COUETTE FLOWS OF A VISCOUS FLUID WITH SLIP EFFECTS AND NON-INTEGER ORDER DERIVATIVE WITHOUT SINGULAR KERNEL

. Couette ﬂows of an incompressible viscous ﬂuid with non-integer order derivative without singular kernel produced by the motion of a ﬂat plate are analyzed under the slip condition at boundaries. An analytical transform approach is used to obtain the exact expressions for velocity and shear stress. Three particular cases from the general results with and without slip at the wall are obtained. These solutions, which are organized in simple forms in terms of exponential and trigonometric functions, can be conveniently engaged to obtain known solutions from the literature. The control of the new non-integer order derivative on the velocity of the ﬂuid moreover a comparative study with an older model, is analyzed for some ﬂows with practical applications. The non-integer order derivative with non-singular kernel is more appropriate for handling mathematical calculations of obtained solutions.

1. Introduction. The nonslip boundary condition is one of the primary principles that provides basis for linearly viscous fluid's mechanics to built on. For a large class of flows, most of the experiments function favourably for the nonslip boundary condition. interesting debates concerning the acceptance of on nonslip condition are traced out. [7]. The success rate for nonslip condition is nonstatic: in case of the great variety of flows, it has proved its efficacy while it falls short of its performance in case of problems involving flows in micro channels or in wavy tubes, multiple interfaces, flows of polymeric liquids or flows of rarefid fluids.
A few years back, Navier [15] suggested a slip boundary condition where relative velocity that is also known as the slip velocity relies linearly on the shear stress. For describing the slip that happens at solid boundaries, a great number of models have been suggested. A list of these modals is found in the reference [17]. Monney's study is a pioneer work done on the slip at the boundary [14]. Many researchers published their work undertaken to discuss the flows of Newtonian or non-Newtonian fluids with slip at the boundary. Khalid and Vafai [10] undertook the research on the effects of the slip condition on Stoke's and Coutte flows because of an oscillating wall, Stoke's flows of a Maxwell fluid with wall slip condition is studied by Vieru and Rauf [20], Abelman et al [1] analysed the Coutte flow of a third grade fluid with rotating frame and slip condition.
A frequent discussion on Fractional calculus is found just as the standard differential and integral calculus are the most studied topics and a list of the application of fractional calculus is huge to be included here. Hence, It is signifacnt to note the fact that fractional derivative generalizations of one dimensional viscoelastic models is of huge usage in modeling the response linear regime [9] and they are in agreement with the second principle of thermodynamics.Furthermore, a suitable agreement of experimental work was attained as an outcome of the work of Makris et al. [13], by using the non-integer order Maxwell model instead of an ordinary one. The aforementioned researchers substantiated that the applied method i.e. fractional method has got a stronger memory of the past than the other method called, ordinary method. It is noticed that the fractional calculus has been frequently used during the last a few years. The researchers solved so many motion problems by using this method [11].
Most frequently, by restoring the integer order time derivatives through formal left hand Liouville or Riemann-Liouville differential operators, the governing equation analogous to motions of ordinary fluid models are adapted [12,16,5]. Nonetheless, a few deficiencies are found in both of these operators as well as the Caputo operator. Their essential core is singular and the nearly all results that have been attained through these methods are displayed in a complex way, encompassing some generalized functions even for Newtonian fluids [18,8].
A contemporary definition of non integer order derivative is given by Caputo and Fabrizio [4] recently with smooth kernel so as to perform a function both for spatial and temporal variables. Ensuing, this derivative has already being applied for the solution of real problems because of its benefit when the Laplace transform is used to do problems with initial conditions [2,6]. Atangana and Baleanu proposed a latest fractional operator whose kernel is also non-singular [3]. It has its foundations liked to the Mittag-Leffer function and is helpful in material and thermal sciences. It is significant to mention that both fractional derivatives namely, Atangana-Baleanu and Caputo-Fabrizio, have all the advantages which are provided by of Riemann-Liouville and Caputo operators. Additionally, their kernel is non-singular as well.
The main objective is to rely on modern definition of non-integer order derivative to attain precise common solutions for an incompressible viscous fluid's Couette flows without singular kernel created by the movement of infinite plate, are analyzed under the slip condition at boundaries; whereas the bottom plate is supposed to be translated in its plane with a given velocity. The flow of the fluid is studied under the assumption that the relative velocity between the fluid at the wall and the wall itself is proportional to the shear rate at the wall. An integral transform determines the accurate expressions for velocity and shear stress. Hence, the velocity fields for viscous fluid analogous to both slip and non slip conditions are derived. The gained results are displayed in the comprehendible way including both trigonometric and exponential functions and can be expediently used to retrieve the connected solutions for ordinary fluids. Finally the results for viscous fluids are viewed in comparison Newtonian fluids under both, slip and non slip conditions. Some properties of the flow are also presented.
2. Statement of the problem. As shown in the Fig. 1 consider Oxyz Cartesian coordinate system with y > 0 in the upward direction and an infinite solid plane wall situated in the (x, z)-plane. The second infinite solid plane wall occupies the plane y = h > 0. An incompressible viscous fluid fills the slab y ∈ (0, h). The fluid and plates are at rest at t = 0. At the moment t = 0 + , the fluid is set into motion due to translation of the bottom plate in the x-axis direction moving with the velocity where τ (y, t) = S xy (y, t) is one of the non zero component of S and ρ is the constant density of the fluid. We assume the presence of slip at the walls and consider that the relative velocity between the velocity of the fluid at the wall and wall is proportional to the shear rate at the wall. The appropriate initial and boundary conditions under the assumption of slip at the wall are; where β is the slip coefficient. We introduce the following dimensionless quantities where T > 0 is a characteristic time, in the above model for the non-dimensionalization of the physical parameters. The non-dimensional flow equations after dropping the asterisks are ∂u(y, t) ∂t and where R = h 2 νT is the Reynolds number and g(t * ) = f (T t * ). Applying Caputo-Fabrizio time Fractional Derivative we get the governing euation Here, unlike the previous published papers, the Caputo-Fabrizio derivative operator of order α [4, 5, 6] will be used. We firstly solve the fractional differential (13) with the initial and boundary conditions (IBC's), and use the obtained results to develop the solution corresponding to the shear stress.
3. Solution of the problem.

