A Time Fractional Model With Non-Singular Kernel the Generalized Couette Flow of Couple Stress Nanofluid

AbstractFractal-fractional derivative is a new class of fractional derivative with power Law kernel which has many applications in real world problems. This operator is used for the first time in such kind of fluid flow. The big advantage of this operator is that one can formulate models describing much better the systems with memory effects. Furthermore, in real world there are many problems where it is necessary to know that how much information the system carries. To explain the memory in a system fractal-fractional derivatives with power law kernel is analyzed in the present work. Keeping these motivation in mind in the present paper new concept of fractal-fractional derivative for the modeling of couple stress fluid (CSF) with the combined effect of heat and mass transfer have been used. The magnetohydrodynamics (MHD) flow of CSF is taken in channel with porous media in the presence of external pressure. The constant motion of the left plate generates the CSF motion while the right plate is kept stationary. The non-dimensional fractal-fractional model of couple stress fluid in Riemann–Liouville sense with power law is solved numerically by using the implicit finite difference method. The obtained solutions for the present problem have been shown through graphs. The effects of various parameters are shown through graphs on velocity, temperature and concentration fields. The velocity, temperature and concentration profiles of the MHD CSF in channel with porous media decreases for the greater values of both fractional parameter $$\alpha$$ α and fractal parameter $$\beta$$ β respectively. From the graphical results it can be noticed that the fractal-fractional solutions are more general as compared to classical and fractional solutions of CSF motion in channel. Furthermore, the fractal-fractional model of CSF explains good memory effect on the dynamics of couple stress fluid in channel as compared to fractional model of CSF. Finally, the skin friction, Nusselt number and Sherwood number are evaluated and presented in tabular form.

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Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles

Symmetry ◽

10.3390/sym12040663 ◽

2020 ◽

Vol 12 (4) ◽

pp. 663 ◽

Cited By ~ 4

Author(s):

Muhammad Saqib ◽

Sharidan Shafie ◽

Ilyas Khan ◽

Yu-Ming Chu ◽

Kottakkaran Sooppy Nisar

Keyword(s):

Mhd Flow ◽

Singular Kernel ◽

Hybrid Nanoparticles ◽

Convection Flow ◽

Hybrid Nanofluid ◽

Fractional Model ◽

Laplace Transform Technique ◽

Local Solutions ◽

Mhd Channel ◽

Fractional Parameter

A nonlocal fractional model of Brinkman type fluid (BTF) containing a hybrid nanostructure was examined. The magnetohydrodynamic (MHD) flow of the hybrid nanofluid was studied using the fractional calculus approach. Hybridized silver (Ag) and Titanium dioxide (TiO2) nanoparticles were dissolved in base fluid water (H2O) to form a hybrid nanofluid. The MHD free convection flow of the nanofluid (Ag-TiO2-H2O) was considered in a microchannel (flow with a bounded domain). The BTF model was generalized using a nonlocal Caputo-Fabrizio fractional operator (CFFO) without a singular kernel of order α with effective thermophysical properties. The governing equations of the model were subjected to physical initial and boundary conditions. The exact solutions for the nonlocal fractional model without a singular kernel were developed via the fractional Laplace transform technique. The fractional solutions were reduced to local solutions by limiting α → 1 . To understand the rheological behavior of the fluid, the obtained solutions were numerically computed and plotted on various graphs. Finally, the influence of pertinent parameters was physically studied. It was found that the solutions were general, reliable, realistic and fixable. For the fractional parameter, the velocity and temperature profiles showed a decreasing trend for a constant time. By setting the values of the fractional parameter, excellent agreement between the theoretical and experimental results could be attained.

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Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0086 ◽

2019 ◽

Vol 22 (6) ◽

pp. 1675-1688 ◽

Cited By ~ 3

Author(s):

Pavan Pranjivan Mehta ◽

Guofei Pang ◽

Fangying Song ◽

George Em Karniadakis

Keyword(s):

Neural Network ◽

Couette Flow ◽

Turbulent Flows ◽

Fractional Order ◽

Scaling Law ◽

Variable Order ◽

Reynolds Stresses ◽

Mean Velocity ◽

Fractional Model ◽

Better Than

Abstract The first fractional model for Reynolds stresses in wall-bounded turbulent flows was proposed by Wen Chen [2]. Here, we extend this formulation by allowing the fractional order α(y) of the model to vary with the distance from the wall (y) for turbulent Couette flow. Using available direct numerical simulation (DNS) data, we formulate an inverse problem for α(y) and design a physics-informed neural network (PINN) to obtain the fractional order. Surprisingly, we found a universal scaling law for α(y+), where y+ is the non-dimensional distance from the wall in wall units. Therefore, we obtain a variable-order fractional model that can be used at any Reynolds number to predict the mean velocity profile and Reynolds stresses with accuracy better than 1%.

