We develop a non-local mathematical model in the current study that uses the time-fractional Fourier’s law to describe the thermal phenomena. The time-fractional generalized Atangana–Baleanu fractional derivative is used to define the new time-fractional constitutive equations. We establish analytical solutions for generalized convection flows of viscous fluid between two parallel plates with general boundary conditions by combining the Laplace transform with the finite sine-Fourier transform. The impacts of fractional and physical factors on the thermal and velocity fields were examined using numerical calculations and graphical representations prepared with the Mathcad program. From the generalized Atangana–Baleanu fractional derivative, we can recover the Atangana–Baleanu fractional derivative, Caputo–Fabrizio fractional derivative, and Caputo fractional derivative as a special case.

The application of nanoparticles in the base fluids strongly influences the presentation of cooling as well as heating techniques. The nanoparticles improve thermal conductivity by fluctuating the heat characteristics in the base fluid. The expertise of nanoparticles in increasing heat transference has captivated several investigators to more evaluate the working fluid. This study disputes the investigation of convection flow for magnetohydrodynamics second-grade nanofluid with an infinite upright heated flat plate. The fractional model is obtained through Fourier law by exploiting Prabhakar fractional approach along with graphene oxide ( GO ) ({\rm{GO}}) and molybdenum disulfide ( Mo S 2 ) ({\rm{Mo}}{{\rm{S}}}_{2}) nanoparticles and engine oil is considered as the base fluid. The equations are solved analytically via the Laplace approach. The temperature and momentum profiles show the dual behavior of the fractional parameters ( α , β , γ ) (\alpha ,\beta ,\gamma ) at different times. The velocity increases as Grashof number {\rm{Grashof\; number}} increases and declines for greater values of magnetic parameter and Prandtl number. In the comparison of different numerical methods, the curves are overlapped, signifying that our attained results are authentic. The numerical investigation of governed profiles comparison shows that our obtained results in percentages of 0.2 0.2 ≤ temperature ≤ 4.36 4.36 and velocity 0.48 ≤ 7.53 0.48\le 7.53 are better than those of Basit et al. The development in temperature and momentum profile, due to engine oil–GO is more progressive, than engine oil–MoS₂.

A methodology is developed to describe time-dependent phenomena associated with nonlocal transport in complex, two-dimensional geometries. It is an extension of the ‘‘iterative method” introduced previously to solve steady-state transport problems [Maggs and Morales, Phys. Rev. E 99, 013307 (2019)], and it is based on the ‘‘jumping particle” concepts associated with the continuous-time random walk (CTRW) model. The method presented explicitly evaluates the time integral contained in the CTRW master equation. A modified version of the Mittag-Leffler function is used for the waiting-time probability distributions to incorporate memory effects. Calculations of the propagation of ‘‘anomalous transport waves” in various systems, with and without memory, illustrate the technique.

This study describes a very efficient and fast numerical solution method for the non-steady free convection flow with radiation of a viscous fluid between two infinite vertical parallel walls. The method of lines (MOL) is used together with the Runge-Kutta ODE Matlab solver to investigate this problem numerically. The presence of radiation adds more stiffness and numerical complexity to the problem. A complete derivation in dimensionless form of the governing equations for momentum and energy is also included. A constant heat flux condition is applied at the left wall and a transient numerical solution is obtained for different values of the radiation parameter (R). The results are presented for dimensionless velocity, dimensionless temperature and Nusselt number for different values of the Prandtl number (Pr), Grashof number (Gr), and the radiation parameter (R). As expected, the results show that the convection heat transfer is high when the Nusselt number is high and the radiation parameter is low. It is also shown that the solution method used is simple and efficient and could be easily adopted to solve more complex problems.

With efficient thermal activities, nanofluids have a novel role in different scientific engineering fields due to their unique and numerous applications. For example, these fluids can be used in magnetic resonance imaging (MRI), magnetic refrigeration (MR), cancer treatment (hyperthermia), and drug delivery. By inspiriting these applications and the significance of electrically conducting nanofluids, in this article, we have studied blood-based nanofluid containing carbon nanotubes (CNTs). The Brinkman type nanofluid modal is established in terms of an efficient mathematical fractional technique namely Prabhakar fractional derivative with ramped temperature and sinusoidal oscillations conditions and for the generalized solutions of temperature and velocity profile, Laplace transformation scheme is utilized. For the heat transfer of nanofluids, the Prabhakar fractional derivative which is based on generalized Fourier’s law of thermal flux is determined. The physical behavior of different parameters is examined by graphical illustrations. As a result, we have concluded that the velocity profile is a bit higher for multi-walled carbon nanotubes (MWCNTs) as compared to single-walled carbon nanotubes (SWCNTs). Furthermore, velocity and temperature fields represent decaying behavior by varying the values of fractional parameters.

