scholarly journals A Class of Exact Solutions for a Variable Viscosity Flow with Body Force for Moderate Peclet Number ViaVon-Mises Coordinates

2019 ◽  
Vol 5 (1) ◽  
pp. 15
Author(s):  
Mushtaq Ahmed
Keyword(s):  
Exact Solutions ◽  
Peclet Number ◽  
Body Force ◽  
Variable Viscosity ◽  
Péclet Number

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 Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface

2021 ◽  
Vol 131 (1) ◽  
Author(s):  
Darren G. Crowdy
Keyword(s):  
Exact Solutions ◽  
Peclet Number ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Implicit Representation ◽  
Péclet Number ◽  
New Exact Solutions ◽  
Kinetics Modelling ◽  
Complex Burgers Equation ◽  
Insoluble Surfactant

AbstractA class of exact solutions is presented describing the time evolution of insoluble surfactant to a stagnant cap equilibrium on the surface of deep water in the Stokes flow regime at zero capillary number and infinite surface Péclet number. This is done by demonstrating, in a two-dimensional model setting, the relevance of the forced complex Burgers equation to this problem when a linear equation of state relates the surface tension to the surfactant concentration. A complex-variable version of the method of characteristics can then be deployed to find an implicit representation of the general solution. A special class of initial conditions is considered for which the associated solutions can be given explicitly. The new exact solutions, which include both spreading and compactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-order reaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leads to a forced complex Burgers equation with linear damping. Generalized exact solutions to the latter equation at infinite surface Péclet number are also found and used to study how reaction effects destroy the surfactant cap equilibrium.

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.

Low Peclet Number Heat and Mass Transfer from a Drop in an Electric Field

1979 ◽  
Vol 101 (3) ◽  
pp. 484-488 ◽  
Author(s):  
S. K. Griffiths ◽  
F. A. Morrison
Keyword(s):  
Mass Transfer ◽  
Electric Field ◽  
Exact Solutions ◽  
Heat And Mass Transfer ◽  
Digital Computer ◽  
Peclet Number ◽  
Perturbation Expansion ◽  
Regular Perturbation ◽  
Péclet Number

An electric field, when applied to a dielectric drop suspended in another such fluid, generates a circulating motion. The low Peclet number transport from the drop is investigated analytically using a regular perturbation expansion. A digital computer is used to obtain exact solutions to the resulting equations. These solutions yield accurate results up to a Peclet number of at least 60.

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. 

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