Riemann-sum Approximation Method Research Articles

Transient generalized Taylor–Couette flow of a dusty fluid: A semi-analytical approach

This paper discusses extensively the time-dependent flow of a dusty viscous, incompressible fluid in rotating horizontal annuli under the influence of an azimuthal pressure gradient. The momentum and continuity equations depicting the flow system alongside the initial and boundary conditions are non-dimensionalized and solved semi-analytically using Laplace transform and Riemann-sum approximation (RSA) method. The velocity, skin frictions, vorticity and mass flow rates are obtained in the Laplace domain and then inverted back to the time domain with the aid of RSA. Steady-state solutions for the velocity, skin frictions, vorticity and mass flow rates are presented analytically to check the validity of the method employed at large values of the time. The governing dimensionless parameters appearing in the flow phenomenon are examined with the aid of line graphs and tables for comparison. From the findings and numerical computations, it is found that at large values of time, t, the velocity, skin frictions, vorticity and mass flow rates reach steady-state. Physically, as the angular velocity of the dusty particles increases, so does the mass concentration of the dust particles, resulting in a reduction in the velocity of the dusty particles.

Memory Response on the Elasto-Thermodiffusive Interaction Subjected to Harmonically Varying Heat Source

Enlightened by the Caputo type of fractional derivative, this research paper is the analysis of a mathematical model of generalized thermoelastic diffusion with memory-dependent derivative (MDD) for an isotropic infinite medium with a cylindrical cavity in the context of dual-phase lag model. The surface of the cavity is traction free and subjected to harmonically heat sources with constant angular frequency of thermal vibration. Laplace transform technique has been used to solve the problem. Later eigenvalue approach is used to obtain the analytical expressions for different physical fields in the transformed domain. Finally to obtain the solutions in the real-time domain, the Riemann-sum approximation method is used. According to the graphical representations corresponding to the numerical results, the effect of heat source speed on temperature, displacement, stress, mass concentration and chemical potential are studied. The effect of memory-dependent parameters, as well as in the absence of MDD on physical quantities are also demonstrated.

Effect of chemical reaction, heat source/sink, and thermal diffusion on transient natural convection through a tube

AbstractA semi‐analytical investigation of the effect of a chemical reaction and thermal diffusion on transient natural convection flow of a viscous incompressible binary fluid with heat source/sink in a vertical and infinite tube is presented. The equations for the model are simplified with the help of the Laplace transform, solved analytically to obtain solutions whose expressions are given in modified Bessel functions. These solutions are then inverted numerically using the Riemann sum approximation method. An exhaustive model analysis is achieved by exhibiting the temperature, concentration, and velocity profiles graphically. Also, numerical values are displayed in tabular form for the Nusselt number, Sherwood number, and the shear stress. Furthermore, a version of the governing equations which are independent of time is solved analytically. The obtained results are found to coincide with the results of the time‐dependent problem at an appreciable value of time. A close examination of the results reveals that raising the generative chemical reaction parameter strengthens both the concentration and momentum of the flow while they are weakened with a surge of the destructive chemical reaction. As the heat source intensifies, both fluid temperature and velocity are elevated while the species concentration is abated with the reversed phenomenon occurring as the heat sink intensifies. An improvement is observed in the concentration and velocity distribution as the Soret parameter increases. This model is useful in petroleum reservoirs, the mining industry, and many industrial processes.

Thermal wave propagation in a two-dimensional problem under gravitational field due to time-dependent thermal loading and memory effect

This paper presents a comprehensive study on developing a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena for an infinite thermoelastic half-space under the action of ramp-type thermal loading and due to the influence of a gravitational field. The bounding plane of the half-space is subjected to rough and rigid foundation so that the rough surface prevents the vertical displacement. Due to the shortcomings of power-law distributions, some other forms of derivatives with few other kernel functions are proposed. The present analysis deals with the heat transport which involves the memory-dependent derivative (MDD) on a slipping interval in the context of Lord–Shulman model to describe the physical phenomena which is defined in the form of convolution with the kernels in the form of power functions. Employing the Laplace and the Fourier transform techniques as tools, the analytical expressions for different physical fields have been obtained on the transformed domain. The numerical inversion of the Fourier transforms have been performed analytically, whereas numerical inversion of the Laplace transform is carried out using the Riemann-sum approximation method. Excellent predictive capability is demonstrated due to the presence of MDD, delay time and gravitational field also.

