If the velocity of an incompressible fluid at a point (x,y,z) is given by (image below) prove that the motion is possible and that the velocity potential is −cosθ/r2. Find the equations of the streamlines where r^2 = x^2 +y^2 +z^2 and θ is the angle between r and the z-axis.

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A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

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A: Introduction:

Q: for the following flows, find the equation of the streamline through(1,1). v= -y^(2)i-6xj

A: Given:Velocity vector, V=-y2 i-6 x jPositio of point, x, y=1, 1Find:The equation of streamline

Q: Consider a three-dimensional, steady velocity field given by V = (u, v, w) = (3.2 + 1.4x)ï + (2.4 –…

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A: Dear Student,In this question, the figure (diagram) is missing. Please reupload the question with…

Q: y = 2r'sin20 (2) 0.5 m A 2D inviscid incompressible fluid flow around a corner is In addition, if…

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Q: For the flow of an incompressible fluid, the velocity in x-direction u = Find velocity component in…

A: GIVEN DATA - VELOCITY IN X-DIRECTION u=ax2+by VELOCITY INZ-DIRECTION ZERO WE HAVE TO FIND VELOCITY…

Q: 3. The two-dimensional velocity field v = (3t- 8xy)i+ 4(y – x')j satisfies the Euler's equation of…

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Q: Investigate the complex potential f(z) = A cosh [π(z/a)],and plot the streamlines inside the region…

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Q: 3 Starting with a small fluid element of volume dx dy dz, derive the continuity equation (Eq. 3.4)…

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Q: A two-dimensional flow field is given by u = 5x 2 − 5y 2v = −10xy (i) Find the streamfunction ψ and…

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Q: please explain everything you did

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Q: QI A/ The inviscid, steady, and incompressible 2D flows are given by (a) o = x-3xy (b) y =x- 2xy-y?…

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Q: Sketch the stream lines represented by y = xy. Also find out the velocity and its direction at point…

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Q: 2. The velocity components expressed in m/s in a fluid flow is known to be: 2 u= 2yzt v=xy W=4xzt²…

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Q: le fluid flow. ntial

A: The velocity potential is-

Q: The 2D flow of a vortex can be expressed as u=y-x and v=-x a) What is the magnitude of the local…

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Q: Q.3Given the x and y velocity components below (A is a constant): u = Axt v = -Ayt (a) Find the…

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Q: velocity field V = xy^2ti + zxj −t^3k exists at the 3D plane along a streamline that passes through…

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Q: In a 2D dimension incompressible flow , if the fluid velocity components are given by u = x-4y , v =…

A: Given:The fluid velocity components are given as:u=x-4yv=-4x

Q: 2-D unsteady flow field has the velocity components: v = - x²y?t u = - Plot the streamline passing…

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Q: 5. The velocity field of an incompressible flow is given by V = (a1x + a2y + az z) i + (b1 x + b2y+…

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Q: Problem 3: Obtaining the stream function from velocity components The steady state, incompressible…

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Q: 2. Consider the two-dimensional time-dependent velocity field u(x, t) = (sint, cost, 0), in the…

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Q: 3 Show that = x² - y² + can represent the velocity potential in an incompressible fluid flow. Also…

A: The velocity potential is-

Q: or the velocity field: u = 3xy, v = -1.5y²+5; for water, ρ = 1000 kg/m³ If the pressure at (0,0) is…

A: The velocity field is .The density of water is .The pressure at  is .

Q: 4. Show that the velocity field V = 2y(2x + 1)î + [x(x+1) – 2y²1j represents a possible…

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Q: A flow is described by velocity V=ay i+bt j ,a=1s-¹ and b =.5m/s².at t=25s,what are the coordinates…

A: we know thatVelocity V=dydtdydt=btdy=bt.dtintegrating both sides we get∫y0ydy=∫t0tbt.dty-yo=br220t

Q: Given the vector equation below for velocity of a given fluid: V=(0.5+1.2x)i + (-2.0-1.2y)j, derive…

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Q: 2. The y component of velocity in a steady, incompressible flow field in the xy plane is 3xy(x²- y?)…

A: Let x component be u     For incompressible flow             ∂u∂x+∂v∂y=0∴∂u∂x=-∂v∂y

Q: Question 4. The stream function is given as w=x³-y³. a. Is this flow possible? b. Is the flow…

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Q: "A steady, incompressible, two-dimensional velocity field is given by V = (u, v) = (2.7-1.75x, 0.7+…

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Q: If u = ax and v = ay and w = -2az obtain equation of streamline passing through the point (2, 2,4)…

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Q: Given velocity fieldV(x,y,z,t) = (5xy² + t)i + (2z + 8)j +18km/s, with x,y,z in meters and it in…

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Q: Q2). Consider the pressure-driven flow E distance b. Coordinate y is measured fro u = U([1 - (21.…

A: Write the expression for the velocity field as shown below.

Q: a. Given the velocity field u=(u,v,w) in Cartesian coordinates with u=2x+y, v=2zt, w=0. . i. Find…

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Q: Consider the steady, two-dimensional velocity field given by: u = 2xy-y²; v=x-y². Show that it is a…

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Q: whether the function o represents the velocity potential %3D x2+y2 lar incompressible 2D flow, and…

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Fluid Kinematics

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid statics) or in motion (posing some velocity). In fluid kinematics, studies focus on the associated motion of fluid particles or groups of particles, and its associated parameters like displacements, velocity, and acceleration without concerning about the forces that caused the motion. All the laws of classical mechanics are applied in the study of fluid kinematics.