In this article, free convection flow of an Oldroyd-B fluid (OBF) through a vertical rectangular channel in the presence of heat generation or absorption subject to generalized boundary conditions is studied. The fractionalized mathematical model is established by Caputo time-fractional derivative through mechanical laws (generalized shear stress constitutive equation and generalized Fourier’s law). Closed form solutions for the velocity and temperature profiles are obtained via Laplace coupled with sine-Fourier transforms and have been embedded with regards to the special functions, namely, the generalized G-functions of Lorenzo and Hartley. Solutions of the known results from recently published work (Nehad et al. Chin. J. Phy., 65, (2020) 367–376) are recovered as limiting cases. Finally, the effects of fractional and various physical parameters are graphically underlined. Furthermore, a comparison between Oldroyd-B, Maxwell and viscous fluids (fractional and ordinary) is depicted. It is found that, for short time, ordinary fluids have greater velocity as compared to the fractional fluids.

The present study provides the heat transfer analysis of a viscous fluid in the presence of bioconvection with a Caputo fractional derivative. The unsteady governing equations are solved by Laplace after using a dimensional analysis approach subject to the given constraints on the boundary. The impact of physical parameters can be seen through a graphical illustration. It is observed that the maximum decline in bioconvection and velocity can be attained for smaller values of the fractional parameter. The fractional approach can be very helpful in controlling the boundary layers of the fluid properties for different values of time. Additionally, it is observed that the model obtained with generalized constitutive laws predicts better memory than the model obtained with artificial replacement. Further, these results are compared with the existing literature to verify the validity of the present results.

In a rectangular region, the multilayered laminar unsteady flow and temperature distribution of the immiscible Maxwell fractional fluids by two parallel moving walls are studied. The flow of the fluid occurs in the presence of Robin’s boundaries and linear fluid-fluid interface conditions due to the motion of the parallel walls on its planes and the time-dependent pressure gradient. The problem is defined as a mathematical model which focuses on the fluid memory, which is represented by a constituent equation with the Caputo time-fractional derivative. The integral transformations approach (the Laplace transform and the finite sine-Fourier transform) is used to determine analytical solutions for velocity, shear stress, and the temperature fields with fluid interface, initial, and boundary conditions. For semianalytical solutions, the algorithms of Talbot are used to calculate the Laplace inverse transformation. We used the Mathcad software for graphical illustration and numerical computation. It has been observed that the memory effect is significant on both fluid motion and temperature flow.

Thermography is a fully noninvasive technique that discerns the thermal profiles of highly viable rheological parameters in heat and mass transference. In this paper, the free convection flow of viscous fluid among two vertical and parallel plates in the existence of a transverse magnetic field is investigated. The Caputo time-fractional derivative is manipulated for introducing a thermal transport equation along with a weak memory. The analytical and closed-form fractional solution for the temperature and velocity profiles are obtained through Laplace paired in conjunction with the finite Sine-Fourier transforms technique. The solution to the classical model is concluded as a special case for the solutions to the fractional modeled problem when the memory factor (the order of fractional derivative) approaches 1. Also, the solutions are stated in connection with the Mittag–Leffler function. The influences of variations of fractional and material parameters are depicted through MathCad15.

This paper studies a new class of fractional differential inclusions involving two Caputo fractional derivatives of different orders and a Riemann–Liouville type integral nonlinearity, supplemented with a combination of fixed and nonlocal (dual) anti-periodic boundary conditions. The existence results for the given problem are obtained for convex and non-convex cases of the multi-valued map by applying the standard tools of the fixed point theory. Examples illustrating the obtained results are presented.

The unsteady magnetohydrodynamic flow of viscoelastic fluids through a parallel plate microchannel under the combined influence of magnetic, electro-osmotic, and pressure gradient forcings is investigated. The fractional Oldroyd-B fluid is used for the constitutive equation to simulate the viscoelastic behavior of fluid in the microchannel. Considering the important role of slip boundary condition in microfluidics, the Navier slip model at wall is adopted. The Laplace and Fourier cosine transforms are performed to derive the analytical expression of velocity distribution. Then, by employing the finite difference method, the numerical solution of the velocity distribution is given. In order to verify the validity of our numerical approach, numerical solutions and analytical solutions of the velocity distribution are contrasted with the exact solutions of the Newtonian fluid in previous work, and the agreements are excellent. Furthermore, based on the values of the velocity distribution for the fully developed flow, the energy equation including volumetric Joule heating, electromagnetic couple effect, and energy dissipation is solved to give the temperature distribution in the microchannel by using the finite difference method. Finally, the influence of fractional parameters and pertinent system parameters on the fluid flow and heat transfer performance and the dependence of the dimensionless Nusselt number Nu on the Hartmann number Ha and Brinkman number Br are discussed graphically.

