Analytical solution of Taylor circulation in a prolate ellipsoid droplet in the frame of 2D Stokes equations

2019 ◽  
Vol 207 ◽  
pp. 145-152 ◽  
Author(s):  
Ilya V. Makeev ◽  
Igor Yu. Popov ◽  
Rufat Sh. Abiev
Keyword(s):  
Analytical Solution ◽  
Stokes Equations ◽  
Prolate Ellipsoid

An analytical solution for the Marangoni mixed convection boundary layer flow

2010 ◽  
Vol 224 (6) ◽  
pp. 1193-1202 ◽  
Author(s):  
M A Moghimi ◽  
A Kimiaeifar ◽  
M Rahimpour ◽  
G H Bagheri
Keyword(s):  
Boundary Layer ◽  
Analytical Solution ◽  
Mixed Convection ◽  
Boundary Layer Flow ◽  
Stokes Equations ◽  
Convection Boundary Layer ◽  
Effective Parameters ◽  
Linear Ordinary Differential Equations ◽  
Convection Boundary ◽  
Layer Flow

In this article, an analytical solution for a Marangoni mixed convection boundary layer flow is presented. A similarity transform reduces the Navier—Stokes equations to a set of non-linear ordinary differential equations, which are solved analytically by means of the homotopy analysis method (HAM). The results obtained in this study are compared with the numerical results released in the literature. A close agreement of the two sets of results indicates the accuracy of the HAM. The method can obtain an expression that is acceptable for all values of effective parameters and is also able to control the convergence of the solution. The numerical solution of the similarity equations is developed and the results are in good agreement with the analytical results based on the HAM.

The Effect of Slip on the Discharge From Partially Filled Circular and Fully Filled Lens and Figure 8 Shaped Pipes

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Samuel Irvine ◽  
Luke Fullard
Keyword(s):  
Free Surface ◽  
Analytical Solution ◽  
Flow Rate ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Symmetry Plane ◽  
Volumetric Flow Rate ◽  
Wall Slip ◽  
Circular Pipe ◽  
Gravity Driven Flow

In this work, we examine the effect of wall slip for a gravity-driven flow of a Newtonian fluid in a partially filled circular pipe. An analytical solution is available for the no-slip case, while a numerical method is used for the case of flow with wall slip. We note that the partially filled circular pipe flow contains a free surface. The solution to the Navier–Stokes equations in such a case is a symmetry of a pipe flow (with no free surface) with the free surface as the symmetry plane. Therefore, we note that the analytical solution for the partially filled case is also the exact solution for fully filled lens and figure 8 shaped pipes, depending on the fill level. We find that the presence of wall slip increases the optimal fill height for maximum volumetric flow rate, brings the “velocity dip” closer to the free surface, and increases the overall flow rate for any fill. The applications of the work are twofold; the analytical solution may be used to verify numerical schemes for flows with a free surface in partially filled circular pipes, or for pipe flows in lens and figure 8 shaped pipes. Second, the work suggests that flows in pipes, particularly shallow filled pipes, can be greatly enhanced in the presence of wall slip, and optimal fill levels must account for the slip phenomenon when present.

On the Immersed Boundary Method: Finite Element Versus Finite Volume Approach

2012 ◽  
Author(s):  
Angelo Frisani ◽  
Yassin A. Hassan
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Finite Volume ◽  
Stokes Equations ◽  
Numerical Results ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Order Accuracy ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow

A projection approach is presented for the coupled system of time-dependent incompressible Navier-Stokes equations in conjunction with the Immersed Boundary Method (IBM) for solving fluid flow problems in the presence of rigid objects not represented by the underlying mesh. The IBM allows solving the flow for geometries with complex objects without the need of generating a body fitted mesh. The no-slip boundary constraint is satisfied applying a boundary force at the immersed body surface. Using projection and interpolation operators from the fluid volume mesh to the solid surface mesh (i.e., the “immersed” boundary) and vice versa, it is possible to impose the extra constraint to the incompressible Navier-Stokes equations as a Lagrange multiplier in a fashion very similar to the effect pressure has on the momentum equations to satisfy the divergence-free constraint. The projection operation removes the immersed boundary surface slip and non-divergence-free components of the velocity field. The boundary force is determined implicitly at the inner iterations of the fractional step method implemented. No constitutive relations for the immersed boundary objects fluid interaction are required, allowing the formulation introduced to use larger CFL numbers compared to previous methodologies. An overview of the immersed boundary approach is presented showing third order accuracy in space and second order accuracy in time when the simulation results for the Taylor-Green decaying vortex are compared to the analytical solution using the Immersed Finite Element Method (IFEM). For the Immersed Finite Volume Method (IFVM) a ghost-cell approach is used. Second order accuracy in space and first order accuracy in time are obtained when the Taylor-Green decaying vortex test case is compared to the analytical solution. The numerical results are compared with the analytical solution also for adaptive mesh refinement (for the IFEM) showing an excellent error reduction. Computations were performed using IFEM and IFVM approaches for the time-dependent incompressible Navier-Stokes equations in a two-dimensional flow past a stationary circular cylinder at Re = 20, and 40, where shedding effects are not present. The drag coefficient and the recirculation length error compared to the experimental data is less than 3–4%. Simulations for the two-dimensional flow past a stationary circular cylinder at Re = 100 were also performed. For Re numbers above 46, unsteadiness generates vortex shedding, and an unsteady flow regime is present. The results shown are in excellent quantitative and qualitative agreement with the flow pattern expected. The numerical results obtained with the discussed IFEM and IFVM were also compared against other immersed boundary methodologies available in literature and simulation performed with the commercial computational fluid dynamics code STAR-CCM+/V5.02.009 for which a body fitted finite volume numerical discretization was used. The benchmark showed that the numerical results obtained with the implemented immersed boundary methods are very close to those obtained from STAR-CCM+ with a very fine mesh and in a good agreement with the other IBM techniques. The IBM based of finite element approach is numerically more accurate than the IBM based on finite volume discretization. In contrast, the latter is computationally more efficient than the former.

