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.

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 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.  

New Exact Solutions of Steady Plane Flows of an In Compressible Fluid of Variable Viscosity

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Sobia Younus
Keyword(s):  
Exact Solutions ◽  
Stream Function ◽  
Compressible Fluid ◽  
Variable Viscosity ◽  
Plane Motion ◽  
Transformation Technique ◽  
Vorticity Distribution ◽  
New Exact Solutions ◽  
Steady Plane ◽  
Uniform Stream

<span>Some new exact solutions to the equations governing the steady plane motion of an in compressible<span> fluid of variable viscosity for the chosen form of the vorticity distribution are determined by using<span> transformation technique. In this case the vorticity distribution is proportional to the stream function<span> perturbed by the product of a uniform stream and an exponential stream<br /><br class="Apple-interchange-newline" /></span></span></span></span>

scholarly journals A class of new exact solutions of the equations governing the steady plane flows of incompressible fluid of variable viscosity

2015 ◽  
Vol 4 (4) ◽  
pp. 429
Author(s):  
Rana Khalid Naeem ◽  
Mushtaq Ahmed
Keyword(s):  
Temperature Distribution ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Energy Function ◽  
Variable Viscosity ◽  
Flow Equations ◽  
New Exact Solutions ◽  
Viscosity Function ◽  
Steady Plane ◽  
Infinite Set

<p>The objective of this paper is to indicate a class of new exact solutions of the equations governing the steady plane flows of incompressible fluid of variable viscosity. The class consists of the stream function characterized by equation (2). Exact solutions are determined for  and  When is arbitrary we can construct an infinite set of streamlines and the velocity components, viscosity function, generalized energy function  and temperature distribution . Therefore, an infinite set of solutions to flow equations. When  is not arbitrary, there are two values of  and therefore, two exact solutions to flow equations. The streamlines are presented through Fig.(1–56) for some chosen from of f(r).</p>

scholarly journals Body-force modeling for aerodynamic analysis of air intake – fan interactions

2016 ◽  
Vol 26 (7) ◽  
pp. 2048-2065 ◽  
Author(s):  
William Thollet ◽  
Guillaume Dufour ◽  
Xavier Carbonneau ◽  
Florian Blanc
Keyword(s):  
Body Force ◽  
Stokes Equations ◽  
The Body ◽  
Test Case ◽  
Force Model ◽  
Strong Impact ◽  
Analytic Model ◽  
Source Terms ◽  
Content Type ◽  
Air Intake

Purpose The purpose of this paper is to explore a methodology that allows to represent turbomachinery rotating parts by replacing the blades with a body force field. The objective is to capture interactions between a fan and an air intake at reduced cost, as compared to full annulus unsteady computations. Design/methodology/approach The blade effects on the flow are taken into account by adding source terms to the Navier-Stokes equations. These source terms give the proper amount of flow turning, entropy, and blockage to the flow. Two different approaches are compared: the source terms can be computed using an analytic model, or they can directly be extracted from RANS computations with the blade’s geometry. Findings The methodology is first applied to an isolated rotor test case, which allows to show that blockage effects have a strong impact on the performance of the rotor. It is also found that the analytic body force model underestimates the mass flow in the blade row for choked conditions. Finally, the body force approach is used to capture the coupling between a fan and an air intake at high angle of attacks. A comparison with full annulus unsteady computations shows that the model adequately captures the potential effects of the fan on the air intake. Originality/value To the authors’ knowledge, it is the first time that the analytic model used in this paper is combined with the blockage source terms. Furthermore, the capability of the model to deal with flows in choked conditions was never assessed.

New Exact Axisymmetric Solutions to the Navier–Stokes Equations

2019 ◽  
Vol 75 (1) ◽  
pp. 29-42
Author(s):  
Oleg Bogoyavlenskij
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Dimensional Space ◽  
Phase Portraits ◽  
Navier Stokes ◽  
Elementary Functions ◽  
Navier Stokes Equations ◽  
New Exact Solutions ◽  
Whole Space ◽  
Dependent Viscosity

AbstractInfinite-dimensional space of axisymmetric exact solutions to the Navier–Stokes equations with time-dependent viscosity $\nu(t)$ is constructed. Inner transformations of the exact solutions are defined that produce an infinite sequence of new solutions from each known one. The solutions are analytic in the whole space ℝ3 and are described by elementary functions. The bifurcations of the instantaneous (for $t={t_{0}}$) phase portraits of the viscous fluid flows are studied for the new exact solutions. Backlund transforms between the axisymmetric Helmholtz equation and a linear case of the Grad–Shafranov equation are derived.

scholarly journals Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces

2019 ◽  
Vol 150 (2) ◽  
pp. 569-606 ◽  
Author(s):  
Dat Cao ◽  
Luan Hoang
Keyword(s):  
Asymptotic Expansion ◽  
Body Force ◽  
Stokes Equations ◽  
Three Dimensional ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Gevrey Space ◽  
Long Time ◽  
The Individual

AbstractThe Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

Some new exact solutions for the two-dimensional Navier–Stokes equations

2007 ◽  
Vol 371 (5-6) ◽  
pp. 438-452 ◽  
Author(s):  
Chiping Wu ◽  
Zhongzhen Ji ◽  
Yongxing Zhang ◽  
Jianzhong Hao ◽  
Xuan Jin
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
New Exact Solutions

Linear Analysis of the Currents in a Pipe

1986 ◽  
Vol 41 (9) ◽  
pp. 1141-1153
Author(s):  
U. Brosa
Keyword(s):  
Exact Solutions ◽  
Cross Section ◽  
Bulk Viscosity ◽  
Stokes Equations ◽  
Simple Procedure ◽  
Circular Cross Section ◽  
Linear Analysis ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Infinite Set

A simple procedure to find solutions of the hydrodynamic Stokes equations is given. The procedure is used to determine the linear modes of a newtonian fluid in a pipe of circular cross section. Compressibility, shear and bulk viscosity are included, and no restrictions on the symmetry of the modes are made. Furthermore an infinite set of exact solutions of the Navier-Stokes equations is presented.

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