Numerical Simulation Of Marangoni-Driven Simulation Analysis
Numerical Simulation of Marangoni-Driven Boundary Layer Flow Over a Flat Plate with an Imposed Temperature Distribution
Abstract: A numerical algorithm is presented for studying Marangoni convection flow over a flat plate with an imposed temperature distribution. Plate temperature varies with x in the following prescribed manner: where A and k are constants. By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a pair of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab environment. Velocity …show more content…
The boundary conditions are then:
3.2 Solution to Initial Value Problems
To solve eqs (12), we denote the two unknown initial values by a1 and a2, the set of initial conditions is then:
If eqs (12) are solved with adaptive Runge-Kutta method using the initial conditions in eq (14), the computed boundary values at depend on the choice of a1 and a2 respectively. We express this dependence as
The correct choice of a1 and a2 yields the given boundary conditions at ; that is, it satisfies the equations These nonlinear equations are solved by the Newton-Raphson method.
3.3 Program Details
This section describes a set of Matlab routines for the solution of eqs (12) along with the boundary conditions (14). They are listed in Table 1. Table 1. A set of Matlab routines used sequentially to solve Equations (12).
Matlab code Brief …show more content…
So, over most of the domain of η, is zero and is . Hence, energy equation (8) may be approximated as
Its solution subject to the boundary conditions (10) is
The temperature gradient at the plate surface is then
Therefore, in the low Prandtl number region, the surface temperature gradient , should be proportional to the Prandtl number.
In the high Prandtl number region, the thermal boundary layer thickness is much smaller than the momentum boundary layer thickness. So, over most of the domain of η, is zero and is . Hence, energy equation (8) may be approximated as
Its solution is
and temperature gradient at the plate surface is
Therefore, in the high Prandtl number region, the surface temperature gradient , should be proportional to the square root of Prandtl number.
The variation of surface temperature gradient with Prandtl number for several values of temperature gradient exponent is drawn
Abstract: A numerical algorithm is presented for studying Marangoni convection flow over a flat plate with an imposed temperature distribution. Plate temperature varies with x in the following prescribed manner: where A and k are constants. By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a pair of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab environment. Velocity …show more content…
The boundary conditions are then:
3.2 Solution to Initial Value Problems
To solve eqs (12), we denote the two unknown initial values by a1 and a2, the set of initial conditions is then:
If eqs (12) are solved with adaptive Runge-Kutta method using the initial conditions in eq (14), the computed boundary values at depend on the choice of a1 and a2 respectively. We express this dependence as
The correct choice of a1 and a2 yields the given boundary conditions at ; that is, it satisfies the equations These nonlinear equations are solved by the Newton-Raphson method.
3.3 Program Details
This section describes a set of Matlab routines for the solution of eqs (12) along with the boundary conditions (14). They are listed in Table 1. Table 1. A set of Matlab routines used sequentially to solve Equations (12).
Matlab code Brief …show more content…
So, over most of the domain of η, is zero and is . Hence, energy equation (8) may be approximated as
Its solution subject to the boundary conditions (10) is
The temperature gradient at the plate surface is then
Therefore, in the low Prandtl number region, the surface temperature gradient , should be proportional to the Prandtl number.
In the high Prandtl number region, the thermal boundary layer thickness is much smaller than the momentum boundary layer thickness. So, over most of the domain of η, is zero and is . Hence, energy equation (8) may be approximated as
Its solution is
and temperature gradient at the plate surface is
Therefore, in the high Prandtl number region, the surface temperature gradient , should be proportional to the square root of Prandtl number.
The variation of surface temperature gradient with Prandtl number for several values of temperature gradient exponent is drawn