectively, in Eq. (71), the vorticity corresponding to the motion in discussion is obtained. The flow vorticity corresponding to the same motion of classical incompressible viscous fluids Ω c ( y , z , t ) can be directly obtained using the expressions of u 0 c ( z , t ) and u n c ( z , t ) from Eqs. (53) and (54), respectively.

4 Numerical results and discussions

The aim of this section is to investigate the flow vorticity in the particular case of the pressure gradient in the flow direction given by the function a ( t ) = 1 − exp ( − t ) and for ω = π / 4 . The vorticity expression (71) enables us to state that the unidirectional flow given by the velocity field (25) is vortex flow, except for several points where the vorticity vector could be zero. This property is highlighted by graphs presented in Figures 2–8.

$Figure 2 The influence of the memory parameter α \alpha on the y-component of the vorticity vector Ω ( y , z , t ) {\boldsymbol{\Omega }}(y,z,t) for three values of the spatial coordinate y.$

Figure 2

The influence of the memory parameter α on the y-component of the vorticity vector Ω ( y , z , t ) for three values of the spatial coordinate y.

$Figure 3 The influence of the memory parameter α \alpha on the y-component of the vorticity vector Ω ( y , z , t ) {\boldsymbol{\Omega }}(y,z,t) for three values of the spatial coordinate z.$

Figure 3

The influence of the memory parameter α on the y-component of the vorticity vector Ω ( y , z , t ) for three values of the spatial coordinate z.

$Figure 4 The influence of the memory parameter α \alpha on the z-component of the vorticity vector Ω ( y , z , t ) {\boldsymbol{\Omega }}(y,z,t) for three values of the spatial coordinate y.$

Figure 4

The influence of the memory parameter α on the z-component of the vorticity vector Ω ( y , z , t ) for three values of the spatial coordinate y.

$Figure 5 The influence of the memory parameter α \alpha on the z-component of the vorticity vector Ω ( y , z , t ) {\boldsymbol{\Omega }}(y,z,t) for three values of the spatial coordinate z.$

Figure 5

The influence of the memory parameter α on the z-component of the vorticity vector Ω ( y , z , t ) for three values of the spatial coordinate z.

$Figure 6 Spatial variation of y-component of the vorticity vector Ω ( y , z , t ) {\boldsymbol{\Omega }}(y,z,t) ( y , z ) ∈ [ 0 , 1.5 ] × [ 0 , 0.5 ] (y,z)\in {[}0,1.5]\times {[}0,0.5] .$

Figure 6

Spatial variation of y-component of the vorticity vector Ω ( y , z , t ) ( y , z ) ∈ [ 0 , 1.5 ] × [ 0 , 0.5 ] .

$Figure 7 Spatial variation of z-component of the vorticity vector Ω ( y , z , t ) {\boldsymbol{\Omega }}(y,z,t) ( y , z ) ∈ [ 0 , 1.5 ] × [ 0 , 0.5 ] (y,z)\in {[}0,1.5]\times {[}0,0.5] .$

Figure 7

Spatial variation of z-component of the vorticity vector Ω ( y , z , t ) ( y , z ) ∈ [ 0 , 1.5 ] × [ 0 , 0.5 ] .

$Figure 8 Profiles of the modulus of the vorticity vector for different values of the fractional parameter and for Re = 6.$

Figure 8

Profiles of the modulus of the vorticity vector for different values of the fractional parameter and for Re = 6.

The variation of y-component of the vorticity with spatial coordinate z in the vertical planes y = 0.5 , y = 1 , and y = 1.5 , at the instant t = 0.1 is presented in Figure 2. It is observed in Figure 2 that near the plan z = 0, the absolute value of the y-component of vorticity is higher for the fluid without shear rate memory (the ordinary fluid), while the presence of shear rate memory leads to decreasing of the flow vorticity. For z > 0.4, the values of the vorticity are decreasing and the difference between vorticity’s values of fractional and ordinary fluid is insignificant.

