Question

A two-dimensional incompressible frictionless flow field is given by \(\vec u = x\hat i - y\hat j\). If ρ is the density of the fluid, the expression for pressure gradient vector at any point in the flow field is given as
1. \(\rho \left( {x\hat i + y\hat j} \right)\)
2. \( - \rho \left( {x\hat i + y\hat j} \right)\)
3. \(\rho \left( {x\hat i - y\hat j} \right)\)
4. \( - \rho \left( {{x^2}\hat i + {y^2}\hat j} \right)\)

Answer 1

Correct Answer - Option 2 : \( - \rho \left( {x\hat i + y\hat j} \right)\)

Concept:

Euler Equation of Motion:

\({\vec F_p} + {\vec F_y} = {\vec F_i}\)

\({\vec F_p} + {\vec F_y} = m\vec a\)

\(\frac{{\partial u}}{{\partial t}} + u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}} = X - \frac{1}{\rho }\frac{{\partial p}}{{\partial x}}\)

\(\frac{{\partial v}}{{\partial t}} + u\frac{{\partial v}}{{\partial x}} + v\frac{{\partial v}}{{\partial y}} + w\frac{{\partial v}}{{\partial z}} = Y - \frac{1}{\rho }\frac{{\partial p}}{{\partial y}}\)

\(\frac{{\partial w}}{{\partial t}} + u\frac{{\partial w}}{{\partial x}} + v\frac{{\partial w}}{{\partial y}} + w\frac{{\partial w}}{{\partial z}} = Z - \frac{1}{\rho }\frac{{\partial p}}{{\partial z}}\)

Where X, Y, and Z are body forces.

\( - ∇ P + \rho ∇ ϕ = \rho \left[ {\frac{{D\vec u}}{{Dt}}} \right]\)

\(\frac{D}{{Dt}} = \frac{\partial }{{\partial t}} + u\frac{\partial }{{\partial x}} + v\frac{\partial }{{\partial y}} + w\frac{\partial }{{\partial z}}\)

Assume: flow is incompressible and friction less

Euler’s equation for vector form:

\( - ∇ P + \rho ∇ ϕ = \rho \left[ {\frac{{D\vec u}}{{Dt}}} \right]\)

\( - ∇ P + \rho ∇ ϕ = \rho \left[ {u\frac{{d\vec u}}{{dx}}\hat i + v\frac{{d\vec u}}{{dy}}\hat j + w\frac{{d\vec u}}{{dz}} + \frac{{d\vec u}}{{dt}}} \right]\)

\(\vec u = x\hat i - y\hat j\)

-∇P + ρ∇ϕ = ρ [xî + (-y) (-1)ĵ + 0 + 0]

Zero body force in x and y direction.

-∇P + ρ [0î + 0ĵ] = ρ [xî + yî] ⇒ ∇P = -ρ [xî + yĵ]