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Understanding generalized Couette flow provides valuable insights into the behavior of fluids under various conditions, contributing to the advancement of more accurate models for real-world applications including tribology and lubrication, polymer and food processing, water conservation and oil exploration, microfluidics, biological fluid dynamics (blood flow in vessels), and electrohydrodynamic, and so on. The present study provided the exact asymptotic solution for the generalized Couette flow of a non-Newtonian Jeffrey fluid in a horizontal channel immersed in a saturated porous medium. The governing partial differential equations were transformed into a dimensionless form using the similarity technique and the resulting system of equations is solved by the Perturbation technique, as well as the method of the separation of variables, and computed on MATLAB (ode15s solver). The behavior of fluid velocity was investigated and presented through 2-D and 3-D graphs for two cases (i) when the implication of the magnetic field was strengthened and (ii) when the magnitude of the magnetic field was fixed but its degree of inclination was altered. The first-order chemical reactions and thermal radiation were also considered. Additionally, the effect of numerous emerging quantities on momentum, temperature, and concentration contours characterizing the fluid flow was depicted graphically and discussed. Furthermore, the skin friction (at different angles of inclination and magnetic strength), Nusselt number, and Sherwood number (at different time intervals) were evaluated at both boundaries and presented tabularly. The findings revealed that there was a decrease in the velocity profile with an increasing degree of inclination and strength of the magnetic field. Moreover, we observed an increment in thermal and mass flux when it was measured over time at both of the channels. Also, the outcomes predicted an oscillatory nature of shear stress at both of the boundries. [ABSTRACT FROM AUTHOR]

This study investigates the complex behavior of Jeffrey nanofluid flow in a porous oscillating microchannel under the influence of magnetohydrodynamic (MHD) effects. The research explores how magnetic field distortions lead to diverse accumulation patterns of nanofluid particles, a phenomenon attributed to homogeneous magnetization in fluid dynamics. Nanoparticles ranging from 0.25 % to 0.5 % (low and inexpensive concentrations) are remarkably consistent for best results in most machining procedures. Different concentrations of various nanomaterial's is utilized (φ1 = φ2 = 0.01, 0.02, 0.03, 0.04) to make the simple nanfluid and hybrid nanofluid suspensions. By employing fractal-fractional derivatives governed by power law, a mathematical model developed to describe the time-varying, compressible MHD flow of Jeffrey nanofluid. The model incorporates the effects of heat transfer, pressure, and magnetic fields on the fluid dynamics. A novel fractional approach utilizing the Laplace transform is applied to solve the fractal MHD hybrid-fluid model integrated into a porous medium. The study reveals velocity flow decrease with increasing Reynolds numbers but increase with channel inclination. Additionally, both the Darcy number and magnetic field orientation enhance heat transfer rates. In addition, the velocity profile enhanced by the hybrid nanofluid suspension as compared to simple nanofluid flow. The research validates its findings by demonstrating the convergence of fractional and numerical solution methods. Furthermore, the study compares the performance of different hybrid nanofluids, concluding that water-based (H2O + GO + MoS2) hybrid fluids exhibit slightly superior characteristics compared to (CMC + GO + MoS2) hybrid nanofluids.

Understanding generalized Couette flow provides valuable insights into the behavior of fluids under various conditions, contributing to the advancement of more accurate models for real-world applications including tribology and lubrication, polymer and food processing, water conservation and oil exploration, microfluidics, biological fluid dynamics (blood flow in vessels), and electrohydrodynamic, and so on. The present study provided the exact asymptotic solution for the generalized Couette flow of a non-Newtonian Jeffrey fluid in a horizontal channel immersed in a saturated porous medium.The governing partial differential equations were transformed into a dimensionless form using the similarity technique and the resulting system of equations is solved by the Perturbation technique, as well as the method of the separation of variables, and computed on MATLAB (ode15s solver).The behavior of fluid velocity was investigated and presented through 2-D and 3-D graphs for two cases (ⅰ) when the implication of the magnetic field was strengthened and (ⅱ) when the magnitude of the magnetic field was fixed but its degree of inclination was altered. The first-order chemical reactions and thermal radiation were also considered. Additionally, the effect of numerous emerging quantities on momentum, temperature, and concentration contours characterizing the fluid flow was depicted graphically and discussed. Furthermore, the skin friction (at different angles of inclination and magnetic strength), Nusselt number, and Sherwood number (at different time intervals) were evaluated at both boundaries and presented tabularly. The findings revealed that there was a decrease in the velocity profile with an increasing degree of inclination and strength of the magnetic field. Moreover, we observed an increment in thermal and mass flux when it was measured over time at both of the channels. Also, the outcomes predicted an oscillatory nature of shear stress at both of the boundries.

