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In a two-dimensional, incompressible flow field, the velocity is V = 2xy ỉ + vj where x, y are in meters and V,v in m/s. a) Determine the y component of velocity, v, if its value on the x-axis is zero. b) Determine the dilatation rate of a fluid element in this flow field. c) Determine if the flow is rotational or not. d) Explain why it is possible to define a stream function for this flow. 1.0 e) Obtain the stream function for this flow. f) Plot the streamline passing through the point (1,1,1) by hand (approximately). State the value of the stream function for this streamline and mark the -1.0 1.0 X direction of the flow on the streamline. g) For this flow field, determine the magnitude of the average velocity of the -1.0 fluid crossing the surface BA shown on the right. ди ne 1V = - əx (əw D xỹ = ду ди 1 j+ əx. дх Hint: U = ду əz, dy) ap + (v - v)p + p(V •V) = a+ ("x at dv + ap др др ду apy + p əz, Dp ди .+ ρ(V. V) Dt + v + w = 0 əx дх ду əz v-(p7)

In a two-dimensional, incompressible flow field, the velocity is V = 2xy ỉ + vj where x, y are in meters and V,v in m/s. a) Determine the y component of velocity, v, if its value on the x-axis is zero. b) Determine the dilatation rate of a fluid element in this flow field. c) Determine if the flow is rotational or not. d) Explain why it is possible to define a stream function for this flow. 1.0 e) Obtain the stream function for this flow. f) Plot the streamline passing through the point (1,1,1) by hand (approximately). State the value of the stream function for this streamline and mark the -1.0 1.0 X direction of the flow on the streamline. g) For this flow field, determine the magnitude of the average velocity of the -1.0 fluid crossing the surface BA shown on the right. ди ne 1V = - əx (əw D xỹ = ду ди 1 j+ əx. дх Hint: U = ду əz, dy) ap + (v - v)p + p(V •V) = a+ ("x at dv + ap др др ду apy + p əz, Dp ди .+ ρ(V. V) Dt + v + w = 0 əx дх ду əz v-(p7)

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In a two-dimensional, incompressible flow field, the velocity is V = 2xy ỉ + vỷ where x, y are in meters and V, v in m/s. a) Determine the y component of velocity, v, if its value on the x-axis is zero. b) Determine the dilatation rate of a fluid element in this flow field. c) Determine if the flow is rotational or not. d) Explain why it is possible to define a stream function for this flow. y 1.0 e) Obtain the stream function for this flow. f) Plot the streamline passing through the point (1,1,1) by hand (approximately). State the value of the stream function for this streamline and mark the -1.0 1.0 x direction of the flow on the streamline. g) For this flow field, determine the magnitude of the average velocity of the -1.0 fluid crossing the surface BA shown on the right. B dv dw j+ (дх ду, ди ne D x = дх ди 1 Hint: U = - V = - ду ðy dz. др др ap ap + w: ду Dp ди + p(V ·V) at + (V · v)p + p(V ·7) at dv + ду + (u + v +p əx = 0 = Dt Əx v-(pV)
Transcribed Image Text:In a two-dimensional, incompressible flow field, the velocity is V = 2xy ỉ + vỷ where x, y are in meters and V, v in m/s. a) Determine the y component of velocity, v, if its value on the x-axis is zero. b) Determine the dilatation rate of a fluid element in this flow field. c) Determine if the flow is rotational or not. d) Explain why it is possible to define a stream function for this flow. y 1.0 e) Obtain the stream function for this flow. f) Plot the streamline passing through the point (1,1,1) by hand (approximately). State the value of the stream function for this streamline and mark the -1.0 1.0 x direction of the flow on the streamline. g) For this flow field, determine the magnitude of the average velocity of the -1.0 fluid crossing the surface BA shown on the right. B dv dw j+ (дх ду, ди ne D x = дх ди 1 Hint: U = - V = - ду ðy dz. др др ap ap + w: ду Dp ди + p(V ·V) at + (V · v)p + p(V ·7) at dv + ду + (u + v +p əx = 0 = Dt Əx v-(pV)
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