Field equations and memory effects in a functionally graded magneto-thermoelastic rod

The main concern of this article is to deal with the thermoelastic interaction in a functionally graded thermoelastic rod being enlightened by the memory-dependent derivative. This article investigates the transient phenomena due to the influence of an induced magnetic field of constant intensity and due to the presence of a moving heat source of constant velocity in the context of three-phase lag model of generalized thermoelasticity. Employing the Laplace and the Fourier transforms as tool, the problem has been constructed in the transformed domain. The inversions of the Fourier transform have been carried out using residual calculus whereas the numerical inversions of the Laplace transform have been performed employing the Riemann sum approximation method. Numerical computations for stress, displacement, and temperature within the rod are carried out and have been demonstrated graphically. The results also demonstrate how the nonhomogeneity parameter and the speed of the moving heat source influence the thermophysical quantities. It is observed that the temperature, thermally induced displacement, and stress of the rod are found to decrease at large source speed. Also, significant differences on the thermophysical quantities are revealed due to the influence of magnetic field, nonhomogeneity, and memory effect also.

Interactions of a Heat Source Moving over a Visco-Thermoelastic Rod kept in a Magnetic Field in the Lord–Shulman Model under a Memory Dependent Derivative

The Fractional derivative is a widely accepted theory to describe physical phenomena and the processes with memory effects which is defined in the form of convolution having kernels as power functions. Due to the shortcomings of power-law distributions, some other forms of derivatives with few other kernel functions are proposed. This present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of the magnetic field and moving heat source in a thermoelastic rod in the context of the Lord–Shulman (LS) theory of generalized thermo-visco-elasticity. Both ends of the rod are fixed and are thermally insulated. Employing the Laplace transform as a tool, the problem has been transformed into the space-domain and solved analytically. Finally, solutions in the real-time domain are obtained on applying the numerical inversion of the Laplace transform, which has been carried out employing the Riemann sum approximation method. Numerical computations for stress, displacement, and temperature within the rod is carried out and have been demonstrated graphically. The results also demonstrate how the speed of the moving heat source influences the thermophysical quantities. It is observed that the temperature, thermally induced displacement and stress of the rod are found to decrease at large source speed. Also, significant differences in the thermophysical quantities are revealed due to the influence of the magnetic field and memory effect

Unsteady Dean flow formation in an annulus with partial slippage: A riemann-sum approximation approach

Abstract This paper investigates time-dependent Dean Flow of an incompressible fluid under homogeneous slip, non-homogeneous slip, and no-slip boundary conditions. The fluid flow is due to the sudden application of an azimuthal pressure gradient. The general solutions of the governing momentum equations are obtained using a two-step approach called Laplace transformation and Riemann-sum approximation method of Laplace inversion. The velocity and skin friction are determined exactly in the Laplace domain and inverted back to time domain using a numerical approach known as Riemann-sum approximation. The steady-state solutions for the velocity and the skin friction are obtained for the validation of the method employed. Graphs are plotted for analysis and numerical values are tabulated for comparison of the Riemann-sum approximation and the exact solution at large values of time. From the analysis, it is observed that the velocity profile of the fluid is higher at the wall with the highest slip coefficients. Finally, the influence of the dimensionless time ( T ) and the slip coefficients is also discussed with the aid of graphical illustrations.