This study investigates the impact of coupled heat and mass transfer on the peristaltic migration of a magnetohydrodynamic (MHD) stress-strain Jeffery-type hybrid nanofluid flowing through an inclined asymmetric micro-channel with a porous medium. The fundamental two-dimensional momentum and energy transport equations are simplified under the assumptions of long wavelength and low Reynolds number. To solve the momentum and heat transfer problems, two advanced fractional derivative approaches are employed: the fractal integral and the Prabhakar fractional derivative. The pressure difference is determined using numerical integration techniques, such as the Stehfest and Tzou's algorithms. The results are presented through graphs and tables, which illustrate the effects of various parameters on the velocity, heat transfer, and trapping phenomena. As a result, we concluded that, pressure gradients grow with higher Reynolds numbers and channel-inclined angles. The heat transfer rate is observed to decrease as the Darcy number and the orientation of the electromagnetic field increase. When comparing the fractional derivative approaches, the fractal operator exhibits a more significant impact on the momentum profiles compared to the Prabhakar fractional operator. This difference is attributed to the distinct characteristics of the integral kernels associated with each fractional derivative definition. Furthermore, when comparing hybrid nanofluids, water-based (H2O + Ag + TiO2) hybrid fluids have a somewhat more significant effect than (C6H9NaO7 + Ag + TiO2) hybrid nanofluids.

In this article, the effects of Ohmic heating, induced magnetic field and Newtonian heating on MHD generalized Couette flow of Jeffrey nanofluid between two horizontal plates with convective cooling has been investigated. The mathematical model used for the magnetohydrodynamic generalized Couette flow of pure water as the fluid phase containing Copper (Cu) as nanoparticles takes into account the effects of viscous dissipation, Soret and Dufour. The numerical solution of the system of nonlinear partial differential equations controlling the flow was performed using the finite difference method. To acquire the flow variable profiles, such as velocity, induced magnetic field, temperature, and concentration profiles graphically, the resulting numerical schemes are simulated in MATLAB software. The findings reveal that an increase in the Schmidt number causes a decrease in the concentration profiles, rise in the Jeffrey parameter causes a decrease in the velocity profiles. The temperature profiles also rise as a result of an increase in the Ohmic heating parameter and the heat generation parameter. Induced magnetic field profiles decreases with an increase in magnetic Prandtl number. The results demonstrate that Cu - water shows significant impact over nanofluid on temperature and velocity in the generalized Couette channel. The results of this study are valuable because they can be applied to the development of many chemical technologies, such as the production of polymers, MHD pumps, food processing, chemical catalytic reactors, MHD flow meters, astronomy, and lubrication. The results are compared with standard literature and show good agreement.

This study provides the exact solutions of couple stress fluid (CSF) in a channel. The CSF is bounded by two plates in which the lower plate is moving with constant velocity Uo and the upper plate is fixed. The influence of the external pressure gradient is considered on the CSF fluid. To transform the classical CSF model, we have introduced three approaches to fractional derivatives, (a) Caputo (b) Caputo–Fabrizio (CF), and (c) Atangana–Baleanu (AB) fractional definitions. Exact solutions have been obtained using the Laplace and finite Fourier sine transforms jointly. Furthermore, the effect of different fractional derivatives is compared, and the results are shown in graphs. Moreover, parameters that affect the CSF motion are discussed in the graphs using the computational software MATHCAD. The most important outcome of the given study is the comparison of Caputo, CF, and AB fractional models with the classical CSF model. From the comparison, it can be noticed that AB fractional model discusses the dynamics of the CSF with a good memory effect as compared to Caputo and CF fractional operators. The CSF model can be reduced to Newtonian fluid as a limiting case and also investigated solutions in the absence of external pressure. Finally, Skin friction is evaluated for lower and upper plates, and the obtained results presented in tabular form. [ABSTRACT FROM AUTHOR]

