Answer to Question #215691 in Differential Equations for Wavie

Question

The velocity distribution in a two dimensional steady flow field is xy plan is v=(Ax-B)i+(C-Ay)j,A=2s^-1,B=5ms^-1,C=3ms^-1. The coordinates are measured in meters and body force distributing is g_x=-gk.Does the velocity field represent an incompressible fluid?Find the stagnation point of the flow field. Obtain expression for the pressures gradient in the flow field. Evaluate the difference at(x,y)=(1,3) and origin if density is 1.2kg/m³

Answer 1

Solution;

a) Incompressible flow

$\frac{du}{dx}+\frac{dv}{dy}=0$

Given:

u=Ax-B

$\frac{du}{dx}$=A

v=C-Ay

$\frac{dv}{dy}$ =-A

$\frac{du}{dx}+\frac{dv}{dx}$ =A+-A=0

Answer

The velocity field represents an incompressible flow.

b) Stagnation point of the flow field;

At stagnation point ;

$\overrightarrow{V}$ =0

Since

$\overrightarrow{V}$ =(Ax-B)i+(C-Ay)j

And A=2,B=5 and C=3

u=2x-5=0

x=5/2

v=3-2y

y=3/2

Answer

Stagnation point

$(\frac52,\frac32)$

c) Expression for the pressure gradient.

From Euler's equation of ideal flow,

Assuming a steady flow;

$\rho$ g_x-∆P=$\rho$ [$u\frac{dv}{dx}+v\frac{dv}{dy}$ ]

From the given data

$-\rho g_xk-∆P=\rho[(Ax-B)Ai+(C-Ay)-Aj]$

Replace the values of A B and C

$-\rho g_xk-∆P=\rho[(4x-10)i+(4y-6)j]$

Which can be rewritten as;

$∆P=-\rho[(4x-10)i+(4y-6)j+g_xk]$

d) Evaluate the difference at (1,3) and Origin(0,0) if $\rho=1.2kg/m^3$

P_1,3-P_0,0=$-\rho(\int_0^1(4x-10)dx+\int_0^3(4y-6)dy+\int_0^0gdz)$

P_(1,3)-P_(0,0)=$-\rho[(2x^2-10x)|_0^1+(2y^2-6y)|_0^3]$

P_(1,3)-P_(0,0)=-1.2(-8)=9.6N/m²

Answer

9.6N/m²