Thermoelastic interaction in a magneto-thermoelastic rod with memory-dependent derivative due to the presence of moving heat source

Fractional derivative is a widely accepted theory to describe the physical phenomena and the processes with memory effects which is defined in the form of convolution-type integrals involving kernels as power functions. Due to the shortcomings of power law distributions, some other forms of derivatives with few other kernel functions have been proposed. This present survey deals with a novel mathematical model of generalized thermoelasticity which investigates the transient phenomena due to the influence of an induced magnetic field and the presence of moving heat source in a thermoelastic rod in the context of Lord–Shulman (LS) theory of generalized thermoelasticity. Both ends of the rod are fixed and are thermally insulated. Employing the Laplace transform, the problem has been transformed to the space-domain have been solved analytically. Finally, solutions in the real-time domain are obtained on applying the numerical inversion of Laplace transform, which has been carried out employing the Riemann-sum approximation method. Numerical computations for stress, displacement and temperature within the rod is carried out and have been demonstrated graphically. The results also demonstrate how the speed of the moving heat source influences the thermophysical quantities. It is observed that the temperature, thermally induced displacement and stress of the rod are found to decrease at large source speed. Also, significant differences on the thermophysical quantities are revealed due to the influence of magnetic field and memory effect also.

A novel model for photothermal excitation of variable thermal conductivity semiconductor elastic medium subjected to mechanical ramp type with two-temperature theory and magnetic field

This article focuses on studying the dependence of the thermal conductivity of a semiconducting medium on temperature in context of photothermal transport process and variable thermal conductivity, chosen to be linear function in temperature. The effect of the initial magnetic field is introduced in the problem governing equations with the two-temperature theory. The complete solution in one dimension is obtained using Laplace transform technique. The thermal heating ramp type and mechanical load throughout the elastic-photothermal excitation are considered in the problem boundary conditions. Some physical fields are obtained by using numerical inversion of the Laplace transform and Riemann-sum approximation method. The thermo-dynamical temperature, conductive temperature, displacement, strain, normal stress and carrier density are discussed and shown graphically with some comparisons.

Transient Dean flow in a channel with suction/injection: A semi-analytical approach

The mathematical model responsible for fully developed laminar transient flow formation between the gaps of two stationary concentric porous tubes due to the imposition of azimuthal pressure gradient (Dean flow) is solved semi-analytically. The tubes walls are porous so that a radial flow can be superimposed. The solution of the momentum and continuity equations are obtained semi-analytically by using the combination of Laplace transform technique and a Laplace inversion method called Riemann sum approximation method. The solutions for skin friction at [Formula: see text] (outer surface of the inner porous tube) and [Formula: see text] (inner surface of the outer porous tube) are presented. The impact of suction/injection parameter and the ratio of the radii of the tubes are examined for the velocity profile and skin friction. Results show that the velocity profile decreases with increase in suction/injection parameter for various values of time, [Formula: see text] and at large value of time, ([Formula: see text]), the velocity and the skin friction attain a steady state.

Effect of fractional derivatives on transient MHD flow and radiative heat transfer in a micro-parallel channel at high zeta potentials

By using the Caputo–Fabrizio time fractional derivative, unsteady flows of an incompressible viscous through a circular tube are developed. Flows are generated due to the combination of electroosmotic, pressure and magnetic forces. The derived fractional momentum equation incorporating the electromagnetic body force resulting from the electric double layer (EDL) field was solved using the Laplace transform technique, Riemann-sum approximation method, and the Stehfest’s algorithm. The solution obtained is validated by assenting comparisons between the Riemann-sum approximation method and Stehfest’s algorithm. Based upon the velocity field solution, the energy equation is analyzed by taking the viscous dissipation, thermal radiative, the volumetric heat generation due to Joule heating effect and electromagnetic couple effect into account. Numerical simulations and graphical illustrations were carried out using the software package Mathcad. The results highlights that, fractional models of fluid-flow and heat transfer with fractional derivatives bring significant differences compared to the ordinary model. The variations of fluid velocity, fluid temperature, skin friction and Nusselt number with related parameters are interpreted and the results are discussed.

Unsteady hydromagnetic-free convection flow with suction/injection

This article analyses the unsteady hydromagnetic free convection flow near impulsive as well as accelerated motion of the infinite vertical porous plate. The governing momentum and energy partial differential equations exhibiting the physics of the flow formation are presented. Using suitable dimensionless parameters, the equations are transformed to their corresponding dimensionless form and solved exactly in the Laplace domain using the well-known Laplace transform technique. However, the Riemann-sum approximation method is used to invert the solution from the Laplace domain to time domain due of the complexity of the solutions obtained in the Laplace domain. The effect of various pertinent parameters such as suction/injection, Hartmann number, Grashof number, and Prandtl number is discussed with the aid of line graphs.