The aim of the present work is to calculate the closed form solutions of the unsteady couple stress nanofluids flow in a channel. Couple stress nanofluids (CSNF) is allowed to pass through the parallel plates separated by a distance h. In this study, we choose blood as base fluid with gold nanoparticles suspension. The lower plate is at rest and the upper plate is suddenly moved with constant velocity U0. Recently, Atangana-Baleanu (AB) introduced a new definition of fractional derivatives. This AB definition of fractional derivative has been applied to the present couple stress nanofluid (CSNF) model. The closed form solutions of present CSNF model via AB approach are obtained by using the Laplace and finite Fourier sine transforms. Exact results of velocity and temperature are displayed and discussed for different parameters of interest. Solutions obtained here are reduced to three different cases in limiting sense i.e. (i) fractional couple stress nanofluid without external pressure gradient. (ii) ordinary couple stress nanofluid. (iii) regular couple stress fluid. Finally, skin friction and Nusselt number are evaluated at lower and upper plates and listed in tabular forms. The results show that increasing external pressure gradient, CSNF velocity increases whereas decreases by increasing Reynolds number. Increasing volume fraction slow down the CSNF velocity. The velocity of Newtonian viscous fluid is higher than CSNF velocity.

The aim of this study is to obtain the closed form solutions for the laminar and unsteady couple stress fluid flow. The fluid is allowed to flow between two infinite parallel plates separated by distance ℓ. Moreover, we have considered that the lower plate is moving with uniform velocity U0 and upper plate is stationary. For this purpose, engine oil is taken as a base fluid and to enhance the efficiency of lubricating oil, Molybdenum disulphide nanoparticles are dispersed uniformly in the engine oil. The flow is formulated mathematically in terms of partial differential equations of order four. Furthermore, the derived system of partial differential equations are fractionalized by using the mostly used definition of Caputo-Fabrizio time fractional derivative. The more general exact solutions for velocity, temperature and concentration distributions are obtained by using the joint applications of Fourier and the Laplace transforms. The effect of different parameters of interest of the obtained general solutions are discussed by sketching graphs. Furthermore, substituting favorable limits of different parameters, four different limiting cases are recovered from our obtained general solutions i.e. (a) Couette flow (b) Classical couple stress fluid (c) Newtonian viscous fluid and (d) in the absence of thermal and concentration. Moreover, the effect of different physical parameters on the velocity, temperature and concentration distributions are discussed graphically. It is worth noting that couple stress parameter corresponds to a decrease in the velocity profile. In order to observe the differences clearly, all the figures are compared for integer order and fractional order which provide a more realistic approach as compared to the classical model. Additionally, skin friction is calculated at lower as well as upper plate. Nusselt number and Sherwood number are also tabulated. It is noticed that the rate of heat transfer of engine oil can be enhanced up to 12.38% and decrease in mass transfer up to 2.14% by adding Molybdenum disulphide nanoparticles in regular engine oil.

Different from a Newtonian fluid, couple stress fluid (CSF) includes a new material constant, which is responsible for couple stress and the lubricant viscosity. This material constant comes with the fourth-order spatial derivative term, and due to this higher-order derivative term in the momentum equation, this fluid (CSF) is comparatively less investigated even for the classical fluid problems. This paper aims to study the fractional model of CSF, based on the Atangana-Baleanu (AB) fractional derivatives definition. Since this AB definition is new, therefore, for the sake of comparison and correctness, this problem is also solved using the Caputo-Fabrizio (CF) fractional derivative definition. The CSF is considered to flow between two parallel plates, one of which is at rest and the other is moving with constant velocity. The external pressure gradient is also applied. This type of flow situations is usually called generalized Couette flow. The problem is first written in dimensionless form and then solved for the exact solution using the Laplace transform and the finite Fourier sine transform. The CSF velocity obtained via an AB fractional derivative is compared with the CSF velocity obtained via a CF fractional derivative approach, and the results obtained are shown graphically. The CSF results for interesting fluid parameters are displayed in various graphs for both the AB and CF fractional derivatives. It is observed that the CSF velocities obtained with the AB and CF fractional derivatives are the same for unit time. For time less than one and greater than one, variation in CSF velocities is observed. In limiting sense, the present CSF solutions are reduced to a similar Newtonian fluid problem solution in the absence of external pressure gradient.