scholarly journals A semi-analytical solution for the mean wind profile in the Atmospheric Boundary Layer: the convective case

2009 ◽  
Vol 9 (5) ◽  
pp. 19817-19844
Author(s):  
L. Buligon ◽  
G. A. Degrazia ◽  
O. C. Acevedo ◽  
C. R. P. Szinvelski ◽  
A. G. O. Goulart
Keyword(s):  
Boundary Layer ◽  
Analytical Solution ◽  
Wind Profile ◽  
Stokes Equations ◽  
Integral Transform ◽  
Integral Transforms ◽  
Navier Stokes ◽  
Convective Boundary ◽  
Navier Stokes Equations ◽  
Average Wind

Abstract. A novel methodology to derive the average wind profile from the Navier-Stokes equations is presented. The development employs the Generalized Integral Transform Technique (GITT), which joints series expansions with Integral Transforms. The new approach provides a solution described in terms of the quantities that control the wind vector with height. Parameters, such as divergence and vorticity, whose magnitudes represent sinoptic patterns are contained in the semi-analytical solution. The results of this new method applied to the convective boundary layer are shown to agree with wind data measured in Wangara experiment.

MHD FLOWS DUE TO NON-COAXIAL ROTATIONS OF POROUS DISK AND A VISCOUS FLUID AT INFINITY: GRAPHICAL SOLUTIONS USING MATLAB

2020 ◽  
Vol 9 (11) ◽  
pp. 9287-9301
Author(s):  
R. Lakshmi ◽  
Santhakumari
Keyword(s):  
Velocity Field ◽  
Analytical Solution ◽  
Viscous Fluid ◽  
Stokes Equations ◽  
Daily Life ◽  
Vital Role ◽  
Navier Stokes ◽  
Porous Disk ◽  
Navier Stokes Equations ◽  
Complex Phenomena

Fluids play a vital role in many aspects of our daily life. We drink water, breath air, fluids run through our bodies and it controls the weather. The study of motion of fluids is a complex phenomena. The equations which govern the flows of Newtonian fluids are Navier-Stokes equations. In this paper, the flows which are due to non – coaxial rotations of porous disk and a fluid at infinity are considered. Analytical solution for the velocity field using Laplace transform is derived. MATLAB coding is written to get the graphical solutions. The results are compared with the existing results. MATLAB software provides accurate results depending on the solution we obtained.

Approximate Analytical Solution for Laminar Flow Over a Backward Step

2015 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Analytical Solution ◽  
Stokes Equations ◽  
Approximate Analytical Solution ◽  
Continuous Functions ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Mesh Free ◽  
Numeric Calculation ◽  
Domain Discretization

Analytical solution of Navier-Stokes equations are extremely difficult and rare. It is one of the unsolved Clay Millennium problems in mathematics. Many solutions that exist are examples of degenerate cases where the nonlinearity is controlled. In this paper we explore the application of Bézier functions to solve the two-dimensional laminar fluid flow over a backward step. The Bézier functions provide a mesh free alternative to domain discretization methods that are currently used to solve such problems. The Navier-Stokes equation are handled directly without transformation and the setup is direct, simple, and involves minimizing the error in the residuals of the differential equations along with the error on the boundary conditions over the domain. The solutions for the velocity and pressure are available in polynomial form. They are single continuous functions over the entire domain. The procedure employs a combination of symbolic and numeric calculation in MATLAB. Two problems are explored. The first is the flow in a 2D channel to illustrate the technique. The second is the flow over the backward step. The solutions are compared to the corresponding finite element solutions from COMSOL Multiphysics software.

A further note on axially symmetric flows of elastico-viscous liquids

1973 ◽  
Vol 73 (1) ◽  
pp. 239-247 ◽  
Author(s):  
J. R. Jones
Keyword(s):  
Analytical Solution ◽  
Closed Form ◽  
Body Force ◽  
Present Note ◽  
Axially Symmetric ◽  
Surfaces Of Revolution ◽  
Small Eccentricity ◽  
Viscous Liquids ◽  
Prolate Ellipsoid ◽  
Arbitrary Section

In earlier papers of the same main title Jones and Lewis(1) (here-after referred to as I†) and Jones (2) studied, using ‘virtual body force’ methods, the axially symmetric flows of elastico-viscous liquids caused by the rotation of surfaces of revolution of arbitrary section. Particular attention was paid in I to the (oblate and prolate) ellipsoid geometry, but, for reasons of mathematical facility, the analysis was restricted to that of small eccentricity, the interesting and degenerate scheme of maximum eccentricity being outside its range of validity. It is the main purpose of the present note to broaden the scope of the analysis of I to cover a wider class of liquids and to present an exact analytical solution (in simple closed form) for the ellipsoid geometry.

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