The combined effects of anisotropic porous medium and stably stratified fluid on free convective flow through an annulus

ABSTRACTThis paper documents the semi-analytical results for transient natural convection fluid flow between two concentric vertical cylinders of infinite lengths in the presence of anisotropic porous medium and stratified fluid. The solutions for the time-dependent coupled second order partial differential equations modelling the flow formation are obtained by using trio-methods: Laplace transform techniques, a numerical procedure based on the Riemann Sum Approximation method and D’Alembert method. Line graphs and contour maps for the fluid velocity and temperature profiles are presented for different physical parameters of interest featuring in the solutions. The inclusion of the stratification and anisotropic parameters in the model has been found to aid the regulation of the fluid flow. Furthermore, from the numerical simulation conducted, it is found that for some saturated stratified fluids, the steady and transient state velocities (as well as the temperatures of the fluid) coincide at large time.

MHD Transient Free Convection Flow in Vertical Concentric Annulus with Isothermal and Adiabatic Boundaries

This paper presents MHD transient flow in an infinite vertical concentric annulus when the fluid is set in motion by free convection current occurring in the annulus as a result of application of isothermal heating on the inner surface of the outer cylinder while the outer surface of the inner cylinder is thermally insulated. The solution of the governing equations are obtained using the well-known Laplace transform technique while the Riemann-sum approximation method has been used to invert the solution from Laplace domain to time domain. The numerical values obtained using Riemann-sum approximation approach is validated by presenting a comparison with the values obtained using the implicit finite difference method as well as the steady-state solution. These comparisons with the steady state solution shows a remarkable agreement at large value of time. The effect of the governing parameters on the velocity field, temperature field, mass flow rate as well as the skin-friction on both surfaces of the annulus have been analysed and presented with the aid of line graph. Generally, we observed that the mass flow rate and skin friction at the isothermally heated surface increases with increase in radius ratio. However, the reverse is seen at the thermally insulated surface as the skin-friction decreases with increase in radius ratio.

MHD Transient Free Convection Flow in Vertical Concentric Annulus with Isothermal and Adiabatic Boundaries

This paper presents MHD transient flow in an infinite vertical concentric annulus when the fluid is set in motion by free convection current occurring in the annulus as a result of application of isothermal heating on the inner surface of the outer cylinder while the outer surface of the inner cylinder is thermally insulated. The solution of the governing equations are obtained using the well-known Laplace transform technique while the Riemann-sum approximation method has been used to invert the solution from Laplace domain to time domain. The numerical values obtained using Riemann-sum approximation approach is validated by presenting a comparison with the values obtained using the implicit finite difference method as well as the steady-state solution. These comparisons with the steady state solution shows a remarkable agreement at large value of time. The effect of the governing parameters on the velocity field, temperature field, mass flow rate as well as the skin-friction on both surfaces of the annulus have been analysed and presented with the aid of line graph. Generally, we observed that the mass flow rate and skin friction at the isothermally heated surface increases with increase in radius ratio. However, the reverse is seen at the thermally insulated surface as the skin-friction decreases with increase in radius ratio.

Generalized thermoelastic functionally graded half space under surface absorption of a laser radiation

The subject of this paper is to study the thermoelastic behavior of a functionally graded semi-infinite medium heated uniformly by a laser beam having temporally Gaussian distribution. The surface of the medium is taken as traction free. The general solution is obtained in the Laplace transform domain. The inverse of the Laplace transform is computed numerically using the Riemann-sum approximation method. The numerical results for temperature, displacement and stress are obtained and presented graphically for the generalized theory of thermo-elasticity with one relaxation time.

