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In this study, we have developed a fourth order compact finite difference method for finding the numerical solution of two-parameter singularly perturbed convection-diffusion boundary value problems. We have used fourth order compact finite difference method on uniform mesh which provides a tridiagonal linear system of equations. The convergence analysis of the proposed method is established through a matrix analysis approach and it is proved that present method gives fourth order convergence results. Present method is implemented on two numerical examples for checking the efficiency and precision of the method. Numerical outcomes are exhibited which supports the theoretical outcomes. Numerical outcomes are compared with other existing methods and found that present method gives more accurate approximate solution as compare to the other existing methods.

The investigation is carried out to study steady Stokes axisymmetrical Reiner-Rivlin streaming flow over a fixed viscous droplet, and this droplet to be deformed sphere in shape. As boundary conditions, vanishing of radial velocities, continuity of tangential velocities and shear stresses at the droplet surface are used. The very common configuration of approximate sphere governed by polar equation $\tilde{r} =a[1 +\alpha_m \vartheta_m(\zeta)]$ has been considered for the study to $o(\varepsilon)$ describing the distortion. Based on the Stokes approximation, an analytical investigation is achieved in the orthogonal curve linear framework in an unbounded region of a Reiner-Rivlin fluid. In constraining cases, some earlier noted outcomes are obtained. Also, the yielded outcomes for the drag have been compared with solution existing in the literature. Further, the change for both force and pressure are evaluated with deflection w.r.t. the parameters of interest and shown through table and graphs.

The selection of interface boundary conditions between porous-medium and clear-fluid regions is vital for the extensive range of applications in engineering. As such, present paper reports an analytically investigating Stokes flow over Reiner–Rivlin liquid sphere embedded in a porous medium filled with micropolar fluid using Brinkman’s model and assuming uniform flow away from the obstacle. The stream function solution of Brinkman equation is obtained for the flow in porous region, while for the inner flow field the solution is obtained by expanding the stream function in a power series of S . The flow fields are determined explicitly by matching the boundary conditions at the interface of porous region and the liquid sphere. Relevant quantities such as velocity and pressure on the surface of the liquid sphere are determined and exhibited graphically. The mathematical expression of separation parameter SEP is also calculated which shows that no flow separation occurs for the considered flow configuration and also validated by its pictorial presentation. The drag coefficient experienced by a liquid sphere embedded in a porous medium is evaluated. The useful features of the Stokes flow for numerous values of parameters are analyzed and discussed. The dependence of the drag force and stream line pattern on permeability parameter( η 2 ), viscosity ratio( λ ), micropolar parameter( m ), coupling number( N ), and dimensionless parameter S is presented graphically and discussed. The analysis also aims at the explanation of velocity overshoot behavior. Some previous noted results are then also obtained from the ongoing analysis.

This paper presents an analytical study of Stokes flow of an incompressible viscous fluid through a swarm of immiscible Reiner–Rivlin liquid droplets-in-cell using the cell model technique. The stream function solution of Stokes equation is obtained for the flow in the fictitious envelope region, while for the inner flow field within the liquid drop, the solution is obtained by expanding the stream function in a power series of S. The proper boundary conditions are taken on the surface of the liquid sphere, while the appropriate conditions applied on the fictitious boundary of the fluid envelope vary depending on the kind of cell-model. The analytical solution of the problem for four models: Happel’s, Kuwabara’s, Kvashnin’s and Mehta–Morse’s model (usually referred to as Cunningham’s) is derived. The velocity profile and the pressure distribution outside of the droplet are shown in numerous graphs for different values of the parameters. Numerical results for the normalized hydrodynamic drag force $$W_{C}$$ acting, in each case, on the spherical droplet-in-cell obtained for different values of the parameters characterizing volume fraction $$\gamma ,$$ the relative viscosity $$\lambda$$ , and the cross-viscosity, i.e., S are presented in tabular and graphical forms as well. It is found that normalized hydrodynamic drag force $$W_{C}$$ is a monotonic increasing function of particle volume fraction $$\gamma .$$ It is also observed that solid sphere in-cell experiences greater drag force $$C_{D}$$ , whereas spherical bubble experiences smaller. One of the important findings of the present investigation is that the cross-viscosity $$\mu _{c}$$ of Reiner–Rivlin fluid decreases $$W_{C}$$ on the liquid droplet-in-cell. Further, the drag coefficient $$C_{D}$$ get reduced to analytical results obtained earlier.