Figure 3 shows the variation with the spatial variable y of the y-component of the vorticity in plans z = 0.15, z = 0.5, and z = 0.8. Large variations of the vorticity are in the vicinity of the plate z = 0. If in the transverse direction, the memory of the shear rate led to the decrease of the vorticity, as shown in Figure 1, in the longitudinal direction an opposite behavior appears, namely, the memory of the shear rate leads to the increase in absolute values of vorticity.

The spatial variation of the z-component of vorticity is shown in Figures 4 and 5. A first important observation is that, in the considered case, the values of z-component of the vorticity are lower compared to the values of the y-component; therefore, the rotations of the fluid particles around the directions parallel to the z-axis are much slower than the rotations around the parallel axes with the y-axis. It is important to note that if in the vicinity of the plate z = 0, the vorticity variations are significant; far from the plate, the vorticity values stabilize and tend to remain constant for each value of time t. Also, it can be seen in the previously figures that there are points in which one or both components of the vorticity are zero.

Figures 6 and 7 highlight the spatial variation of the vorticity components. The intersection of the surfaces representing the components of the vorticity vector with the z = 0 plan were also highlighted. It is clearly seen that in the area close to the plate z = 0, the values of the vorticity components have significant variations. These variations tend to a constant value at large distances from the plate.

The variation of the modulus of the vorticity vector for different values of the fractional parameter is shown in Figure 8. It can be seen that the intensity of the vorticity vector is greater in the area close of the horizontal plane z = 0. Let’s note that near this plane the vorticity is higher for the ordinary case corresponding to the zero value of the fractional parameter. In the positions located further from this plane, the vorticity increases with the increase of the fractional parameter.

5 Conclusions

The isothermal unsteady Poiseuille flow of a fractional incompressible viscous fluid over an infinite flat plate has been analytically studied using the Laplace transform technique.

Exact solutions have been established for the dimensionless velocity field u ( y , z , t ) , the corresponding shear stresses τ x y ( y , z , t ) , τ x z ( y , z , t ) , and the flow vorticity Ω ( y , z , t ) .

These solutions, unlike those of Burmasheva and Prosviryakov [6], which can become endless for large values of the horizontal coordinate y, are bounded or tend to zero at infinity.

In addition, the advantage of fractional models consists in the fact that the value of the fractional parameter α can be chosen so that the material properties predicted by such a model to be in agreement with the experimental data.

Acknowledgments

This project was supported by Researchers Supporting Project number (RSPD2024R909), King Saud University, Riyadh, Saudi Arabia.

Funding information: This project was supported by Researchers Supporting Project number (RSPD2024R909), King Saud University, Riyadh, Saudi Arabia.
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.

Appendix

(A1) L − 1 { e − b s } = b 2 t π t exp − b 2 4 t if Re ( b 2 ) > 0 ,

where Re(z) is the real part of that which follows

(A2) L − 1 1 s β e − b s σ = t β − 1 Φ ( β , − σ , − b t − σ ) if β ≥ 0 , b > 0 , 0 < σ < 1 ,

where Φ ( β , − σ , z ) = ∑ n = 0 ∞ z n n ! Γ ( β − n σ ) is the Wright function [15].

(A3) L − 1 1 s e − b s = erfc b 2 t , Re ( b ) ≥ 0 ,

(A4) L − 1 s α − β s α − b = t β − 1 E α , β ( b t α ) , Re ( α ) > 0 , Re ( β ) > 0 ,

(A5) L − 1 1 s exp ( − b s ) = 1 π t exp − b 2 4 t , Re ( b 2 ) ≥ 0 .

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Received: 2023-12-13

Revised: 2024-02-24

Accepted: 2024-03-06

Published Online: 2024-04-17

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

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0006

Keywords for this article

fractional viscous fluid; unsteady Poiseuille flow; flow vorticity; exact solutions

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