Differential Equation For Stream Function Research Articles

Peristaltic Transport of Visco-Elasto-Plastic Fluids in a Planar Channel

Abstract We numerically investigate peristaltic transport of incompressible visco-elasto-plastic fluids in a two-dimensional symmetric channel. The constitutive equation used for extra stress tensor is of more general form as it includes a number of well-known models like Maxwell A and B, Johnson–Segalman, Oldroyd-B, and Bingham models as its special cases. A detailed mathematical modelling of the problem is presented. The flow equations in the wave frame reduce to a single nonlinear ordinary differential equation in stream function by the implication of widely taken assumptions of long wavelength and low Reynolds number. The solution of the problem is obtained by two ways; namely, shooting method and Matlab built in routine bvp4c, and their comparison shows an excellent agreement. A parametric study based on bvp4c solution is performed to see the effects of parameters on velocity profile, pressure rise per wavelength, frictional forces, and trapping phenomenon.

NUMERICAL PREDICTION OF FORCE ON RECTANGULAR CYLINDERS IN OSCILLATING VISCOUS FLOW

Numerical results are presented for an oscillating viscous flow past a square cylinder with square and rounded corners and a diamond cylinder with square corners at Keulegan–Carpenter numbers up to 5. This unsteady flow problem is formulated by the two-dimensional Navier–Stokes equations in vorticity and stream-function form on body-fitted coordinates and solved by a finite-difference method. Second-order Adams-Bashforth and central-difference schemes are used to discretize the vorticity transport equation while a third-order upwinding scheme is incorporated to represent the nonlinear convective terms. Since the vorticity distribution has a mathematical singularity at a sharp corner and since the force coefficients are found in experiments to be sensitive to the corner radius of rectangular cylinders, a grid-generation technique is applied to provide an efficient mesh system for this complex flow. Local grid concentration near the sharp corners, instead of any artificial treatment of the sharp corners being introduced, is used in order to obtain high numerical resolution. The elliptic partial differential equation for stream function and vorticity in the transformed plane is solved by a multigrid iteration method. For an oscillating flow past a rectangular cylinder, vortex detachment occurs at irregular high frequency modes at KC numbers larger than 3 for a square cylinder, larger than 1 for a diamond cylinder and larger than 3 for a square cylinder with rounded corners. The calculated drag and inertia coefficients are in very good agreement with the experimental data. The calculated vortex patterns are used to explain some of the force coefficient behavior.

Stream functions in three‐dimensional groundwater flow

Development of the partial differential equation of stream function ψ in two‐dimensional groundwater flow is based on the assumption that the curl of hydraulic gradient vector (∇×∇ϕ) is equal to zero. The equation of ψ is expressed in terms of hydraulic conductivity. This equation is valid for steady state groundwater flow only. In this paper, two partial differential equations of stream functions χ and λ in three‐dimensional groundwater flow are developed. In development of these equations, it is assumed that the curl of pseudopotential gradient (∇×∇β) is equal to zero. Pseudopotential gradient is the hydraulic gradient in the direction of flow. The equations of χ and λ are expressed in terms of pseudopotential conductivity. Pseudopotential conductivity is the hydraulic conductivity in the direction of flow. It is a scalar function and can be defined in terms of hydraulic gradient and hydraulic conductivity. Therefore besides being valid for steady state flow, the equations of χ and λ can be applied to evaluate stream functions in anisotropic porous media at a given point in time, provided that the hydraulic gradient field is known at the point in time. Application of the χ and λ equations is demonstrated by considering steady state groundwater flow under the Borden landfill in Ontario, Canada. The two equations will be valuable in visualization of groundwater flow and contaminant transport, design of numerical solution grids for contaminant transport modeling, and design of contaminant remediation well systems.

Numerical studies of the flow around a circular cylinder by a finite element method

Numerical solutions of the steady, incompressible, viscous flow past a circular cylinder are presented for Reynolds numbers R ranging from 1 to 100. The governing Navier-Stokes equations in the form of a single, fourth order differential equation for stream function and the boundary conditions are replaced by an equivalent variational principle. The numerical method is based on a finite element approximation of this principle. The resulting non-linear system is solved by the Newton-Raphson process. The pressure field is obtained from a finite element solution of the Poisson equation once the stream function is known. The results are compared with those determined by other numerical techniques and experiments. In particular, the discussion is concerned with the development of the closed wake with Reynolds number, and the tendency of R ≥ 40 flow toward instability.