scholarly journals A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force

2018 ◽  
Vol 7 (3) ◽  
pp. 77
Author(s):  
Mushtaq Ahmed ◽  
Waseem Ahmed Khan ◽  
S M. Shad Ahsen
Keyword(s):  
Exact Solutions ◽  
Body Force ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Steady Motion ◽  
Incompressible Fluids ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
First Case

This paper determines a class of exact solutions for plane steady motion of incompressible fluids of variable viscosity with body force term in the Navier-Stokes equations. The class consists of stream function  characterized by equation , in polar coordinates ,  where ,  and  are continuously differentiable functions, derivative of  is non-zero but double derivative of  is zero. We find exact solutions, for a suitable component of body force, considering two cases based on velocity profile. The first case fixes both the functions ,  and provides viscosity as function of temperature. Where as the second case fixes the function , leaves  arbitrary and provides viscosity and temperature for the arbitrary function . In both the cases, we can create infinite set of expressions for streamlines, viscosity function, generalized energy function and temperature distribution in the presence of body force.  

scholarly journals A Class of New Exact Solutions of Navier-Stokes Equations with Body Force For Viscous Incompressible Fluid

2018 ◽  
Vol 7 (1) ◽  
pp. 20
Author(s):  
Mushtaq Ahmed ◽  
Rana Khalid Naeem ◽  
Syed Anwer Ali
Keyword(s):  
Incompressible Fluid ◽  
Exact Solutions ◽  
Body Force ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Steady Motion ◽  
The Body ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
New Exact Solutions

This paper is to indicate a class of new exact solutions of the equations governing the two-dimensional steady motion of incompressible fluid of variable viscosity in the presence of body force. The class consists of the stream function $\psi$ characterized by equation $\theta=f(r)+ a \psi + b $ in polar coordinates $r$, $\theta$ , where a continuously differentiable function is $f(r)$ and $a\neq 0 , b $ are constants. The exact solutions are determined for given one component of the body force, for both the cases when $f(r)$ is arbitrary and when it is not. When $f(r)$ is arbitrary, we find $a=1$ and we can construct an infinite set of streamlines and the velocity components, viscosity function, generalized energy function and temperature distribution for the cases when $R_{e}P_{r}=1$ and when $R_{e}P_{r}\neq 1$ where $R_{e}$ represents Reynolds number and $P_{r}$Prandtl number. For the case when $f(r)$ is not arbitrary we can find solutions for the cases $R_{e}P_{r}\neq a$ and $R_{e}P_{r}=a$ where $"a"$ remains arbitrary. 

scholarly journals A class of new exact solutions of the system of PDE for the plane motion of viscous incompressible fluids in the presence of body force

2018 ◽  
Vol 7 (2) ◽  
pp. 42
Author(s):  
Mushtaq Ahmed ◽  
Waseem Ahmed Khan
Keyword(s):  
Exact Solutions ◽  
Body Force ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Plane Motion ◽  
The Body ◽  
Polar Coordinates ◽  
Temperature Function ◽  
New Exact Solutions ◽  
Infinite Set

The purpose of this paper is to indicate a class of exact solutions of the system of partial differential equations governing the steady, plane motion of incompressible fluid of variable viscosity with body force term to the right-hand side of Navier-Stokes equations. The class consists of the stream function characterized by the equation  in polar coordinates  and  where  and  are continuously differentiable functions and the function  is such that  where a non-zero constant is  and overhead prime represents derivative with respect to . When  or  we show exact solutions for given one component of the body force for both the cases when the function  is arbitrary and when it is not. For the arbitrary function case,  appears in the coefficient of a linear second order ordinary differential equation showing a large numbers of solutions of this equation. This in turn establishes an infinite set of exact solutions to the problem concerned however; we show three examples of such exact solutions. The alternate case fixes  and provides viscosity as derivative of temperature function for  and . Anyhow, we find an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution.

Numerical Study of Viscous Flows Inside Partially Filled Spinning and Coning Cylinders

1996 ◽  
Vol 118 (2) ◽  
pp. 335-340 ◽  
Author(s):  
Mohamed Selmi
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Center Of Mass ◽  
Steady Motion ◽  
Practical Interest ◽  
Nonlinear Effects ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Practical Applications ◽  
Analysis And Design

This paper is concerned with the solution of the 3-D-Navier-Stokes equations describing the steady motion of a viscous fluid inside a partially filled spinning and coning cylinder. The cylinder contains either a single fluid of volume less than that of the cylinder or a central rod and a single fluid of combined volume (volume of the rod plus volume of the fluid) equal to that of the cylinder. The cylinder rotates about its axis at the spin rate ω and rotates about an axis that passes through its center of mass at the coning rate Ω. In practical applications, as in the analysis and design of liquid-filled projectiles, the parameter ε = τ sin θ, where τ = Ω/ω and θ is the angle between spin axis and coning axis, is small. As a result, linearization of the Navier-Stokes equations with this parameter is possible. Here, the full and linearized Navier-Stokes equations are solved by a spectral collocation method to investigate the nonlinear effects on the moments caused by the motion of the fluid inside the cylinder. In this regard, it has been found that nonlinear effects are negligible for τ ≈ 0.1, which is of practical interest to the design of liquid-filled projectiles, and the solution of the linearized Navier-Stokes equations is adequate for such a case. However, as τ increases, nonlinear effects increase, and become significant as ε surpasses about 0.1. In such a case, the nonlinear problem must be solved. Complete details on how to solve such a problem is presented.

On the Generalized Navier-Stokes Equations

2003 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Ahmed Salem
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Hankel Transforms ◽  
Fourier Sine ◽  
Special Cases

In this paper, we present a general Inodel of the classical Navier-Stokes equations. With the help of Laplace, Fourier Sine transforms, finite Fourier Sine transforms, and finite Hankel transforms, an exact solutions for three different special cases have been obtained.

On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes equations in polar coordinates

2018 ◽  
Vol 33 (3) ◽  
pp. 199-210 ◽  
Author(s):  
Alexander Zlotnik
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Physical Property ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Uniform Mesh ◽  
Viscous Stress ◽  
Spatial Discretization ◽  
Navier Stokes Equations ◽  
Viscous Stress Tensor

Abstract The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier–Stokes system involving additional 2nd order terms with a regularizing parameter τ > 0. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier–Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced. The main results are valid for τ = 0 as well, i.e., for the barotropic compressible Navier–Stokes system.

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