A Class of Exact Solutions of Equations for Plane Steady Motion of Incompressible Fluids of Variable Viscosity for finite Peclet Number through Von-Mises Coordinates

2019 ◽  
Vol 65 (3) ◽  
pp. 216-225
Author(s):  
Mushtaq Ahmed
Keyword(s):  
Exact Solutions ◽  
Peclet Number ◽  
Variable Viscosity ◽  
Steady Motion ◽  
Incompressible Fluids ◽  
Péclet Number ◽  
Solutions Of Equations ◽  
Von Mises

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.  

Thermocapillary migration of bubbles: convective effects at low Péclet number

2001 ◽  
Vol 443 ◽  
pp. 377-401 ◽  
Author(s):  
A. M. LESHANSKY ◽  
O. M. LAVRENTEVA ◽  
A. NIR
Keyword(s):  
Peclet Number ◽  
Large Time ◽  
Sufficient Conditions ◽  
Steady Motion ◽  
Separation Distance ◽  
Péclet Number ◽  
Thermocapillary Migration ◽  
Initial Separation ◽  
The Individual ◽  
Two Bubbles

The effect of a weak convective heat transfer on the thermocapillary interaction of two bubbles migrating in an externally imposed temperature gradient is examined. It is shown that, for short and moderate separation distances, the corrections to the individual migration velocities of the bubbles are of O(Pe), where Pe is Péclet number. For separation distances larger than O(Pe−1/2) the correction is of O(Pe2) as previously found for an isolated drop. The perturbations to the bubble velocities have opposite signs: the motion of the leading bubble is enhanced while the motion of the trailing one is retarded. A newly found feature is that equal-sized bubbles, which otherwise would move with equal velocities, acquire a relative motion apart from each other under the influence of convection. For slightly unequal bubbles there are three different regimes of large-time asymptotic behaviour: attraction up to the collision, infinite growth of the separation distance, and a steady migration with equal velocities, the steady motion separation distance being a function of the parameters of the problem. Sufficient conditions for the realization of each regime are given in terms of the Péclet number, initial separation and radii ratio.

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.

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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