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A flow velocity field between two plates is given by the following expression: u(y)=C(h2−y2) & v=0 m/s. Where C is a constant and h is the distance between the plates. If present, determine the current function. (During your analysis, consider that any integration constant is negligible.)

User Amit S
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Answer:

To determine the values of b and c such that the velocity field represents a non-rotational flow, we need to ensure that the curl of the velocity field is zero. The curl of a vector field can be calculated using the cross product of the del operator (∇) with the vector field.

Given the velocity field:

V = (1.35 + 2.78x + 0.754y + 4.21z) + (3.45 + cx - 2.78y + z) + (-4.21x - 1.89y)

Let's calculate the curl of V:

∇ × V = (∂/∂y)(-4.21x - 1.89y) - (∂/∂z)(3.45 + cx - 2.78y + z)

+ (∂/∂z)(1.35 + 2.78x + 0.754y + 4.21z) - (∂/∂x)(-4.21x - 1.89y)

+ (∂/∂x)(3.45 + cx - 2.78y + z) - (∂/∂y)(1.35 + 2.78x + 0.754y + 4.21z)

Simplifying the above expression, we have:

∇ × V = (-∂(4.21x)/∂y - ∂(1.89y)/∂y) - (∂(3.45)/∂z + ∂(cx)/∂z - ∂(2.78y)/∂z + ∂(z)/∂z)

+ (∂(1.35)/∂z + ∂(2.78x)/∂z + ∂(0.754y)/∂z + ∂(4.21z)/∂z) - (-∂(4.21x)/∂x - ∂(1.89y)/∂x)

+ (∂(3.45)/∂x + ∂(cx)/∂x - ∂(2.78y)/∂x + ∂(z)/∂x) - (∂(1.35)/∂y - ∂(2.78x)/∂y - ∂(0.754y)/∂y - ∂(4.21z)/∂y)

Now, we can compute each partial derivative term:

∂(4.21x)/∂y = 0

∂(1.89y)/∂y = 1.89

∂(3.45)/∂z = 0

∂(cx)/∂z = 0

∂(2.78y)/∂z = 0

∂(z)/∂z = 1

∂(1.35)/∂z = 0

∂(2.78x)/∂z = 0

∂(0.754y)/∂z = 0.754

∂(4.21z)/∂z = 4.21

∂(4.21x)/∂x = 4.21

∂(1.89y)/∂x = 0

∂(3.45)/∂x = 0

∂(cx)/∂x = c

∂(2.78y)/∂x = 2.78

∂(z)/∂x = 0

User Mauro
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