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H = p ρ + 1 2 v 2 = constant p = - 1 2 ρ v 2 + constant , since ρ is also constant

The pressure relative to some datum can be determined by the square of the magnitude of velocity. This is easily calculated from the complex potential.

d w d z = v x - i v y d w d z ¯ = v x + i v y thus d w d z d w d z ¯ = v x 2 + v y 2 = v 2

There are some theorems that facilitate the integration of pressure around bodies in the complex plane, but they will not be discussed here. The pressure and tangential velocity profiles for the inviscid flow around an object are needed for calculation of the viscous flow in the boundary layer between the solid boundary and the inviscid outer flow.

Assignment 7.6

Pressure profiles Calculate the pressure field or the square of the velocity field for the flow in or around a corner and the flow past a circular cylinder. Look at the expression for the corner flow. Under what conditions is there a flow singularity? Show the pressure or velocity squared pseudo-color for wall angles of π / 2 , π , 3 π / 2 , and 2 π . Which cases are physically realistic and what do you think happens in the unrealistic cases? What is special about the pressure profile around the circular cylinder? What value of form drag will it predict? Is it realistic and if not, why not? Add the following code to the code for corner flow and flow around a circular cylinder. pause

% Calculate pressure distribution from square of velocity (your code here to calculate pressure field)pcolor(x,y,p) colormap(hot)shading flat axis image

Solution of hyperbolic systems

The conservation equations for material, momentum, and energy reduce to first order PDE in the absence of diffusivity, dispersion, viscosity, and heat conduction. In thin films, viscosity may be a dominant effect in the velocity profile normal to the surface but the continuity equation integrated over the film thickness will have only first order spatial derivatives unless the effects of interfacial curvature become important. In one dimension, the system of first order partial differential equations can be calculated by the method of characteristics [A. Jeffrey (1976), H.-K. Rhee, R. Aris, and N. R. Amundson (1986, 1987)]. Here we will only consider the case of a single dependent variable with constant initial and boundary conditions. Denote the dependent variable as S and the independent variables as x and t . The differential equation with its initial and boundary conditions are as follows.

S t + f ( S ) x = 0 , t > 0 , x > 0 S ( x , 0 ) = S I C f ( 0 , t ) = f B C

The dependent variable can be normalized such that the initial condition is equal to zero and the boundary condition is equal to unity. Thus, henceforth it is assumed such a transformation has been made. The dependent variable will be called 'saturation' and the flux called 'fractional flow' to use the nomenclature for multiphase flow in porous media. However, the dependent variable could be film thickness in film drainage or height of a free surface as in water waves. The PDE, IC, and BC are rewritten as follows.

S t + d f ( S ) d S S x = 0 , t > 0 , x > 0 S ( x , 0 ) = 0 f ( 0 , t ) = 1

The differential, d f / d S is easily calculated since there is only one independent saturation. If there were three or more phases this differential would be a Jacobian matrix. The locus of constant saturation will be sought by taking the total differential of S ( x , t ) .

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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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