Perform the convective on velocity vectors u in cylindrical coordinates : Du/Dt
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- Consider a two-dimensional flow which varies in time and is defined by the velocity field, u = 1 and v = 2yt. Compute the convective derivative of each velocity component: Du/Dt and Dv/Dt.derive the equation of stream line in cylinderical coordinates in 2 dimensions.consider the 2 dimensional velocity field V= -Ayi +Axj where in this flow field does the speed equal to A? Where does the speed equal to 2A?
- The compressible form of the continuity equation is (∂?/∂t) + ∇-›·(?V-›) = 0. Expand this equation as far as possible in Cartesian coordinates (x, y, z) and (u, ?, w).Consider a steady, two-dimensional flow field in the xy-plane whose x-component of velocity is given by u = a + b(x − c)2 where a, b, and c are constants with appropriate dimensions. Of what form does the y-component of velocity need to be in order for the flow field to be incompressible? In other words, generate an expression for ? as a function of x, y, and the constants of the given equation such that the flow is incompressibleThe velocity field of a flow is defined through the vector v =-ayi+axj; where "a" is a constant. It is desired to determine a) the stream function and the equation of the streamlines; b) if the flow is rotational
- By using the expression for the shear stress derived in class (and in BSL), show that the shear force on asphere spinning at a constant angular velocity in a Stokes’ flow, is zero.This means that a neutrally buoyant sphere (weight equal buoyancy force) that is made to spin in aStokes’ flow, will neither rise nor fall, nor translate in any preferential direction in the (x-y) plane. expressions for velocity are: v_r (r,θ)= U_∞ [1-3R/2r+R^3/(2r^3 )] cosθ v_θ (r,θ)= -U_∞ [1-3R/4r-R^3/(4r^3 )] sinθ Where v_r and v_θ are the radial and angle velocity, U_∞ is the velocity of fluid coming to sphere which very faar away from the sphere. And R is the radius of sphere.Consider the steady, two-dimensional, incompressible velocity field, namely, V-›= (u, ?) = (ax + b) i-›+ (−ay + cx) j-›. Calculate the pressure as a function of x and y.3.3 Starting with a small fluid element of volume dx dy dz, derive the continuity equation (Eq. 3.4) in rectangular cartesian coordinates.
- Consider a velocity field where the x and y components of velocity aregiven by u = cy/(x2 + y2) and v = −cx/(x2 + y2), where c is a constant. For vortex flow, calculate: a. The time rate of change of the volume of a fluid element per unitvolume.b. The vorticity.Find the two-dimensional velocity potential ϕ ( r , θ ) for thepolar coordinate fl ow pattern υ r = Q / r , υ θ = K / r , where Qand K are constants.For the velocity field that is linear in both spatial directions (x and y) is V-›= (u, ? ) = (U + a1x + b1y) i-›+ (V + a2x + b2y) j-›where U and V and the coefficients are constants, calculate the shear strain rate in the xy-plane.