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expression for analytic function for ferro-fluid

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In "magnetic drug" problem, if i actually use the ferro-fluid instead of taking it's mass fraction as 0.1, what would be the expression for analytic function? According to my understanding, we don't need pump for flow of ferro-fluid under the effect of magnetic field and it's velocity depends on the temperature gradient of the fluid.For that i m providing a constant flux at the lower surface of the channel and eventually ferro-fluid will take away this heat

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