Appendix 3: Notes on Three-Dimensional Differential Calculus and the Fundamental Equations of Electrostatics

2005 ◽  
pp. 179-179
Keyword(s):  
Differential Calculus ◽  
Three Dimensional ◽  
Fundamental Equations

scholarly journals Quantum Physics as Dynamic Space-Time Differential Calculus

2017 ◽  
Vol 1 (1) ◽  
Keyword(s):  
Differential Calculus ◽  
Quantum Physics ◽  
Three Dimensional ◽  
Truncated Normal Distribution ◽  
Two Dimensional Image ◽  
Time Differential ◽  
Inverse Radius ◽  
Truncated Normal ◽  
Log Normal ◽  
First Time

For the first time, a unified quantum metric system has been developed analytically without any artifacts, such as m, s, and kg without measurements at all. Energy diagrams of Feynman are replaced by calculations of relative spacetime differentials. The main constants of quantum physics are, in fact, the dynamic gradients of the normal, halfnormal, log-normal and truncated normal distribution of the inverse radius of the pulsating spiral. Quantum physics as a whole is a logarithmically compressed two-dimensional image of the three-dimensional motion of wave fronts.

scholarly journals Three-Dimensional Rotating Flow of MHD Jeffrey Fluid Flow between Two Parallel Plates with Impact of Hall Current

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Mehreen Fiza ◽  
Abdelaziz Alsubie ◽  
Hakeem Ullah ◽  
Nawaf N. Hamadneh ◽  
Saeed Islam ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Hall Current ◽  
Three Dimensional ◽  
Velocity Fields ◽  
Rotating Flow ◽  
Jeffrey Fluid ◽  
Physical Parameters ◽  
Rotating Frame ◽  
Parallel Plates ◽  
Fundamental Equations

This article deals with three-dimensional non-Newtonian Jeffrey fluid in rotating frame in the presence of magnetic field. The flow is studied in the application of Hall current, where the flow is assumed in steady states. The upper plate is considered fixed, and the lower is kept stretched. The fundamental equations are transformed into a set of ordinary differential equations (ODEs). A homotopy technique is practiced for a solution. The variation in the skin friction and its effects on the velocity fields have been examined numerically. The effects of physical parameters are discussed in various plots.

LES of Three-Dimensional Turbulent Separated and Reattached Flow Around a Surface-Mounted Square Plate

2002 ◽  
Author(s):  
Terukazu Ota ◽  
Eishi Takeuchi ◽  
Hiroyuki Yoshikawa
Keyword(s):  
Three Dimensional ◽  
Narrow Channel ◽  
Complicated Structure ◽  
Smagorinsky Model ◽  
Structure Of Flow ◽  
Horseshoe Vortices ◽  
Square Plate ◽  
The Finite Difference Method ◽  
Separated And Reattached Flow ◽  
Fundamental Equations

LES method is applied to simulate numerically a three-dimensional turbulent separated and reattached flow around a surface-mounted square plate. Smagorinsky model is used in the analysis and fundamental equations are discretized by mean of the finite difference method. The calculations are made on the flow in a relatively narrow channel width at Re = 105. It is found that the flow around the plate separates and reattaches onto the surface and the horseshoe vortices are formed, resulting in a complicated structure of flow.

A Novel Approach in the Direct Kinematics of Stewart Platform Mechanisms with Planar Platforms

2004 ◽  
Vol 128 (1) ◽  
pp. 252-263 ◽  
Author(s):  
İ. D. Akçali ◽  
H. Mutlu
Keyword(s):  
Three Dimensional ◽  
Stewart Platform ◽  
Rigid Bodies ◽  
Dimensional Problem ◽  
Solution Sets ◽  
Novel Approach ◽  
Efficiency And Effectiveness ◽  
To Come ◽  
Fundamental Equations ◽  
Direct Kinematics