This paper investigates the influence of spatial inhomogeneity of fluid viscosity in an unsteady channel flow of a viscous second grade fluid. The viscosity model is assumed to follow a power-law spatial relation which may arise from concentration gradient, thermal gradient or boundary induced phase transition in the weakly elastic fluid. Two flow problems are considered (i) the Generalized Couette and (ii) the time-periodic plane Poiseuille flows. The associated partial differential equation governing each flow setup is decoupled into steady and transient state problems which are analyzed for possible closed form solutions. The analytical technique explored in this study is premised on theoretical analysis of the resulting ordinary differential equations. Corresponding results to the deduced ordinary differential equations with variable coefficient are presented as trigonometric and hyperbolic functions. In cases where analytical results are not attainable, numerical solutions are obtained via finite volume techniques. On comparing instances with the same parameter values, both the numerical and analytical solutions show good agreement. The spatial variation in the viscosity results in either a plug flow for cases where the viscosity index is negative or a fast flow throw the axis in instances where the index is positive. With graphical and tabular illustrations the influence of associated material parameters on the flow are presented and discussed. [ABSTRACT FROM AUTHOR]

Bingham fluid through porous parallel plates with Ion-slip and Hall currents has been studied numerically. The non-linear PDEs, governing the problem under assumptions, have been transformed into dimensionless non-linear PDEs by using usual transformations. The obtained dimensionless governing equations have been solved numerically by applying the explicit finite difference method (FDM) with the help of MATLAB R2015a tool. The time sensitivity test is performed for the steady-state solution and is obtained at dimensionless time t=4.00. It is also observed that the secondary velocity reaches the steady-state more gradually than the primary velocity and temperature profiles. The appropriate mesh space (m=40 and n=40) is obtained by the mesh sensitivity test. The impact of various interesting parameters on the primary velocity, secondary velocity and temperature profiles, also on the local Nusselt number and shear stress have been analyzed and discussed through the graph in details. Finally, a qualitative and quantitative comparison with the published results has been discussed. [ABSTRACT FROM AUTHOR]

This paper addresses the mixed Couette-Poiseuille problem, that is the flow between two parallel plates, in the presence of simultaneous pressure gradient and wall motion. Instead of the wall-normal coordinate y, we use the local shear stress as our primary variable and rewrite the corresponding formulae for the velocity profile, flow rate, etc. This gives rise to a graphical technique for solving the problem in the case of arbitrary (possibly measured) generalized Newtonian fluid rheology. We demonstrate the use of the proposed technique on two problems: (a) Bingham fluid and (b) a non-Newtonian fluid with general, nonmonotonous viscosity function. [ABSTRACT FROM AUTHOR]

In the present article, the exact solutions for fundamental flows namely Couette, Poiseuille and generalized Couette flows of an incompressible couple stress fluid between parallel plates are obtained using slip boundary conditions. The effect of various parameters on velocity for each problem is discussed. It is found that, for each of the problems, the solution in the limiting case as couple stresses approaches to zero is similar to that of classical viscous Newtonian fluid. The results indicate that, the presence of couple stresses decreases the velocity of the fluid.

The lubrication theory based approximation of the Navier–Stokes equations represents a widely used approach for the computation of viscous force dominated flows through narrow channels or gaps. Its inherent neglect of advective transport becomes increasingly questionable when considering strongly converging geometries and particularly when considering heat transfer as well. The present work develops an extended lubrication theory based model, which accounts for inertia and convective heat transfer in terms of first-order perturbations. The extended model is applied to generalized Couette flow through strongly converging annular gaps typically met in wire coating dies, where the moving wall-driven motion is characterized by very high local shear rates and steep axial pressure gradients. The evaluation of the analytical solutions against numerical results from CFD simulations demonstrates the scope and the limits of the original lubrication theory approximation in describing the local flow and thermal conditions inside the gap. Including first-order convective contributions the extended model is shown to provide a significantly improved description, especially of the redistribution of the generated viscous heat. The most pronounced improvements are shown by the temperature profiles predicted at axial positions with rapidly changing cross-section for fluids with higher Prandtl numbers. The observed gain in accuracy is also reflected by very accurate predictions of the total transfer rates at the boundaries in the global balances of heat and momentum. Using the presently proposed extension the lubrication theory based model represents a reliable and computationally efficient approach for investigating the flow and heat transfer inside narrow gaps, which is well applicable to a wide range of different gap geometries. [ABSTRACT FROM AUTHOR]