This paper presents an analytical study of an infinite expanse of uniform flow of steady axisymmetric Stokes flow of an incompressible Newtonian fluid around the spherical drop of Reiner-Rivlin liquid coated with the permeable layer with the assumption that the liquid located outside the capsule penetrates into the permeable layer, but it is not mingled with the liquid located in the internal concave of capsule. The flow inside the permeable layer is described by the Brinkman equation. The viscosity of the permeable medium is assumed to be same as pure liquid. The stream function solution for the outer flow field is obtained in terms of modified Bessel functions and Gegenbauer functions, and for the inner flow field, the stream function solution is obtained by expanding the stream function in terms of S. The flow fields are determined explicitly by matching the boundary conditions at the pure liquid-porous interface, porous-Reiner-Rivlin liquid interface, and uniform velocity at infinity. The drag force experienced by the capsule is evaluated, and its variation with regard to permeability parameter a, dimensionless parameter S, ratio of viscosities l 2 , and thickness of permeable layer d is studied and graphs plotted against these parameters. Several cases of interest are deduced from the present analysis. It is observed that the cross-viscosity increases the drag force, whereas the thickness d decreases the drag on capsule. It is also observed that the drag force is increasing or decreasing function of permeability parameter for l 2 < 1.

This paper presents an analytical study of Stokes flow of an incompressible viscous fluid past an immiscible Reiner–Rivlin liquid sphere embedded in porous medium using the validity of Brinkman’s model. The stream function solution of Brinkman equation is obtained for the flow in porous region, while for the inner flow field, the solution is obtained by expanding the stream function in a power series of $$S$$ . The flow fields are determined explicitly by matching the boundary conditions at the interface of porous region and the liquid sphere. Relevant quantities such as shearing stresses and velocities on the surface of the liquid sphere are obtained and presented graphically. It is found that dimensionless shearing stress on the surface is of periodic nature and its absolute value decreases with permeability parameter $$\alpha $$ and almost constant for all the representative values of $$S$$ ; on the other hand, the permeability parameter increases the velocity in the vicinity of the liquid sphere. The mathematical expression of separation parameter SEP is also calculated which shows that no flow separation occurs for the considered flow configuration and also validated by its pictorial depiction. The drag coefficient experienced by a liquid sphere embedded in a porous medium is evaluated. The dependence of the drag coefficient on permeability parameter, viscosity ratio and dimensionless parameter $$S$$ is presented graphically and discussed. Some previous well-known results are then also deduced from the present analysis.

The creeping motion of a non-Newtonian (Reiner-Rivlin) liquid sphere at the instant it passes the center of an approximate spherical container is discussed. The flow in the spheroidal container is governed by the Stokes equation, while for the flow inside the Reiner-Rivlin liquid sphere, the expression for the stream function is obtained by expressing it in the power series of a parameter S , characterizing the cross-viscosity. Both the flow fields are then determined explicitly by matching the boundary conditions at the interface of Newtonian fluid and non-Newtonian fluid, and also the condition of imperviousness and no-slip on the outer surface. As an application, we have considered an oblate spheroidal container. The drag and wall effects on the liquid spherical body are evaluated. Their variations with regard to the separation parameter l , viscosity ratio $ \lambda$ , cross-viscosity S, and deformation parameter $ \varepsilon$ are studied and demonstrated graphically. Several renowned cases are derived from the present analysis. It is observed that the drag not only varies with $ \varepsilon$ , but as l increases, the rate of change in behavior of drag force also increases. The influences of these parameters on the wall effects has also been studied and presented in a table.