In this paper, we study the existence of solutions for nonlocal single and multi-valued boundary value problems involving right-Caputo and left-Riemann–Liouville fractional derivatives of different orders and right-left Riemann–Liouville fractional integrals. The existence of solutions for the single-valued case relies on Sadovskii’s fixed point theorem. The first existence results for the multi-valued case are proved by applying Bohnenblust-Karlin’s fixed point theorem, while the second one is based on Martelli’s fixed point theorem. We also demonstrate the applications of the obtained results.

In this paper, we study a coupled system of Caputo-Hadamard type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. The sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are obtained. We make use of the Leray-Schauder alternative and contraction mapping principle to derive the desired results. Illustrative examples for the main results are also presented.

In this article, the influence of a magnetic field is studied on a generalized viscous fluid model with double convection, due to simultaneous effects of heat and mass transfer induced by temperature and concentration gradients. The fluid is considered over an exponentially accelerated vertical plate with time-dependent boundary conditions. Additional effects of heat generation and chemical reaction are also considered. A generalized viscous fluid model consists of three partial differential equations of momentum, heat, and mass transfer with corresponding initial and boundary condition. The idea of non-integer order Caputo time-fractional derivatives is used, and exact solutions for velocity, temperature, and concentration in terms of Wright function and function of Lorenzo–Hartley are developed for ordinary cases. Graphical analysis of flow and fractional parameters is made by using computational software MathCad, and discussed. The results obtained are also in good agreement with the published results from the literature. As a result, it is found that temperature and fluid velocity can be enhanced for smaller values of fractional parameters.

In the present study, the oscillatory flow of Maxwell fluid in a long tube with a rectangular cross section is considered. The analytical expressions for velocity profile and phase difference are obtained, and particularly, the singularities of the exact solution are discussed. Furthermore, the convenient expressions of velocity and phase difference are given explicitly for calculations. The effects of the relaxation time and Deborah number on the velocity profile and phase difference are discussed numerically and graphically.

Natural convection on a convectively heated vertical wall, one of the fundamental issues of heat and mass transfer in many engineering applications, is investigated in this work. The configuration is governed by the Rayleigh number (Ra_L or Ra), the Prandtl number (Pr), and the non-dimensional convective heat transfer coefficient (Ci_L or Ci). A scaling analysis for the dynamics of the boundary layer flow and heat transfer is carried out. The scales of the velocity/thickness of the boundary layer flow and the temperature/thickness of the thermal boundary layer related to the non-dimensional governing parameters are obtained. The scales are validated using the numerical results by large eddy simulation. The results show that the non-dimensional velocity of the boundary layer flow is proportional to Ci_L^2/5Ra_L^2/5; the thickness from the wall to the layer of the maximal velocity is inversely proportional Ci_L^1/5Ra_L^1/5; the non-dimensional thickness of the thermal boundary layer is inversely proportional Ci_L^1/5Ra_L^1/5; the non-dimensional temperature in the thermal boundary layer is proportional to Ci_L^4/5Ra_L^−1/5. The reduction factor describing the thermal resistance of the thermal boundary layer is further discussed, which is proportional to Ci^4/5Ra^−1/5.

We develop a model to study the natural convection of a nanofluid between a square enclosure and a circular, an elliptical, or a rectangular cylinder. Using super elliptic functions, the dimensionless governing equations of two-dimensional rectangular coordinates have been transformed into a system of equations valid for the above geometry. The resulting equations are then solved utilizing finite difference technique. We illustrate the flow and heat transfer characteristics of nanofluids with streamlines and isotherms as well as the Nusselt number at the inner and outer cylinders. It is found that the intensity of streamlines becomes stronger with the increase in the volume fraction of nanoparticles and the Rayleigh number. The Nusselt number at the inner and outer cylinders is almost linearly increased for higher values of the volume fraction of nanoparticles while an exponentially increasing tendency is observed with the increase in the Rayleigh number. The distinct findings are that the intensity of the streamlines increases with rectangular, circular, and elliptical inner shapes. Moreover, the Nusselt number at the inner and outer cylinders diminishes with circular, elliptical, and rectangular inner shapes. The acquired knowledge from the results could be used to augment or control the heat transfer of nanofluids and for the advancement of existing technology. Moreover, the present concept of introducing super elliptic functions might be useful to formulate a model for more complex geometry.