In handling the kinematic analysis of two rigid bodies connected to each other by six legs through the use of six double spherical joints, methods have been implemented both in the formulation and solution phases of the problem. A three-dimensional problem has been viewed, in fact, as a multitude of two-dimensional works on several planes, the intersections of which yield relationships allowing transition between adjacent planes. Thus formulation is purely based on the geometric structure consisting of eight planes of interest, ending in the three fundamental equations involving three angles between the base and side triangular planes. In solving the resulting three equations, an efficient strategy has been established to come up with 16 solution sets effectively. Extensions of the theory have been shown to include the analyses of other Stewart platform models. Efficiency and effectiveness of the approach has been verified on numerical examples.

scholarly journals Study on the theory of three-dimensional limit analysis of soil and practical calculation

2018 ◽  
Vol 175 ◽  
pp. 03013
Author(s):  
Bin Lia ◽  
Jianbao Fu
Keyword(s):  
Stress Analysis ◽  
Equilibrium Equation ◽  
Stress Component ◽  
Three Dimensional ◽  
Yield Condition ◽  
Computation Method ◽  
Practical Calculation ◽  
Equilibrium Equations ◽  
Fundamental Equations ◽  
The Relationship

Based on the stress analysis of a point in space, the stress analysis of a point on a surface is performed, and then the relationship among the normal stress, the shear stress, and the stress component of a point on a surface and the first order partial derivative of the surface equation is deduced. Equilibrium equations of soil columns between the sliding surface and the top surface of the slope are established, which include differential equilibrium equation of force, equilibrium equation of force, and equilibrium equation of moment. These equilibrium equations and Coulomb yield condition can form the fundamental equations of three-dimensional slope stability analysis. Applying the supposition similar to that applied in the simplified Bishop method, a kind of three-dimensional slope analysis method can be obtained. An example is presented to show that the computation method is reasonable and applicable.

scholarly journals On a Modified Form of Navier-Stokes Equations for Three-Dimensional Flows

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
J. Venetis
Keyword(s):  
Stokes Equations ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Vector Form ◽  
Navier Stokes Equations ◽  
First Order ◽  
Modified Equations ◽  
Cartesian Coordinate ◽  
Fundamental Equations

A rephrased form of Navier-Stokes equations is performed for incompressible, three-dimensional, unsteady flows according to Eulerian formalism for the fluid motion. In particular, we propose a geometrical method for the elimination of the nonlinear terms of these fundamental equations, which are expressed in true vector form, and finally arrive at an equivalent system of three semilinear first order PDEs, which hold for a three-dimensional rectangular Cartesian coordinate system. Next, we present the related variational formulation of these modified equations as well as a general type of weak solutions which mainly concern Sobolev spaces.

Elasticity Theory of Plates and a Refined Theory

1979 ◽  
Vol 46 (3) ◽  
pp. 644-650 ◽  
Author(s):  
Shun Cheng
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Three Dimensional ◽  
Fundamental Equation ◽  
Present Theory ◽  
Refined Theory ◽  
Refined Plate Theory ◽  
Elasticity Equations ◽  
Theory Of Plates ◽  
Fundamental Equations

A method for the solution of three-dimensional elasticity equations is presented and is applied to the problem of thick plates. Through this method three governing differential equations, the well-known biharmonic equation, a shear equation and a third governing equation, are deduced directly and systematically from Navier’s equations. It is then shown that the solution of the second fundamental equation (the shear equation) is in fact related to the shear deformation in the bending of plates, hence it may be appropriately called the shear solution and the equation the shear equation. Moreover, it is found that the solution of the third fundamental equation does not yield transverse shearing forces. Because of these results, a refined plate theory which takes into account the transverse shear deformation can now be explicitly established without employing assumptions. With the present theory three boundary conditions at each edge of the plate and all the fundamental equations of elasticity can be satisfied. As an illustrative example, the present theory is applied to the problem of torsion resulting in exactly the same solution as the Saint Venant’s solution of torsion, although the two approaches are appreciably different. The second example also illustrates that accurate solutions, as compared with exact solutions, can be obtained by means of the refined plate theory.

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