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공학수학 | 보존력(conservative force)과 유체의 흐름 |

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작성자 cemtool 작성일14-06-16 16:40 조회7,896회 댓글4건

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이상적인 유체가 속도 V로 흐르는 것을 생각해보자. 비회전 유동(irrotational flow)일 경우의 식 은 다음과 같다.
 
여기서, u(x,y)는 포텐셜(potential)을 나타내는 스칼라 함수이다. 예를 들어, 다음과 같은 포텐셜 함수를 생각해보자.
 
이것의 속도는 다음과 같다.
 
그러면, 이므로, 와도(vorticity)가 0임을 쉽게 알 수 있다. “u(x,y)=상수”를 만족하는 곡선들을 등위선(equipotential curve)이라 하는데, 속도 벡터에 해당하 는 경사도(gradient)는 등위선에 수직이다. 접선 방정식은 관계식
 
를 만족하므로, 주어진 함수 u(x,y)에 대한 접선 방정식은
 
가 된다. 이것을 유선(streamline, flow line)이라고 한다. 유선은 유체가 이동하는 모양을 나타낸 다. CEMTool을 사용하여, 주어진 함수에 대한 등위선과 유선을 그려라.
ex14_19.cem
/*
 Example 14.19 Conservative forces and fluid flow
*/
 
// Plot the function u=xy and the streamlines
 
xline=0:3:0.001;                // 45 deg. lineyline=xline;
c=[1,3,5];
x1=1:5:0.001;
x2=1.7:5:0.001;
x3=2:5:0.001;
u_c1=c(1)./x1;
u_c2=c(2)./x2;
u_c3=c(3)./x3;
plot(xline,yline,"-")holdon
plot(x1,u_c1,"--",x2,u_c2,"--",x3,u_c3,"--")// plot the field lines x^2-y^2=const
c=[1 2 3];
x1=1:5:0.005;
y1=sqrt(x1.^2 - c(1)^2);
x2=2:5:0.005;
y2=sqrt(x2.^2 - c(2)^2);
x3=3:5:0.005;
y3=sqrt(x3.^2 - c(3)^2);
plot(x1,y1,x2,y2,x3,y3)title("Potential u=xy and Field v=y*i+x*j")xtitle("x")ytitle("y")
 

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