Unsteady Hartmann Two-Phase Flow: The Riemann-Sum Approximation Approach

We consider the time dependent Hartmann flow of a conducting fluid in a channel formed by two horizontal parallel plates of infinite extent, there being a layer of a non-conducting fluid between the conducting fluid and the upper channel wall. The flow formation of conducting and non-conducting fluids is coupled by equating the velocity and shear stress at the interface. The unsteady flow formation inside the channel is caused by a sudden change in the pressure gradient. The relevant partial differential equations capturing the present physical situation are transformed into ordinary differential equations using the Laplace transform technique. The ordinary differential equations are then solved analytically and the Riemann-sum approximation method is used to invert the Laplace domain into time domain. The solution obtained is validated by comparisons with the closed form solutions obtained for steady states which have been derived separately and also by the implicit finite difference method. The variation of velocity, mass flow rate and skin-friction on both plates for various physical parameters involved in the problem are reported and discussed with the help of line graphs. It was found that the effect of changes of the electric load parameter is to aid or oppose the flow as compared to the short-circuited case.

Transient free convective flow in an annular porous medium: A semi-analytical approach

The transient fully developed free-convective flow of viscous incompressible fluid between two concentric vertical cylinders filled with porous material and saturated with the same fluid have been analysed due to isothermal or isoflux heating of the outer surface of inner cylinder. The governing partial differential equations of motion and energy are transformed into ordinary differential equations using Laplace transform technique. The ordinary differential equations are then solved analytically in Laplace domain. The Riemann-sum approximation method is used to invert the Laplace domain into time domain. The solution obtained is validated by comparison with closed form solution obtained for steady states which has been obtained separately. An excellent agreement was found for transient and steady state solution at large value of time. The governing partial differential equations are also solved by implicit finite difference method to verify the present proposed method. The variation of temperature, velocity, skin-frictions and mass flow rate with dimensionless parameter controlling the present physical situation are illustrated graphically and discussed. It is observed that velocity and temperature increases with time and finally attains its steady state status. Furthermore, both velocity as well as temperature of the fluid is higher in case of isothermal heating of the outer surface of the inner cylinder compared with the isoflux heating case when the gap between the cylinder is less or equal to the radius of inner cylinder while reversed trend is observed when the gap between the cylinders is greater than the radius of inner cylinder for all considered values of time.

Transient magnetohydrodynamic free convective flow in vertical micro-concentric annuli

This article analyses the transient magnetohydrodynamic free convective flow in vertical micro-concentric annuli in the presence of velocity slip and temperature jump at the outer surface of the inner cylinder and the inner surface of the outer cylinder. The Laplace transform technique has been used to find the solutions for the velocity and temperature fields by solving the governing partial differential equations in Laplace domain. However, the Riemann-sum approximation method is used to invert the Laplace domain to the time domain. The solution derived is validated by assenting comparison with exact solutions derived for the steady state which has been derived separately. An excellent agreement was found for transient and steady state at large value of time. The solution obtained for the velocity has been used to compute the skin friction, while the temperature has been used to compute the Nusselt number. The effect of various flow parameters entering into the problem such as time, Prandtl number, curvature radius ratio, Hartmann number, rarefaction parameter, and fluid–wall interaction parameter are discussed with the aid of line graphs.

A Semi-analytical Solution for Start-Up Flow in an Annulus Partially Filled with Porous Material

This study investigates the start-up flow in a horizontal annulus partially filled with clear fluid and partially with a fluid-saturated porous material (composite channel) as a result of the sudden application of constant pressure gradient in horizontal direction. Momentum transfer in the porous medium is simulated using the Brinkman-extended Darcy model. The fluid and porous regions are linked together by equating their velocities and incorporating the shear stress jump conditions at the interface. With the application of Laplace transform technique, the governing partial differential equations are transformed into ordinary differential equations which are solved in the Laplace domain to obtain exact solutions. Inversion of the exact solution in Laplace domain to time domain is done using the Riemann-sum approximation approach. Validation of the Riemann-sum approximation method is achieved by comparing numerical values obtained with those of exact solution obtained for steady state flow and transient solution obtained by implicit finite difference method. At large values of time, transient solution obtained using Riemann-sum approximation approach and implicit finite difference method coincide with closed form solution obtain exactly for steady state flow showing excellent agreement between these methods. Line graphs are used to discuss the effect of the different flow parameters involved in the flow formation.