3.1.
Velocity field. Applying the Laplace transform to (13) and using the initial conditions from (9) we obtain, the following set of equations For constant C 1 and C 2 use boundary conditions after using Laplace transform and where Laplce transform of g(t) is G(q). Using these values in (16), we get where In order to obtain the inverse Laplace transform of function F 1 (y, q), we consider the auxiliary function whose singular points are simple poles located at where p n = 0 are the real roots of the equation By using the residue theorem and after some simplifications we get, the Laplace inverse transform of F 1 (y, q) as follows Res[F 1 (y, q)e qt , q n ], where A n (y) = 2Rp n α βp n cosp n + sinp n cosp n y β 2 p 2 n − 2β − 1 cosp n + 2βp n β + 1 sinp n R + p 2 n (1 − α) and B n (y) = −2Rp n α cosp n − βp n sinp n sinp n y we can also write in the following equivalent form where C n = βp n cos(1 − y)p n + sin(1 − y)p n β 2 p 2 n − 2β − 1 cosp n + 2βp n β + 1 sinp n • For flows of generlized Newtonian fluids with a no slip boundary condition, that is β = 0, the transform domain solution is • For flows of viscous Newtonian fluids with a no slip boundary condition, that is α → 1 and β = 0, the transform domain solution is which is similar to [19], Eq. (37) • The relative velocity between the fluid at the bottom wall and wall itself for fractional order Couette flow fluids and for a viscous Newtonian fluid is given by for α → 1 3.2. Shear stress. As we have expression for shear stress using the value of velocity from (23), we have the final expression as follow 4. Numerical results and discussion. With a view to get a little insight of the results that have been obtained, three special cases with engineering applications are considered along with different graphical representations and are presented and discussed, as well as for comparison, the graphical representations are prepared only for the velocity fields corresponding to the three kinds of motions.

4.1.
Translation of the plate with a constant velocity. The motion of the bottom plate is given by the function g(t) = H(t) and the velocity u(y, t) is obtained from (23) and (24) with g(t − τ ) = 1 for s ∈ (0, t). The velocity field corresponding to this type of the motion has the following expressions: • For fractional order Newtonian fluid with slip at the boundary: • For generalized Newtonian fluids with no slip at the boundary: • The relative velocity for fraction order is given by 4.2. Translation of the plate with a constant acceleration. The motion of the bottom plate is given by the function g(t) = H(t)t and the velocity u(y, t) is obtained from (23) and (24) with g(t − τ ) = t − τ for s ∈ (0, t). The velocity field corresponding to this type of the motion has the following expressions: • Solution with slip at the boundary: (33) • Solution with no slip at the boundary: • The relative velocity is given by • The Couette flow of a generalized viscous fluid with no slip boundary condition • The relative velocity is given by