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MHD Flow and Heat Transfer in Sodium Alginate Fluid with Thermal Radiation and Porosity Effects: Fractional Model of Atangana–Baleanu Derivative of Non-Local and Non-Singular Kernel

Symmetry ◽

10.3390/sym11101295 ◽

2019 ◽

Vol 11 (10) ◽

pp. 1295 ◽

Cited By ~ 3

Author(s):

Arshad Khan ◽

Dolat Khan ◽

Ilyas Khan ◽

Muhammad Taj ◽

Imran Ullah ◽

...

Keyword(s):

Heat Transfer ◽

Temperature Profile ◽

Sodium Alginate ◽

Mhd Flow ◽

Casson Fluid ◽

Singular Kernel ◽

Heat Transfer Analysis ◽

Fractional Model ◽

The Laplace Transform ◽

Non Local

Heat transfer analysis in an unsteady magnetohydrodynamic (MHD) flow of generalized Casson fluid over a vertical plate is analyzed. The medium is porous, accepting Darcy’s resistance. The plate is oscillating in its plane with a cosine type of oscillation. Sodium alginate (SA–NaAlg) is taken as a specific example of Casson fluid. The fractional model of SA–NaAlg fluid using the Atangana–Baleanu fractional derivative (ABFD) of the non-local and non-singular kernel has been examined. The ABFD definition was based on the Mittag–Leffler function, and promises an improved description of the dynamics of the system with the memory effects. Exact solutions in the case of ABFD are obtained via the Laplace transform and compared graphically. The influence of embedded parameters on the velocity field is sketched and discussed. A comparison of the Atangana–Baleanu fractional model with an ordinary model is made. It is observed that the velocity and temperature profile for the Atangana–Baleanu fractional model are less than that of the ordinary model. The Atangana–Baleanu fractional model reduced the velocity profile up to 45.76% and temperature profile up to 13.74% compared to an ordinary model.

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Comparative study of generalized couette flow of couple stress fluid using optimal homotopy asymptotic method and new iterative method

Scientific Reports ◽

10.1038/s41598-021-82746-8 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Muhammad Farooq ◽

Alamgeer Khan ◽

Rashid Nawaz ◽

Saeed Islam ◽

Muhammad Ayaz ◽

...

Keyword(s):

Differential Equations ◽

Iterative Method ◽

Couette Flow ◽

Asymptotic Method ◽

Couple Stress ◽

Research Work ◽

Couple Stress Fluid ◽

Parallel Plates ◽

Optimal Homotopy Asymptotic Method ◽

New Iterative Method

AbstractIn this research work, we have studied the steady generalized Couette flow of couple stress fluid between two parallel plates considering the non-isothermal effects. The governing equations that are, continuity, momentum and energy equations are reduced to ordinary differential equations. The Optimal Homotopy Asymptotic Method (OHAM) and New Iterative Method (NIM) are used to solve this coupled system of differential equations. Using the said methods, we have obtained expressions for velocity profile, temperature distribution, volume flux, average velocity and shear stress. The results of OHAM and NIM are compared numerically as well as graphically and a tremendous agreement is attained.

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Time Fractional Analysis of Channel Flow of Couple Stress Casson Fluid using Fick’s and Fourier’s Laws

10.21203/rs.3.rs-701466/v1 ◽

2021 ◽

Author(s):

Shafiq Ahmad ◽

Sami Ul Haq ◽

Farhad Ali ◽

Ilyas Khan ◽

Kottakkaran Sooppy Nisar

Keyword(s):

Channel Flow ◽

Couple Stress ◽

Integral Transformation ◽

Physical Phenomenon ◽

Classical Model ◽

Fourier Integral ◽

Casson Fluid ◽

Physical Parameters ◽

Fractional Model ◽

Stress Parameter

Abstract This study aim to examine the channel flow of a couple stress Casson fluid. The flow is generated due to the motion of the plate at y = o, while the plate at y = d is at rest. This physical phenomenon is derived in terms of partial differential equations. The subjected governing PDE’s are non-dimensionalized with the help of dimensionless variables. The dimensionless classical model is generalized by transforming it to the time fractional model using Fick’s and Fourier’s Laws. The general fractional model is solved by applying the Laplace and Fourier integral transformation. Furthermore, the parametric influence of various physical parameters like Casson parameter, couple stress parameter, Grashof number, Schmidt number and Prandtl number on velocity, temperature, and concentration distributions is shown graphically and discussed. The heat transfer rate, skin friction, and Sherwood number are calculated and presented in tabular form. It is worth noting that the increasing values of the couple stress parameter λ deaccelerate the velocity of Couple stress Casson fluid.

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