This theoretical study investigates three types of basic flows of viscous incompressible Herschel-Bulkley fluid such as (i) plane Couette flow, (ii) Poiseuille flow and (iii) generalized Couette flow with slip velocity at the boundary. The analytic solutions to the nonlinear boundary value problems have been obtained. The effects of various physical parameters on the velocity, flow rate, wall shear stress and frictional resistance to flow are analyzed through appropriate graphs. It is observed that in plane Poiseuille flow and generalized Couette flow, the velocity and flow rate of the fluid increase considerably with the increase of the slip parameter, power law index, pressure gradient. The fluid velocity is significantly higher in plane Poiseuille flow than in plane Couette flow. The wall shear stress and frictional resistance to flow decrease considerably with the increase of the power law index and increase significantly with the increase of the yield stress of the fluid. The wall shear stress and frictional resistance to flow are considerably higher in plane Poiseuille flow than in generalized Couette flow. [ABSTRACT FROM AUTHOR]

In their article (Fundamental flows with nonlinear slip conditions: exact solutions, R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 () 877-888.), the authors considered three simple cases of the steady flow of a third grade fluid between parallel plates with slip conditions; namely, Couette flow, Poiseuille flow, and generalized Couette flow. They obtained exact solutions, which were utilized in a way that did not lead to useful results. Their conclusion that the Couette flow cannot be obtained from the generalized Couette flow, by dropping the pressure gradient, is incorrect. Meaningful results based on their solution are herein presented. [ABSTRACT FROM AUTHOR]

Abstract: The steady flow and heat transfer of an electrically conducting fluid with variable viscosity and electrical conductivity between two parallel plates in the presence of a transverse magnetic field is investigated. It is assumed that the flow is driven by combined action of axial pressure gradient and uniform motion of the upper plate. The governing nonlinear equations of momentum and energy transport are solved numerically using a shooting iteration technique together with a sixth-order Runge-Kutta integration algorithm. Solutions are presented in graphical form and given in terms of fluid velocity, fluid temperature, skin friction and heat transfer rate for various parametric values. Our results reveal that the combined effect of magnetic field, viscosity, exponents of variable properties, various fluid and heat transfer dimensionless quantities and the electrical conductivity variation, have significant impact on the hydromagnetic and electrical properties of the fluid. [Copyright &y& Elsevier]

Abstract: We study the thermal transition of a reactive flow of a third-grade fluid with viscous heating and chemical reaction between two horizontal flat plates, where the top is moving with a uniform speed and the bottom plate is fixed in the presence of imposed pressure gradient. This study is a natural continuation of earlier work on rectilinear shear flows. The governing equations are non-dimensionalized and the resulting system of equations are not coupled. An approximate explicit solution is found for the flow velocity using homotopy-perturbation technique and the range of validity is determined. After the velocity is known, the heat transport may be analyzed. It is found that the temperature solution depends on the non-Newtonian material parameter of the fluid, Λ, viscous heating parameter, Γ, and an exponent, m. Attention is focused upon the disappearance of criticality of the solution set {β, δ, θ max} for various values of Λ, Γ and m, and the numerical computations are presented graphically to show salient features of the solution set. [Copyright &y& Elsevier]

In the present work, we report heat transfer during the unidirectional laminar flow of ferrofluids in a parallel plate channel in the presence of a homogeneously applied magnetic field perpendicular to the flow direction. We have considered generalized Couette flow which results from the combination of shear stress, due to the relative motion between the parallel plates, and applied pressure gradient. For a comprehensive investigation, four different thermal boundary conditions have been considered. The application of magnetic field modifies the viscosity of the colloidal ferrofluid and thereby causes variations in the local flow field. Further, it also modifies the thermal conductivity of the fluid by changing particle orientation. As a consequence, the temperature distribution between the parallel plate channel is uniquely modified. The complex interplay between the hydrodynamics and the magnetics results in implicitly coupled governing differential equations, which have been solved numerically. We report that the local temperature in the domain could be reduced by increasing the applied field strength irrespective of the imposed thermal boundary conditions. However, the influence of the applied field finally gets saturated. We also discuss the effect of Brinkman number, loading of magnetic particles and pressure gradient on temperature distribution. Finally, analytical solutions, which presents closed-form expressions of the temperature profiles for four different types of boundary conditions, have been proposed. [ABSTRACT FROM AUTHOR]