Exact analytical solutions of the stokes equation for testing the equations of mantle convection with a variable viscosity

2006 ◽  
Vol 42 (7) ◽  
pp. 537-545 ◽  
Author(s):  
V. P. Trubitsyn ◽  
A. A. Baranov ◽  
A. N. Eyseev ◽  
A. P. Trubitsyn
Keyword(s):  
Stokes Equation ◽  
Analytical Solutions ◽  
Mantle Convection ◽  
Variable Viscosity ◽  
Exact Analytical Solutions ◽  
The Stokes Equation

scholarly journals Practical analytical solutions for benchmarking of 2-D and 3-D geodynamic Stokes problems with variable viscosity

Solid Earth ◽  
2014 ◽  
Vol 5 (1) ◽  
pp. 461-476 ◽  
Author(s):  
I. Yu. Popov ◽  
I. S. Lobanov ◽  
S. I. Popov ◽  
A. I. Popov ◽  
T. V. Gerya
Keyword(s):  
Analytical Solutions ◽  
Mantle Convection ◽  
Variable Viscosity ◽  
Test Problems ◽  
Continuity Equations ◽  
Stokes Problems ◽  
Vertical Gravity ◽  
Dynamics Problems ◽  
Benchmark Solutions ◽  
Lithospheric Dynamics

Abstract. Geodynamic modeling is often related with challenging computations involving solution of the Stokes and continuity equations under the condition of highly variable viscosity. Based on a new analytical approach we have developed particular analytical solutions for 2-D and 3-D incompressible Stokes flows with both linearly and exponentially variable viscosity. We demonstrate how these particular solutions can be converted into 2-D and 3-D test problems suitable for benchmarking numerical codes aimed at modeling various mantle convection and lithospheric dynamics problems. The Main advantage of this new generalized approach is that a large variety of benchmark solutions can be generated, including relatively complex cases with open model boundaries, non-vertical gravity and variable gradients of the viscosity and density fields, which are not parallel to the Cartesian axes. Examples of respective 2-D and 3-D MatLab codes are provided with this paper.

scholarly journals Modeling tissue growth with the Stokes equation

2019 ◽  
Author(s):  
Teemu J. Häkkinen ◽  
Jukka Jernvall ◽  
Antti Hannukainen
Keyword(s):  
Stokes Equation ◽  
Variable Viscosity ◽  
Reaction Diffusion ◽  
Tissue Growth ◽  
Test Cases ◽  
Model Dynamics ◽  
A Cell ◽  
Tissue Interface ◽  
Bulk Tissue ◽  
The Stokes Equation

AbstractWe present a cell-free continuum model for simulating generalized bulk tissue growth in 3D. We assume that the tissue behaves mechanically as viscous fluid so that its behavior can be described with the Stokes equation with mass sources. The growth is directed by a diffusing morphogen produced by specialized signaling centers, whose positions are established through a reaction-diffusion system coupled with differentiation. We further assume that the tissue interface may be stiff (modeled as surface tension), and that tissue adhesion can vary (modeled as variable viscosity). The numerical validity of the implementation is investigated using test cases with known solutions, and the model dynamics are demonstrated in simulations of idealized tissue growth. The combination of Stokes equation and diffusing morphogens allow the integration of patterning and growth as in real organs systems such as limbs and teeth. We propose that the presented techniques could be useful for simulating and exploring mechanistic principles of tissue growth in various developing organs.

scholarly journals A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Ji Juan-Juan ◽  
Guo Ye-Cai ◽  
Zhang Lan-Fang ◽  
Zhang Chao-Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Space Time ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Table Lookup ◽  
Exact Analytical Solutions ◽  
Function Solution

A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

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