The x and y components of a velocity field are given by u = x2y and v = -xy2. Determine the equation for the streamlines of this flow and compare it with those in Example 4.2. Is the flow in this problem the same as that in Example 4.2? Explain.

Transcribed Image Text:

EXAMPLE 4.2 Streamlines for a Given Velocity Field GIVEN Consider the two-dimensional steady flow discussed FIND Determine the streamlines for this flow. in Example 4.1, V- (V₁/€)(-xi+yj). SOLUTION Since u-(-V₁/€)x and v= (V/¤)y (1) it follows that streamlines are given by solution of the equation (V/E)y dx 14 -(VJE)x in which variables can be separated and the equation integrated to give or √²--√ ª In y = -In x + constant Thus, along the streamline xy = C, where C is a constant (Ans) By using different values of the constant C, we can plot various lines in the x-y plane-the streamlines. The streamlines for x 20 are plotted in Fig. E4.2. A comparison of this figure with Fig. E4. la illustrates the fact that streamlines are lines tangent to the velocity field. COMMENT Note that a flow is not completely specified by the shape of the streamlines alone. For example, the streamlines for the flow with V/= 10 have the same shape as those for the flow with V/= -10. However, the direction of the flow is op- posite for these two cases. The arrows in Fig. E4.2 representing the flow direction are correct for V/= 10 since, from Eq. 1, = -10x and v= 10y. That is, the flow is from right to left. For V/l = -10 the arrows are reversed. The flow is from left to right. 2 0 -2 Figure E4.2 2 C=9 C=4 C-1 -4-C=-1 C=-4 EXAMPLE 4.2 Streamlines for a Given Velocity Field GIVEN Consider the two-dimensional steady flow discussed FIND Determine the streamlines for this flow. in Example 4.1, V- (V₁/€)(-xi+yj). SOLUTION Since u-(-V₁/€)x and v= (V/¤)y (1) it follows that streamlines are given by solution of the equation (V/E)y dx 14 -(VJE)x in which variables can be separated and the equation integrated to give or √²--√ ª In y = -In x + constant Thus, along the streamline xy = C, where C is a constant (Ans) By using different values of the constant C, we can plot various lines in the x-y plane-the streamlines. The streamlines for x 20 are plotted in Fig. E4.2. A comparison of this figure with Fig. E4. la illustrates the fact that streamlines are lines tangent to the velocity field. COMMENT Note that a flow is not completely specified by the shape of the streamlines alone. For example, the streamlines for the flow with V/= 10 have the same shape as those for the flow with V/= -10. However, the direction of the flow is op- posite for these two cases. The arrows in Fig. E4.2 representing the flow direction are correct for V/= 10 since, from Eq. 1, = -10x and v= 10y. That is, the flow is from right to left. For V/l = -10 the arrows are reversed. The flow is from left to right. 2 0 -2 Figure E4.2 2 C=9 C=4 C-1 -4-C=-1 C=-4

Answer rating: 100% (QA)

Assume the flow field coordinates are x and y and u and v a... View the full answer